Tuples[{1,-1},5]
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odm=DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot]
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MatrixPlot[%]
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hg=Graph[Position[DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot],1]]
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hdm=GraphDistanceMatrix[hg]
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MatrixPlot[%]
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odm-hdm
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EdgeCount[%127]
In[]:=
640
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Case of size 3
Case of size 3
Tuples[{1,-1},3]
In[]:=
{{1,1,1},{1,1,-1},{1,-1,1},{1,-1,-1},{-1,1,1},{-1,1,-1},{-1,-1,1},{-1,-1,-1}}
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DistanceMatrix[Tuples[{1,-1},3],DistanceFunctionDot]
In[]:=
{{3,1,1,1,1,1,1,3},{1,3,1,1,1,1,3,1},{1,1,3,1,1,3,1,1},{1,1,1,3,3,1,1,1},{1,1,1,3,3,1,1,1},{1,1,3,1,1,3,1,1},{1,3,1,1,1,1,3,1},{3,1,1,1,1,1,1,3}}
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Position[%,1]
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Graph[%]
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SimpleGraph[%]
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GraphDistanceMatrix
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{{0,1,1,1,1,1,1,2},{1,0,1,1,1,1,2,1},{1,1,0,1,1,2,1,1},{1,1,1,0,2,1,1,1},{1,1,1,2,0,1,1,1},{1,1,2,1,1,0,1,1},{1,2,1,1,1,1,0,1},{2,1,1,1,1,1,1,0}}
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%135
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{{3,1,1,1,1,1,1,3},{1,3,1,1,1,1,3,1},{1,1,3,1,1,3,1,1},{1,1,1,3,3,1,1,1},{1,1,1,3,3,1,1,1},{1,1,3,1,1,3,1,1},{1,3,1,1,1,1,3,1},{3,1,1,1,1,1,1,3}}
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Transitivity
Transitivity
(First/@Gather[ParallelMapMonitored[Catch[Module[{u},Do[u=MultiwaySystem[#,"A",t,"BranchialGraphStructure"];If[VertexCount[u]>150||EdgeCount[u]>150,Throw[u]],{t,9}];u]]#&,Catenate[allrules]],IsomorphicGraphQ[First[#],First[#2]]&])
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TransitiveClosureGraph
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FindClique
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{{AAAAAAAAAAB,AAAAAAABB,AAAAAABAB,AAAAABAAB,AAAABAAAB,AAABAAAAB,AABAAAAAB,ABAAAAAAB,BAAAAAAAB}}
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FindClique
,8,All
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FindClique
,{7},All
In[]:=
{{AABAAAAAB,BAAAAAAAB,BAAAABB,BAAABAB,BAABAAB,BABAAAB,BBAAAAB},{AAABAAAAB,ABAAAAAAB,ABAAABB,ABAABAB,ABABAAB,ABBAAAB,BBAAAAB},{AAAAAABAB,AAAABAAAB,AAAABBB,AAABBAB,AABABAB,ABAABAB,BAAABAB},{AAAAAAABB,AAAAABAAB,AAAABBB,AAABABB,AABAABB,ABAAABB,BAAAABB},{AAAAABAAB,AAABAAAAB,AAABABB,AAABBAB,AABBAAB,ABABAAB,BAABAAB},{AAAABAAAB,AABAAAAAB,AABAABB,AABABAB,AABBAAB,ABBAAAB,BABAAAB}}
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HighlightGraph
,%
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Models of Hilbert Space
Models of Hilbert Space
Graphics3D[Point[Tuples[{-1,1},3]]]
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Tuples[{-1,1},5]
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GridGraph[Table[2,5]]
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Position[DistanceMatrix[Tuples[{-1,1},5],DistanceFunction(Norm[#1-#2]^2&)],4]
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Graph[%]
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SimpleGraph[%]
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GlobalClusteringCoefficient
In[]:=
0
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RaggedMeanAroundValuesGraphNeighborhoodVolumes
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{1,6,16,26,31,32}
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Differences[%]
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{5,10,10,5,1}
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Table[First@Values[GraphNeighborhoodVolumes[GridGraph[Table[2,n]],{1},Automatic]],{n,10}]
In[]:=
{{1,2},{1,3,4},{1,4,7,8},{1,5,11,15,16},{1,6,16,26,31,32},{1,7,22,42,57,63,64},{1,8,29,64,99,120,127,128},{1,9,37,93,163,219,247,255,256},{1,10,46,130,256,382,466,502,511,512},{1,11,56,176,386,638,848,968,1013,1023,1024}}
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https://oeis.org/A008949
ListLinePlot[%151]
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ListLinePlot[Table[First@Values[GraphNeighborhoodVolumes[GridGraph[Table[2,n]],{1},Automatic]]/2^n,{n,10}],PlotRangeAll]
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ListLinePlot[Log[10,%151]]
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Ternary n-cube:
Table[First@Values[GraphNeighborhoodVolumes[GridGraph[Table[3,n]],{1},Automatic]],{n,10}]
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ListLinePlot[%]
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ListLinePlot[Log[10,%158]]
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T(n, 0) = 1, T(n, n) = 2^n, T(n, k) = T(n-1, k-1) + T(n-1, k)
Accumulate/@Table[Binomial[n,i],{n,0,20},{i,0,n}]
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Table[First@Values[GraphNeighborhoodVolumes[GridGraph[Table[2,n]],{1},Automatic]],{n,10}]
In[]:=
{{1,2},{1,3,4},{1,4,7,8},{1,5,11,15,16},{1,6,16,26,31,32},{1,7,22,42,57,63,64},{1,8,29,64,99,120,127,128},{1,9,37,93,163,219,247,255,256},{1,10,46,130,256,382,466,502,511,512},{1,11,56,176,386,638,848,968,1013,1023,1024}}
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This distance was Euclidean distance....
