matrix switching Markov sequence

• To: mathgroup at smc.vnet.net
• Subject: [mg51609] matrix switching Markov sequence
• From: Roger Bagula <tftn at earthlink.net>
• Date: Wed, 27 Oct 2004 01:54:20 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```These four matrices give minimal Pisot type eigenvalues.
I call them pseudopermutations because they act
very much like permutations and are Determinant one.
They don't really form a matrix multiplication group, but are an
SL(3,R) set of matrices.
The second version below is the vector array version set up to give a
single
output integer.
I had considered them as a marriage group type that would take a
{1/4,1/2,1/4} type gene array of four types of marriage types and
permutate it
so that lethal crosses wouldn't occur. Most such marriage groups are
based on 4by4
marriage group permutations not 3by3. I've exchanged the matrices for
the types
and made the operations on the genetic types.
I have no way of proving
they will work better than the ones the primitive societies used!

(* all real 3by3 Markov sequence of  a four element pseudopermutation *)
digits=25
(* each of these matrices gives a Minimal Pisot type of eigenvalue*)
s1={{0,1,0},{1,0,1},{1,0,0}}
s2={{0,0,1},{1,0,1},{0,1,0}}
s3={{0,1,0},{0,0,1},{1,1,0}}
s4={{0,1,1},{1,0,0},{0,1,0}}
s={s1,s2,s3,s4}
A[n_]:=s[[1+Mod[n,4]]].A[n-1];
A[0]:=s[[1]];
(* sequence of 3by3 matrices made with  a four element pseudopermutation *)
b=Flatten[Table[A[n],{n,0,digits}]]

{0,1,0,1,0,1,1,0,0,1,0,0,1,1,0,1,0,1,1,1,0,1,0,1,2,1,0,3,1,1,1,1,0,1,0,1,1,1,

0,4,1,2,3,1,1,3,1,1,4,2,1,4,1,2,4,2,1,4,1,2,7,3,2,11,4,4,4,2,1,4,1,2,4,2,1,

15,5,6,11,4,4,11,4,4,15,6,5,15,5,6,15,6,5,15,5,6,26,10,9,41,15,15,15,6,5,15,
5,6,15,6,5,56,20,21,41,15,15,41,15,15,56,21,20,56,20,21,56,21,20,56,20,21,

97,36,35,153,56,56,56,21,20,56,20,21,56,21,20,209,76,77,153,56,56,153,56,56,
209,77,76,209,76,77,209,77,76,209,76,77,362,133,132,571,209,209,209,77,76,

209,76,77,209,77,76,780,285,286,571,209,209,571,209,209,780,286,285,780,285,
286,780,286,285,780,285,286,1351,495,494,2131,780,780,780,286,285,780,285,
286,780,286,285,2911,1065,1066,2131,780,780,2131,780,780,2911,1066,1065,
2911,1065,1066}

(* all real 3by3 Markov sequence of  a four element pseudopermutation *)
digits=50
(* each of these matrices gives a Minimal Pisot type of eigenvalue*)
s1={{0,1,0},{1,0,1},{1,0,0}}
s2={{0,0,1},{1,0,1},{0,1,0}}
s3={{0,1,0},{0,0,1},{1,1,0}}
s4={{0,1,1},{1,0,0},{0,1,0}}
s={s1,s2,s3,s4}
A[n_]:=s[[1+Mod[n,4]]].A[n-1];
A[0]:={0,1,1};
(* sequence of 3 arrays made with  a four element pseudopermutation *
b=Table[A[n][[1]],{n,0,digits}]

{{0,1,1,3,1,3,4,11,4,11,15,41,15,41,56,153,56,153,209,571,209,571,780,2131,780,

2131,2911,7953,2911,7953,10864,29681,10864,29681,40545,110771,40545,110771,
151316,413403,151316,413403,564719,1542841,564719,1542841,2107560,5757961,
2107560,5757961,7865521}
Respectfully, Roger L. Bagula