Algebraic properties of external numbers

Dinis, Bruno; Berg, Imme van den

doi:10.4115/jla.2011.3.9

Cited by 10 publications

(23 citation statements)

References 18 publications

(18 reference statements)

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“…Yet distributivity can be entirely characterized [2]. Let α = a+A, β and γ be external numbers, where a ∈ R and A is a neutrix.…”

Section: Elamentioning

confidence: 99%

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Cramer's rule applied to flexible systems of linear equations

Justino¹,

Berg²

2012

ELA

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Abstract. Systems of linear equations, called flexible systems, with coefficients having uncertainties of type o (.) or O (.) are studied. In some cases an exact solution may not exist but a general theorem that guarantees the existence of an admissible solution, in terms of inclusion, is presented. This admissible solution is produced by Cramer's Rule; depending on the size of the uncertainties appearing in the matrix of coefficients and in the constant term vector some adaptations may be needed.

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“…Yet distributivity can be entirely characterized [2]. Let α = a+A, β and γ be external numbers, where a ∈ R and A is a neutrix.…”

Section: Elamentioning

confidence: 99%

“…, n} , let x = [x j ] be a solution of the system n j=1 a ij x j = b i . Then by distributivity regarding multiplication by real numbers [2] and Part 1 of Lemma 5.7…”

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confidence: 99%

Cramer's rule applied to flexible systems of linear equations

Justino¹,

Berg²

2012

ELA

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“…As such the individualized neutral elements are unique. The proof is similar to the proof of the uniqueness of neutral elements in groups (see [10]). Often it is convenient to use the functional notation e(x) to indicate the individualized neutral element of the element x.…”

Section: The Axiomsmentioning

confidence: 82%

“…Axioms 2.1-2.29, extend the axioms originally presented in [10] and were shown to be consistent in [11] by the construction of a direct model in the language of ZF C. This was given in the form of a set of cosets of a non-Archimedean field. Allowing for definable classes, it was shown in [10] that the external numbers of [19] and [20] satisfy the axioms for addition and for multiplication, together with a modified form of the distributivity axiom; this modified form was shown to be equivalent to Axiom 2.22 in [12]. The remaining algebraic axioms deal with multiplication of magnitudes, and are in fact taken from calculation rules of the external numbers observed in [19] and [20].…”

Section: On Consistencymentioning

confidence: 84%

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Axiomatics for the external numbers of nonstandard analysis

Dinis¹,

Berg

2017

JLA

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Neutrices are additive subgroups of a nonstandard model of the real numbers. An external number is the algebraic sum of a nonstandard real number and a neutrix. Due to the stability by some shifts, external numbers may be seen as mathematical models for orders of magnitude. The algebraic properties of external numbers gave rise to the so-called solids, which are extensions of ordered fields, having a restricted distributivity law. However, necessary and sufficient conditions can be given for distributivity to hold. In this article we develop an axiomatics for the external numbers. The axioms are similar to, but mostly somewhat weaker than the axioms for the real numbers and deal with algebraic rules, Dedekind completeness and the Archimedean property. A structure satisfying these axioms is called a complete arithmetical solid. We show that the external numbers form a complete arithmetical solid, implying the consistency of the axioms presented. We also show that the set of precise elements (elements with minimal magnitude) has a built-in nonstandard model of the rationals. Indeed the set of precise elements is situated between the nonstandard rationals and the nonstandard reals whereas the set of non-precise numbers is completely determined.2010 Mathematics Subject Classification. 03H05, 03C65, 26E35.

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On Non-linear Optimization with a Perturbed Objective Function

Tran

Berg

2022

Dynamic Control and Optimization

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Algebraic properties of external numbers

Cited by 10 publications

References 18 publications

Cramer's rule applied to flexible systems of linear equations

Cramer's rule applied to flexible systems of linear equations

Axiomatics for the external numbers of nonstandard analysis

On Non-linear Optimization with a Perturbed Objective Function

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