eu APPROXIMATION OF THE CUT FUNCTION BY STANNARD AND RICHARD SIGMOID FUNCTIONS

Several sigmoidal functions (Stannard [10], [16], [19], [23], Richards [13], [17], [20], [22], Chapman–Richards [5]) find numerous applications in various fields related to life sciences, chemistry, physics, artificial intelligence, population dynamics, plant biology, fuzzy set theory, etc. A practically important class of sigmoid functions is the class of cut functions [6], [7], [9] including the Heaviside step function as a limiting case.

A practically important class of sigmoid functions is the class of cut functions [6], [7], [9] including the Heaviside step function as a limiting case.
Approximation of the cut function by a squashing functions is discussed from various computational and modelling aspects in [15].
We study the uniform approximation of the cut function by Stannard and Richard growth functions.We find an expression for the error of the best uniform approximation.
Present approximation task is extremely actual in connection with detailed precision of the lag phase, growth phase and plateau phase in the growth process [3], [18].

Sigmoid Functions
In this work we consider sigmoid functions of a single variable defined on the real line, that is functions of the form R −→ R. Sigmoid functions can be defined as bounded monotone non-decreasing functions on R. One usually makes use of normalized sigmoid functions defined as monotone non-decreasing functions s(t), t ∈ R, such that lim s(t) t→−∞ = 0 and lim s(t) t→∞ = 1 (in some applications the left asymptote is assumed to be −1: lim s(t) t→−∞ = −1).

The Cut and the Stannard Functions
The cut (ramp) function is the simplest piece-wise linear sigmoid function.Let ∆ = [γ − δ, γ + δ] be an interval on the real line R with centre γ ∈ R and radius δ ∈ R. A cut function is defined as follows: Note that the slope of function c γ,δ (t) on the interval ∆ is 1/(2δ) (the slope is constant in the whole interval ∆).
Two special cases and a limiting case are of interest for our discussion in the sequel.
Definition.Define the shifted Stannard function S γ (t) with jump at point γ as: Special case.

Approximation of the Cut Function by Stannard Function
We next focus on the approximation of the cut function (1) by shifted Stannard growth function S γ (t) defined by (( 6)-( 7)).
In addition chose c and S to have same slopes at their coinciding centres.
Then, noticing that the largest uniform distance ρ between the cut and Stannard functions is achieved at the endpoints of the underlying interval [0, 2δ] we have: The above can be summarized in the following Theorem 1.The function S γ (t) defined by ( 6)-( 7): i) is the Stannard function of best uniform one-sided approximation to function c γ,δ in the interval [γ, ∞) (as well as in the interval (−∞, γ]); ii) approximates the cut function c γ,δ (t) in uniform metric with an error Remark.We note that the uniform distance ρ = ρ(m) is an absolute constant that does depends only on the growth parameter m (see Fig. 1 and Fig. 3).Some computational examples using relation (8)

Approximation of the Cut Function by Richard Function
Definition.Define the special shifted Richard growth function R γ (t) with jump at point γ as: Then we have R γ (γ) = 1 2 .We next focus on the approximation of the cut function (1) by shifted Richard growth function R γ (t) defined by (9).
In addition chose c and R to have same slopes at their coinciding centres.
Then, noticing that the largest uniform distance ρ 1 between the cut and Richard functions is achieved at the endpoints of the underlying interval [0, 2δ] we have: The above can be summarized in the following Theorem 2. The function R γ (t) defined by ( 9): i) is the Richard function of best uniform one-sided approximation to function c γ,δ in the interval [γ, ∞) (as well as in the interval (−∞, γ]); ii) approximates the cut function c γ,δ (t) in uniform metric with an error    (10) for various rates m.
Some computational examples using relation (10) are presented in Table 2.
Remarks.Usually, the numerical solution of differential equations that describe them, for example, Schnute, Stannard and Richards growth curves is associated with sensitive unstable solutions.
In this sense, could be useful results related to the theory and practice of differential equations [4], [8].
Such views could prove very useful in detail refinement of Lag phase and Plateau phase in elongation processes.

Table 2 :
Bounds for ρ 1 computed by