https://mizugadro.mydns.jp/t/index.php?title=Zooming_equation&feed=atom&action=history Zooming equation - Revision history 2021-04-22T02:10:29Z Revision history for this page on the wiki MediaWiki 1.31.14 https://mizugadro.mydns.jp/t/index.php?title=Zooming_equation&diff=28000&oldid=prev T: Text replacement - "\$([^\$]+)\$" to "\$$\1\$$" 2019-07-30T09:26:12Z <p>Text replacement - &quot;\$([^\$]+)\$&quot; to &quot;\$$\1\$$&quot;</p> <a href="https://mizugadro.mydns.jp/t/index.php?title=Zooming_equation&amp;diff=28000&amp;oldid=14942">Show changes</a> T https://mizugadro.mydns.jp/t/index.php?title=Zooming_equation&diff=14942&oldid=prev Maintenance script at 22:07, 30 November 2018 2018-11-30T22:07:17Z <p></p> <a href="https://mizugadro.mydns.jp/t/index.php?title=Zooming_equation&amp;diff=14942&amp;oldid=962">Show changes</a> Maintenance script https://mizugadro.mydns.jp/t/index.php?title=Zooming_equation&diff=962&oldid=prev Maintenance script at 06:01, 20 June 2013 2013-06-20T06:01:52Z <p></p> <p><b>New page</b></p><div>[[Zooming equation]] is tentative name for the equation<br /> <br /> (1) $~ ~ ~ T\big(f(z)\big)= f( s\, z)~$<br /> <br /> where $~T~$ is some given function and $~s~$ is some given constant. Usually it is assumed that $~T~$ is holomorphic at least in some vicinity of zero, and $~s\!=\!T'(0)~$. Function $~f~$ is requested to built-up.<br /> <br /> The physical sense of function $~T~$ is transfer function; it describes the variation of value of function $~f~$ at the scaling, zooming of its argument with factor $~s~$.<br /> <br /> The tentative name for the solution $~f~$ of the [[zooming equation]] is [[zooming function]].<br /> <br /> ==Schroeder equaiton==<br /> The solution $~f$ of the zooming equation is related to the solution $~g~$ of the [[Schroeder equation]]<br /> <br /> (2) $~ ~ ~ g\big(T(z)\big)= s \, g(z)$<br /> <br /> it is assumed that $~f=g^{-1}$ and $~g=f^{-1}$, id est, in wide ranges of values of $z$, the relations below hold:<br /> <br /> (3) $~ ~ ~ f(g(z))=z$<br /> <br /> (4) $~ ~ ~ g(f(z))=z$<br /> <br /> ==[[Regular iteration]]==<br /> <br /> Assuming that zero is [[fixed point]] of the transfer function $~T~$ is holomorphioc (regular) in vicinity of zero, <br /> the [[zooming function]] $~f~$ and the [[Schroeder function]] $~g~$ can be used to construct the non–integer [[iterate]] $~T^r~$, that is regular in vicinity of zero, id est, the [[regular iterate]]<br /> <br /> (5) $\displaystyle ~ ~ ~ T^r(z)=f\Big(r\,s\, g(z)\Big)$<br /> <br /> Such a rule can be extended to the cases of other fixed points $~L~$, performing the corresponding transform of the transfer function from $~T~$ to $~t~$,<br /> assuming that<br /> <br /> (6) $~ ~ ~ t(z)=T(z\!+\!L)-L$<br /> <br /> and treating $~t~$ as a new transfer function with fixed point zero.<br /> After to construct iterates of function $~t~$, regular at zero, the $~r$th iteration of function $~T~$, regular at $~L~$, can be expressed as<br /> <br /> (7) $~ ~ ~ T^t(z)=t^r(z\!-\!L)+L$<br /> <br /> If the transfer function $~T~$ has more than one [[fixed point]]s, then, the non-integer iterate $~T^r~$, regular at one of these fixed points, has no need to be regular at another [[fixed point]]; another fixed point may be [[branchpoint]] &lt;ref&gt;<br /> http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br /> http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf <br /> D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.<br /> &lt;/ref&gt;. The non-integer iterates of various functions can be constructed also with the [[superfunction]] and [[Abel function]], considering the [[transfer equation]] instead of the [[zooming equation]]<br /> &lt;ref&gt;<br /> http://tori.ils.uec.ac.jp/PAPERS/2010logistie.pdf<br /> http://www.springerlink.com/content/u712vtp4122544x4 <br /> D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31) <br /> DOI 10.3103/S0027134910020049 <br /> &lt;/ref&gt;&lt;ref&gt;<br /> http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf <br /> http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1<br /> D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Russian version: p.8-14)<br /> &lt;/ref&gt;.<br /> <br /> ==References==<br /> &lt;references/&gt; <br /> <br /> [[Category:Zooming equation]]<br /> [[Category:Schroeder equation]]<br /> [[Category:Transfer equation]]<br /> [[Category:Zooming function]]<br /> [[Category:Superfunctions]]<br /> [[Category:Fractional iterate]]<br /> [[Category:Regular iterate]]<br /> [[Category:Articles in English]]</div> Maintenance script