A New Methodology for Extending the Domain of Mathematical Successive Operations, with Applications to Combinatorial Analysis
This publication presents the author’s master’s thesis, marking his debut as a published academic author. The dissertation is available through the LNCC library collection and in a finely crafted, hardbound limited edition of just twenty individually assembled copies.
Original Title in Portuguese: Uma nova metodologia para a extensão de domínio de operações matemáticas sucessivas, com aplicações na análise combinatória
ABSTRACT:
This work stablishes the principles, definitions and properties that allow recursive functions creation using binary operations acting on the elements of sequences.
The methodology is developed in the domain of algebraic structures (semigroups, commutative groups and rings), which will demonstrate the potential of these operations to address issues in Elementary Algebra, Number Theory and Abstract Algebra fields.
As a consequence of this approach, it is obtained an algebraic structure that allows to create operations similar to the ones that can be conceived through usual products and summations, however, more generic.
Thus, it will be possible to address issues involving successive operations with integer domain following a general methodology, where the rules are derived from the basic elements of the presented theory.
To highlight the potential of the developed generalization, applications will be shown in the representation of binomial numbers and their possible extensions.
Finally, in order to prove the applicability of the new methodology, comparisons were made with results from the literature.
