an [0,1] iterative based on the Farey tree that isn't b-normal
- To: mathgroup at smc.vnet.net
- Subject: [mg52089] an [0,1] iterative based on the Farey tree that isn't b-normal
- From: Roger Bagula <tftn at earthlink.net>
- Date: Wed, 10 Nov 2004 04:45:38 -0500 (EST)
- Reply-to: tftn at earthlink.net
- Sender: owner-wri-mathgroup at wolfram.com
This is an iteration based on two transcental numbers E and Pi. Any other pair of rational transcendentals can be used. The result is limited ( not very well here!) to the interval [0,1] It isn't actually a very good way to manufacture transcendentals, but it works most of the time. The idea is to present a [0,1] iterative that doesn't depend on the Modulo one fractional part and isn't b-normal by the Bailey definition. Clear[x,y,a,b,s,g,a0,f] (* transcendental Farey tree recursive function*) t1=Pi; t2=E; f[a_,b_]:=((a/b)/(1-a/b))/;0<=a/b<=1/2 f[a_,b_]:=((1-a/b)/(a/b))/;1/2<a/b<=1 g[n_]:=g[n]=If[( g[n-1]<1/2)&&(g[n-1]>0),N[g[n-1]/(1-g[n-1]),20],If[ g[n-1]<1,N[(1-g[n-1])/g[n-1],20],N[(E/Pi)^2,20]]] g[0]=N[f[t2,t1],20]; a0=Table[If [g[n]<1,g[n],0],{n,0,400}] ListPlot[a0,PlotJoined->True,PlotRange->All] a1=Sort[a0] ListPlot[a1,PlotJoined->True,PlotRange->All] Respectfully, Roger L. Bagula tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn at netscape.net URL : http://home.earthlink.net/~tftn