Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains
By Yaroslav D. Sergeyev. 2009
82 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Mathematical Physics, Theoretical Computer Science
View Paper: https://www.semanticscholar.org/paper/d7a5b857a74279be5e0c7efc22c0f3f4346d6eb8
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In Computational Theory and Mathematics
In Mathematical Physics
In Theoretical Computer Science
Citations in the corpus
- Counting systems and the First Hilbert problem (2010, background): “For example, if we observe a table byye, then we see it as being continuous.”
- Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems (2017, methodology): “In traditional Mathematics this is impossible whereas the new approach (see [90,91]) allows us to execute the required evaluations by using the following method.”