Merge pull request #4 from kalmarek/enh/AA_MatrixAlgebra
Use Matrix Algebras
This commit is contained in:
GitHub
commit
38bf6605ca
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@ -6,6 +6,7 @@ os:
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julia:
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- 1.0
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- 1.1
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- 1.2
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- nightly
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notifications:
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email: true
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@ -2,9 +2,9 @@
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[[AbstractAlgebra]]
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deps = ["InteractiveUtils", "LinearAlgebra", "Markdown", "Random", "SparseArrays", "Test"]
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git-tree-sha1 = "c0d57a3f0618bfbb214005860b9b5e5bceafa61c"
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git-tree-sha1 = "0d4f7283435bd7e12a703a3fd58aa11224a96019"
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uuid = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
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version = "0.5.0"
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version = "0.5.2"
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[[Base64]]
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uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"
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@ -16,10 +16,10 @@ uuid = "9e28174c-4ba2-5203-b857-d8d62c4213ee"
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version = "0.8.10"
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[[BinaryProvider]]
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deps = ["Libdl", "SHA"]
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git-tree-sha1 = "c7361ce8a2129f20b0e05a89f7070820cfed6648"
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deps = ["Libdl", "Logging", "SHA"]
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git-tree-sha1 = "8153fd64131cd00a79544bb23788877741f627bb"
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uuid = "b99e7846-7c00-51b0-8f62-c81ae34c0232"
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version = "0.5.4"
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version = "0.5.5"
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[[Blosc]]
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deps = ["BinaryProvider", "CMakeWrapper", "Compat", "Libdl"]
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@ -124,19 +124,19 @@ version = "0.10.3"
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[[GroupRings]]
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deps = ["AbstractAlgebra", "LinearAlgebra", "Markdown", "SparseArrays"]
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git-tree-sha1 = "6d43558d0e7946d4e0e3d94a108344b2514e95b7"
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git-tree-sha1 = "9f41bad54217ce2605c13f10b79d1221c259ed32"
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repo-rev = "master"
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repo-url = "https://github.com/kalmarek/GroupRings.jl"
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uuid = "0befed6a-bd73-11e8-1e41-a1190947c2f5"
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version = "0.2.1"
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version = "0.3.0"
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[[Groups]]
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deps = ["AbstractAlgebra", "LinearAlgebra", "Markdown"]
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git-tree-sha1 = "64dcaa46affb4568501d35e6fae36a71a7ae5cbb"
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git-tree-sha1 = "bf267f82f4c313a6deb3db64efc7893b139d4340"
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repo-rev = "master"
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repo-url = "https://github.com/kalmarek/Groups.jl"
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uuid = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
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version = "0.2.1"
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version = "0.2.2"
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[[HDF5]]
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deps = ["BinDeps", "Blosc", "Homebrew", "Libdl", "Mmap", "WinRPM"]
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@ -192,9 +192,9 @@ version = "0.4.1"
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[[LibCURL]]
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deps = ["BinaryProvider", "Libdl"]
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git-tree-sha1 = "5ee138c679fa202ebe211b2683d1eee2a87b3dbe"
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git-tree-sha1 = "fd5fc15f2a04608fe1435a769dbbfc7959ff1daa"
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uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"
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version = "0.5.1"
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version = "0.5.2"
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[[LibExpat]]
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deps = ["Compat"]
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@ -246,12 +246,6 @@ git-tree-sha1 = "ce3b85e484a5d4c71dd5316215069311135fa9f2"
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uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
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version = "0.3.2"
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[[Nemo]]
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deps = ["AbstractAlgebra", "BinaryProvider", "InteractiveUtils", "Libdl", "LinearAlgebra", "Markdown", "Test"]
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git-tree-sha1 = "b359c25b708c3edcc2f17345d59da7169c207385"
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uuid = "2edaba10-b0f1-5616-af89-8c11ac63239a"
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version = "0.14.0"
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[[OrderedCollections]]
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deps = ["Random", "Serialization", "Test"]
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git-tree-sha1 = "c4c13474d23c60d20a67b217f1d7f22a40edf8f1"
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@ -1,7 +1,7 @@
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name = "PropertyT"
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uuid = "03b72c93-0167-51e2-8a1e-eb4ff1fb940d"
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authors = ["Marek Kaluba <kalmar@amu.edu.pl>"]
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version = "0.2.1"
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version = "0.3.0"
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[deps]
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AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
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@ -14,12 +14,11 @@ JuMP = "4076af6c-e467-56ae-b986-b466b2749572"
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LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
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Markdown = "d6f4376e-aef5-505a-96c1-9c027394607a"
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MathProgBase = "fdba3010-5040-5b88-9595-932c9decdf73"
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Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
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Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7"
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SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
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[compat]
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GroupRings = "^0.2.0"
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GroupRings = "^0.3.0"
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Groups = "^0.2.1"
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JuMP = "^0.19.0"
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@ -10,9 +10,9 @@ abstract type Settings end
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struct Naive{El} <: Settings
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name::String
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G::Group
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G::Union{Group, NCRing}
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S::Vector{El}
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radius::Int
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halfradius::Int
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upper_bound::Float64
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solver::JuMP.OptimizerFactory
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@ -21,10 +21,10 @@ end
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struct Symmetrized{El} <: Settings
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name::String
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G::Group
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G::Union{Group, NCRing}
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S::Vector{El}
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autS::Group
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radius::Int
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halfradius::Int
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upper_bound::Float64
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solver::JuMP.OptimizerFactory
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@ -32,15 +32,15 @@ struct Symmetrized{El} <: Settings
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end
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function Settings(name::String,
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G::Group, S::Vector{<:GroupElem}, solver::JuMP.OptimizerFactory;
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radius::Integer=2, upper_bound::Float64=1.0, warmstart=true)
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return Naive(name, G, S, radius, upper_bound, solver, warmstart)
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G::Union{Group, NCRing}, S::AbstractVector{El}, solver::JuMP.OptimizerFactory;
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halfradius::Integer=2, upper_bound::Float64=1.0, warmstart=true) where El <: Union{GroupElem, NCRingElem}
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return Naive(name, G, S, halfradius, upper_bound, solver, warmstart)
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end
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function Settings(name::String,
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G::Group, S::Vector{<:GroupElem}, autS::Group, solver::JuMP.OptimizerFactory;
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radius::Integer=2, upper_bound::Float64=1.0, warmstart=true)
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return Symmetrized(name, G, S, autS, radius, upper_bound, solver, warmstart)
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G::Union{Group, NCRing}, S::AbstractVector{El}, autS::Group, solver::JuMP.OptimizerFactory;
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halfradius::Integer=2, upper_bound::Float64=1.0, warmstart=true) where El <: Union{GroupElem, NCRingElem}
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return Symmetrized(name, G, S, autS, halfradius, upper_bound, solver, warmstart)
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end
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prefix(s::Naive) = s.name
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@ -79,12 +79,13 @@ end
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###############################################################################
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function warmstart(sett::Settings)
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warmstart_fname = filename(sett, :warmstart)
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try
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ws = load(filename(sett, :warmstart), "warmstart")
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@info "Loaded $(filename(sett, :warmstart))."
