Sum-Difference Model for Production of a Star-Like Harmonic Oscillator Transformation Semigroup
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Date
2025-05-20
Journal Title
Journal ISSN
Volume Title
Publisher
NIPES Pub.
Abstract
This study introduces a novel sum-difference model for generating a
star-like harmonic oscillator transformation semigroup, based on
combinatorial properties of partial transformations over the finite
set ?? = {1, 2, 3… , ?}. A partial transformation on ?? is defined as
a mapping from a subset ?∗ . Using this framework, we construct a
generalized two-dimensional star-like harmonic structure to produce
ordered sequences and visualizable transformations. Specifically, for
transformations ?∗, ?∗, ?∗ ∈ ???
∗ , the cardinality |???
∗ | is modeled as
1
2 ??∗ = ??∗−??∗
2 4?∗?∗ which governs the generation of star-like
sequences and image mappings. The paper presents a unified
analytical approach to deriving these results using a star-like sumdifference
operator. We further establish new harmonic oscillator
relations within the semigroup, supported by proofs of key
combinatorial functions. This research contributes to the development
of algebraic methods in harmonic transformation theory and lays a
foundation for future applications in signal modeling, automata
theory, and abstract algebraic systems.