File:Elutin1a4tori.jpg: Difference between revisions

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Summary

Title / Description	Iterations of the logistic transfer function $f(x)=4x(1\!-\!x)$ (shown with thick black line) $y=f^{c}(x)$ for $c=$ 0.2, 0.5, 0.8, 1, 1,5 . Function $f$ is iterated $c$ times; however, the number $c$ of iterations has no need to be integer. This pic was generated with the "universal" algorithm that evaluates the iterations of more general function $f_{u}(x)=u~x~(1\!-\!x)$ ; see ^[1]. Namely for $u\!=\!4$ , the iterates can be expressed through the elementary function, and such a plot can be generated, for example, in Mathematica with very simple code: F[c_, z_] = 1/2 (1 - Cos[2^c ArcCos[1 - 2 z]]) Plot[{F[1.5, x], F[1, x], F[.8, x], F[.5, x], F[.2, x]}, {x, 0, 1}] In order to keep the code short, the colors are not adjusted. The code above can be obtained from the representation of the superfunction $F$ and the Abel function $G$ $f^{c}(z)=F(c+G(z))$ at Failed to parse (syntax error): {\displaystyle F(z)= \frac{1}{2}(1−\cos(2z))} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(z)=F^{-1}(z)=\log_2(\arccos(1\!−\!2z))}
Citizendium author	Dmitrii Kouznetsov
Date created	March 2011
Country of first publication	Japan
Notes	More superfunctions represented through elementary functions can be found in ^[2]. Expressions for the more general case are suggested in the article Logistic sequence. Copyleft 2011 by Dmitrii Kouznetsov. Previously, this image appeared at http://tori.ils.uec.ac.jp/TORI/index.php/File:Elutin1a4tori.jpg ; the free use is allowed. References ↑ http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. ↑ http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1/ D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
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[logistic-1] ttp://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.

[factorial-2] ttp://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1/ D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.

[1]

[2]

@@ Line 1: / Line 1: @@
 == Summary ==
 {{Image_Details|user-pd
-|description  = Iterations of the [[logistic transfer function]] $f(x)=4x(1\!-\!x)$ (shown qith thick black line) $y=f^c(x)$ for $c=$ 0.2, 0.5, 0.8, 1, 1,5 .   Function $f$ is iterated $c$ times; however, the number $c$ of iterations has no need to be integer.  This pic was generated with the "universal" algorithm that evaluates the iterations of more general function $f_u(x)=u~x~ (1\!-\!x)$; see <ref name="logistic"> http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. </ref>. Namely for $u\!=\!4$, the iterates can be expressed through the elementary function, and such a plot can be generated, for example, in [[Mathematica]] with very simple code:  F[c_, z_] = 1/2 (1 - Cos[2^c ArcCos[1 - 2 z]])  Plot[{F[1.5, x], F[1, x], F[.8, x], F[.5, x], F[.2, x]}, {x, 0, 1}] In order to keep the code short, the colors are not adjusted. The representation above can be obtained from the representation of the [[superfunction]] $F$ and the [[Abel function]] $G$: : $f^c(z)=F(c+G(z))$ at : $F(z)= \frac{1}{2}(1−\cos(2z))$ : $G(z)=F^{-1}(z)=\log_2(\arccos(1\!−\!2z))$
+|description  = Iterations of the [[logistic transfer function]] <math>f(x)=4x(1\!-\!x)</math> (shown with thick black line) <math>y=f^c(x)</math> for <math>c=</math> 0.2, 0.5, 0.8, 1, 1,5 .   Function <math>f</math> is iterated <math>c</math> times; however, the number <math>c</math> of iterations has no need to be [[integer]].  This pic was generated with the "universal" algorithm that evaluates the iterations of more general function <math>f_u(x)=u~x~ (1\!-\!x)</math>; see <ref name="logistic"> http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. </ref>. Namely for <math>u\!=\!4</math>, the iterates can be expressed through the elementary function, and such a plot can be generated, for example, in [[Mathematica]] with very simple code:
+  F[c_, z_] = 1/2 (1 - Cos[2^c ArcCos[1 - 2 z]])
+  Plot[{F[1.5, x], F[1, x], F[.8, x], F[.5, x], F[.2, x]}, {x, 0, 1}]
+In order to keep the code short, the colors are not adjusted. The code above can be obtained from the representation of the [[superfunction]] <math>F</math> and the [[Abel function]] <math>G</math>
+ <math>f^c(z)=F(c+G(z))</math>
+at
+ <math>F(z)= \frac{1}{2}(1−\cos(2z))</math>
+ <math>G(z)=F^{-1}(z)=\log_2(\arccos(1\!−\!2z))</math>
 |author       = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
 |date-created = March 2011
@@ Line 7: / Line 14: @@
 |notes        = More superfunctions represented through [[elementary function]]s can be found in
 <ref name="factorial">
-http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
+http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1/ D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
-</ref>.
+</ref>. Expressions for the more general case are suggested in the article [[Logistic sequence]].
 <b>Copyleft</b> 2011 by Dmitrii Kouznetsov.
-The free use is allowed.
+Previously, this image appeared at http://tori.ils.uec.ac.jp/TORI/index.php/File:Elutin1a4tori.jpg ; the free use is allowed.
 ==References==
@@ Line 18: / Line 25: @@
 |versions     = http://tori.ils.uec.ac.jp/TORI/index.php/File:Elutin1a4tori.jpg
 }}
 == Licensing ==
 {{CC|zero|1.0}}