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@info "Loaded $warmstart_fname."
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return ws
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catch
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@info "Couldn't load $(filename(sett, :warmstart))."
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catch ex
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@warn "$(ex.msg). Not providing a warmstart to the solver."
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return nothing
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end
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end
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@ -128,10 +129,12 @@ function approximate_by_SOS(sett::Symmetrized,
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orbit_data = try
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orbit_data = load(filename(sett, :OrbitData), "OrbitData")
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@info "Loaded Orbit Data"
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@info "Loaded orbit data."
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orbit_data
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catch
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catch ex
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@warn ex.msg
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isdefined(parent(orderunit), :basis) || throw("You need to define basis of Group Ring to compute orbit decomposition!")
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@info "Computing orbit and Wedderburn decomposition..."
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orbit_data = OrbitData(parent(orderunit), sett.autS)
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save(filename(sett, :OrbitData), "OrbitData", orbit_data)
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orbit_data
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@ -245,7 +248,7 @@ function print_summary(sett::Settings)
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separator = "="^76
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info_strs = [separator,
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"Running tests for $(sett.name):",
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"Upper bound for λ: $(sett.upper_bound), on radius $(sett.radius).",
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"Upper bound for λ: $(sett.upper_bound), on halfradius $(sett.halfradius).",
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"Warmstart: $(sett.warmstart)",
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"Results will be stored in ./$(PropertyT.prepath(sett))",
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"Solver: $(typeof(sett.solver()))",
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@ -275,26 +278,39 @@ function spectral_gap(sett::Settings)
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isdir(fp) || mkpath(fp)
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Δ = try
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@info "Loading precomputed Δ..."
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loadGRElem(filename(sett,:Δ), sett.G)
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catch
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# compute
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Δ = Laplacian(sett.S, sett.radius)
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Δ = loadGRElem(filename(sett,:Δ), sett.G)
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@info "Loaded precomputed Δ."
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Δ
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catch ex
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@warn ex.msg
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@info "Computing Δ..."
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Δ = Laplacian(sett.S, sett.halfradius)
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saveGRElem(filename(sett, :Δ), Δ)
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Δ
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end
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λ, P = try
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sett.warmstart && error()
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load(filename(sett, :solution), "λ", "P")
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catch
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function compute(sett, Δ)
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@info "Computing λ and P..."
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λ, P = approximate_by_SOS(sett, Δ^2, Δ;
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solverlog=filename(sett, :solverlog))
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save(filename(sett, :solution), "λ", λ, "P", P)
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λ < 0 && @warn "Solver did not produce a valid solution!"
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λ, P
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return λ, P
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end
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if sett.warmstart
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λ, P = compute(sett, Δ)
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else
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λ, P =try
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λ, P = load(filename(sett, :solution), "λ", "P")
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@info "Loaded existing λ and P."
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λ, P
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catch ex
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@warn ex.msg
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compute(sett, Δ)
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end
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end
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info_strs = ["Numerical metrics of matrix solution:",
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@ -307,7 +323,7 @@ function spectral_gap(sett::Settings)
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@warn "The solution matrix doesn't seem to be positive definite!"
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@time Q = real(sqrt(Symmetric( (P.+ P')./2 )))
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certified_sgap = spectral_gap(Δ, λ, Q, R=sett.radius)
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certified_sgap = spectral_gap(Δ, λ, Q, R=sett.halfradius)
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return certified_sgap
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end
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@ -0,0 +1,26 @@
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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 EltaryMat(M::MatAlgebra, 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::MatAlgebra, n=nrows(M))
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elts = [EltaryMat(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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include("sqadjop.jl")
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@ -13,15 +13,19 @@ using GroupRings
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using JLD
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using JuMP
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import AbstractAlgebra: Group, Ring, perm
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import AbstractAlgebra: Group, NCRing, perm
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import MathProgBase.SolverInterface.AbstractMathProgSolver
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AbstractAlgebra.one(G::Group) = G()
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include("laplacians.jl")
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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("1712.07167.jl")
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include("1812.03456.jl")
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end # module Property(T)
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@ -70,7 +70,7 @@ end
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function central_projection(RG::GroupRing, chi::AbstractCharacter, T::Type=Rational{Int})
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result = RG(zeros(T, length(RG.basis)))
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dim = chi(RG.group())
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dim = chi(one(RG.group))
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ord = Int(order(RG.group))
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for g in RG.basis
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@ -187,8 +187,8 @@ function rankOne_projections(RBn::GroupRing{G}, T::Type=Rational{Int}) where {G<
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r = collect(1:N)
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for i in 1:N-1
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first_emb = g->Bn(Generic.emb!(Bn.P(), g, view(r, 1:i)))
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last_emb = g->Bn(Generic.emb!(Bn.P(), g, view(r, (i+1):N)))
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first_emb = g->Bn(Generic.emb!(one(Bn.P), g, view(r, 1:i)))
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last_emb = g->Bn(Generic.emb!(one(Bn.P), g, view(r, (i+1):N)))
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Sk_first = (RBn(first_emb, p) for p in Sn_rankOnePr[i])
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Sk_last = (RBn(last_emb, p) for p in Sn_rankOnePr[N-i])
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@ -238,7 +238,7 @@ end
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> The identity element `Id` and binary operation function `op` can be supplied
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> to e.g. take advantage of additive group structure.
