next updates to README

.gitignore

@@ -1,15 +1,4 @@
-Articles
-Higman
-MCG*
-notebooks
-Oldies
-oSAutF*
-oSL*
-SAutF*
-SL*_*
 *ipynb*
 *.gws
-.*
+*/*.jld
-tests*
+*/*.log
-*.py
-*.pyc

AutF4.jl

@@ -0,0 +1,120 @@
+using Combinatorics
+
+using JuMP
+import SCS: SCSSolver
+import Mosek: MosekSolver
+
+push!(LOAD_PATH, "./")
+using SemiDirectProduct
+using GroupAlgebras
+include("property(T).jl")
+
+const N = 4
+
+const VERBOSE = true
+
+function permutation_matrix(p::Vector{Int})
+    n = length(p)
+    sort(p) == collect(1:n) || throw(ArgumentError("Input array must be a permutation of 1:n"))
+    A = eye(n)
+    return A[p,:]
+end
+
+SymmetricGroup(n) = [nthperm(collect(1:n), k) for k in 1:factorial(n)]
+
+# const SymmetricGroup = [permutation_matrix(x) for x in SymmetricGroup_perms]
+
+function E(i, j; dim::Int=N)
+    @assert i≠j
+    k = eye(dim)
+    k[i,j] = 1
+    return k
+end
+
+function eltary_basis_vector(i; dim::Int=N)
+    result = zeros(dim)
+    if 0 < i ≤ dim
+        result[i] = 1
+    end
+    return result
+end
+
+v(i; dim=N) = eltary_basis_vector(i,dim=dim)
+
+ϱ(i,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), v(j,dim=n))
+λ(i,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), -v(j,dim=n))
+
+function ɛ(i, n::Int=N)
+    result = eye(n)
+    result[i,i] = -1
+    return SemiDirectProductElement(result)
+end
+
+σ(permutation::Vector{Int}) =
+     SemiDirectProductElement(permutation_matrix(permutation))
+
+# Standard generating set: 103 elements
+
+function generatingset_ofAutF(n::Int=N)
+    indexing = [[i,j] for i in 1:n for j in 1:n if i≠j]
+    ϱs = [ϱ(ij...) for ij in indexing]
+    λs = [λ(ij...) for ij in indexing]
+    ɛs = [ɛ(i) for i in 1:N]
+    σs = [σ(perm) for perm in SymmetricGroup(n)]
+    S = vcat(ϱs, λs, ɛs, σs);
+    S = unique(vcat(S, [inv(x) for x in S]));
+    return S
+end
+
+#=
+Note that the element
+    α(i,j,k) = ϱ(i,j)*ϱ(i,k)*inv(ϱ(i,j))*inv(ϱ(i,k)),
+which surely belongs to ball of radius 4 in Aut(F₄) becomes trivial under the representation
+    Aut(F₄) → GL₄(ℤ)⋉ℤ⁴ → GL₅(ℂ).
+Moreover, due to work of Potapchik and Rapinchuk [1] every real representation of Aut(Fₙ) into GLₘ(ℂ) (for m ≤ 2n-2) factors through GLₙ(ℤ)⋉ℤⁿ, so will have the same problem.
+
+We need a different approach!
+=#
+
+const ID = eye(N+1)
+
+const S₁ = generatingset_ofAutF(N)
+
+matrix_S₁ = [matrix_repr(x) for x in S₁]
+
+const TOL=10.0^-7
+
+matrix_S₁[1:10,:][:,1]
+
+Δ, cm = prepare_Laplacian_and_constraints(matrix_S₁)
+
+#solver = SCSSolver(eps=TOL, max_iters=ITERATIONS, verbose=true);
+solver = MosekSolver(MSK_DPAR_INTPNT_CO_TOL_REL_GAP=TOL,
+#                      MSK_DPAR_INTPNT_CO_TOL_PFEAS=1e-15,
+#                      MSK_DPAR_INTPNT_CO_TOL_DFEAS=1e-15,
+#                      MSK_IPAR_PRESOLVE_USE=0,
+                  QUIET=!VERBOSE)
+
+# κ, A = solve_for_property_T(S₁, solver, verbose=VERBOSE)
+
+product_matrix = readdlm("SL₃Z.product_matrix", Int)
+L = readdlm("SL₃Z.Δ.coefficients")[:, 1]
+Δ = GroupAlgebraElement(L, product_matrix)
+
+A = readdlm("matrix.A.Mosek")
+κ = readdlm("kappa.Mosek")[1]
+
+# @show eigvals(A)
+@assert isapprox(eigvals(A), abs(eigvals(A)), atol=TOL)
+@assert A == Symmetric(A)
+
+
+const A_sqrt = real(sqrtm(A))
+
+SOS_EOI_fp_L₁, Ω_fp_dist = check_solution(κ, A_sqrt, Δ)
+
+κ_rational = rationalize(BigInt, κ;)
+A_sqrt_rational = rationalize(BigInt, A_sqrt)
+Δ_rational = rationalize(BigInt, Δ)
+
+SOS_EOI_rat_L₁, Ω_rat_dist = check_solution(κ_rational, A_sqrt_rational, Δ_rational)

