Sum-Difference Model for Production of a Star-Like Harmonic Oscillator Transformation Semigroup

Abstract

This work presents a new sum-difference model based oncombinatorial features of partial transformations over the finite set forcreating a star-like harmonic oscillator transformation semigroup. Amapping from a subset is called a partial transformation. Using thisframework, we generate ordered sequences and visualizabletransformations by building a generalized two-dimensional star-likeharmonic structure. The cardinality is specifically designed to controlthe creation of picture mappings and star-like sequences fortransformations. In this study, a star-like sum-difference operator isused to derive these conclusions in a unified analytical manner. Wealso prove important combinatorial functions to support our novelharmonic oscillator relations within the semigroup. This studyestablishes the groundwork for upcoming applications in signalmodeling, automata theory, and abstract algebraic systems while alsoadvancing algebraic techniques in harmonic transformation theory.