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"""
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function generateGroup(gens::Vector{T}, r=2, Id::T=parent(first(gens))(), op=*) where {T<:GroupElem}
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function generateGroup(gens::Vector{T}, r=2, Id::T=one(parent(first(gens))), op=*) where {T<:GroupElem}
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n = 0
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R = 1
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elts = gens
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@ -4,16 +4,7 @@
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#
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###############################################################################
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function spLaplacian(RG::GroupRing, S, T::Type=Float64)
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result = RG(T)
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result[RG.group()] = T(length(S))
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for s in S
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result[s] -= one(T)
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end
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return result
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end
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function spLaplacian(RG::GroupRing, S::Vector{REl}, T::Type=Float64) where {REl<:AbstractAlgebra.ModuleElem}
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function spLaplacian(RG::GroupRing, S::AbstractVector, T::Type=Float64)
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result = RG(T)
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result[one(RG.group)] = T(length(S))
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for s in S
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@ -22,26 +13,17 @@ function spLaplacian(RG::GroupRing, S::Vector{REl}, T::Type=Float64) where {REl<
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return result
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end
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function Laplacian(S::Vector{E}, radius) where E<:AbstractAlgebra.ModuleElem
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R = parent(first(S))
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return Laplacian(S, one(R), radius)
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end
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function Laplacian(S::Vector{E}, radius) where E<:AbstractAlgebra.GroupElem
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function Laplacian(S::AbstractVector{REl}, halfradius) where REl<:Union{NCRingElem, GroupElem}
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G = parent(first(S))
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return Laplacian(S, G(), radius)
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end
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function Laplacian(S, Id, radius)
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@info "Generating metric ball of radius" radius=2radius
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@time E_R, sizes = Groups.generate_balls(S, Id, radius=2radius)
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@info "Generating metric ball of radius" radius=2halfradius
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@time E_R, sizes = Groups.generate_balls(S, one(G), radius=2halfradius)
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@info "Generated balls:" sizes
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@info "Creating product matrix..."
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rdict = GroupRings.reverse_dict(E_R)
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@time pm = GroupRings.create_pm(E_R, rdict, sizes[radius]; twisted=true)
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@time pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=true)
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RG = GroupRing(parent(Id), E_R, rdict, pm)
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RG = GroupRing(G, E_R, rdict, pm)
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Δ = spLaplacian(RG, S)
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return Δ
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end
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@ -56,7 +38,7 @@ function loadGRElem(fname::String, RG::GroupRing)
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return GroupRingElem(coeffs, RG)
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end
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function loadGRElem(fname::String, G::Group)
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function loadGRElem(fname::String, G::Union{Group, NCRing})
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pm = load(fname, "pm")
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RG = GroupRing(G, pm)
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return loadGRElem(fname, RG)
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@ -28,7 +28,7 @@ function OrbitData(RG::GroupRing, autS::Group, verbose=true)
|
|||
@time Uπs = [orthSVD(matrix_repr(p, mreps)) for p in autS_mps]
|
||||
|
||||
multiplicities = size.(Uπs,2)
|
||||