AutFN.jl

@@ -0,0 +1,232 @@
+using ArgParse
+
+using Groups
+using GroupAlgebras
+using PropertyT
+
+import SCS.SCSSolver
+
+#=
+Note that the element
+    α(i,j,k) = ϱ(i,j)*ϱ(i,k)*inv(ϱ(i,j))*inv(ϱ(i,k)),
+which surely belongs to ball of radius 4 in Aut(F₄) becomes trivial under the representation
+    Aut(F₄) → GL₄(ℤ)⋉ℤ⁴ → GL₅(ℂ).
+Moreover, due to work of Potapchik and Rapinchuk [1] every real representation of Aut(Fₙ) into GLₘ(ℂ) (for m ≤ 2n-2) factors through GLₙ(ℤ)⋉ℤⁿ, so will have the same problem.
+
+We need a different approach: Here we actually compute in Aut(𝔽₄)
+=#
+
+import Combinatorics.nthperm
+SymmetricGroup(n) = [nthperm(collect(1:n), k) for k in 1:factorial(n)]
+
+function generating_set_of_AutF(N::Int)
+
+    indexing = [[i,j] for i in 1:N for j in 1:N if i≠j]
+    σs = [symmetric_AutSymbol(perm) for perm in SymmetricGroup(N)[2:end]];
+    ϱs = [rmul_AutSymbol(i,j) for (i,j) in indexing]
+    λs = [lmul_AutSymbol(i,j) for (i,j) in indexing]
+    ɛs = [flip_AutSymbol(i) for i in 1:N];
+
+    S = vcat(ϱs,λs)
+    S = vcat(S..., σs..., ɛs)
+    S = vcat(S..., [inv(g) for g in S])
+    return Vector{AutWord}(unique(S)), one(AutWord)
+end
+
+function generating_set_of_OutF(N::Int)
+    indexing = [[i,j] for i in 1:N for j in 1:N if i≠j]
+    ϱs = [rmul_AutSymbol(i,j) for (i,j) in indexing]
+    λs = [lmul_AutSymbol(i,j) for (i,j) in indexing]
+    ɛs = [flip_AutSymbol(i) for i in 1:N];
+
+    S = ϱs
+    push!(S, λs..., ɛs...)
+    push!(S,[inv(g) for g in S]...)
+
+    return Vector{AutWord}(unique(S)), one(AutWord)
+end
+
+function generating_set_of_Sym(N::Int)
+    σs = [symmetric_AutSymbol(perm) for perm in SymmetricGroup(N)[2:end]];
+
+    S = σs
+    push!(S, [inv(s) for s in S]...)
+    return Vector{AutWord}(unique(S)), one(AutWord)
+end
+
+
+function products(S1::Vector{AutWord}, S2::Vector{AutWord})
+    result = Vector{AutWord}()
+    seen = Set{Vector{FGWord}}()
+    n = length(S1)
+    for (i,x) in enumerate(S1)
+        for y in S2
+            z::AutWord = x*y
+            v::Vector{FGWord} = z(domain)
+            if !in(v, seen)
+                push!(seen, v)
+                push!(result, z)
+            end
+        end
+    end
+    return result
+end
+
+function products_images(S1::Vector{AutWord}, S2::Vector{AutWord})
+    result = Vector{Vector{FGWord}}()
+    seen = Set{Vector{FGWord}}()
+    n = length(S1)
+    for (i,x) in enumerate(S1)
+        z = x(domain)
+        for y in S2
+            v = y(z)
+            if !in(v, seen)
+                push!(seen, v)
+                push!(result, v)
+            end
+        end
+    end
+    return result
+end
+
+function hashed_product{T}(image::T, B, images_dict::Dict{T, Int})
+    n = size(B,1)
+    column = zeros(Int,n)
+    Threads.@threads for j in 1:n
+        w = (B[j])(image)
+        k = images_dict[w]
+        k ≠ 0 || throw(ArgumentError(
+            "($i,$j): $(x^-1)*$y don't seem to be supported on basis!"))
+        column[j] = k
+    end
+    return column
+end
+
+function create_product_matrix(basis::Vector{AutWord}, images)
+    n = length(basis)
+    product_matrix = zeros(Int, (n, n));
+    print("Creating hashtable of images...")
+    @time images_dict = Dict{Vector{FGWord}, Int}(x => i
+        for (i,x) in enumerate(images))
+
+    for i in 1:n
+        z = (inv(basis[i]))(domain)
+        product_matrix[i,:] = hashed_product(z, basis, images_dict)
+    end
+    return product_matrix
+end
+
+function ΔandSDPconstraints(identity::AutWord, S::Vector{AutWord})
+
+    println("Generating Balls of increasing radius...")
+    @time B₁ = vcat([identity], S)
+    @time B₂ = products(B₁,B₁);
+    @show length(B₂)
+    if length(B₂) != length(B₁)
+        @time B₃ = products(B₁, B₂)
+        @show length(B₃)
+        if length(B₃) != length(B₂)
+            @time B₄_images = products_images(B₁, B₃)
+        else
+            B₄_images = unique([f(domain) for f in B₃])
+        end
+    else
+        B₃ = B₂
+        B₄ = B₂
+        B₄_images = unique([f(domain) for f in B₃])
+    end
+
+    @show length(B₄_images)
+    # @assert length(B₄_images) == 3425657
+
+    println("Creating product matrix...")
+    @time pm = create_product_matrix(B₂, B₄_images)
+    println("Creating sdp_constratints...")
+    @time sdp_constraints = PropertyT.constraints_from_pm(pm)
+
+    L_coeff = PropertyT.splaplacian_coeff(S, B₂, length(B₄_images))
+    Δ = PropertyT.GroupAlgebraElement(L_coeff, Array{Int,2}(pm))
+
+    return Δ, sdp_constraints
+end
+
+const symbols =  [FGSymbol("x₁",1), FGSymbol("x₂",1), FGSymbol("x₃",1), FGSymbol("x₄",1), FGSymbol("x₅",1), FGSymbol("x₆",1)]
+
+const TOL=1e-8
+const N = 4
+const domain = Vector{FGWord}(symbols[1:N])
+
+function cpuinfo_physicalcores()
+    maxcore = -1
+    for line in eachline("/proc/cpuinfo")
+        if startswith(line, "core id")
+            maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
+        end
+    end
+    maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
+    return maxcore + 1
+end
+
+function parse_commandline()
+    s = ArgParseSettings()
+
+    @add_arg_table s begin
+        "--tol"
+            help = "set numerical tolerance for the SDP solver (default: 1e-5)"
+            arg_type = Float64
+            default = 1e-5
+        "--iterations"
+            help = "set maximal number of iterations for the SDP solver (default: 20000)"
+            arg_type = Int
+            default = 20000
+        "--upper-bound"
+            help = "Set an upper bound for the spectral gap (default: Inf)"
+            arg_type = Float64
+            default = Inf
+        "--cpus"
+            help = "Set number of cpus used by solver (default: auto)"
+            arg_type = Int
+            required = false
+        "-N"
+            help = "Consider automorphisms of free group on N generators (default: N=3)"
+            arg_type = Int
+            default = 3
+    end
+
+    return parse_args(s)
+end
+
+
+# const name = "SYM$N"
+# const upper_bound=factorial(N)-TOL^(1/5)
+# S() = generating_set_of_Sym(N)
+
+# name = "AutF$N"
+# S() = generating_set_of_AutF(N)
+
+function main()
+    parsed_args = parse_commandline()
+
+    tol = parsed_args["tol"]
+    iterations = parsed_args["iterations"]
+
+    solver = SCSSolver(eps=tol, max_iters=iterations, verbose=true, linearsolver=SCS.Indirect)
+
+    N = parsed_args["N"]
+    upper_bound = parsed_args["upper-bound"]
+
+    name = "OutF$N"
+    name = name*"-$(string(upper_bound))"
+    S() = generating_set_of_OutF(N)
+
+    if parsed_args["cpus"] ≠ nothing
+        if parsed_args["cpus"] > cpuinfo_physicalcores()
+            warn("Number of specified cores exceeds the physical core cound. Performance will suffer.")
+        end
+        Blas.set_num_threads(parsed_args["cpus"])
+    end
+    @time PropertyT.check_property_T(name, S, solver, upper_bound, tol)
+    return 0
+end
+
+main()

CPUselect.jl

@@ -1,35 +0,0 @@
-function cpuinfo_physicalcores()
-    maxcore = -1
-    for line in eachline("/proc/cpuinfo")
-        if startswith(line, "core id")
-            maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
-        end
-    end
-    maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
-    return maxcore + 1
-end
-
-function set_parallel_mthread(N::Int, workers::Bool)
-    if N > cpuinfo_physicalcores()
-        warn("Number of specified cores exceeds the physical core count. Performance may suffer.")
-    end
-
-    if workers
-        addprocs(N)
-        info("Using $N cpus in @parallel code.")
-    end
-    info("Using $(Threads.nthreads()) threads in @threads code.")
-    BLAS.set_num_threads(N)
-    info("Using $N threads in BLAS.")
-
-end
-
-function set_parallel_mthread(parsed_args::Dict; workers=false)
-    if parsed_args["cpus"] == nothing
-        N = cpuinfo_physicalcores()
-    else
-        N = parsed_args["cpus"]
-    end
-
-    set_parallel_mthread(N, workers)
-end

FPGroups_GAP.jl

@@ -1,157 +0,0 @@
-using JLD
-
-function GAP_code(group_code, dir, R; maxeqns=10_000, infolevel=2)
-    code = """
-LogTo("$(dir)/GAP.log");
-RequirePackage("kbmag");
-SetInfoLevel(InfoRWS, $infolevel);
-
-
-MetricBalls := function(rws, R)
-    local l, basis, sizes, i;
-    l := EnumerateReducedWords(rws, 0, R);;
-    SortBy(l, Length);
-    sizes := [1..R];
-    Apply(sizes, i -> Number(l, w -> Length(w) <= i));
-    return [l, sizes];
-end;;
-
-ProductMatrix := function(rws, basis, len)
-    local result, dict, g, tmpList, t;
-    result := [];
-    dict := NewDictionary(basis[1], true);
-    t := Runtime();
-    for g in [1..Length(basis)] do;
-        AddDictionary(dict, basis[g], g);
-    od;
-    Print("Creating dictionary: \t\t", StringTime(Runtime()-t), "\\n");
-    for g in basis{[1..len]} do;
-        tmpList := List(Inverse(g)*basis{[1..len]}, w->ReducedForm(rws, w));
-        #t := Runtime();
-        tmpList := List(tmpList, x -> LookupDictionary(dict, x));
-        #Print(Runtime()-t, "\\n");
-        Assert(1, ForAll(tmpList, x -> x <> fail));
-        Add(result, tmpList);
-    od;
-    return result;
-end;;
-
-SaveCSV := function(fname, pm)
-   local file, i, j, k;
-   file := OutputTextFile(fname, false);;
-   for i in pm do;
-      k := 1;
-      for j in i do;
-         if k < Length(i) then
-            AppendTo(file, j, ", ");
-         else
-            AppendTo(file, j, "\\n");
-         fi;
-         k := k+1;
-      od;
-   od;
-   CloseStream(file);
-end;;
-
-$group_code
-
-# G:= SimplifiedFpGroup(G);
-RWS := KBMAGRewritingSystem(G);
-# ResetRewritingSystem(RWS);
-O:=OptionsRecordOfKBMAGRewritingSystem(RWS);;
-O.maxeqns := $maxeqns;
-O.maxstates := 1000*$maxeqns;
-#O.maxstoredlen := [100,100];
-
-before := Runtimes();;
-KnuthBendix(RWS);
-after := Runtimes();;
-delta := after.user_time_children - before.user_time_children;;
-Print("Knuth-Bendix completion: \t", StringTime(delta), "\\n");
-
-t := Runtime();
-res := MetricBalls(RWS,$(2*R));;
-Print("Metric-Balls generation: \t", StringTime(Runtime()-t), "\\n");
-B := res[1];; sizes := res[2];;
-Print("Sizes of generated Balls: \t", sizes, "\\n");
-
-t := Runtime();
-pm := ProductMatrix(RWS, B, sizes[$R]);;
-Print("Computing ProductMatrix: \t", StringTime(Runtime()-t), "\\n");
-
-S := EnumerateReducedWords(RWS, 1, 1);
-S := List(S, s -> Position(B,s));
-
-SaveCSV("$(dir)/pm.csv", pm);
-SaveCSV("$(dir)/S.csv", [S]);
-SaveCSV("$(dir)/sizes.csv", [sizes]);
-
-Print("DONE!\\n");
-
-quit;""";
-    return code
-end
-
-function GAP_groupcode(S, rels)
-    F = "FreeGroup("*join(["\"$v\""for v in S], ", ") *");"
-    m = match(r".*(\[.*\])$", string(rels))
-    rels = replace(m.captures[1], " ", "\n")
-    code = """
-F := $F;
-AssignGeneratorVariables(F);;
-relations := $rels;;
-G := F/relations;
-    """
-    return code
-end
-
-function GAP_execute(gap_code, dir)
-   isdir(dir) || mkdir(dir)
-   GAP_file = joinpath(dir, "GAP_code.g")
-   @show dir
-   @show GAP_file;
-
-   open(GAP_file, "w") do io
-      write(io, gap_code)
-   end
-   run(pipeline(`cat $(GAP_file)`, `gap -q`))
-end
-
-function prepare_pm_delta_csv(name, group_code, R; maxeqns=10_000, infolevel=2)
-   info("Preparing multiplication table using GAP (via kbmag)")
-   gap_code = GAP_code(group_code, name, R, maxeqns=maxeqns, infolevel=infolevel)
-   GAP_execute(gap_code, name)
-end
-
-function prepare_pm_delta(name, group_code, R; maxeqns=100_000, infolevel=2)
-
-   pm_fname = joinpath(name, "pm.csv")
-   S_fname = joinpath(name, "S.csv")
-   sizes_fname = joinpath(name, "sizes.csv")
-   delta_fname = joinpath(name, "delta.jld")
-
-   if !isfile(pm_fname) || !isfile(S_fname) || !isfile(sizes_fname)
-      prepare_pm_delta_csv(name, group_code, R, maxeqns=maxeqns, infolevel=infolevel)
-   end
-
-   if isfile(sizes_fname)
-      sizes = readcsv(sizes_fname, Int)[1,:]
-      if 2R > length(sizes)
-         prepare_pm_delta_csv(name, group_code, R, maxeqns=maxeqns, infolevel=infolevel)
-      end
-   end
-
-   pm = readcsv(pm_fname, Int)
-   S = readcsv(S_fname, Int)[1,:]
-   sizes = readcsv(sizes_fname, Int)[1,:]
-
-   Δ = spzeros(sizes[2R])
-   Δ[S] .= -1
-   Δ[1] = length(S)
-
-   pm = pm[1:sizes[R], 1:sizes[R]]
-
-   save(joinpath(name, "pm.jld"), "pm", pm)
-   save(joinpath(name, "delta.jld"), "Δ", Δ)
-
-end

