On Flexible Sequences
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On Flexible Sequences Bruno Dinis1
· Nam Van Tran2 · Imme van den Berg3
Received: 8 January 2018 / Revised: 4 July 2018 / Accepted: 17 September 2018 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019
Abstract In the setting of nonstandard analysis, we introduce the notion of flexible sequence. The terms of flexible sequences are external numbers. These are a sort of analogue for the classical O(·) and o(·) notation for functions, and have algebraic properties similar to those of real numbers. The flexibility originates from the fact that external numbers are stable under some shifts, additions, and multiplications. We introduce two forms of convergence and study their relation. We show that the usual properties of convergence of sequences hold or can be adapted to these new notions of convergence and give some applications. Keywords External numbers · Flexible sequences · Convergence · Nonstandard analysis Mathematics Subject Classification (2010) 03H05 · 40A05
1 Introduction In this article, we introduce the notion of flexible sequence and study a general form as well as convergence properties and behavior under operations. In our context, flexibility deals with stability under some shifts, additions, and multiplications, and is a matter of order of magnitude, or degree. The term “flexible” is borrowed from [9], where the propagation of errors in matrix calculus is modeled similarly. Our objectives are to extend this error analysis Bruno Dinis
[email protected] Nam Van Tran [email protected] Imme van den Berg [email protected] 1
Departamento de Matem´atica, Faculdade de Ciˆencias da Universidade de Lisboa, Campo Grande, Ed. C6, 1749-016, Lisboa, Portugal
2
Faculty of Applied Sciences, Ho Chi Minh City University of Technology and Education, 01 Vo Van Ngan Street, Linh Chieu Ward, Thu Duc District, Ho Chi Minh City, Vietnam
3
´ ´ Departamento de Matem´atica, Universidade de Evora, Evora, Portugal
B. Dinis et al.
to sequences and to make rigorous and precise certain ways of speaking in asymptotics, such as “converging to o(ε)” where ε is arbitrarily small, or neglecting small fluctuations. We were inspired by the ars negligendi of Van der Corput [23]; this is the art of neglecting some values which are small or unimportant with respect to other aspects of some problem. Van der Corput introduced as a tool neutrices, referring to additive commutative groups of functions without unity. Here, we use the (scalar) neutrices introduced in [11, 12] in the setting of nonstandard analysis. A neutrix is an additive convex subgroup of the (nonstandard) real numbers. Except for the obvious neutrices {0} and the set of real numbers itself, neutrices are external sets. Some simple examples of neutrices are , the external set of all real infinitesimals, and £, the external set of all limited real numbers (numbers bounded in absolute value by a standard number). An external number is the algebraic sum of a neutrix and a real number. A flexible sequ
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