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8 changed files with 444 additions and 20 deletions

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@ -2,19 +2,13 @@
[[AbstractAlgebra]] [[AbstractAlgebra]]
deps = ["InteractiveUtils", "LinearAlgebra", "Markdown", "Random", "SparseArrays", "Test"] deps = ["InteractiveUtils", "LinearAlgebra", "Markdown", "Random", "SparseArrays", "Test"]
git-tree-sha1 = "a45d75b3cad41fafe2642d0f64e17866873869c8" git-tree-sha1 = "56bdd9e9bb2cfdf0876d0846b337c8f395d9a4b7"
uuid = "c3fe647b-3220-5bb0-a1ea-a7954cac585d" uuid = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
version = "0.4.4" version = "0.4.6"
[[Base64]] [[Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"
[[BenchmarkTools]]
deps = ["JSON", "Printf", "Statistics", "Test"]
git-tree-sha1 = "5d1dd8577643ba9014574cd40d9c028cd5e4b85a"
uuid = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
version = "0.4.2"
[[BinDeps]] [[BinDeps]]
deps = ["Compat", "Libdl", "SHA", "URIParser"] deps = ["Compat", "Libdl", "SHA", "URIParser"]
git-tree-sha1 = "12093ca6cdd0ee547c39b1870e0c9c3f154d9ca9" git-tree-sha1 = "12093ca6cdd0ee547c39b1870e0c9c3f154d9ca9"
@ -74,9 +68,9 @@ version = "0.2.0"
[[Compat]] [[Compat]]
deps = ["Base64", "Dates", "DelimitedFiles", "Distributed", "InteractiveUtils", "LibGit2", "Libdl", "LinearAlgebra", "Markdown", "Mmap", "Pkg", "Printf", "REPL", "Random", "Serialization", "SharedArrays", "Sockets", "SparseArrays", "Statistics", "Test", "UUIDs", "Unicode"] deps = ["Base64", "Dates", "DelimitedFiles", "Distributed", "InteractiveUtils", "LibGit2", "Libdl", "LinearAlgebra", "Markdown", "Mmap", "Pkg", "Printf", "REPL", "Random", "Serialization", "SharedArrays", "Sockets", "SparseArrays", "Statistics", "Test", "UUIDs", "Unicode"]
git-tree-sha1 = "195a3ffcb8b0762684b6821de18f83a16455c6ea" git-tree-sha1 = "84aa74986c5b9b898b0d1acaf3258741ee64754f"
uuid = "34da2185-b29b-5c13-b0c7-acf172513d20" uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
version = "2.0.0" version = "2.1.0"
[[DataStructures]] [[DataStructures]]
deps = ["InteractiveUtils", "OrderedCollections", "Random", "Serialization", "Test"] deps = ["InteractiveUtils", "OrderedCollections", "Random", "Serialization", "Test"]
@ -122,9 +116,9 @@ version = "0.2.0"
[[FileIO]] [[FileIO]]
deps = ["Pkg", "Random", "Test"] deps = ["Pkg", "Random", "Test"]
git-tree-sha1 = "c94b0787956629036fb2b20fccde9e52b89d079a" git-tree-sha1 = "da32159d4a2e526338506685e280e39ed2f18961"
uuid = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549" uuid = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549"
version = "1.0.5" version = "1.0.6"
[[ForwardDiff]] [[ForwardDiff]]
deps = ["CommonSubexpressions", "DiffResults", "DiffRules", "InteractiveUtils", "LinearAlgebra", "NaNMath", "Random", "SparseArrays", "SpecialFunctions", "StaticArrays", "Test"] deps = ["CommonSubexpressions", "DiffResults", "DiffRules", "InteractiveUtils", "LinearAlgebra", "NaNMath", "Random", "SparseArrays", "SpecialFunctions", "StaticArrays", "Test"]
@ -201,10 +195,10 @@ uuid = "1b4a561d-cfcb-5daf-8433-43fcf8b4bea3"
version = "0.4.1" version = "0.4.1"
[[LibCURL]] [[LibCURL]]