LICENSE.md

@@ -1,634 +0,0 @@
-> Copyright (c) 2017: Marek Kaluba.
-> This program is free software: you can redistribute it and/or modify
-> it under the terms of the GNU General Public License as published by
-> the Free Software Foundation, either version 3 of the License, or
-> (at your option) any later version.
->
-> This program is distributed in the hope that it will be useful,
-> but WITHOUT ANY WARRANTY; without even the implied warranty of
-> MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-> GNU General Public License for more details.
->
->
->                     GNU GENERAL PUBLIC LICENSE
->                        Version 3, 29 June 2007
->
->  Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
->  Everyone is permitted to copy and distribute verbatim copies
->  of this license document, but changing it is not allowed.
->
->                             Preamble
->
->   The GNU General Public License is a free, copyleft license for
-> software and other kinds of works.
->
->   The licenses for most software and other practical works are designed
-> to take away your freedom to share and change the works.  By contrast,
-> the GNU General Public License is intended to guarantee your freedom to
-> share and change all versions of a program--to make sure it remains free
-> software for all its users.  We, the Free Software Foundation, use the
-> GNU General Public License for most of our software; it applies also to
-> any other work released this way by its authors.  You can apply it to
-> your programs, too.
->
->   When we speak of free software, we are referring to freedom, not
-> price.  Our General Public Licenses are designed to make sure that you
-> have the freedom to distribute copies of free software (and charge for
-> them if you wish), that you receive source code or can get it if you
-> want it, that you can change the software or use pieces of it in new
-> free programs, and that you know you can do these things.
->
->   To protect your rights, we need to prevent others from denying you
-> these rights or asking you to surrender the rights.  Therefore, you have
-> certain responsibilities if you distribute copies of the software, or if
-> you modify it: responsibilities to respect the freedom of others.
->
->   For example, if you distribute copies of such a program, whether
-> gratis or for a fee, you must pass on to the recipients the same
-> freedoms that you received.  You must make sure that they, too, receive
-> or can get the source code.  And you must show them these terms so they
-> know their rights.
->
->   Developers that use the GNU GPL protect your rights with two steps:
-> (1) assert copyright on the software, and (2) offer you this License
-> giving you legal permission to copy, distribute and/or modify it.
->
->   For the developers' and authors' protection, the GPL clearly explains
-> that there is no warranty for this free software.  For both users' and
-> authors' sake, the GPL requires that modified versions be marked as
-> changed, so that their problems will not be attributed erroneously to
-> authors of previous versions.
->
->   Some devices are designed to deny users access to install or run
-> modified versions of the software inside them, although the manufacturer
-> can do so.  This is fundamentally incompatible with the aim of
-> protecting users' freedom to change the software.  The systematic
-> pattern of such abuse occurs in the area of products for individuals to
-> use, which is precisely where it is most unacceptable.  Therefore, we
-> have designed this version of the GPL to prohibit the practice for those
-> products.  If such problems arise substantially in other domains, we
-> stand ready to extend this provision to those domains in future versions
-> of the GPL, as needed to protect the freedom of users.
->
->   Finally, every program is threatened constantly by software patents.
-> States should not allow patents to restrict development and use of
-> software on general-purpose computers, but in those that do, we wish to
-> avoid the special danger that patents applied to a free program could
-> make it effectively proprietary.  To prevent this, the GPL assures that
-> patents cannot be used to render the program non-free.
->
->   The precise terms and conditions for copying, distribution and
-> modification follow.
->
->                        TERMS AND CONDITIONS
->
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README.md

@@ -1,10 +1,4 @@
-# DEPRECATED!
+This repository contains code for computations in [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680).
-
-This repository has not been updated for a while!
-If You are interested in replicating results for [1712.07167](https://arxiv.org/abs/1712.07167) please check [these instruction](https://kalmar.faculty.wmi.amu.edu.pl/post/1712.07176/)
-Also [this notebook](https://nbviewer.jupyter.org/gist/kalmarek/03510181bc1e7c98615e86e1ec580b2a) could be of some help. If everything else fails the [zenodo dataset](https://zenodo.org/record/1133440) should contain the last-resort instructions.
-
-This repository contains some legacy code for computations in [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680).
 
 # Installing
 To run the code You need `julia-v0.5` (should work on `v0.6`, but with warnings).
@@ -29,8 +23,6 @@ cd GroupswithPropertyT
 
 # Running
 
-## Naive implementation
-
 To check that $\Delta^2-\lambda\Delta$ is not decomposable to a sum of hermitian squares of elements in the ball of radius $2$ in $SL(2,7)$ run
 ```shell
 julia SL.jl -N 2 -p 7 --radius 2 --iterations 100000
@@ -47,24 +39,7 @@ If You see in the output (or in `full.log`) that the upper end of the interval w
 ```
 julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000 --tol 1e-9
 ```
-to achieve a better estimate (the residuals $\ell_1$-norm should be around $\|B_d(e))\|\cdot tol$)
+to achieve a better estimate (the residuals $\ell_1$-norm should be around $\|B_d(e))\|*tol$)
-
-## Symmetrization enhanced implementation
-
-A newer version of the software uses orbit and Wedderburn decomposition to effecitively find a (much) smaller optimisation problem to compute the spectral gap $\lambda$. In particular the solution to the original (naive) optimisation problem can be reconstructed from the solution of the symmetrised one.
-
-E.g. Run
-```shell
-julia SL_orbit.jl -N 4 --radius 2 --upper-bound 1.3
-```
-to find (and certify) the spectral gap for $SL(4, \mathbb{Z})$ is at least `1.2999...` in just under $2$ minutes time (for comparison this result requires over `5` hours in the old implementation on the same hardware).
-
-To replicate the results of _$\operatorname{Aut}(\textbf{F}_5)$ has property (T)_ You neet to run (on a `4`-core CPU)
-```shell
-julia ../AutFN_orbit.jl -N 5 --upper-bound 1.2 --iterations 24000000 --cpus 4
-```
-
-Note that this computation took more than `12` days and required at least `32`GB of ram (and possible more).
 
 # Help
 
@@ -95,7 +70,7 @@ optional arguments:
   -h, --help            show this help message and exit
 ```
 
-# Specific version of [1703.09680](https://arxiv.org/abs/1703.09680)
+# Specific version of the article
 