dimensions = [Int(p[autS()]*Int(order(autS))) for p in autS_mps]
|
||||
dimensions = [Int(p[one(autS)]*Int(order(autS))) for p in autS_mps]
|
||||
if verbose
|
||||
info_strs = ["",
|
||||
lpad("multiplicities", 14) * " =" * join(lpad.(multiplicities, 4), ""),
|
||||
|
|
@ -213,12 +213,12 @@ function (g::perm)(y::GroupRingElem{T, <:SparseVector}) where T
|
|||
return result
|
||||
end
|
||||
|
||||
function (p::perm)(A::MatElem)
|
||||
function (p::perm)(A::MatAlgElem)
|
||||
length(p.d) == size(A, 1) == size(A,2) || throw("Can't act via $p on matrix of size $(size(A))")
|
||||
result = similar(A)
|
||||
@inbounds for i in 1:size(A, 1)
|
||||
for j in 1:size(A, 2)
|
||||
result[i, j] = A[p[i], p[j]] # action by permuting rows and colums/conjugation
|
||||
result[p[i], p[j]] = A[i,j] # action by permuting rows and colums/conjugation
|
||||
end
|
||||
end
|
||||
return result
|
||||
|
|
@ -226,7 +226,7 @@ end
|
|||
|
||||
###############################################################################
|
||||
#
|
||||
# WreathProductElems action on Nemo.MatElem
|
||||
# WreathProductElems action on MatAlgElem
|
||||
#
|
||||
###############################################################################
|
||||
|
||||
|
|
@ -242,7 +242,7 @@ function (g::WreathProductElem)(y::GroupRingElem)
|
|||
return result
|
||||
end
|
||||
|
||||
function (g::WreathProductElem{N})(A::MatElem) where N
|
||||
function (g::WreathProductElem{N})(A::MatAlgElem) where N
|
||||
# @assert N == size(A,1) == size(A,2)
|
||||
flips = ntuple(i->(g.n[i].d[1]==1 && g.n[i].d[2]==2 ? 1 : -1), N)
|
||||
result = similar(A)
|
||||
|
|
@ -251,11 +251,11 @@ function (g::WreathProductElem{N})(A::MatElem) where N
|
|||
|
||||
@inbounds for i = 1:size(A,1)
|
||||
for j = 1:size(A,2)
|
||||
x = A[g.p[i], g.p[j]]
|
||||
x = A[i, j]
|
||||
if flips[i]*flips[j] == 1
|
||||
result[i, j] = x
|
||||
result[g.p[i], g.p[j]] = x
|
||||
else
|
||||
result[i, j] = -x
|
||||
result[g.p[i], g.p[j]] = -x
|
||||
end
|
||||
# result[i, j] = AbstractAlgebra.mul!(x, x, flips[i]*flips[j])
|
||||
# this mul! needs to be separately defined, but is 2x faster
|
||||
|
|
@ -273,8 +273,8 @@ end
|
|||
function AutFG_emb(A::AutGroup, g::WreathProductElem)
|
||||
isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
|
||||
parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
|
||||
elt = A()
|
||||
Id = parent(g.n.elts[1])()
|
||||
elt = one(A)
|
||||
Id = one(parent(g.n.elts[1]))
|
||||
flips = Groups.AutSymbol[Groups.flip_autsymbol(i) for i in 1:length(g.p.d) if g.n.elts[i] != Id]
|
||||
Groups.r_multiply!(elt, flips, reduced=false)
|
||||
Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
|
||||
|
|
|
|||
|
|
@ -0,0 +1,98 @@
|
|||
isopposite(σ::perm, τ::perm, i=1, j=2) =
|
||||
σ[i] ≠ τ[i] && σ[i] ≠ τ[j] &&
|
||||
σ[j] ≠ τ[i] && σ[j] ≠ τ[j]
|
||||
|
||||
isadjacent(σ::perm, τ::perm, i=1, j=2) =
|
||||
(σ[i] == τ[i] && σ[j] ≠ τ[j]) || # first equal, second differ
|
||||
(σ[j] == τ[j] && σ[i] ≠ τ[i]) || # sedond equal, first differ
|
||||
(σ[i] == τ[j] && σ[j] ≠ τ[i]) || # first σ equal to second τ
|
||||
(σ[j] == τ[i] && σ[i] ≠ τ[j]) # second σ equal to first τ
|
||||
|
||||
Base.div(X::GroupRingElem, x::Number) = parent(X)(X.coeffs.÷x)
|
||||
|
||||
function Sq(RG::GroupRing, N::Integer)
|
||||
S₂ = generating_set(RG.group, 2)
|
||||
ℤ = Int64
|
||||
Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
|
||||
|
||||
Alt_N = [g for g in PermutationGroup(N) if parity(g) == 0]
|
||||
|
||||
sq = RG()
|
||||
for σ in Alt_N
|
||||
GroupRings.addeq!(sq, *(σ(Δ₂), σ(Δ₂), false))
|
||||
end
|
||||
return sq÷factorial(N-2)
|
||||
end
|
||||
|
||||
function Adj(RG::GroupRing, N::Integer)
|
||||
S₂ = generating_set(RG.group, 2)
|
||||
ℤ = Int64
|
||||
Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
|
||||
|
||||
Alt_N = [g for g in PermutationGroup(N) if parity(g) == 0]
|
||||
Δ₂s = Dict(σ=>σ(Δ₂) for σ in Alt_N)
|
||||
adj = RG()
|
||||
|
||||
for σ in Alt_N
|
||||
for τ in Alt_N
|
||||
if isadjacent(σ, τ)
|
||||
GroupRings.addeq!(adj, *(Δ₂s[σ], Δ₂s[τ], false))
|
||||
end
|
||||
end
|
||||
end
|
||||
return adj÷factorial(N-2)^2
|
||||
end
|
||||
|
||||
function Op(RG::GroupRing, N::Integer)
|
||||
if N < 4
|
||||
return RG()
|
||||
end
|
||||
S₂ = generating_set(RG.group, 2)
|
||||
ℤ = Int64
|
||||
Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
|
||||
|
||||
Alt_N = [g for g in PermutationGroup(N) if parity(g) == 0]
|
||||
Δ₂s = Dict(σ=>σ(Δ₂) for σ in Alt_N)
|
||||
op = RG()
|
||||
|
||||
for σ in Alt_N
|
||||
for τ in Alt_N
|
||||
if isopposite(σ, τ)
|
||||
GroupRings.addeq!(op, *(Δ₂s[σ], Δ₂s[τ], false))
|
||||
end
|
||||
end
|
||||
end
|
||||
return op÷factorial(N-2)^2
|
||||
end
|
||||
|
||||
for Elt in [:Sq, :Adj, :Op]
|
||||
@eval begin
|
||||
$Elt(RG::GroupRing{AutGroup{N}}) where N = $Elt(RG, N)
|
||||
$Elt(RG::GroupRing{<:MatAlgebra}) = $Elt(RG, nrows(RG.group))
|
||||
end
|
||||
end
|
||||
|
||||
function SqAdjOp(RG::GroupRing, N::Integer)
|
||||
S₂ = generating_set(RG.group, 2)
|
||||
ℤ = Int64
|
||||
Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
|
||||
|
||||
Alt_N = [σ for σ in PermutationGroup(N) if parity(σ) == 0]
|
||||
sq, adj, op = RG(), RG(), RG()
|
||||
|
||||
Δ₂s = Dict(σ=>σ(Δ₂) for σ in Alt_N)
|
||||
|
||||
for σ in Alt_N
|
||||
GroupRings.addeq!(sq, *(Δ₂s[σ], Δ₂s[σ], false))
|
||||
for τ in Alt_N
|
||||
if isopposite(σ, τ)
|
||||