deps = ["BinaryProvider", "Compat", "Libdl", "Printf"] deps = ["BinaryProvider", "Libdl", "Printf", "Test"]
git-tree-sha1 = "6339c87cb76923a3cf947fcd213cbc364355c9c9" git-tree-sha1 = "d051c8057512ca38a273aaa514145a0b25f24d46"
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21" uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"
version = "0.4.1" version = "0.5.0"
[[LibExpat]] [[LibExpat]]
deps = ["Compat"] deps = ["Compat"]
@ -257,10 +251,10 @@ uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.2" version = "0.3.2"
[[Nemo]] [[Nemo]]
deps = ["AbstractAlgebra", "InteractiveUtils", "Libdl", "LinearAlgebra", "Markdown", "Test"] deps = ["AbstractAlgebra", "BinaryProvider", "InteractiveUtils", "Libdl", "LinearAlgebra", "Markdown", "Test"]
git-tree-sha1 = "a41d652f0eefc6e08956bde976a10fed3535db57" git-tree-sha1 = "24b361effd69b4ca78014a7bab1b81d1de7f35c0"
uuid = "2edaba10-b0f1-5616-af89-8c11ac63239a" uuid = "2edaba10-b0f1-5616-af89-8c11ac63239a"
version = "0.12.3" version = "0.13.2"
[[OrderedCollections]] [[OrderedCollections]]
deps = ["Random", "Serialization", "Test"] deps = ["Random", "Serialization", "Test"]
@ -288,7 +282,7 @@ uuid = "9abbd945-dff8-562f-b5e8-e1ebf5ef1b79"
[[PropertyT]] [[PropertyT]]
deps = ["AbstractAlgebra", "Dates", "GroupRings", "Groups", "IntervalArithmetic", "JLD", "JuMP", "LinearAlgebra", "Markdown", "MathProgBase", "Nemo", "Printf", "SparseArrays"] deps = ["AbstractAlgebra", "Dates", "GroupRings", "Groups", "IntervalArithmetic", "JLD", "JuMP", "LinearAlgebra", "Markdown", "MathProgBase", "Nemo", "Printf", "SparseArrays"]
git-tree-sha1 = "4c4a5086b0c4dee30bba27ddf42e7ebb403765c2" git-tree-sha1 = "1307df01fc484b44fd24328f30304912ebd9bf2a"
repo-rev = "master" repo-rev = "master"
repo-url = "https://github.com/kalmarek/PropertyT.jl" repo-url = "https://github.com/kalmarek/PropertyT.jl"
uuid = "03b72c93-0167-51e2-8a1e-eb4ff1fb940d" uuid = "03b72c93-0167-51e2-8a1e-eb4ff1fb940d"

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@ -1,9 +1,23 @@
name = "1712.07167"
uuid = "c0ba6078-cfcc-57dc-8f34-6df289db1335"
authors = ["Marek Kaluba <kalmar@amu.edu.pl>"]
version = "0.1.0"
[deps] [deps]
BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
GroupRings = "0befed6a-bd73-11e8-1e41-a1190947c2f5" GroupRings = "0befed6a-bd73-11e8-1e41-a1190947c2f5"
Groups = "5d8bd718-bd84-11e8-3b40-ad14f4a32557" Groups = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253" IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
JLD = "4138dd39-2aa7-5051-a626-17a0bb65d9c8" JLD = "4138dd39-2aa7-5051-a626-17a0bb65d9c8"
Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
PropertyT = "03b72c93-0167-51e2-8a1e-eb4ff1fb940d" PropertyT = "03b72c93-0167-51e2-8a1e-eb4ff1fb940d"
SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13" SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"
SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
[extras]
SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"
Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
[targets]
test = ["Test", "SCS"]

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bin/check_SpNs.jl Normal file
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using Pkg
Pkg.activate(".")