 To checkout the specific versions of packages used for [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680) run (inside the cloned `GroupswithPropertyT`)
 ```shell
@@ -108,36 +83,3 @@ Pkg.checkout("GroupRings", "1703.09680v1")
 Pkg.checkout("PropertyT", "1703.09680v1")
 Pkg.resolve()
 ```
-
-# Specific version of [1712.07167](https://arxiv.org/abs/1712.07167)
-
-You need to run `julia-0.6`.
-
-Clone `https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT` and checkout the `1712.07167` branch:
-```
-git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git
-cd ./GroupsWithPropertyT
-git checkout 1712.07167
-```
-
-In `julia`s REPL execute
-```julia
-Pkg.add("ArgParse")
-Pkg.add("Nemo")
-Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git")
-Pkg.checkout("Groups", "1712.07167")
-Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git")
-Pkg.checkout("GroupRings", "1712.07167")
-Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git")
-Pkg.checkout("PropertyT", "1712.07167")
-Pkg.checkout("SCS")
-Pkg.build("SCS")
-```
-
-This should resolve all the dependencies. Quit `julia` and place the `oSAutF5_r2` folder downloaded from [here](https://cloud.impan.pl/s/fGIpxvxdTYYkUxK) inside `GroupsWithPropertyT` folder. To verify the decomposition of $\Delta^2 - \lambda \Delta$ for the group run (if You have a `4`-core CPU at Your disposal)
-```julia
-julia AutFN_orbit.jl -N 5 --upper-bound=1.2 --cpus 4
-```
-If You want to generate `pm` and other files on Your own delete everything from the `oSAutF5_r2` folder but `1.2` folder and its contents and run the same command again.
-
-Note: You need at least `32`GB of RAM and spare `24`h of Your CPU.

SL.jl

@@ -0,0 +1,170 @@
+using ArgParse
+using GroupAlgebras
+using PropertyT
+
+using Nemo
+
+import SCS.SCSSolver
+
+
+function E(i::Int, j::Int, M::Nemo.MatSpace)
+    @assert i≠j
+    m = one(M)
+    m[i,j] = m[1,1]
+    return m
+end
+
+function SL_generatingset(n::Int)
+    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
+    G = Nemo.MatrixSpace(Nemo.ZZ, n,n)
+    S = [E(i,j,G) for (i,j) in indexing];
+    S = vcat(S, [transpose(x) for x in S]);
+    S = vcat(S, [inv(x) for x in S]);
+    return unique(S), one(G)
+end
+
+function SLsize(n,p)
+    result = 1
+    for k in 0:n-1
+        result *= p^n - p^k
+    end
+    return div(result, p-1)
+end
+
+function SL_generatingset(n::Int, p::Int)
+    p == 0 && return SL_generatingset(n)
+    (p > 1 && n > 0) || throw(ArgumentError("Both n and p should be positive integers!"))
+    println("Size(SL(n,p)) = $(SLsize(n,p))")
+    F = Nemo.ResidueRing(Nemo.ZZ, p)
+    G = Nemo.MatrixSpace(F, n,n)
+    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
+    S = [E(i, j, G) for (i,j) in indexing]
+    S = vcat(S, [transpose(x) for x in S])
+    S = vcat(S, [inv(s) for s in S])
+    return unique(S), one(G)
+end
+
+function products{T}(U::AbstractVector{T}, V::AbstractVector{T})
+    result = Vector{T}()
+    for u in U
+        for v in V
+            push!(result, u*v)
+        end
+    end
+    return unique(result)
+end
+
+function ΔandSDPconstraints(Id, S, radius)
+    sizes = Vector{Int}()
+    S = vcat([Id], S)
+    B = S
+    push!(sizes,length(B))
+    for i in 2:2*radius
+        B = products(S, B);
+        push!(sizes, length(B))
+    end
+    println("Generated balls of sizes $sizes")
+    basis = B[1:sizes[radius]]
+
+    product_matrix = PropertyT.create_product_matrix(B, sizes[radius]);
+    sdp_constraints = PropertyT.constraints_from_pm(product_matrix, length(B))
+    L_coeff = PropertyT.splaplacian_coeff(S, basis, length(B));
+    Δ = GroupAlgebraElement(L_coeff, product_matrix)
+
+    return Δ, sdp_constraints
+end
+
+#=
+To use file property(T).jl (specifically: check_property_T function)
+You need to define:
+
+function ΔandSDPconstraints(identity, S):: (Δ, sdp_constraints)
+
+=#
+
+function cpuinfo_physicalcores()
+    maxcore = -1
+    for line in eachline("/proc/cpuinfo")
+        if startswith(line, "core id")
+            maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
+        end
+    end
+    maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
+    return maxcore + 1
+end
+
+function parse_commandline()
+    s = ArgParseSettings()
+
+    @add_arg_table s begin
+        "--tol"
+            help = "set numerical tolerance for the SDP solver (default: 1e-5)"
+            arg_type = Float64
+            default = 1e-5
+        "--iterations"
+            help = "set maximal number of iterations for the SDP solver (default: 20000)"
+            arg_type = Int
+            default = 20000
+        "--upper-bound"
+            help = "Set an upper bound for the spectral gap (default: Inf)"
+            arg_type = Float64
+            default = Inf
+        "--cpus"
+            help = "Set number of cpus used by solver (default: auto)"
+            arg_type = Int
+            required = false
+        "-N"
+            help = "Consider matrices of size N (default: N=3)"
+            arg_type = Int
+            default = 3
+        "-p"
+            help = "Matrices over filed of p-elements (default: p=0 => over ZZ)"
+            arg_type = Int
+            default = 0
+        "--radius"
+            help = "Find the decomposition over B_r(e,S)"
+            arg_type = Int
+            default = 0
+    end
+
+    return parse_args(s)
+end
+
+function main()
+
+
+    parsed_args = parse_commandline()
+
+    tol = parsed_args["tol"]
+    iterations = parsed_args["iterations"]
+
+    solver = SCSSolver(eps=tol, max_iters=iterations, verbose=true)
+
+    N = parsed_args["N"]
+    upper_bound = parsed_args["upper-bound"]
+    p = parsed_args["p"]
+    if p == 0
+        name = "SL$(N)Z"
+    else
+        name = "SL$(N)_$p"
+    end
+    radius = parsed_args["radius"]
+    if radius == 0
+        name = name*"-$(string(upper_bound))"
+        radius = 2
+    else
+        name = name*"-$(string(upper_bound))-r=$radius"
+    end
+    S() = SL_generatingset(N, p)
+
+    if parsed_args["cpus"] ≠ nothing
+        if parsed_args["cpus"] > cpuinfo_physicalcores()
+            warn("Number of specified cores exceeds the physical core cound. Performance will suffer.")
+        end
+        BLAS.set_num_threads(parsed_args["cpus"])
+    end
+    @time PropertyT.check_property_T(name, S, solver, upper_bound, tol, radius)
+    return 0
+end
+
+main()

SemiDirectProduct.jl

@@ -0,0 +1,88 @@
+module SemiDirectProduct
+
+import Base: convert, show, isequal, ==, size, inv
+import Base: +, -, *, //
+
+export SemiDirectProductElement, matrix_repr
+
+"""
+Implements elements of a semidirect product of groups H and N, where N is normal in the product. Usually written as H ⋉ N.
+The multiplication inside semidirect product is defined as
+    (h₁, n₁) ⋅ (h₂, n₂) = (h₁h₂, n₁φ(h₁)(n₂)),
+where φ:H → Aut(N) is a homomorphism.
+
+In the case below we implement H = GL(n,K) and N = Kⁿ, the Affine Group (i.e. GL(n,K) ⋉ Kⁿ) where elements of GL(n,K) act on vectors in Kⁿ via matrix multiplication.
+# Arguments:
+* `h::Array{T,2}` : square invertible matrix (element of GL(n,K))
+* `n::Vector{T,1}` : vector in Kⁿ
+* `φ = φ(h,n) = φ(h)(n)` :2-argument function which defines the action of GL(n,K) on Kⁿ; matrix-vector multiplication by default.
+"""
+immutable SemiDirectProductElement{T<:Number}
+    h::Array{T,2}
+    n::Vector{T}
+    φ::Function
+
+    function SemiDirectProductElement(h::Array{T,2},n::Vector{T},φ::Function)
+        # size(h,1) == size(h,2)|| throw(ArgumentError("h has to be square matrix"))
+        det(h) ≠ 0 || throw(ArgumentError("h has to be invertible!"))
+        new(h,n,φ)
+    end
+end
+
+SemiDirectProductElement{T}(h::Array{T,2}, n::Vector{T}, φ) =
+    SemiDirectProductElement{T}(h,n,φ)
+
+SemiDirectProductElement{T}(h::Array{T,2}, n::Vector{T}) =
+    SemiDirectProductElement(h,n,*)
+
+SemiDirectProductElement{T}(h::Array{T,2}) =
+    SemiDirectProductElement(h,zeros(h[:,1]))
+
+SemiDirectProductElement{T}(n::Vector{T}) =
+        SemiDirectProductElement(eye(eltype(n), n))
+
+convert{T<:Number}(::Type{T}, X::SemiDirectProductElement) =
+    SemiDirectProductElement(convert(Array{T,2},X.h),
+                             convert(Vector{T},X.n),
+                             X.φ)
+
+size(X::SemiDirectProductElement) = (size(X.h), size(X.n))
+
+matrix_repr{T}(X::SemiDirectProductElement{T}) =
+    [X.h X.n; zeros(T, 1, size(X.h,2)) [1]]
+
+show{T}(io::IO, X::SemiDirectProductElement{T}) = print(io,
+    "Element of SemiDirectProduct over $T of size $(size(X)):\n",
+    matrix_repr(X))
+
+function isequal{T}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{T})
+    X.h == Y.h || return false
+    X.n == Y.n || return false
+    X.φ == Y.φ || return false
+    return true
+end
+
+function isequal{T,S}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{S})
+    W = promote_type(T,S)
+    warn("Comparing elements with different coefficients! trying to promoting to $W")
+    X = convert(W, X)
+    Y = convert(W, Y)
+    return isequal(X,Y)
+end
+
+(==)(X::SemiDirectProductElement, Y::SemiDirectProductElement) = isequal(X, Y)
+
+function semidirect_multiplication{T}(X::SemiDirectProductElement{T},
+                                      Y::SemiDirectProductElement{T})
+    size(X) == size(Y) || throw(ArgumentError("trying to multiply elements from different groups!"))
+    return SemiDirectProductElement(X.h*Y.h, X.n + X.φ(X.h, Y.n))
+end
+
+(*){T}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{T}) =
+    semidirect_multiplication(X,Y)
+
+inv{T}(X::SemiDirectProductElement{T}) =
+    SemiDirectProductElement(inv(X.h), X.φ(inv(X.h), -X.n))
+
+
+end

