imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user-pd
| | Importing file |
| |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 : $F(z)= \frac{1}{2}(1−\cos(2z))$ : $G(z)=F^{-1}(z)=\log_2(\arccos(1\!−\!2z))$
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = March 2011
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| |pub-country = Japan
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| |notes = More superfunctions represented through [[elementary function]]s can be found in
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| <ref name="factorial">
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| 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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| </ref>.
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| <b>Copyleft</b> 2011 by Dmitrii Kouznetsov.
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| The free use is allowed.
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| ==References==
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| <references/>
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| |versions = http://tori.ils.uec.ac.jp/TORI/index.php/File:Elutin1a4tori.jpg
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| }}
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| == Licensing ==
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| {{CC|zero|1.0}}
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