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add sqadjop.jl and unit tests
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@ -22,6 +22,9 @@ include("RGprojections.jl")
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include("orbitdata.jl")
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include("sos_sdps.jl")
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include("checksolution.jl")
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include("sqadjop.jl")
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include("1712.07167.jl")
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end # module Property(T)
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125
src/sqadjop.jl
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125
src/sqadjop.jl
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@ -0,0 +1,125 @@
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indexing(n) = [(i,j) for i in 1:n for j in 1:n if i≠j]
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function generating_set(G::AutGroup{N}, n=N) where N
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rmuls = [Groups.rmul_autsymbol(i,j) for (i,j) in indexing(n)]
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lmuls = [Groups.lmul_autsymbol(i,j) for (i,j) in indexing(n)]
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gen_set = G.([rmuls; lmuls])
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return [gen_set; inv.(gen_set)]
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end
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function E(M::MatSpace, i::Integer, j::Integer, val=1)
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@assert i ≠ j
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@assert 1 ≤ i ≤ nrows(M)
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@assert 1 ≤ j ≤ ncols(M)
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m = one(M)
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m[i,j] = val
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return m
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end
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function generating_set(M::MatSpace, n=nrows(M))
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@assert nrows(M) == ncols(M)
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elts = [E(M, i,j) for (i,j) in indexing(n)]
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return elem_type(M)[elts; inv.(elts)]
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end
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isopposite(σ::perm, τ::perm, i=1, j=2) =
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σ[i] ≠ τ[i] && σ[i] ≠ τ[j] &&
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σ[j] ≠ τ[i] && σ[j] ≠ τ[j]
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isadjacent(σ::perm, τ::perm, i=1, j=2) =
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(σ[i] == τ[i] && σ[j] ≠ τ[j]) || # first equal, second differ
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(σ[j] == τ[j] && σ[i] ≠ τ[i]) || # sedond equal, first differ
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(σ[i] == τ[j] && σ[j] ≠ τ[i]) || # first σ equal to second τ
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(σ[j] == τ[i] && σ[i] ≠ τ[j]) # second σ equal to first τ
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Base.div(X::GroupRingElem, x::Number) = parent(X)(X.coeffs.÷x)
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function Sq(RG::GroupRing, N::Integer)
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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Alt_N = [g for g in PermutationGroup(N) if parity(g) == 0]
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sq = RG()
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for σ in Alt_N
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GroupRings.addeq!(sq, *(σ(Δ₂), σ(Δ₂), false))
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end
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return sq÷factorial(N-2)
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end
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function Adj(RG::GroupRing, N::Integer)
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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Alt_N = [g for g in PermutationGroup(N) if parity(g) == 0]
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Δ₂s = Dict(σ=>σ(Δ₂) for σ in Alt_N)
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adj = RG()
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for σ in Alt_N
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for τ in Alt_N
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if isadjacent(σ, τ)
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GroupRings.addeq!(adj, *(Δ₂s[σ], Δ₂s[τ], false))
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end
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end
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end
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return adj÷factorial(N-2)^2
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end
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function Op(RG::GroupRing, N::Integer)
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if N < 4
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return RG()
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end
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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Alt_N = [g for g in PermutationGroup(N) if parity(g) == 0]
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Δ₂s = Dict(σ=>σ(Δ₂) for σ in Alt_N)
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op = RG()
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for σ in Alt_N
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for τ in Alt_N
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if isopposite(σ, τ)
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GroupRings.addeq!(op, *(Δ₂s[σ], Δ₂s[τ], false))
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end
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end
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end
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return op÷factorial(N-2)^2
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end
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for Elt in [:Sq, :Adj, :Op]
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@eval begin
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$Elt(RG::GroupRing{AutGroup{N}}) where N = $Elt(RG, N)
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$Elt(RG::GroupRing{<:MatSpace}) = $Elt(RG, nrows(RG.group))
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end
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end
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function SqAdjOp(RG::GroupRing, N::Integer)
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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Alt_N = [σ for σ in PermutationGroup(N) if parity(σ) == 0]
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sq, adj, op = RG(), RG(), RG()
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Δ₂s = Dict(σ=>σ(Δ₂) for σ in Alt_N)
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for σ in Alt_N
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GroupRings.addeq!(sq, *(Δ₂s[σ], Δ₂s[σ], false))
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for τ in Alt_N
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if isopposite(σ, τ)