groups/Allgroups.jl

@@ -1,56 +0,0 @@
-module PropertyTGroups
-
-using PropertyT
-using AbstractAlgebra
-using Nemo
-using Groups
-using GroupRings
-
-export PropertyTGroup, SymmetrizedGroup, GAPGroup,
-    SpecialLinearGroup,
-    SpecialAutomorphismGroup,
-    HigmanGroup,
-    CapraceGroup,
-    MappingClassGroup
-
-export PropertyTGroup
-
-abstract type PropertyTGroup end
-
-abstract type SymmetrizedGroup <: PropertyTGroup end
-
-abstract type GAPGroup <: PropertyTGroup end
-
-function PropertyTGroup(args)
-    if haskey(args, "SL")
-        G = PropertyTGroups.SpecialLinearGroup(args)
-    elseif haskey(args, "SAut")
-        G = PropertyTGroups.SpecialAutomorphismGroup(args)
-    elseif haskey(args, "MCG")
-        G = PropertyTGroups.MappingClassGroup(args)
-    elseif haskey(args, "Higman")
-        G = PropertyTGroups.HigmanGroup(args)
-    elseif haskey(args, "Caprace")
-        G = PropertyTGroups.CapraceGroup(args)
-    else
-        throw("You must provide one of --SL, --SAut, --MCG, --Higman, --Caprace")
-    end
-    return G
-end
-
-include("autfreegroup.jl")
-include("speciallinear.jl")
-
-Comm(x,y) = x*y*x^-1*y^-1
-
-function generatingset(G::GAPGroup)
-    S = gens(group(G))
-    return unique([S; inv.(S)])
-end
-
-include("mappingclassgroup.jl")
-include("higman.jl")
-include("caprace.jl")
-include("actions.jl")
-
-end # of module PropertyTGroups

groups/actions.jl

@@ -1,92 +0,0 @@
-function (p::perm)(A::GroupRingElem)
-    RG = parent(A)
-    result = zero(RG, eltype(A.coeffs))
-    
-    for (idx, c) in enumerate(A.coeffs)
-        if c!= zero(eltype(A.coeffs))
-            result[p(RG.basis[idx])] = c
-        end
-    end
-    return result
-end
-
-###############################################################################
-#
-#  Action of WreathProductElems on Nemo.MatElem
-#
-###############################################################################
-
-function matrix_emb(n::DirectProductGroupElem, p::perm)
-    Id = parent(n.elts[1])()
-    elt = diagm([(-1)^(el == Id ? 0 : 1) for el in n.elts])
-    return elt[:, p.d]
-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
-
-import Base.*
-
-doc"""
-    *(x::AbstractAlgebra.MatElem, P::Generic.perm)
-> Apply the pemutation $P$ to the rows of the matrix $x$ and return the result.
-"""
-function *(x::AbstractAlgebra.MatElem, P::Generic.perm)
-   z = similar(x)
-   m = rows(x)
-   n = cols(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
-
-###############################################################################
-#
-#  Action of WreathProductElems on AutGroupElem
-#
-###############################################################################
-
-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])()
-    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)])
-    return elt
-end
-
-function AutFG_emb(A::AutGroup, p::perm)
-    isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
-    parent(p).n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(p)) into $A")
-    return A(Groups.perm_autsymbol(p))
-end
-
-function (g::WreathProductElem)(a::Groups.Automorphism)
-    A = parent(a)
-    g = AutFG_emb(A,g)
-    res = A()
-    Groups.r_multiply!(res, g.symbols, reduced=false)
-    Groups.r_multiply!(res, a.symbols, reduced=false)
-    Groups.r_multiply!(res, [inv(s) for s in reverse!(g.symbols)])
-    return res
-end
-
-function (p::perm)(a::Groups.Automorphism)
-    g = AutFG_emb(parent(a),p)
-    return g*a*inv(g)
-end

groups/autfreegroup.jl

@@ -1,21 +0,0 @@
-struct SpecialAutomorphismGroup{N} <: SymmetrizedGroup
-    group::AutGroup
-end
-
-function SpecialAutomorphismGroup(args::Dict)
-    N = args["SAut"]
-    return SpecialAutomorphismGroup{N}(AutGroup(FreeGroup(N), special=true))
-end
-
-name(G::SpecialAutomorphismGroup{N}) where N = "SAutF$(N)"
-
-group(G::SpecialAutomorphismGroup) = G.group
-
-function generatingset(G::SpecialAutomorphismGroup)
-    S = gens(group(G));
-    return unique([S; inv.(S)])
-end
-
-function autS(G::SpecialAutomorphismGroup{N}) where N
-    return WreathProduct(PermutationGroup(2), PermutationGroup(N))
-end

groups/caprace.jl

@@ -1,35 +0,0 @@
-struct CapraceGroup <: GAPGroup end
-
-name(G::CapraceGroup) = "CapraceGroup"
-
-function group(G::CapraceGroup)
-
-   caprace_group = Groups.FPGroup(["x","y","z","t","r"])
-
-   x,y,z,t,r = gens(caprace_group)
-
-   relations = [
-   x^7,
-   y^7,
-   t^2,
-   r^73,
-   t*r*t*r,
-   Comm(x,y)*z^-1,
-   Comm(x,z),
-   Comm(y,z),
-   Comm(x^2*y*z^-1, t),
-   Comm(x*y*z^3, t*r),
-   Comm(x^3*y*z^2, t*r^17),
-   Comm(x, t*r^-34),
-   Comm(y, t*r^-32),
-   Comm(z, t*r^-29),
-   Comm(x^-2*y*z, t*r^-25),
-   Comm(x^-1*y*z^-3, t*r^-19),
-   Comm(x^-3*y*z^-2, t*r^-11)
-   ];
-
-   relations = [relations; [inv(rel) for rel in relations]]
-
-   Groups.add_rels!(caprace_group, Dict(rel => caprace_group() for rel in relations))
-   return caprace_group
-end

groups/higman.jl

@@ -1,22 +0,0 @@
-struct HigmanGroup <: GAPGroup end
-
-name(G::HigmanGroup) = "HigmanGroup"
-
-function group(G::HigmanGroup)
-
-   higman_group = Groups.FPGroup(["a","b","c","d"]);
-
-   a,b,c,d = gens(higman_group)
-
-   relations = [
-   b*Comm(b,a),
-   c*Comm(c,b),
-   d*Comm(d,c),
-   a*Comm(a,d)
-   ];
-
-   relations = [relations; [inv(rel) for rel in relations]]
-
-   Groups.add_rels!(higman_group, Dict(rel => higman_group() for rel in relations))
-   return higman_group
-end

