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

Abstract

“Arithmetic Calculus” (AC), introduced recently by the author, is explored further 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 be laid out in this network can be classified into one of five types of sequences, which can be expressed by algebraic polynomial or exponential functions but AC can reveal information concealed by algebra. The paper defines these sequences and refers to each family member as the parent sequence. These sequences make up their own universe as: (i) the population of the terms in each parent sequence is infinite; (ii) each parent sequence forms a hierarchy, in which their number of levels can extend into infinity; and (ii) there are infinitely diverse parent sequences. A glimpse of AC is illustrated through examples but their analytical capability is applicable in their universe. The paper shows that small sets of building blocks are operated by the convolution theorem or its variations, which is embedded “all-pervasively” in each and every member of the universe. Sufficient details are presented to ensure the emergence of 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. These operations reveal that the new network can parallel Cartesian coordinates and that for problems with no noise, the deconvolution problem is well-posed against common myth of it being ill-posed.