using Test
using PropertyT
using Nemo
using PropertyT.LinearAlgebra
using PropertyT.SparseArrays
using PropertyT.JuMP
using PropertyT.AbstractAlgebra
using PropertyT.Groups
using PropertyT.GroupRings
using PropertyT.JLD
# include("../1712.07167/Roots.jl")
include(joinpath("..", "src", "SpNs.jl"))
using .SpNs
using .SpNs.Roots
BLAS.set_num_threads(Threads.nthreads())
ENV["OMP_NUM_THREADS"] = Threads.nthreads()
const N = 2
const RADIUS = 3
const UPPER_BOUND = 0.801
root_system = SpNs.gens_roots(N)
S = [g.matrix for g in root_system]
G = parent(S[1])
autS = WreathProduct(PermGroup(2), PermGroup(N))
using SCS
with_SCS = with_optimizer(SCS.Optimizer, linear_solver=SCS.Direct, max_iters=400000, eps=1e-11, alpha=1.95, acceleration_lookback=1, warm_start=true)
sett = PropertyT.Settings("Sp($(2N),Z)_r$RADIUS", G, S, autS, with_SCS;
radius=RADIUS, upper_bound=UPPER_BOUND, warmstart=true)
PropertyT.print_summary(sett)
fp = PropertyT.fullpath(sett)
isdir(fp) || mkpath(fp)
Δ = PropertyT.Laplacian(S, RADIUS)
RG = parent(Δ)
Δs = SpNs.laplacians(RG, root_system)
Sq = sum( Δα^2 for (α, Δα) in Δs );
# Op = sum( Δα * sum(Δβ for (β, Δβ) in Δs if isorthogonal(α, β)) for (α, Δα) in Δs );
Adj = sum( Δα * sum(Δβ for (β, Δβ) in Δs if !isproportional(α, β)) for (α, Δα) in Δs );
const elt = Adj
solver_logfile(sett) = joinpath(PropertyT.fullpath(sett), "Adj_solver_$(PropertyT.Dates.now()).log")
λ, P = PropertyT.approximate_by_SOS(sett, elt, Δ,
solverlog=solver_logfile(sett))
save(PropertyT.filename(sett, :solution), "λ", λ, "P", P)
λ < 0 && @warn "Solver did not produce a valid solution!"
Q = real(sqrt(P));
certified_λ = PropertyT.certify_SOS_decomposition(elt, Δ, λ, Q, R=RADIUS)
@info "The obtained SOS can be use to certify λ ≥" certified_λ
#
# od = PropertyT.OrbitData(RG, WreathProduct(PermGroup(2), PermGroup(N)))
# orbit_data = PropertyT.decimate(od);
#
# using SCS
# using JuMP
# warmstart = nothing
#
# UB = 0.88 # for elt = Δ²;
# SDP_problem, varP = PropertyT.SOS_problem(elt, Δ, orbit_data, upper_bound=UB);
# # SDP_problem, varP = PropertyT.SOS_problem(elt, Δ, orbit_data);
#
# sett = PropertyT.Settings("Sp($(2N),Z)", G, S, autS, solver(2000, accel=20);
# upper_bound=1.3, warmstart=false)
#
# with_SCS = with_optimizer(SCS.Optimizer, linear_solver=SCS.Direct, max_iters=100000, eps=1e-12, alpha=1.95, acceleration_lookback=1, warm_start=true)
#
# status, warmstart = PropertyT.solve(SDP_problem, with_SCS, warmstart)

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src/1712.07167.jl Normal file
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src/Roots.jl Normal file
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module Roots
using StaticArrays
using LinearAlgebra
export Root, isproportional, isorthogonal, ~,
abstract type AbstractRoot end
struct Root{N} <: AbstractRoot
coord::SVector{N, Float64}
Root(a::SVector{N}) where N = new{N}(a)
end
Root(a) = Root(SVector(a...))