groups/mappingclassgroup.jl

@@ -1,83 +0,0 @@
-struct MappingClassGroup{N} <: GAPGroup end
-
-MappingClassGroup(args::Dict) = MappingClassGroup{args["MCG"]}()
-
-name(G::MappingClassGroup{N}) where N = "MCG(N)"
-
-function group(G::MappingClassGroup{N}) where N
-
-    if N < 2
-        throw("Genus must be at least 2!")
-    elseif N == 2
-        MCGroup = Groups.FPGroup(["a1","a2","a3","a4","a5"]);
-        S = gens(MCGroup)
-
-        n = length(S)
-        A = prod(reverse(S))*prod(S)
-
-        relations = [
-           [Comm(S[i], S[j]) for i in 1:n for j in 1:n if abs(i-j) > 1]...,
-           [S[i]*S[i+1]*S[i]*inv(S[i+1]*S[i]*S[i+1]) for i in 1:G.n-1]...,
-           (S[1]*S[2]*S[3])^4*inv(S[5])^2,
-           Comm(A, S[1]),
-           A^2
-           ]
-
-       relations = [relations; [inv(rel) for rel in relations]]
-       Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
-       return MCGroup
-
-    else
-        MCGroup = Groups.FPGroup(["a$i" for i in 0:2N])
-        S = gens(MCGroup)
-
-        a0 = S[1]
-        A = S[2:end]
-        k = length(A)
-
-        relations = [
-            [Comm(A[i], A[j]) for i in 1:k for j in 1:k if abs(i-j) > 1]...,
-            [Comm(a0, A[i]) for i in 1:k if i != 4]...,
-            [A[i]*A[(i+1)]*A[i]*inv(A[i+1]*A[i]*A[i+1]) for i in 1:k-1]...,
-            A[4]*a0*A[4]*inv(a0*A[4]*a0)
-            ]
-
-        # 3-chain relation
-        c = prod(reverse(A[1:4]))*prod(A[1:4])
-        b0 = c*a0*inv(c)
-        push!(relations, (A[1]*A[2]*A[3])^4*inv(a0*b0))
-
-        # Lantern relation
-        b1 = inv(A[4]*A[5]*A[3]*A[4])*a0*(A[4]*A[5]*A[3]*A[4])
-        b2 = inv(A[2]*A[3]*A[1]*A[2])*b1*(A[2]*A[3]*A[1]*A[2])
-        u  = inv(A[6]*A[5])*b1*(A[6]*A[5])
-        x  = prod(reverse(A[2:6]))*u*prod(inv.(A[1:4]))
-        b3 = x*a0*inv(x)
-        push!(relations, a0*b2*b1*inv(A[1]*A[3]*A[5]*b3))
-
-        # Hyperelliptic relation
-        X = prod(reverse(A))*prod(A)
-
-        function n(i::Int, b=b0)
-            if i == 1
-                return A[1]
-            elseif i == 2
-                return b
-            else
-                return w(i-2)*n(i-2)*w(i-2)
-            end
-        end
-
-        function w(i::Int)
-           (A[2i+4]*A[2i+3]*A[2i+2]* n(i+1))*(A[2i+1]*A[2i]  *A[2i+2]*A[2i+1])*
-           (A[2i+3]*A[2i+2]*A[2i+4]*A[2i+3])*( n(i+1)*A[2i+2]*A[2i+1]*A[2i]  )
-        end
-
-        # push!(relations, X*n(N)*inv(n(N)*X))
-
-        relations = [relations; [inv(rel) for rel in relations]]
-        Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
-
-        return MCGroup
-    end
-end

groups/speciallinear.jl

@@ -1,62 +0,0 @@
-struct SpecialLinearGroup{N} <: SymmetrizedGroup
-    group::AbstractAlgebra.Group
-    p::Int
-    X::Bool
-end
-
-function SpecialLinearGroup(args::Dict)
-    N = args["SL"]
-    p = args["p"]
-    X = args["X"]
-
-    if p == 0
-        G = MatrixSpace(Nemo.ZZ, N, N)
-    else
-        R = Nemo.NmodRing(UInt(p))
-        G = MatrixSpace(R, N, N)
-    end
-    return SpecialLinearGroup{N}(G, p, X)
-end
-
-function name(G::SpecialLinearGroup{N}) where N
-    if G.p == 0
-       R = (G.X ? "Z[x]" : "Z")
-    else
-       R = "F$(G.p)"
-    end
-    return SL($(G.N),$R)
-end
-
-group(G::SpecialLinearGroup) = G.group
-
-function generatingset(G::SpecialLinearGroup{N}) where N
-    G.p > 0 && G.X && throw("SL(n, F_p[x]) not implemented")
-    SL = group(G)
-    return generatingset(SL, G.X)
-end
-
-# r is the injectivity radius of
-# SL(n, Z[X]) -> SL(n, Z) induced by X -> 100
-
-function generatingset(SL::MatSpace, X::Bool=false, r=5)
-    n = SL.cols
-    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
-
-    if !X
-        S = [E(idx[1],idx[2],SL) for idx in indexing]
-    else
-        S = [E(i,j,SL,v) for (i,j) in indexing for v in [1, 100*r]]
-    end
-    return unique([S; inv.(S)])
-end
-
-function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring))
-    @assert i≠j
-    m = one(M)
-    m[i,j] = val
-    return m
-end
-
-function autS(G::SpecialLinearGroup{N}) where N
-    return WreathProduct(PermutationGroup(2), PermutationGroup(N))
-end

logging.jl

@@ -1,58 +0,0 @@
-using Memento
-
-function setup_logging(filename::String, handlername::Symbol=:log)
-    isdir(dirname(filename)) || mkdir(dirname(filename))
-    logger = Memento.config!("info", fmt="{date}| {msg}")
-    handler = DefaultHandler(filename, DefaultFormatter("{date}| {msg}"))
-    logger.handlers[String(handlername)] = handler
-    return logger
-end
-
-macro logtime(logger, ex)
-    quote
-        local stats = Base.gc_num()
-        local elapsedtime = Base.time_ns()
-        local val = $(esc(ex))
-        elapsedtime = Base.time_ns() - elapsedtime
-        local diff = Base.GC_Diff(Base.gc_num(), stats)
-        local ts = time_string(elapsedtime,
-            diff.allocd,
-            diff.total_time,
-            Base.gc_alloc_count(diff)
-            )
-        $(esc(info))($(esc(logger)), ts)
-        val
-    end
-end
-
-function time_string(elapsedtime, bytes, gctime, allocs)
-    str = @sprintf("%10.6f seconds", elapsedtime/1e9)
-    if bytes != 0 || allocs != 0
-        bytes, mb = Base.prettyprint_getunits(bytes, length(Base._mem_units), Int64(1024))
-        allocs, ma = Base.prettyprint_getunits(allocs, length(Base._cnt_units), Int64(1000))
-        if ma == 1
-            str*= @sprintf(" (%d%s allocation%s: ", allocs, Base._cnt_units[ma], allocs==1 ? "" : "s")
-        else
-            str*= @sprintf(" (%.2f%s allocations: ", allocs, Base._cnt_units[ma])
-        end
-        if mb == 1
-            str*= @sprintf("%d %s%s", bytes, Base._mem_units[mb], bytes==1 ? "" : "s")
-        else
-            str*= @sprintf("%.3f %s", bytes, Base._mem_units[mb])
-        end
-        if gctime > 0
-            str*= @sprintf(", %.2f%% gc time", 100*gctime/elapsedtime)
-        end
-        str*=")"
-    elseif gctime > 0
-        str*= @sprintf(", %.2f%% gc time", 100*gctime/elapsedtime)
-    end
-    return str
-end
-
-import Base: info, @time
-
-Base.info(x) = info(getlogger(), x)
-macro time(x)
-    return :(@logtime(getlogger(Main), $(esc(x))))
-end

main.jl

@@ -1,61 +0,0 @@
-using PropertyT
-
-include("FPGroups_GAP.jl")
-
-include("groups/Allgroups.jl")
-using PropertyTGroups
-
-import PropertyT.Settings
-
-function summarize(sett::PropertyT.Settings)
-    info("Threads:     $(Threads.nthreads())")
-    info("Workers:     $(workers())")
-    info("GroupDir:    $(PropertyT.prepath(sett))")
-    info(string(sett.G))
-    info("with generating set of size $(length(sett.S))")
-
-    info("Radius:      $(sett.radius)")
-    info("Precision:   $(sett.tol)")
-    info("Upper bound: $(sett.upper_bound)")
-    info("Solver:      $(sett.solver)")
-end
-
-function Settings(Gr::PropertyTGroup, args, solver)
-    r = get(args, "radius", 2)
-    gr_name = PropertyTGroups.name(Gr)*"_r$r"
-    G = PropertyTGroups.group(Gr)
-    S = PropertyTGroups.generatingset(Gr)
-
-    sol = solver
-    ub = get(args,"upper-bound", Inf)
-    tol = get(args,"tol", 1e-10)
-    ws = get(args, "warmstart", false)
-
-    if get(args, "nosymmetry", false)
-        return PropertyT.Settings(gr_name, G, S, r, sol, ub, tol, ws)
-    else
-        autS = PropertyTGroups.autS(Gr)
-        return PropertyT.Settings(gr_name, G, S, r, sol, ub, tol, ws, autS)
-    end
-end
-
-function main(::PropertyTGroup, sett::PropertyT.Settings)
-    isdir(PropertyT.fullpath(sett)) || mkpath(PropertyT.fullpath(sett))
-
-    summarize(sett)
-
-    return PropertyT.check_property_T(sett)
-end
-
-function main(::GAPGroup, sett::PropertyT.Settings)
-    isdir(PropertyT.fullpath(sett)) || mkpath(PropertyT.fullpath(sett))
-
-    summarize(sett)
-
-    S = [s for s in sett.S if s.symbols[1].pow == 1]
-    relations = [k*inv(v) for (k,v) in sett.G.rels]
-
-    prepare_pm_delta(PropertyT.prepath(sett), GAP_groupcode(S, relations), sett.radius)
-
-    return PropertyT.check_property_T(sett)
-end

