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WOLFRAM NOTEBOOK
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NestGraphTagged[n|->{2n,n+1},0,10]
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Each colored subgraph represents a particular “number system”.
Is the red subgraph connected? [No]
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NestGraphTagged[n|->{2n},1,10]
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NestGraphTagged[n|->{n+1},1,10,EdgeStyle->Blue]
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AdjacencyMatrix[%32]//MatrixPlot
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MatrixPlot[AdjacencyMatrix[NestGraphTagged[n|->{2n,n+1},0,10]]]
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[These two rules are (presumably) not straightforward tensor products]
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Table[{Count[VertexOutDegree[#],2],Count[VertexInDegree[#],2]}&[NestGraphTagged[n|->{2n,n+1},0,t]],{t,10}]
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{{1,0},{2,1},{3,1},{5,2},{8,3},{13,5},{21,8},{34,13},{55,21},{89,34}}
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Table[{Count[VertexOutDegree[#],2],Count[VertexInDegree[#],2]}&[NestGraphTagged[n|->{2n+1,3n+1},0,t]],{t,10}]
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{{1,1},{2,1},{4,1},{8,1},{16,2},{31,2},{61,3},{120,3},{238,4},{473,6}}
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Table[{Count[VertexOutDegree[#],2],Count[VertexInDegree[#],2]}&[NestGraphTagged[n|->{2n+1,3n+1},0,t]],{t,14}]
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{{1,1},{2,1},{4,1},{8,1},{16,2},{31,2},{61,3},{120,3},{238,4},{473,6},{941,8},{1875,10},{3741,13},{7470,22}}
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Monitor[Table[{Count[VertexOutDegree[#],2],Count[VertexInDegree[#],2]}&[NestGraphTagged[n|->{2n+1,3n+1},0,t]],{t,20}],t]
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{{1,1},{2,1},{4,1},{8,1},{16,2},{31,2},{61,3},{120,3},{238,4},{473,6},{941,8},{1875,10},{3741,13},{7470,22},{14919,33},{29806,51},{59562,81},{119044,122},{237967,184},{475751,289}}
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Ratios[First/@%]//N
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{2.,2.,2.,2.,1.9375,1.96774,1.96721,1.98333,1.98739,1.98943,1.99256,1.9952,1.99679,1.99719,1.99786,1.99832,1.99866,1.99898,1.99923}
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Last/@%%
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{1,1,1,1,2,2,3,3,4,6,8,10,13,22,33,51,81,122,184,289}
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Ratios[%]//N
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{1.,1.,1.,2.,1.,1.5,1.,1.33333,1.5,1.33333,1.25,1.3,1.69231,1.5,1.54545,1.58824,1.50617,1.5082,1.57065}
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ListLinePlot[%]
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gr10=NestGraphTagged[n|->{2n+1,3n+1},0,10]
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MatrixPlot[AdjacencyMatrix[gr10]]
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ArrayPlot[AdjacencyMatrix[gr10]]
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How many paths, starting e.g. at 3 of length n end in 13?
This can be thought of as being covered by the multiway graph ... and supports additional paths that aren’t “real”
(it’s a Tali-style quiver covering, because it preserves the colors)
(it’s a Tali-style quiver covering, because it preserves the colors)
For a given path, we do “homotopy lifting” to find out what the path would have been in the multiway graph.....
Lift through a series of moduli ...
Lift through a series of moduli ...
Combined DFA: every vertex is a list of the n mod {m1,m2,m3,...}
Combined DFA: every vertex is a list of the n mod {m1,m2,m3,...}
Goal is to find the “intersection word” which is a possible word to actually reach a merge point over the integers.....
Cf irreducibility of representations ?