GroupRings.addeq!(op, *(Δ₂s[σ], Δ₂s[τ], false))
|
||||
elseif isadjacent(σ, τ)
|
||||
GroupRings.addeq!(adj, *(Δ₂s[σ], Δ₂s[τ], false))
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
k = factorial(N-2)
|
||||
return sq÷k, adj÷k^2, op÷k^2
|
||||
end
|
||||
|
|
@ -2,38 +2,50 @@
|
|||
|
||||
@testset "SL(2,Z)" begin
|
||||
N = 2
|
||||
G = MatrixSpace(Nemo.ZZ, N,N)
|
||||
S = Groups.gens(G)
|
||||
S = [S; inv.(S)]
|
||||
G = MatrixAlgebra(zz, N)
|
||||
S = PropertyT.generating_set(G)
|
||||
|
||||
rm("SL($N,Z)", recursive=true, force=true)
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, solver(20000, accel=20); upper_bound=0.1)
|
||||
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, with_SCS(20000, accel=20); upper_bound=0.1)
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
λ = PropertyT.spectral_gap(sett)
|
||||
@test λ < 0.0
|
||||
@test PropertyT.interpret_results(sett, λ) == false
|
||||
end
|
||||
|
||||
@testset "SL(3,Z)" begin
|
||||
N = 3
|
||||
G = MatrixSpace(Nemo.ZZ, N,N)
|
||||
S = Groups.gens(G)
|
||||
S = [S; inv.(S)]
|
||||
|
||||
G = MatrixAlgebra(zz, N)
|
||||
S = PropertyT.generating_set(G)
|
||||
|
||||
rm("SL($N,Z)", recursive=true, force=true)
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, solver(1000, accel=20); upper_bound=0.1)
|
||||
|
||||
@test PropertyT.check_property_T(sett) == true
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, with_SCS(1000, accel=20); upper_bound=0.1)
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
λ = PropertyT.spectral_gap(sett)
|
||||
@test λ > 0.0999
|
||||
@test PropertyT.interpret_results(sett, λ) == true
|
||||
|
||||
@test PropertyT.check_property_T(sett) == true #second run should be fast
|
||||
end
|
||||
|
||||
|
||||
@testset "SAut(F₂)" begin
|
||||
N = 2
|
||||
G = SAut(FreeGroup(N))
|
||||
S = Groups.gens(G)
|
||||
S = [S; inv.(S)]
|
||||
|
||||
S = PropertyT.generating_set(G)
|
||||
|
||||
rm("SAut(F$N)", recursive=true, force=true)
|
||||
sett = PropertyT.Settings("SAut(F$N)", G, S, solver(20000);
|
||||
sett = PropertyT.Settings("SAut(F$N)", G, S, with_SCS(10000);
|
||||
upper_bound=0.15, warmstart=false)
|
||||
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
λ = PropertyT.spectral_gap(sett)
|
||||
@test λ < 0.0
|
||||
@test PropertyT.interpret_results(sett, λ) == false
|
||||
|
||||
end
|
||||
end
|
||||
|
|
|
|||
|
|
@ -2,58 +2,97 @@
|
|||
|
||||
@testset "oSL(3,Z)" begin
|
||||
N = 3
|
||||
G = MatrixSpace(Nemo.ZZ, N,N)
|
||||
S = Groups.gens(G)
|
||||
S = [S; inv.(S)]
|
||||
G = MatrixAlgebra(zz, N)
|
||||
S = PropertyT.generating_set(G)
|
||||
autS = WreathProduct(PermGroup(2), PermGroup(N))
|
||||
|
||||
rm("oSL($N,Z)", recursive=true, force=true)
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, autS, solver(2000, accel=20);
|
||||
upper_bound=0.27, warmstart=false)
|
||||
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
#second run just checks the solution
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, autS, solver(2000, accel=20);
|
||||
rm("oSL($N,Z)", recursive=true, force=true)
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, autS, with_SCS(2000, accel=20);
|
||||
upper_bound=0.27, warmstart=false)
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
λ = PropertyT.spectral_gap(sett)
|
||||
@test λ < 0.0
|
||||
@test PropertyT.interpret_results(sett, λ) == false
|
||||
|
||||
# second run just checks the solution due to warmstart=false above
|
||||
@test λ == PropertyT.spectral_gap(sett)
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, autS, with_SCS(2000, accel=20);
|
||||
upper_bound=0.27, warmstart=true)
|
||||
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
λ = PropertyT.spectral_gap(sett)
|
||||
@test λ > 0.269999
|
||||
@test PropertyT.interpret_results(sett, λ) == true
|
||||
|
||||
# this should be very fast due to warmstarting:
|
||||
@test λ ≈ PropertyT.spectral_gap(sett) atol=1e-5
|
||||
@test PropertyT.check_property_T(sett) == true
|
||||
|
||||
##########
|
||||
# Symmetrizing by PermGroup(3):
|
||||
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, PermGroup(N), with_SCS(4000, accel=20);
|
||||
upper_bound=0.27, warmstart=true)
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
λ = PropertyT.spectral_gap(sett)
|
||||
@test λ > 0.269999
|
||||
@test PropertyT.interpret_results(sett, λ) == true
|
||||
end
|
||||
|
||||
|
||||
@testset "oSL(4,Z)" begin
|
||||
N = 4
|
||||
G = MatrixSpace(Nemo.ZZ, N,N)
|
||||
S = Groups.gens(G)
|
||||
S = [S; inv.(S)]
|
||||
G = MatrixAlgebra(zz, N)
|
||||
S = PropertyT.generating_set(G)
|
||||
autS = WreathProduct(PermGroup(2), PermGroup(N))
|
||||
|
||||
rm("oSL($N,Z)", recursive=true, force=true)
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, autS, solver(2000, accel=20);
|
||||
upper_bound=1.3, warmstart=false)
|
||||
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
#second run just checks the obtained solution
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, autS, solver(5000, accel=20);
|
||||
rm("oSL($N,Z)", recursive=true, force=true)
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, autS, with_SCS(2000, accel=20);
|
||||
upper_bound=1.3, warmstart=false)
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
λ = PropertyT.spectral_gap(sett)
|
||||
@test λ < 0.0
|
||||
@test PropertyT.interpret_results(sett, λ) == false
|
||||
|
||||
# second run just checks the solution due to warmstart=false above
|