positivity/check_positivity.jl

@@ -1,197 +0,0 @@
-using AbstractAlgebra
-using Groups
-using GroupRings
-using PropertyT
-
-using SCS
-solver(tol, iterations) =
-    SCSSolver(linearsolver=SCS.Direct,
-        eps=tol, max_iters=iterations,
-        alpha=1.95, acceleration_lookback=1)
-
-include("../main.jl")
-
-using PropertyTGroups
-
-args = Dict("SAut" => 5, "upper-bound" => 50.0, "radius" => 2, "nosymmetry"=>false, "tol"=>1e-12, "iterations" =>200000, "warmstart" => true)
-
-Gr = PropertyTGroups.PropertyTGroup(args)
-sett = PropertyT.Settings(Gr, args,
-    solver(args["tol"], args["iterations"]))
-
-@show sett
-
-fullpath = PropertyT.fullpath(sett)
-isdir(fullpath) || mkpath(fullpath)
-# setup_logging(PropertyT.filename(fullpath, :fulllog), :fulllog)
-
-function small_generating_set(RG::GroupRing{AutGroup{N}}, n) where N
-    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
-
-    rmuls = [Groups.rmul_autsymbol(i,j) for (i,j) in indexing]
-    lmuls = [Groups.lmul_autsymbol(i,j) for (i,j) in indexing]
-    gen_set = RG.group.([rmuls; lmuls])
-
-    return [gen_set; inv.(gen_set)]
-end
-
-function computeX(RG::GroupRing{AutGroup{N}}) where N
-    Tn = small_generating_set(RG, N-1)
-
-    ℤ = Int64
-    Δn = length(Tn)*one(RG, ℤ) - RG(Tn, ℤ);
-
-    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
-
-    @time X = sum(σ(Δn)*sum(τ(Δn) for τ ∈ Alt_N if τ ≠ σ) for σ in Alt_N);
-    return X
-end
-
-function Sq(RG::GroupRing{AutGroup{N}}) where N
-    T2 = small_generating_set(RG, 2)
-    ℤ = Int64
-    Δ₂ = length(T2)*one(RG, ℤ) - RG(T2, ℤ);
-
-    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
-    elt = sum(σ(Δ₂)^2 for σ in Alt_N)
-    return elt
-end
-
-function Adj(RG::GroupRing{AutGroup{N}}) where N
-    T2 = small_generating_set(RG, 2)
-
-    ℤ = Int64
-    Δ₂ = length(T2)*one(RG, ℤ) - RG(T2, ℤ);
-
-    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
-
-    adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) ∩ Set([τ[i], τ[j]])
-    adj(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 1]
-
-    @time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in adj(σ)) for σ in Alt_N);
-    # return RG(elt.coeffs÷factorial(N-2)^2)
-    return elt
-end
-
-function Op(RG::GroupRing{AutGroup{N}}) where N
-    T2 = small_generating_set(RG, 2)
-
-    ℤ = Int64
-    Δ₂ = length(T2)*one(RG, ℤ) - RG(T2, ℤ);
-
-    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
-
-    adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) ∩ Set([τ[i], τ[j]])
-    adj(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 0]
-
-    @time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in adj(σ)) for σ in Alt_N);
-    # return RG(elt.coeffs÷factorial(N-2)^2)
-    return elt
-end
-
-const ELT_FILE = joinpath(dirname(PropertyT.filename(sett, :Δ)), "SqAdjOp_coeffs.jld")
-const WARMSTART_FILE = PropertyT.filename(sett, :warmstart)
-
-if isfile(PropertyT.filename(sett,:Δ)) && isfile(ELT_FILE) &&
-    isfile(PropertyT.filename(sett, :OrbitData))
-    # cached
-    Δ = PropertyT.loadGRElem(PropertyT.filename(sett,:Δ), sett.G)
-    RG = parent(Δ)
-    orbit_data = load(PropertyT.filename(sett, :OrbitData), "OrbitData")
-    sq_c, adj_c, op_c = load(ELT_FILE, "Sq", "Adj", "Op")
-    # elt = ELT_FILE, sett.G)
-    sq = GroupRingElem(sq_c, RG)
-    adj = GroupRingElem(adj_c, RG)
-    op = GroupRingElem(op_c, RG);
-else
-    info("Compute Laplacian")
-    Δ = PropertyT.Laplacian(sett.S, sett.radius)
-    RG = parent(Δ)
-
-    info("Compute Sq, Adj, Op")
-    @time sq, adj, op = Sq(RG), Adj(RG), Op(RG)
-
-    PropertyT.saveGRElem(PropertyT.filename(sett, :Δ), Δ)
-    save(ELT_FILE, "Sq", sq.coeffs, "Adj", adj.coeffs, "Op", op.coeffs)
-
-    info("Compute OrbitData")
-    if !isfile(PropertyT.filename(sett, :OrbitData))
-        orbit_data = PropertyT.OrbitData(parent(Y), sett.autS)
-        save(PropertyT.filename(sett, :OrbitData), "OrbitData", orbit_data)
-    else
-        orbit_data = load(PropertyT.filename(sett, :OrbitData), "OrbitData")
-    end
-end;
-
-orbit_data = PropertyT.decimate(orbit_data);
-
-elt = adj+2op;
-
-const SOLUTION_FILE = PropertyT.filename(sett, :solution)
-
-if !isfile(SOLUTION_FILE)
-    
-    SDP_problem, varλ, varP = PropertyT.SOS_problem(elt, Δ, orbit_data; upper_bound=sett.upper_bound)
-
-    begin
-        using SCS
-        scs_solver = SCS.SCSSolver(linearsolver=SCS.Direct,
-            eps=sett.tol,
-            max_iters=args["iterations"],
-            alpha=1.95,
-            acceleration_lookback=1)
-
-        JuMP.setsolver(SDP_problem, scs_solver)
-    end
-
-    λ = Ps  = nothing
-    ws = PropertyT.warmstart(sett)
-
-    # using ProgressMeter
-
-    # @showprogress "Running SDP optimization step... " for i in 1:args["repetitions"]
-    while true
-        λ, Ps, ws = PropertyT.solve(PropertyT.filename(sett, :solverlog),
-        SDP_problem, varλ, varP, ws);
-
-        if all((!isnan).(ws[1]))
-            save(WARMSTART_FILE, "warmstart", ws, "λ", λ, "Ps", Ps)
-            save(WARMSTART_FILE[1:end-4]*"_$(now()).jld", "warmstart", ws, "λ", λ, "Ps", Ps)
-        else
-            warn("No valid solution was saved!")
-        end
-    end
-    
-    info("Reconstructing P...")
-    @time P = PropertyT.reconstruct(Ps, orbit_data);
-    save(SOLUTION_FILE, "λ", λ, "P", P)
-end
-
-P, λ = load(SOLUTION_FILE, "P", "λ")
-@show λ;
-
-@time const Q = real(sqrtm(P));
-
-function SOS_residual(eoi::GroupRingElem, Q::Matrix)
-    RG = parent(eoi)
-    @time sos = PropertyT.compute_SOS(RG, Q);
-    return eoi - sos
-end
-
-info("Floating Point arithmetic:")
-EOI = elt - λ*Δ
-b = SOS_residual(EOI, Q)
-@show norm(b, 1);
-
-info("Interval arithmetic:")
-using IntervalArithmetic
-Qint = PropertyT.augIdproj(Q);
-@assert all([zero(eltype(Q)) in sum(view(Qint, :, i)) for i in 1:size(Qint, 2)])
-
-EOI_int = elt - @interval(λ)*Δ;
-Q_int = PropertyT.augIdproj(Q);
-@assert all([zero(eltype(Q)) in sum(view(Q_int, :, i)) for i in 1:size(Q_int, 2)])
-b_int = SOS_residual(EOI_int, Q_int)
-@show norm(b_int, 1);
-
-info("λ is certified to be > ", (@interval(λ) - 2^2*norm(b_int,1)).lo)

