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    STUDIA MATHEMATICA - Issue no. 1 / 2020  
         
  Article:   THE CRITICAL POINT OF A SIGMOIDAL CURVE.

Authors:  AYSE HUMEYRA BILGE, YUNUS OZDEMIR.
 
       
         
  Abstract:  
DOI: 10.24193/subbmath.2020.1.07

Published Online: 2020-03-06
Published Print: 2020-03-30
pp. 77-91
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ABSTRACT: Let $y(t)$ be a monotone increasing curve with $displaystyle lim_{t o pminfty}y^{(n)}(t)=0$ for all $n$ and let $t_n$ be the location of the global extremum of the $n$th derivative $y^{(n)}(t)$. Under certain assumptions on the Fourier and Hilbert transforms of $y(t)$, we prove that the sequence ${t_n}$ 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 cite{BP2013}.

Key words: Sigmoidal curve; critical point; Fourier transform; Hilbert transform
 
         
     
         
         
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