||||
@test λ == PropertyT.spectral_gap(sett)
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
|
||||
sett = PropertyT.Settings("SL($N,Z)", G, S, autS, with_SCS(5000, accel=20);
|
||||
upper_bound=1.3, warmstart=true)
|
||||
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
λ = PropertyT.spectral_gap(sett)
|
||||
@test λ > 1.2999
|
||||
@test PropertyT.interpret_results(sett, λ) == true
|
||||
|
||||
# this should be very fast due to warmstarting:
|
||||
@test λ ≈ PropertyT.spectral_gap(sett) atol=1e-5
|
||||
@test PropertyT.check_property_T(sett) == true
|
||||
end
|
||||
|
||||
@testset "SAut(F₃)" begin
|
||||
N = 3
|
||||
G = SAut(FreeGroup(N))
|
||||
S = Groups.gens(G)
|
||||
S = [S; inv.(S)]
|
||||
S = PropertyT.generating_set(G)
|
||||
autS = WreathProduct(PermGroup(2), PermGroup(N))
|
||||
|
||||
|
||||
rm("oSAut(F$N)", recursive=true, force=true)
|
||||
|
||||
sett = PropertyT.Settings("SAut(F$N)", G, S, autS, solver(5000);
|
||||
|
||||
sett = PropertyT.Settings("SAut(F$N)", G, S, autS, with_SCS(1000);
|
||||
upper_bound=0.15, warmstart=false)
|
||||
|
||||
|
||||
PropertyT.print_summary(sett)
|
||||
|
||||
@test PropertyT.check_property_T(sett) == false
|
||||
end
|
||||
end
|
||||
|
|
|
|||
|
|
@ -0,0 +1,245 @@
|
|||
@testset "Sq, Adj, Op" begin
|
||||
function isconstant_on_orbit(v, orb)
|
||||
isempty(orb) && return true
|
||||
k = v[first(orb)]
|
||||
return all(v[o] == k for o in orb)
|
||||
end
|
||||
|
||||
@testset "unit tests" begin
|
||||
for N in [3,4]
|
||||
M = MatrixAlgebra(zz, N)
|
||||
|
||||
@test PropertyT.EltaryMat(M, 1, 2) isa MatAlgElem
|
||||
e12 = PropertyT.EltaryMat(M, 1, 2)
|
||||
@test e12[1,2] == 1
|
||||
@test inv(e12)[1,2] == -1
|
||||
|
||||
S = PropertyT.generating_set(M)
|
||||
@test e12 ∈ S
|
||||
|
||||
@test length(PropertyT.generating_set(M)) == 2N*(N-1)
|
||||
@test all(inv(s) ∈ S for s in S)
|
||||
|
||||
A = SAut(FreeGroup(N))
|
||||
@test length(PropertyT.generating_set(A)) == 4N*(N-1)
|
||||
S = PropertyT.generating_set(A)
|
||||
@test all(inv(s) ∈ S for s in S)
|
||||
end
|
||||
|
||||
@test PropertyT.isopposite(perm"(1,2,3)(4)", perm"(1,4,2)")
|
||||
@test PropertyT.isadjacent(perm"(1,2,3)", perm"(1,2)(3)")
|
||||
|
||||
@test !PropertyT.isopposite(perm"(1,2,3)", perm"(1,2)(3)")
|
||||
@test !PropertyT.isadjacent(perm"(1,4)", perm"(2,3)(4)")
|
||||
|
||||
@test isconstant_on_orbit([1,1,1,2,2], [2,3])
|
||||
@test !isconstant_on_orbit([1,1,1,2,2], [2,3,4])
|
||||
end
|
||||
|
||||
@testset "Sq, Adj, Op" begin
|
||||
|
||||
N = 4
|
||||
M = MatrixAlgebra(zz, N)
|
||||
S = PropertyT.generating_set(M)
|
||||
Δ = PropertyT.Laplacian(S, 2)
|
||||
RG = parent(Δ)
|
||||
|
||||
autS = WreathProduct(PermGroup(2), PermGroup(N))
|
||||
orbits = PropertyT.orbit_decomposition(autS, RG.basis)
|
||||
|
||||
@test PropertyT.Sq(RG) isa GroupRingElem
|
||||
sq = PropertyT.Sq(RG)
|
||||
@test all(isconstant_on_orbit(sq, orb) for orb in orbits)
|
||||
|
||||
@test PropertyT.Adj(RG) isa GroupRingElem
|
||||
adj = PropertyT.Adj(RG)
|
||||
@test all(isconstant_on_orbit(adj, orb) for orb in orbits)
|
||||
|
||||
@test PropertyT.Op(RG) isa GroupRingElem
|
||||
op = PropertyT.Op(RG)
|
||||
@test all(isconstant_on_orbit(op, orb) for orb in orbits)
|
||||
|
||||
sq, adj, op = PropertyT.SqAdjOp(RG, N)
|
||||
@test sq == PropertyT.Sq(RG)
|
||||
@test adj == PropertyT.Adj(RG)
|
||||
@test op == PropertyT.Op(RG)
|
||||
|
||||
e = one(M)
|
||||
g = PropertyT.EltaryMat(M, 1,2)
|
||||
h = PropertyT.EltaryMat(M, 1,3)
|
||||
k = PropertyT.EltaryMat(M, 3,4)
|
||||
|
||||
edges = N*(N-1)÷2
|
||||
@test sq[e] == 20*edges
|
||||
@test sq[g] == sq[h] == -8
|
||||
@test sq[g^2] == sq[h^2] == 1
|
||||
@test sq[g*h] == sq[h*g] == 0
|
||||
|
||||
# @test adj[e] == ...
|
||||
@test adj[g] == adj[h] # == ...
|
||||
@test adj[g^2] == adj[h^2] == 0
|
||||
@test adj[g*h] == adj[h*g] # == ...
|
||||
|
||||
|
||||
# @test op[e] == ...
|
||||
@test op[g] == op[h] # == ...
|
||||
@test op[g^2] == op[h^2] == 0
|
||||
@test op[g*h] == op[h*g] == 0
|
||||
@test op[g*k] == op[k*g] # == ...
|
||||
@test op[h*k] == op[k*h] == 0
|
||||
end
|
||||
end
|
||||
|
||||
|
||||
@testset "1812.03456 examples" begin
|
||||
|
||||
function SOS_residual(x::GroupRingElem, Q::Matrix)
|
||||
RG = parent(x)
|
||||
@time sos = PropertyT.compute_SOS(RG, Q);
|
||||
return x - sos
|
||||
end
|
||||
|
||||
function check_positivity(elt, Δ, orbit_data, upper_bound, warm=nothing; with_solver=with_SCS(20_000, accel=10))
|
||||
SDP_problem, varP = PropertyT.SOS_problem(elt, Δ, orbit_data; upper_bound=upper_bound)
|
||||
|
||||
status, warm = PropertyT.solve(SDP_problem, with_solver, warm);
|
||||
Base.Libc.flush_cstdio()
|
||||
@info "Optimization status:" status
|
||||
|
||||
λ = value(SDP_problem[:λ])
|
||||
Ps = [value.(P) for P in varP]
|
||||
|
||||
Qs = real.(sqrt.(Ps));
|
||||
Q = PropertyT.reconstruct(Qs, orbit_data);
|
||||
|
||||
b = SOS_residual(elt - λ*Δ, Q)
|
||||
return b, λ, warm
|
||||
end
|
||||
|
||||
@testset "SL(3,Z)" begin
|
||||
N = 3
|
||||
halfradius = 2
|
||||
M = MatrixAlgebra(zz, N)
|
||||
S = PropertyT.generating_set(M)
|
||||
Δ = PropertyT.Laplacian(S, halfradius)
|
||||
RG = parent(Δ)
|
||||
orbit_data = PropertyT.OrbitData(RG, WreathProduct(PermGroup(2), PermGroup(N)))
|
||||
orbit_data = PropertyT.decimate(orbit_data);
|
||||
|
||||
@testset "Sq₃ is SOS" begin
|
||||
elt = PropertyT.Sq(RG)
|
||||
UB = 0.05 # 0.105?
