Determining the Critical Point of a Sigmoidal Curve via its Fourier Transform

[摘要] A sigmoidal curve y(t) is a monotone increasing curve such that all derivatives vanish at infinity. Let tnbe the point where the nth derivative of y(t) reaches its global extremum. In the previous work on sol-gel transition modelled by the Susceptible-Infected- Recovered (SIR) system, we observed that the sequence {tn} seemed to converge to a point that agrees qualitatively with the location of the gel point [2]. In the present work we outline a proof that for sigmoidal curves satisfying fairly general assumptions on their Fourier transform, the sequence {tn} is convergent and we call it "the critical point of the sigmoidal curve". In the context of phase transitions, the limit point is interpreted as a junction point of two different regimes where all derivatives undergo their highest rate of change.

[发布机构] Faculty of Engineering and Natural Sciences, Kadir Has University, Department of Mathematics, Anadolu University, Istanbul, Eskisehir, Turkey^1

[效力级别] 数学

[关键词] A-monotone;Gel point;Junction point;Limit points;Rate of change;Sigmoidal curves;Sol-gel transitions