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Heuristic Contextualisation of Arithmetic Calculus by a New Network Based on the Difference Table

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Document pages: 40 pages

Abstract: “Arithmetic Calculus” (AC), introduced recently by the author, is exploredfurther in this paper by giving a new lease of life to the age-old differences table by transforming it into a new kind of network. Any sequence that can belaid out in this network can be classified into one of five types of sequences,which can be expressed by algebraic polynomial or exponential functions butAC can reveal information concealed by algebra. The paper defines these sequencesand refers to each family member as the parent sequence. These sequencesmake up their own universe as: (i) the population of the terms in eachparent sequence is infinite; (ii) each parent sequence forms a hierarchy, inwhich their number of levels can extend into infinity; and (ii) there are infinitelydiverse parent sequences. A glimpse of AC is illustrated through examplesbut their analytical capability is applicable in their universe. The papershows that small sets of building blocks are operated by the convolution theoremor its variations, which is embedded “all-pervasively” in each and everymember of the universe. Sufficient details are presented to ensure the emergenceof the mosaic image of AC through problems including: (i) differentiation(reducement), (ii) integration (conducement), (iii) diagonal operations(reminiscent of gradient methods), and (iv) structure of hierarchies. Theseoperations reveal that the new network can parallel Cartesian coordinates andthat for problems with no noise, the deconvolution problem is well-posedagainst common myth of it being ill-posed.

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