HOSOYA-PELL CYCLIC & PROXIMITY GROUPS. APPLICATION OF HP SINUSOIDAL POLYNOMIALS IN MODULATING OSCILLATORY WAVEFORMS EMANATING FROM VIBRATING SYSTEMS

This paper introduces the periodicity (repeating subsequences) of Hosoya-type Pell orbits as a source of important results for the generation of cyclic groups. Application of time-constrained Hosoya-Pell sinusoidal polynomials HPsp(t) (introduced in this paper) is given in terms of modulating motion waveforms emanating from vibrating systems. Methods: This paper demonstrates how periodicity in Hosoya-Pell subsequences are a source of cyclic groups and in Hosoya-Pell sinusoidal polynomials as a means of modulating highly oscillatory waveforms enclosed in Hilbert envelopes, leading to proximal groups, which provide concise representation of modulating motion waveforms emanating from vibrating systems. Results: A main result is the concise representation of the periodicity of Hosoya-type Pell orbits in cyclic groups. This application of HPsp(t) is threefold, namely, (i) use of Pell triangle vertex numbers as HPsp(t) coefficients, (ii) modulation (smoothing) of highly oscillatory motion waveforms as a means of diminishing energy consumption in vibrating systems, and (iii) concise proximal group representation of complex HPsp-modulated waveforms .