This distance was Euclidean distance....
But in Hilbert space, distance is dot product....
But in Hilbert space, distance is dot product....
DistanceMatrix[Tuples[{-1,1},5],DistanceFunction(Norm[#1-#2]^2&)]//ArrayPlot
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Table[ArrayPlot[DistanceMatrix[Tuples[{0,1},n],DistanceFunction(Norm[#1-#2]^2&)],ImageSizeTiny],{n,1,6}]
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GraphDistanceMatrix[GridGraph[Table[2,4]]]//MatrixPlot
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DistanceMatrix[Tuples[{0,1},3],DistanceFunction(Norm[#1-#2]^2&)]
In[]:=
{{0,1,1,2,1,2,2,3},{1,0,2,1,2,1,3,2},{1,2,0,1,2,3,1,2},{2,1,1,0,3,2,2,1},{1,2,2,3,0,1,1,2},{2,1,3,2,1,0,2,1},{2,3,1,2,1,2,0,1},{3,2,2,1,2,1,1,0}}
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GraphDistanceMatrix[GridGraph[Table[2,3]]]
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{{0,1,1,2,1,2,2,3},{1,0,2,1,2,1,3,2},{1,2,0,1,2,3,1,2},{2,1,1,0,3,2,2,1},{1,2,2,3,0,1,1,2},{2,1,3,2,1,0,2,1},{2,3,1,2,1,2,0,1},{3,2,2,1,2,1,1,0}}
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Comparisons
Comparisons
ParallelTable[RaggedMeanAround[Values[GraphNeighborhoodVolumes[MultiwaySystem[{"""A","""B"},"A",t,"BranchialGraphStructure"]]]],{t,2,10}]
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ListLinePlot[Log[10,%]]
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Finite Hilbert space
Finite Hilbert space
Approximating Hilbert Space [WRONG]
Approximating Hilbert Space [WRONG]
odm=(5-DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot])/2
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MatrixPlot[%]
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SimpleGraph[Position[(5-DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot])/2,1]]
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GraphDistanceMatrix[%]
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MatrixPlot[%]
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%168-%172
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RaggedMeanAround[Values[GraphNeighborhoodVolumes[SimpleGraph[Position[(5-DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot])/2,1]]]]]
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{1,11,32}
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Table[RaggedMeanAround[Values[GraphNeighborhoodVolumes[SimpleGraph[Position[(n-DistanceMatrix[Tuples[{1,-1},n],DistanceFunctionDot])/2,1]]]]],{n,2,10}]
In[]:=
{{1,3,4},{1,7,8},{1,9,16},{1,11,32},{1,13,44,64},{1,15,58,128},{1,17,74,186,256},{1,19,92,260,512},{1,21,112,352,772,1024}}
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N[Ratios[#]]&/@%
In[]:=
{{3.,1.33333},{7.,1.14286},{9.,1.77778},{11.,2.90909},{13.,3.38462,1.45455},{15.,3.86667,2.2069},{17.,4.35294,2.51351,1.37634},{19.,4.84211,2.82609,1.96923},{21.,5.33333,3.14286,2.19318,1.32642}}
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ListLinePlot[%178]
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Distance 1 apart is like sphere packing....
Distance 1 apart is like sphere packing....
E8 lattice ?
E8 lattice ?
Next try
Next try
Tuples[{1,-1},4]
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DistanceMatrix[Tuples[{1,-1},4],DistanceFunction
Non-Flat Hilbert Space
Non-Flat Hilbert Space
Projective Hilbert space
Projective Hilbert space
Mod out by projective transformation
GridGraph[Table[3,5]]
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GridGraph[Table[2,5]]
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GridGraph[Table[2,2]]
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GridGraph[Table[5,2]]
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There is a finite-dimensional projective Hilbert space, that has a Kahler metric (CP^n)
There is a finite-dimensional projective Hilbert space, that has a Kahler metric (CP^n)
SimpleGraph[TorusGraph[Table[3,5]]]
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RaggedMeanAround[Values[GraphNeighborhoodVolumes[%]]]
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{1,11,51,131,211,243}
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Accumulate/@Table[Binomial[n,i],{n,0,10},{i,0,n}]
In[]:=
{{1},{1,2},{1,3,4},{1,4,7,8},{1,5,11,15,16},{1,6,16,26,31,32},{1,7,22,42,57,63,64},{1,8,29,64,99,120,127,128},{1,9,37,93,163,219,247,255,256},{1,10,46,130,256,382,466,502,511,512},{1,11,56,176,386,638,848,968,1013,1023,1024}}
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Table[Graph[TorusGraph[Table[3,n]],ImageSizeTiny],{n,4}]
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GraphDiameter
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4
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RaggedMeanAround[Values[GraphNeighborhoodVolumes[#]]]&/@%
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{{1,3},{1,5,9},{1,7,19,27},{1,9,33,65,81}}
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Ratios/@%//N
In[]:=
{{3.},{5.,1.8},{7.,2.71429,1.42105},{9.,3.66667,1.9697,1.24615}}
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