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MathGroup Archive 2004

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Martix reconstruction by tensors of nonlinearity in an interger sequence

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50410] Martix reconstruction by tensors of nonlinearity in an interger sequence
  • From: "Roger L. Bagula" <rlbtftn at netscape.net>
  • Date: Thu, 2 Sep 2004 04:34:43 -0400 (EDT)
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

(* using a Hofstadter type (1,4) chaotic recuring sequence as a pattern*)
(* set of solved Markov matrices are used to reconstruct to sequentual 
domain*)
(* using an average Matrix to bridge extremes on nonlinearity*)
(* the reconstruction is extended to 60 from 32 where the first "bad 
spot" happens*)
(* periodic "bad spots" that run for several matrices in a row exist*)
Clear[fib,A,M,Hofstadter,ave]
digits=60
Hofstadter[n_Integer?Positive] :=Hofstadter[n] =Hofstadter[n - 
Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-4]]
Hofstadter[1] = Hofstadter[2] =1;Hofstadter[3] =2;  Hofstadter[0]= 0;
Hofstadter[4]= 3;
n0=4
(* pattern matrices of the chaotic sequence*)
A[k_]=Table[Hofstadter[k+i+j-2],{i,1,n0},{j,1,n0}]
M=Array[f,{n0,n0}]
m1=Flatten[M]
(* linear Markov matrix solutions *)
a=Table[Flatten[M/.Solve[A[n]-M.A[n-1]==0 ,m1],1],{n,1, digits}];
(* function for average matrix calculation*)
ave[n_Integer?Positive] :=ave[n] =	(ave[n-1]*(n-1)+a[[n]])/n
ave[1]=a[[1]];
c=Table[ave[n],{n,1,32}];
c[[32]]
(* Martix reconstruction tensors skipping the "bad spots" by 
substitution of an average matrix*)
B[n_]:=If[(n<33||n>33)&&(n<42||n>45)&&(n<56||n>59),a[[n]].B[n-1],c[[32]].B[n-1]]
  B[0] = {{0, 1, 1, 2}, {1, 1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}};
(* output sequence of the reconstruction*)
b=Flatten[Table[Floor[B[n][[1,1]]],{n,0,digits}]]
ListPlot[b,PlotJoined->True,PlotRange->All]

{0,1,1,2,3,4,5,5,6,6,7,8,8,9,9,10,11,11,11,12,12,13,14,14,15,15,16,17,17,17,
 
18,18,19,20,20,21,22,20,22,22,21,24,24,25,25,24,27,28,24,26,31,27,30,31,27,
   32,30,33,31,27,38}
Respectfully, Roger L. Bagula
tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 
619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/


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