The critical point of a sigmoidal curve
DOI:
https://doi.org/10.24193/subbmath.2020.1.07Keywords:
Sigmoidal curve, critical point, Fourier transform, Hilbert transform.Abstract
Let y(t) be a monotone increasing curve with lim t→±∞y(n)(t) = 0 for alln and let tn be the location of the global extremum of the nth derivative y(n)(t). Under certain assumptions on the Fourier and Hilbert transforms of y(t), we prove that the sequence {tn} is convergent. This implies in particular a preferred choice of the origin of the time axis and an intrinsic definition of the even and odd components of a sigmoidal function. In the context of phase transitions, the limit point has the interpretation of the critical point of the transition as discussed in previous work [3].
Mathematics Subject Classification (2010): 34A99.
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