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GroupRings.addeq!(op, *(Δ₂s[σ], Δ₂s[τ], false))
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elseif isadjacent(σ, τ)
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GroupRings.addeq!(adj, *(Δ₂s[σ], Δ₂s[τ], false))
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end
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end
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end
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k = factorial(N-2)
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return sq÷k, adj÷k^2, op÷k^2
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end
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93
test/1812.03456.jl
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93
test/1812.03456.jl
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@ -0,0 +1,93 @@
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@testset "Sq, Adj, Op" begin
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function isconstant_on_orbit(v, orb)
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isempty(orb) && return true
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k = v[first(orb)]
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return all(v[o] == k for o in orb)
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end
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@testset "unit tests" begin
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for N in [3,4]
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M = MatrixSpace(Nemo.ZZ, N,N)
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A = SAut(FreeGroup(N))
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@test length(PropertyT.generating_set(M)) == 2N*(N-1)
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S = PropertyT.generating_set(M)
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@test all(inv(s) ∈ S for s in S)
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@test length(PropertyT.generating_set(A)) == 4N*(N-1)
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S = PropertyT.generating_set(A)
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@test all(inv(s) ∈ S for s in S)
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end
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N = 4
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M = MatrixSpace(Nemo.ZZ, N,N)
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S = PropertyT.generating_set(M)
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@test PropertyT.E(M, 1, 2) isa MatElem
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e12 = PropertyT.E(M, 1, 2)
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@test e12[1,2] == 1
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@test inv(e12)[1,2] == -1
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@test e12 ∈ S
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@test PropertyT.isopposite(perm"(1,2,3)(4)", perm"(1,4,2)")
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@test PropertyT.isadjacent(perm"(1,2,3)", perm"(1,2)(3)")
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@test !PropertyT.isopposite(perm"(1,2,3)", perm"(1,2)(3)")
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@test !PropertyT.isadjacent(perm"(1,4)", perm"(2,3)(4)")
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@test isconstant_on_orbit([1,1,1,2,2], [2,3])
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@test !isconstant_on_orbit([1,1,1,2,2], [2,3,4])
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end
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@testset "Sq, Adj, Op" begin
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N = 4
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M = MatrixSpace(Nemo.ZZ, N,N)
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S = PropertyT.generating_set(M)
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Δ = PropertyT.Laplacian(S, 2)
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RG = parent(Δ)
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autS = WreathProduct(PermGroup(2), PermGroup(N))
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orbits = PropertyT.orbit_decomposition(autS, RG.basis)
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@test PropertyT.Sq(RG) isa GroupRingElem
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sq = PropertyT.Sq(RG)
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@test all(isconstant_on_orbit(sq, orb) for orb in orbits)
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@test PropertyT.Adj(RG) isa GroupRingElem
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adj = PropertyT.Adj(RG)
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@test all(isconstant_on_orbit(adj, orb) for orb in orbits)
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@test PropertyT.Op(RG) isa GroupRingElem
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op = PropertyT.Op(RG)
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@test all(isconstant_on_orbit(op, orb) for orb in orbits)
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sq, adj, op = PropertyT.SqAdjOp(RG, N)
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@test sq == PropertyT.Sq(RG)
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@test adj == PropertyT.Adj(RG)
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@test op == PropertyT.Op(RG)
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e = one(M)
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g = PropertyT.E(M, 1,2)
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h = PropertyT.E(M, 1,3)
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k = PropertyT.E(M, 3,4)
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edges = N*(N-1)÷2
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@test sq[e] == 20*edges
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@test sq[g] == sq[h] == -8
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@test sq[g^2] == sq[h^2] == 1
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@test sq[g*h] == sq[h*g] == 0
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# @test adj[e] == ...
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@test adj[g] == adj[h] # == ...
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@test adj[g^2] == adj[h^2] == 0
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@test adj[g*h] == adj[h*g] # == ...
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# @test op[e] == ...
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@test op[g] == op[h] # == ...
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@test op[g^2] == op[h^2] == 0
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@test op[g*h] == op[h*g] == 0
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@test op[g*k] == op[k*g] # == ...
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@test op[h*k] == op[k*h] == 0
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end
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end
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@ -23,3 +23,4 @@ solver(iters; accel=1) =
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include("1703.09680.jl")
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include("1712.07167.jl")
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include("SOS_correctness.jl")
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include("1812.03456.jl")
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