function Base.:(==)(r::Root{N}, s::Root{M}) where {M, N}
M == N || return false
r.coord == s.coord || return false
return true
end
Base.hash(r::Root, h::UInt) = hash(Root, hash(r.coord, h))
Base.:+(r::Root{N}, s::Root{N}) where N = Root(r.coord+s.coord)
Base.:-(r::Root{N}, s::Root{N}) where N = Root(r.coord-s.coord)
Base.:-(r::Root{N}) where N = Root(-r.coord)
Base.:*(a::Number, r::Root) = Root(a*r.coord)
Base.:*(r::Root, a::Number) = a*r
Base.length(r::Root) = norm(r, 2)
LinearAlgebra.norm(r::Root, p::Real=2) = norm(r.coord, p)
LinearAlgebra.dot(r::Root{N}, s::Root{N}) where N = dot(r.coord, s.coord)
cos_angle(a,b) = dot(a, b) / (norm(a) * norm(b))
function isproportional(α::Root{N}, β::Root{N}) where N
return isapprox(abs(cos_angle(α, β)), 1.0, atol=eps(1.0))
end
Base.:~(α::Root{N}, β::Root{N}) where N = isproportional(α, β)
function isorthogonal(α::Root{N}, β::Root{N}) where N
return isapprox(cos_angle(α, β), 0.0, atol=eps(1.0))
end
(α::Root{N}, β::Root{N}) where N = isorthogonal(α, β)
function Base.show(io::IO, r::Root{N}) where N
print(io, "$(r.coord): root of length $(norm(r))")
end
E(N, i::Integer) = Root(ntuple(k -> k == i ? 1 : 0, N))
end # of module RootSystems

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function blkdiag(A, B)
new_size = (size(A,1) + size(B,1), size(A,2) + size(B,2))
res = zeros(eltype(A), new_size)
res[1:size(A,1), 1:size(A,2)] .= A
res[size(A,1)+1:end, size(A,2)+1:end] .= B
return res
end
function matrix_emb(n::DirectPowerGroupElem, p::perm)
Id = parent(n.elts[1])()
elt = Diagonal([(-1)^(el == Id ? 0 : 1) for el in n.elts])
elt = elt[:, p.d]
return blkdiag(elt, elt)
end
function (g::WreathProductElem)(A::MatElem)
g_inv = inv(g)
G = matrix_emb(g.n, g_inv.p)
G_inv = matrix_emb(g_inv.n, g.p)
M = parent(A)
return M(G)*A*M(G_inv)
end
function Base.:*(x::AbstractAlgebra.MatElem, P::Generic.perm)
z = similar(x)
m = nrows(x)
n = ncols(x)
for i = 1:m
for j = 1:n
z[i, j] = x[i,P[j]]
end
end
return z
end
function (p::perm)(A::MatElem)
length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))")
return p*A*inv(p)
end

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module SpNs
using LinearAlgebra
using AbstractAlgebra
using Nemo
using Groups
using GroupRings
include("Roots.jl")
using .Roots
export Sp, isproportional, ~, isorthogonal, , positive
# two families of generators
function a(i::Int, j::Int, M::MatSpace, val=one(M.base_ring)) # short roots
@assert i j
m = one(M)
N = size(m,2)÷2
m[i,j] = val
m[N+j, N+i] = -val
@assert issymplectic(m)
return m
end
function b(i::Int, j::Int, M::MatSpace, val=one(M.base_ring)) # long roots
m = one(M)
N = size(m,2)÷2
m[i, N+j] = val
m[j, N+i] = val
@assert issymplectic(m)
return m
end
# symplectic generator with associated root datum:
struct Sp{N}
root::Root{N}
matrix::Nemo.fmpz_mat
end
function Sp{N}(typ::Val{:a}, i::Int, j::Int) where N
@assert i≠j
r = Roots.E(N, i) - Roots.E(N, j)
M = Nemo.MatrixSpace(Nemo.ZZ, 2N, 2N)
m = a(i,j, M)
return Sp{N}(r, m)
end
function Sp{N}(typ::Val{:b}, i::Int, j::Int=i) where N
r = Roots.E(N, i) + Roots.E(N, j)
M = Nemo.MatrixSpace(Nemo.ZZ, 2N, 2N)
m = b(i,j, M)
return Sp{N}(r, m)
end
Sp{N}(s::Symbol, i::Int, j::Int=i) where N = Sp{N}(Val(s), i, j)
Base.inv(s::Sp) = Sp(s.root, inv(s.matrix))
Base.:-(s::Sp) = Sp(-s.root, transpose(s.matrix))
Roots.isproportional(x::Sp, y::Sp) = isproportional(x.root, y.root)
Roots.:~(x::Sp,y::Sp) = x.root ~ y.root
Roots.isorthogonal(x::Sp, y::Sp) = isorthogonal(x.root, y.root)
Roots.:(x::Sp,y::Sp) = x.root y.root
function issymplectic(M)
r = nrows(M)
c = ncols(M)
@assert r == c
@assert iseven(r)
n = r÷2
I = Diagonal(ones(Int, n))
O = zeros(Int,n,n);
Ω = parent(M)([O I; -I O])
return Ω == transpose(M)*Ω*M
end
indexing(n) = [(i,j) for i in 1:n for j in i+1:n]
function gens_roots(N)
a_ijs = [Sp{N}(:a, i,j) for (i,j) in indexing(N)]
b_is = [Sp{N}(:b, i) for i in 1:N]
c_ijs = [Sp{N}(:b, i,j) for (i,j) in indexing(N)]
root_system = [a_ijs; b_is; c_ijs]
root_system = [root_system; [-r for r in root_system]];
root_system = [root_system; inv.(root_system)]
return root_system
end
function positive(root_system::Vector{SP}) where SP
roots = [r.root for r in root_system]
pos_roots = eltype(roots)[]
for (idx,γ) in enumerate(roots)
if any(isproportional(s, γ) for s in roots[1:idx-1])
continue
end
push!(pos_roots, γ)
