Re: neat sums and pattered randomness

• To: mathgroup at smc.vnet.net
• Subject: [mg52248] Re: neat sums and pattered randomness
• From: Roger Bagula <tftn at earthlink.net>
• Date: Wed, 17 Nov 2004 02:20:16 -0500 (EST)
• References: <cnbn5l\$9vq\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```I thought to use a method I just developed for maps on this pair of
iterations:
(* Pair Iteration gives a  Line :y=-x+1*)
x[n_]:=x[n]=Mod[x[n-1]*2+If[Mod[n,2]==1,n/(n+1),1/(n+1)],1]
y[n_]:=y[n]=Mod[y[n-1]*2+If[Mod[n,2]==1,1/(n+1),n/(n+1)],1]
x[0]=0;y[0]=0;
a0=Table[{x[n],y[n]},{n,0, 200}];
ListPlot[a0,PlotJoined->True, PlotRange->All]
Plot[-x+1,{x,0,1}]

Roger Bagula wrote:

>In my fractal nonlinear IFS work I have used the rational pair
>(n/(n+1),1/(n+1))
>to produce several new fractals.
>I know that it behaves very much in IFS like a nonlinear Cantor set.
>I made up Log[2] like sums alternating the pairs.
>The result is two irrational numbers that are summed to one.
>In the iterations based on these sum functions,
>I get patterned noise, but they still give a sorted slope of one.
>The result appears to be a paired noise pattern.
>
>(* a pair of sums from rational pairs (n/(1+n),1/(n+1))*)
>(*
>1st=0.5224031171170045693773071024046350601893524864083449381053044765826974398161552455727317173783003561708929280568165560107397662133885113895083716587179298436322129249418632659176904330363338074199274*)
>(*
>2nd=0.4775968828829954306226928975953649398106475135916550618946949011157747740696740400208629046092755848038901998264701338775987548167280850721475099467997575850060913783991812218340970953593189635761664*)
>(*1st+2nd=1*)
>f[n_]=If[Mod[n,2]==1,1/((n+1)*2^n),n/((n+1)*2^n)]
>digits=200
>a=Table[N[f[n],digits],{n,1,digits}];
>b=N[Apply[Plus,a],digits]
>Clear[f,a,b]
>f[n_]=If[Mod[n,2]==1,n/((n+1)*2^n),1/((n+1)*2^n)]
>a=Table[N[f[n],digits],{n,1,digits}];
>b=N[Apply[Plus,a],digits]
>
>(* iterations based on these that have patterns in them*)
>x[n_]:=x[n]=Mod[x[n-1]*2+If[Mod[n,2]==1,1/(n+1),n/(n+1)],1]
>    x[0]=0
>Clear[a,b]
>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]
>Clear[x]
>x[n_]:=x[n]=Mod[x[n-1]*2+If[Mod[n,2]==1,n/(n+1),1/(n+1)],1]
>    x[0]=0
>Clear[a,b]
>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]
>Respectfully, Roger L. Bagula
>
>alternative email: rlbtftn at netscape.net
>
>
>

--
Respectfully, Roger L. Bagula