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MathGroup Archive 2004

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matrix method for minimal pisot theta sequences as matrices

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50387] matrix method for minimal pisot theta sequences as matrices
  • From: "Roger L. Bagula" <rlbtftn at netscape.net>
  • Date: Wed, 1 Sep 2004 01:49:28 -0400 (EDT)
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

The Markov matrix sequence of :
Mf={{0,1},{1,1}]--> Fibonacci
M0={{0,1,0},{0,0,1},{1,1,0}}--> Minmal cubic Pisot eigenvalues/ theta0 : 
1-> {2,3}
M1={{0,1,0,0},{0,0,1,0},{0,0,0,1},{1,0,0,1}}-->Minmal quartic Pisot 
eigenvalues/ theta1
1-> {1,4}
Produce tensor like sequences when used in a Markov
recursive manner.

(* Minimal Pisot 3by3 Markov sequence*)
digits=21
M={{0,1,0},{0,0,1},{1,1,0}}
i=IdentityMatrix[3]
Det[-M+x*i]
A[n_]:=M.A[n-1];
A[0]:={{0,1,1},{1,1,2},{1,2,2}};
(* flattened sequence of 3by3matrices made with a Minimal Pisot recurrence*)
b=Flatten[Table[M.A[n],{n,0,digits}]]
Dimensions[b][[1]]
ListPlot[b,PlotJoined->True]

(* Minimal Pisot theta 1 4by4 Markov sequence*)
digits=15
M={{0,1,0,0},{0,0,1,0},{0,0,0,1},{1,0,0,1}}

A[n_]:=M.A[n-1];
A[0]:={{1,1,1,1},{1,1,1,2},{1,1,2,3},{1,2,3,4}};
i=IdentityMatrix[4]
Det[M-x*i]
(* flattened sequence of 4by4 matrices made with a theta1 Minimal Pisot
     recurrence*)
b=Flatten[Table[M.A[n],{n,0,digits}]]
Dimensions[b][[1]]
b=Flatten[Table[M.A[n],{n,0,digits}]]

Mathematica solver for Minmal Pisot matrix Markov:
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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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