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Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains

Sergeyev, Yaroslav (2009) Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains. Nonlinear Analysis Series A: Theory, Methods & Applications, 71 (12). e1688-e1707.

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Abstract

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind of a computer – the Infinity Computer – able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given.


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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Sergeyev, Yaroslavyaro@dimes.unical.it0000-0002-1429-069X
Keywords: Infinite and infinitesimal numbers and numerals; infinite and infinitesimal functions and derivatives; physical and mathematical notions of continuity.
Subjects: Specific Sciences > Cognitive Science > Computation
Specific Sciences > Cognitive Science > Concepts and Representations
Specific Sciences > Mathematics
Specific Sciences > Cognitive Science > Perception
General Issues > Theory/Observation
Depositing User: Prof. Yaroslav Sergeyev
Date Deposited: 23 Sep 2024 18:28
Last Modified: 23 Sep 2024 18:28
Item ID: 23932
Journal or Publication Title: Nonlinear Analysis Series A: Theory, Methods & Applications
Publisher: Elsevier
Official URL: https://www.sciencedirect.com/science/article/pii/...
DOI or Unique Handle: 10.1016/j.na.2009.02.030
Subjects: Specific Sciences > Cognitive Science > Computation
Specific Sciences > Cognitive Science > Concepts and Representations
Specific Sciences > Mathematics
Specific Sciences > Cognitive Science > Perception
General Issues > Theory/Observation
Date: 15 December 2009
Page Range: e1688-e1707
Volume: 71
Number: 12
URI: https://philsci-archive.pitt.edu/id/eprint/23932

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