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Introducing “Arithmetic Calculus” with Some Applications: New Terms, Definitions, Notations and Operators

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New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements; and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a “Tree of Numbers” reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned; and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.
Cite this paper
Khatibi, R. (2014) Introducing “Arithmetic Calculus” with Some Applications: New Terms, Definitions, Notations and Operators. Applied Mathematics, 5, 2909-2934. doi: 10.4236/am.2014.519277.
 

[1] Wilf, H.S. (1994) Generating Functionology.
http://www.math.upenn.edu/~wilf/gfology2.pdf
[2] Khatibi, R. (2013) Chapter 1: Learning from Natural Selection in Biology: Reinventing Existing Science to Generalise Theory of Evolution—Evolutionary Systemics. In: Lynch, J.R., Derek, T. and Williamson, D.T., Eds., Natural Selection: Biological Processes, Theory and Role in Evolution, 1-96. https://www.novapublishers.com/catalog/product_info.php?products_id=41521
[3] Avigad, J. (2010) Understanding, Formal Verification, and the Philosophy of Mathematics.                                             eww141106lx
http://www.andrew.cmu.edu/user/avigad/Papers/understanding2.pdf
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