@info "" positive_root = γ
end
# TODO: what about their sums??
return pos_roots
end
function laplacians(RG::GroupRing, root_system::Vector{SpNs.Sp{N}}) where N
positive_roots = positive(root_system)
Sαs = Dict(α => [w.matrix for w in root_system if isproportional(w.root, α)] for α in positive_roots)
Δs = Dict(α => RG(length(S)) - RG(S) for (α, S) in Sαs)
return Δs
end
include("SpN_actions.jl")
end # of module SPNs

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using Test
using PropertyT
include("../src/SpNs.jl")
using .SpNs
using .SpNs.Roots
@testset "Roots" begin
@test Root((2,1)) isa Root
r = Root((2,1))
@test -r isa Root
@test 2r isa Root
@test length(2r) == 2length(r)
@test r+r isa Root
@test length(r+r) == 2length(r)
@test r-r isa Root;
@test length(r-r) == 0.0
r = Root((1,1));
s = Root((-1,1));
@test string(Root((1,1))) == "[1.0, 1.0]: root of length 1.4142135623730951"
@test isproportional(r, r)
@test isproportional(r, -r)
@test isproportional(r, 2r)
@test isproportional(-r, 2r)
@test isproportional(r+s, -s-r)
@test isorthogonal(r, s)
@test isorthogonal(r, -s)
@test isorthogonal(r, 2s)
@test isorthogonal(-r, 2s)
@test !isorthogonal(r, r+s)
@test !isorthogonal(r, -r)
@test !isorthogonal(r+s, 2s)
@test !isorthogonal(-r, 2s+r)
@test isorthogonal(r+s, -s+r)
N = 3
@test length(Roots.E(N, 1) - Roots.E(N,2)) == sqrt(2)
end
@testset "Symplectic Generators" begin
s = Sp{2}(:a, 1, 2)
@test -s isa Sp
@test inv(s) isa Sp
t = Sp{2}(:b, 1, 2)
@test -t isa Sp
@test inv(t) isa Sp
@test isproportional(s, -s)
@test isproportional(s, -s)
@test isproportional(s, inv(s))
@test isproportional(t, -t)
@test isproportional(t, inv(t))
@test !isproportional(s, t)
@test !isproportional(inv(s), -t)
end
@testset "Symplectic Group" begin
@testset "Symplectic Root System" begin
rs = SpNs.gens_roots(2)
S = [r.matrix for r in rs];
@test all(SpNs.issymplectic.(S))
@test length(S) == 16
rs = SpNs.gens_roots(3)
S = [r.matrix for r in rs];
@test all(SpNs.issymplectic.(S))
@test length(S) == 36
end
@testset "Sp(4,Z)" begin
N = 2
root_system = SpNs.gens_roots(N)
S = [g.matrix for g in root_system]
Δ = PropertyT.Laplacian(S, 2)
RG = parent(Δ)
Δs = SpNs.laplacians(RG, root_system)
@test sum(Δα for (α, Δα) in Δs) == Δ
@testset "Orthogonality & commutation" begin
r = Root([2, 0])
s = Root([0, 2])
a = Δs[r]
b = Δs[s]
@test a*b == b*a
end
Sq = sum( Δα^2 for (α, Δα) in Δs );
Adj = sum( Δα * sum(Δβ for (β, Δβ) in Δs if !isproportional(α, β)) for (α, Δα) in Δs );
@test Sq + Adj == Δ^2
end
end