|
||||
|
||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
||||
Base.Libc.flush_cstdio()
|
||||
@info "obtained λ and residual" λ norm(residual, 1)
|
||||
|
||||
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
||||
|
||||
@test 2^2*norm(residual, 1) < λ/100
|
||||
end
|
||||
|
||||
@testset "Adj₃ is SOS" begin
|
||||
elt = PropertyT.Adj(RG)
|
||||
UB = 0.1 # 0.157?
|
||||
|
||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
||||
Base.Libc.flush_cstdio()
|
||||
@info "obtained λ and residual" λ norm(residual, 1)
|
||||
|
||||
@test 2^2*norm(residual, 1) < λ
|
||||
@test 2^2*norm(residual, 1) < λ/100
|
||||
end
|
||||
|
||||
@testset "Op₃ is empty, so can not be certified" begin
|
||||
elt = PropertyT.Op(RG)
|
||||
UB = Inf
|
||||
|
||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
||||
Base.Libc.flush_cstdio()
|
||||
@info "obtained λ and residual" λ norm(residual, 1)
|
||||
|
||||
@test 2^2*norm(residual, 1) > λ
|
||||
end
|
||||
end
|
||||
|
||||
@testset "SL(4,Z)" begin
|
||||
N = 4
|
||||
halfradius = 2
|
||||
M = MatrixAlgebra(zz, N)
|
||||
S = PropertyT.generating_set(M)
|
||||
Δ = PropertyT.Laplacian(S, halfradius)
|
||||
RG = parent(Δ)
|
||||
orbit_data = PropertyT.OrbitData(RG, WreathProduct(PermGroup(2), PermGroup(N)))
|
||||
orbit_data = PropertyT.decimate(orbit_data);
|
||||
|
||||
@testset "Sq₄ is SOS" begin
|
||||
elt = PropertyT.Sq(RG)
|
||||
UB = 0.2 # 0.3172
|
||||
|
||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
||||
Base.Libc.flush_cstdio()
|
||||
@info "obtained λ and residual" λ norm(residual, 1)
|
||||
|
||||
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
||||
@test 2^2*norm(residual, 1) < λ/100
|
||||
end
|
||||
|
||||
@testset "Adj₄ is SOS" begin
|
||||
elt = PropertyT.Adj(RG)
|
||||
UB = 0.3 # 0.5459?
|
||||
|
||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
||||
@info "obtained λ and residual" λ norm(residual, 1)
|
||||
|
||||
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
||||
@test 2^2*norm(residual, 1) < λ/100
|
||||
end
|
||||
|
||||
@testset "we can't cerify that Op₄ SOS" begin
|
||||
elt = PropertyT.Op(RG)
|
||||
UB = 2.0
|
||||
|
||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB,
|
||||
with_solver=with_SCS(20_000, accel=10, eps=2e-10))
|
||||
Base.Libc.flush_cstdio()
|
||||
@info "obtained λ and residual" λ norm(residual, 1)
|
||||
|
||||
@test 2^2*norm(residual, 1) > λ # i.e. we can certify positivity
|
||||
end
|
||||
|
||||
@testset "Adj₄ + Op₄ is SOS" begin
|
||||
elt = PropertyT.Adj(RG) + PropertyT.Op(RG)
|
||||
UB = 0.6 # 0.82005
|
||||
|
||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
||||
Base.Libc.flush_cstdio()
|
||||
@info "obtained λ and residual" λ norm(residual, 1)
|
||||
|
||||
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
||||
@test 2^2*norm(residual, 1) < λ/100
|
||||
end
|
||||
end
|
||||
|
||||
# @testset "Adj₄ + 100 Op₄ ∈ ISAut(F₄) is SOS" begin
|
||||
# N = 4
|
||||
# halfradius = 2
|
||||
# M = SAut(FreeGroup(N))
|
||||
# S = PropertyT.generating_set(M)
|
||||
# Δ = PropertyT.Laplacian(S, halfradius)
|
||||
# RG = parent(Δ)
|
||||
# orbit_data = PropertyT.OrbitData(RG, WreathProduct(PermGroup(2), PermGroup(N)))
|
||||
# orbit_data = PropertyT.decimate(orbit_data);
|
||||
#
|
||||
# @time elt = PropertyT.Adj(RG) + 100PropertyT.Op(RG)
|
||||
# UB = 0.05
|
||||
#
|
||||
# warm = nothing
|
||||
#
|
||||
# residual, λ, warm = check_positivity(elt, Δ, orbit_data, UB, warm, with_solver=with_SCS(20_000, accel=10))
|
||||
# @info "obtained λ and residual" λ norm(residual, 1)
|
||||
#
|
||||
# @test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
||||
# @test 2^2*norm(residual, 1) < λ/100
|
||||
# end
|
||||
end
|
||||
|
|
@ -1,5 +1,3 @@
|
|||
using PropertyT.GroupRings
|
||||
|
||||
@testset "Correctness of HPC SOS computation" begin
|
||||
|
||||
function prepare(G_name, λ, S_size)
|
||||
|
|
@ -35,7 +33,7 @@ using PropertyT.GroupRings
|
|||
@time sos_sqr = PropertyT.compute_SOS_square(pm, Q)
|
||||
@time sos_hpc = PropertyT.compute_SOS(pm, Q)
|
||||
|
||||
@test norm(sos_sqr - sos_hpc, 1) < 3e-12
|
||||
@test norm(sos_sqr - sos_hpc, 1) < 4e-12
|
||||
@info "$NAME:\nDifference in l₁-norm between square and hpc sos decompositions:" norm(eoi-sos_sqr,1) norm(eoi-sos_hpc,1) norm(sos_sqr - sos_hpc, 1)
|
||||
|
||||
#########################################################
|
||||
|
|
|
|||
|
|
@ -0,0 +1,104 @@
|
|||
@testset "actions on Group[Rings]" begin
|
||||
Eij = PropertyT.EltaryMat
|
||||
ssgs(M::MatAlgebra, i, j) = (S = [Eij(M, i, j), Eij(M, j, i)];
|
||||
