iterative b -normal examples from one of Bailey's papers
- To: mathgroup at smc.vnet.net
- Subject: [mg52088] iterative b -normal examples from one of Bailey's papers
- From: Roger Bagula <tftn at earthlink.net>
- Date: Wed, 10 Nov 2004 04:45:35 -0500 (EST)
- Reply-to: tftn at earthlink.net
- Sender: owner-wri-mathgroup at wolfram.com
These are some examples that demonstrate what Dr. Bailey means when he says that Pi is b normal to base 16. It means that a certain type of iteration is equal probably on the interval [0,1] ( Like the classical pseudorandom variable) These b normal formulations are by their definition a Markov type of chaotic iteration. (* Bailey's Log[2] b-normal example*) Clear[x,a,digits] digits=200 x[n_]:=x[n]=Mod[2*x[n-1]+1/n,1] x[0]=0 a=Table[N[x[n],digits],{n,0,digits}]; ListPlot[a,PlotJoined->True,PlotRange->All] b=Sort[Table[N[x[n],digits],{n,0,digits}]]; ListPlot[b,PlotJoined->True,PlotRange->All] Fit[digits*b,{1,x},x] (* Log[9/10] example sum*) Clear[f,a] f[n_]=1/(n*10^n) digits=200 a=Table[N[f[n],digits],{n,1,digits}]; b=N[-Apply[Plus,a],digits] b-N[Log[9/10],digits] (* second example of b normal sequences: 1/Sqrt[E]*) Clear[x,a,digits] digits=200 x[n_]:=x[n]=Mod[4*x[n-1]+(1/(2*n)!)*((4*n+1)/(4*n+2)),1] x[0]=0 a=Table[N[x[n],digits],{n,0,digits}]; ListPlot[a,PlotJoined->True,PlotRange->All] b=Sort[Table[N[x[n],digits],{n,0,digits}]]; ListPlot[b,PlotJoined->True,PlotRange->All] Fit[digits*b,{1,x},x] N[Sum[(1/(4^n))*(1/(2*n)!)*((4*n+1)/(4*n+2)),{n,0, digits}],digits] 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