imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user-pd
| | Importing file |
| |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:
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| F[c_, z_] = 1/2 (1 - Cos[2^c ArcCos[1 - 2 z]])
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| Plot[{F[1.5, x], F[1, x], F[.8, x], F[.5, x], F[.2, x]}, {x, 0, 1}]
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| 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>
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| <math>f^c(z)=F(c+G(z))</math>
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| at
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| <math>F(z)= \frac{1}{2}(1−\cos(2z))</math>
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| <math>G(z)=F^{-1}(z)=\log_2(\arccos(1\!−\!2z))</math>
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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 = Expressions for the more general case are suggested in the article [[Logistic sequence]].
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| 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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| Previously, this image appeared at http://tori.ils.uec.ac.jp/TORI/index.php/File:Elutin1a4tori.jpg ; 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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