S = unique([S; inv.(S)]); S)
|
||||
|
||||
rmul = Groups.rmul_autsymbol
|
||||
lmul = Groups.lmul_autsymbol
|
||||
|
||||
function ssgs(A::AutGroup, i, j)
|
||||
rmuls = [rmul(i,j), rmul(j,i)]
|
||||
lmuls = [lmul(i,j), lmul(j,i)]
|
||||
gen_set = A.([rmuls; lmuls])
|
||||
return unique([gen_set; inv.(gen_set)])
|
||||
end
|
||||
|
||||
@testset "actions on SL(3,Z) and its group ring" begin
|
||||
N = 3
|
||||
halfradius = 2
|
||||
M = MatrixAlgebra(zz, N)
|
||||
S = PropertyT.generating_set(M)
|
||||
E_R, sizes = Groups.generate_balls(S, one(M), radius=2halfradius);
|
||||
|
||||
rdict = GroupRings.reverse_dict(E_R)
|
||||
pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=false);
|
||||
RG = GroupRing(M, E_R, rdict, pm)
|
||||
|
||||
@testset "correctness of actions" begin
|
||||
Δ = length(S)*RG(1) - sum(RG(s) for s in S)
|
||||
@test Δ == PropertyT.spLaplacian(RG, S)
|
||||
|
||||
elt = S[5]
|
||||
x = RG(1) - RG(elt)
|
||||
elt2 = E_R[rand(sizes[1]:sizes[2])]
|
||||
y = 2RG(elt2) - RG(elt)
|
||||
|
||||
for G in [PermGroup(N), WreathProduct(PermGroup(2), PermGroup(N))]
|
||||
@test all(g(one(M)) == one(M) for g in G)
|
||||
@test all(rdict[g(m)] <= sizes[1] for g in G for m in S)
|
||||
@test all(g(m)*g(n) == g(m*n) for g in G for m in S for n in S)
|
||||
|
||||
@test all(g(Δ) == Δ for g in G)
|
||||
@test all(g(x) == RG(1) - RG(g(elt)) for g in G)
|
||||
|
||||
@test all(2RG(g(elt2)) - RG(g(elt)) == g(y) for g in G)
|
||||
end
|
||||
end
|
||||
|
||||
@testset "small Laplacians" begin
|
||||
for (i,j) in PropertyT.indexing(N)
|
||||
Sij = ssgs(M, i,j)
|
||||
Δij= PropertyT.spLaplacian(RG, Sij)
|
||||
|
||||
@test all(p(Δij) == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in PermGroup(N))
|
||||
|
||||
@test all(g(Δij) == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(PermGroup(2), PermGroup(N)))
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@testset "actions on SAut(F_3) and its group ring" begin
|
||||
N = 3
|
||||
halfradius = 2
|
||||
M = SAut(FreeGroup(N))
|
||||
S = PropertyT.generating_set(M)
|
||||
E_R, sizes = Groups.generate_balls(S, one(M), radius=2halfradius);
|
||||
|
||||
rdict = GroupRings.reverse_dict(E_R)
|
||||
pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=false);
|
||||
RG = GroupRing(M, E_R, rdict, pm)
|
||||
|
||||
|
||||
@testset "correctness of actions" begin
|
||||
|
||||
Δ = length(S)*RG(1) - sum(RG(s) for s in S)
|
||||
@test Δ == PropertyT.spLaplacian(RG, S)
|
||||
|
||||
elt = S[5]
|
||||
x = RG(1) - RG(elt)
|
||||
elt2 = E_R[rand(sizes[1]:sizes[2])]
|
||||
y = 2RG(elt2) - RG(elt)
|
||||
|
||||
for G in [PermGroup(N), WreathProduct(PermGroup(2), PermGroup(N))]
|
||||
@test all(g(one(M)) == one(M) for g in G)
|
||||
@test all(rdict[g(m)] <= sizes[1] for g in G for m in S)
|
||||
@test all(g(m)*g(n) == g(m*n) for g in G for m in S for n in S)
|
||||
|
||||
@test all(g(Δ) == Δ for g in G)
|
||||
@test all(g(x) == RG(1) - RG(g(elt)) for g in G)
|
||||
|
||||
@test all(2RG(g(elt2)) - RG(g(elt)) == g(y) for g in G)
|
||||
end
|
||||
end
|
||||
|
||||
for (i,j) in PropertyT.indexing(N)
|
||||
Sij = ssgs(M, i,j)
|
||||
Δij= PropertyT.spLaplacian(RG, Sij)
|
||||
|
||||
@test all(p(Δij) == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in PermGroup(N))
|
||||
|
||||
@test all(g(Δij) == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(PermGroup(2), PermGroup(N)))
|
||||
end
|
||||
end
|
||||
|
||||
end
|
||||
|
|
@ -1,25 +1,18 @@
|
|||
using AbstractAlgebra, Nemo, Groups, SCS
|
||||
using SparseArrays
|
||||
using JLD
|
||||
using PropertyT
|
||||
using Test
|
||||
using JuMP
|
||||
using LinearAlgebra, SparseArrays
|
||||
using AbstractAlgebra, Groups, GroupRings
|
||||
using PropertyT
|
||||
using JLD
|
||||
|
||||
indexing(n) = [(i,j) for i in 1:n for j in (i+1):n]
|
||||
function Groups.gens(M::MatSpace)
|
||||
@assert ncols(M) == nrows(M)
|
||||
N = ncols(M)
|
||||
E(i,j) = begin g = M(1); g[i,j] = 1; g end
|
||||
S = [E(i,j) for (i,j) in indexing(N)]
|
||||
S = [S; transpose.(S)]
|
||||
return S
|
||||
end
|
||||
using JuMP, SCS
|
||||
|
||||
solver(iters; accel=1) =
|
||||
with_SCS(iters; accel=1, eps=1e-10) =
|
||||
with_optimizer(SCS.Optimizer,
|
||||
linear_solver=SCS.Direct, max_iters=iters,
|
||||
acceleration_lookback=accel, eps=1e-10, warm_start=true)
|
||||
acceleration_lookback=accel, eps=eps, warm_start=true)
|
||||
|
||||
include("1703.09680.jl")
|
||||
include("actions.jl")
|
||||
include("1712.07167.jl")
|
||||
include("SOS_correctness.jl")
|
||||
include("1812.03456.jl")
|
||||
|
|
|
|||
Loading…
Reference in New Issue