85 lines
2.1 KiB
Julia
85 lines
2.1 KiB
Julia
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using LinearAlgebra
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using AbstractAlgebra
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function printgood(msg)
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printstyled("✓", color = :green)
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println(" ", msg)
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end
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function printbad(msg)
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printstyled("✗", color = :red)
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println(" ", msg)
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end
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F, gens = rational_function_field(AbstractAlgebra.Rationals{BigInt}(), ["a₁", "a₂", "b₁", "b₂", "c₁", "c₂"])
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a = gens[1:2]
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b = gens[3:4]
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c = gens[5:6]
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# three mutually tangent spheres which are all perpendicular to the x, y plane
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gram = [
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-1 1 1;
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1 -1 1;
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1 1 -1
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]
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eig = eigen(gram)
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n_pos = count(eig.values .> 0.5)
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n_neg = count(eig.values .< -0.5)
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if n_pos + n_neg == size(gram, 1)
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printgood("Non-degenerate subspace")
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else
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printbad("Degenerate subspace")
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end
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sig_rem = Int64[ones(1-n_pos); -ones(4-n_neg)]
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unk = hcat(a, b, c)
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M = matrix_space(F, 5, 5)
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big_gram = M(F.([
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diagm(sig_rem) unk;
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transpose(unk) gram
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]))
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r, p, L, U = lu(big_gram)
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if isone(p)
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printgood("Found a solution")
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else
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printbad("Didn't find a solution")
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end
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solution = transpose(L)
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mform = U * inv(solution)
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vals = [0, 0, 0, 1, 0, -3//4]
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solution_ex = [evaluate(entry, vals) for entry in solution]
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mform_ex = [evaluate(entry, vals) for entry in mform]
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std_basis = [
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0 0 0 1 1;
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0 0 0 1 -1;
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1 0 0 0 0;
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0 1 0 0 0;
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0 0 1 0 0
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]
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std_solution = M(F.(std_basis)) * solution
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std_solution_ex = std_basis * solution_ex
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println("Minkowski form:")
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display(mform_ex)
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big_gram_recovered = transpose(solution_ex) * mform_ex * solution_ex
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valid = all(iszero.(
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[evaluate(entry, vals) for entry in big_gram] - big_gram_recovered
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))
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if valid
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printgood("Recovered Gram matrix:")
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else
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printbad("Didn't recover Gram matrix. Instead, got:")
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end
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display(big_gram_recovered)
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# this should be a solution
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hand_solution = [0 0 1 0 0; 0 0 -1 2 2; 0 0 0 1 -1; 1 0 0 0 0; 0 1 0 0 0]
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unmix = Rational{Int64}[[1//2 1//2; 1//2 -1//2] zeros(Int64, 2, 3); zeros(Int64, 3, 2) Matrix{Int64}(I, 3, 3)]
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hand_solution_diag = unmix * hand_solution
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big_gram_hand_recovered = transpose(hand_solution_diag) * diagm([1; -ones(Int64, 4)]) * hand_solution_diag
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println("Gram matrix from hand-written solution:")
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display(big_gram_hand_recovered)
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