TITLE:
Heuristic Contextualisation of Arithmetic Calculus by a New Network Based on the Difference Table
AUTHORS:
Rahman Khatibi
KEYWORDS:
Arithmetic Calculus, Convolution, Conducemental Sequences, Deconvolution, Replication
JOURNAL NAME:
Applied Mathematics,
Vol.8 No.10,
October
19,
2017
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.