run.jl

@@ -1,108 +0,0 @@
-using ArgParse
-
-###############################################################################
-#
-#  Parsing command line
-#
-###############################################################################
-
-function parse_commandline()
-    settings = ArgParseSettings()
-
-    @add_arg_table settings begin
-        "--tol"
-            help = "set numerical tolerance for the SDP solver"
-            arg_type = Float64
-            default = 1e-6
-        "--iterations"
-            help = "set maximal number of iterations for the SDP solver"
-            arg_type = Int
-            default = 50000
-        "--upper-bound"
-            help = "Set an upper bound for the spectral gap"
-            arg_type = Float64
-            default = Inf
-        "--cpus"
-            help = "Set number of cpus used by solver"
-            arg_type = Int
-            required = false
-        "--radius"
-            help = "Radius of ball B_r(e,S) to find solution over"
-            arg_type = Int
-            default = 2
-        "--warmstart"
-            help = "Use warmstart.jld as the initial guess for SCS"
-            action = :store_true
-        "--nosymmetry"
-            help = "Don't use symmetries of the Laplacian"
-            action = :store_true
-
-        "--SL "
-            help = "GROUP: the group generated by elementary matrices of size n by n"
-            arg_type = Int
-            required = false
-        "-p"
-            help = "Matrices over field of p-elements (p=0 => over ZZ) [only with --SL]"
-            arg_type = Int
-            default = 0
-        "-X"
-            help = "Consider EL(N, ZZ⟨X⟩) [only with --SL]"
-            action = :store_true
-
-        "--SAut"
-            help = "GROUP: the automorphisms group of the free group on N generators"
-            arg_type = Int
-            required = false
-
-        "--MCG"
-            help = "GROUP: mapping class group of surface of genus N"
-            arg_type = Int
-            required = false
-
-        "--Higman"
-            help = "GROUP: the Higman Group"
-            action = :store_true
-
-        "--Caprace"
-            help = "GROUP: for Caprace Group"
-            action = :store_true
-    end
-    return parse_args(settings)
-end
-
-const PARSEDARGS = parse_commandline()
-
-set_parallel_mthread(PARSEDARGS, workers=false)
-
-include("CPUselect.jl")
-include("logging.jl")
-include("main.jl")
-
-using SCS.SCSSolver
-# using Mosek
-# using CSDP
-# using SDPA
-
-solver(tol, iterations) =
-    SCSSolver(linearsolver=SCS.Direct,
-        eps=tol, max_iters=iterations,
-        alpha=1.95, acceleration_lookback=1)
-
-# Mosek.MosekSolver(
-#    MSK_DPAR_INTPNT_CO_TOL_REL_GAP=tol,
-#    MSK_IPAR_INTPNT_MAX_ITERATIONS=iterations,
-#    QUIET=false)
-
-# CSDP.CSDPSolver(axtol=tol, atytol=tol, objtol=tol, minstepp=tol*10.0^-1, minstepd=tol*10.0^-1)
-
-# SDPA.SDPASolver(epsilonStar=tol, epsilonDash=tol)
-
-const Gr = PropertyTGroups.PropertyTGroup(PARSEDARGS)
-const sett = PropertyT.Settings(Gr, PARSEDARGS,
-    solver(PARSEDARGS["tol"], PARSEDARGS["iterations"]))
-
-fullpath = PropertyT.fullpath(sett)
-isdir(fullpath) || mkpath(fullpath)
-setup_logging(PropertyT.filename(fullpath, :fulllog), :fulllog)
-
-main(Gr, sett)

runtests.jl

@@ -1,222 +0,0 @@
-using Base.Test
-
-include("main.jl")
-
-testdir = "tests_"*string(now())
-mkdir(testdir)
-include("logging.jl")
-logger=setup_logging(joinpath(testdir, "tests.log"))
-info(testdir)
-
-cd(testdir)
-
-# groupname = name(G)
-# ub = PARSEDARGS["upper-bound"]
-#
-# fullpath = joinpath(groupname, string(ub))
-# isdir(fullpath) || mkpath(fullpath)
-
-separator(n=60) = info("\n"*("\n"*"="^n*"\n"^3)*"\n")
-
-
-function SL_tests(args)
-
-
-    args["SL"] = 2
-    args["p"] = 3
-    G = PropertyTGroup(args)
-    @test main(G) == true
-    separator()
-
-    let args = args
-        args["SL"] = 2
-        args["p"] = 5
-        G = PropertyTGroup(args)
-        @test main(G) == false
-        separator()
-
-        args["warmstart"] = true
-        G = PropertyTGroup(args)
-        @test main(G) == false
-        separator()
-
-        args["upper-bound"] = 0.1
-        G = PropertyTGroup(args)
-        @test main(G) == true
-        separator()
-    end
-
-    args["SL"] = 2
-    args["p"] = 7
-    G = PropertyTGroup(args)
-    @test main(G) == false
-    separator()
-
-    args["SL"] = 3
-    args["p"] = 7
-    G = PropertyTGroup(args)
-    @test main(G) == true
-    separator()
-
-    # begin
-    #     args["iterations"] = 25000
-    #     args["N"] = 3
-    #     args["p"] = 0
-    #     args["upper-bound"] = Inf
-    #
-    #     G = PropertyTGroups.SpecialLinearGroup(args)
-    #     @test main(G) == false
-    #     separator()
-    #
-	#     args["warmstart"] = false
-    #     args["upper-bound"] = 0.27
-    #     G = PropertyTGroups.SpecialLinearGroup(args)
-    #     @test main(G) == false
-    #     separator()
-    #
-    #     args["warmstart"] = true
-    #     G = PropertyTGroups.SpecialLinearGroup(args)
-    #     @test main(G) == true
-    #     separator()
-    # end
-
-    return 0
-end
-
-function SAut_tests(args)
-
-    G = PropertyTGroup(args)
-    @test main(G) == false
-    separator()
-
-    args["warmstart"] = true
-    G = PropertyTGroup(args)
-    @test main(G) == false
-    separator()
-
-    args["upper-bound"] = 0.1
-    G = PropertyTGroup(args)
-    @test main(G) == false
-    separator()
-
-    return 0
-end
-
-@testset "Groups with(out) (T)" begin
-
-    @testset "GAPGroups" begin
-        args = Dict(
-            "Higman" => true,
-            "iterations"=>5000,
-            "tol"=>1e-7,
-            "upper-bound"=>Inf,
-            "cpus"=>2,
-            "radius"=>2,
-            "warmstart"=>false,
-            "nosymmetry"=>true,
-        )
-
-        G = PropertyTGroup(args)
-        @test main(G) == false
-
-        args = Dict(
-            "Caprace" => true,
-            "iterations"=>5000,
-            "tol"=>1e-7,
-            "upper-bound"=>Inf,
-            "cpus"=>2,
-            "radius"=>2,
-            "warmstart"=>false,
-            "nosymmetry"=>true,
-        )
-
-        G = PropertyTGroup(args)
-        @test main(G) == false
-
-        args = Dict(
-            "MCG" => 3,
-            "iterations"=>5000,
-            "tol"=>1e-7,
-            "upper-bound"=>Inf,
-            "cpus"=>2,
-            "radius"=>2,
-            "warmstart"=>false,
-            "nosymmetry"=>true,
-        )
-
-        G = PropertyTGroup(args)
-        @test main(G) == false
-    end
-
-    @testset "SLn's" begin
-        @testset "Non-Symmetrized" begin
-
-            args = Dict(
-                "SL" => 2,
-                "p" => 3,
-                "X" => false,
-                "iterations"=>50000,
-                "tol"=>1e-7,
-                "upper-bound"=>Inf,
-                "cpus"=>2,
-                "radius"=>2,
-                "warmstart"=>false,
-                "nosymmetry"=>true,
-            )
-
-            SL_tests(args)
-        end
-
-        @testset "Symmetrized" begin
-
-            args = Dict(
-                "SL" => 2,
-                "p" => 3,
-                "X" => false,
-                "iterations"=>20000,
-                "tol"=>1e-7,
-                "upper-bound"=>Inf,
-                "cpus"=>2,
-                "radius"=>2,
-                "warmstart"=>false,
-                "nosymmetry"=>false,
-            )
-
-            SL_tests(args)
-        end
-    end
-
-    @testset "SAutF_n's" begin
-
-        @testset "Non-Symmetrized" begin
-
-            args = Dict(
-                "SAut" => 2,
-                "iterations"=>5000,
-                "tol"=>1e-7,
-                "upper-bound"=>Inf,
-                "cpus"=>2,
-                "radius"=>2,
-                "warmstart"=>false,
-                "nosymmetry"=>true,
-            )
-            SAut_tests(args)
-        end
-
-        @testset "Symmetrized" begin
-            args = Dict(
-                "SAut" => 3,
-                "iterations"=>500,
-                "tol"=>1e-7,
-                "upper-bound"=>Inf,
-                "cpus"=>2,
-                "radius"=>2,
-                "warmstart"=>false,
-                "nosymmetry"=>false,
-            )
-            SAut_tests(args)
-        end
-    end
-
-
-end