text:
module Manifolds

using ComputedFieldTypes
using DifferentialForms
using LinearAlgebra
using NearestNeighbors
using SparseArrays
using StaticArrays

using ..Algorithms
using ..Defs
using ..Meshing
using ..SparseOps
using ..ZeroOrOne

################################################################################

export PrimalDual, Pr, Dl
@enum PrimalDual::Bool Pr Dl

Base.:!(P::PrimalDual) = PrimalDual(!(Bool(P)))

################################################################################

export DualKind, BarycentricDuals, CircumcentricDuals
@enum DualKind BarycentricDuals CircumcentricDuals

################################################################################

const VecDict{K,T} = Dict{K,IDVector{<:Any,T}} where {T}
const OpDict{K,T} = Dict{K,SparseOp{<:Any,<:Any,T}} where {K,T}

export Manifold
"""
A discrete manifold
"""
mutable struct Manifold{D,C,S}
    name::String

    signature::Int              # [+1, -1]

    dualkind::DualKind
    use_weighted_duals::Bool

    # TODO: Implement also a "cube complex" representation

    # If `simplices[R][i,j]` is present, then vertex `i` is part of
    # the `R`-simplex `j`. `R ∈ 0:D`. We could omit `R=0`.
    # TODO: Replace this by Vector{SVector{R+1,Int}}
    _simplices::OpDict{Int,One}
    # simplices::Dict{Int,Array{Int,2}}

    # The boundary ∂ of `0`-forms vanishes and is not stored. If
    # `boundaries[R][i,j] = s`, then `R`-simplex `j` has
    # `(R-1)`-simplex `i` as boundary with orientation `s`. `R ∈ 1:D`.
    _boundaries::OpDict{Int,Int8}
    _isboundary::OpDict{Int,One}

    # Lookup tables from `Ri`-simplices to `Rj`-simplices. If
    # `lookup[(Ri,Rj)][i,j]` is present, then `Rj`-simplex `j`
    # contains `Ri`-simplex `i`. `Ri ∈ 0:D, Rj ∈ 0:D`. Many of these
    # are trivial and could be omitted.
    _lookup::OpDict{Tuple{Int,Int},One}
    # lookup::Dict{Tuple{Int,Int},Array{Int,2}}

    # coords::Array{S,2}
    _coords::VecDict{Int,SVector{C,S}}
    _volumes::VecDict{Int,S}

    # Vertex weights [arXiv:math/0508188], [DOI:10.1145/2602143]
    _weights::IDVector{0,S}

    # Coordinates of vertices of dual grid, i.e.
    # barycentres/circumcentres of primal top-simplices
    # dualcoords::Array{S,2}
    _dualcoords::VecDict{Int,SVector{C,S}}
    # `dualvolumes[R]` are the volumes of the simplices dual to the
    # primal `R`-simplices
    _dualvolumes::VecDict{Int,S}

    # Nearest neighbour tree for simplex vertices
    _simplex_tree::Union{Nothing,KDTree{SVector{C,S},Euclidean,S}}

    function Manifold{D,C,S}(name::String, signature::Int, dualkind::DualKind,
                             use_weighted_duals::Bool,
                             simplices::OpDict{Int,One},
                             boundaries::OpDict{Int,Int8},
                             coords0::IDVector{0,SVector{C,S}},
                             weights::IDVector{0,S}) where {D,C,S}
        D::Int
        @assert 0 ≤ D ≤ C
        mfd = new{D,C,S}(name, signature, dualkind, use_weighted_duals,
                         simplices, boundaries, OpDict{Int,One}(),
                         OpDict{Tuple{Int,Int},One}(),
                         VecDict{Int,SVector{C,S}}(0 => coords0),
                         VecDict{Int,S}(), weights, VecDict{Int,SVector{C,S}}(),
                         VecDict{Int,S}(), nothing)
        #TODO @assert invariant(mfd)
        return mfd
    end
end

function Manifold(name::String, signature::Int, dualkind::DualKind,
                  use_weighted_duals::Bool, simplices::OpDict{Int,One},
                  boundaries::OpDict{Int,Int8},
                  coords0::IDVector{0,SVector{C,S}},
                  weights::IDVector{0,S}) where {C,S}
    D = maximum(keys(simplices))
    return Manifold{D,C,S}(name, signature, dualkind, use_weighted_duals,
                           simplices, boundaries, coords0, weights)
end

################################################################################

function Base.show(io::IO, mfd::Manifold{D}) where {D}
    println(io)
    println(io, "Manifold{$D}(")
    println(io, "    name=$(mfd.name)")
    for R in 0:D
        println(io, "    simplices[$R]=$(mfd._simplices[R])")
    end
    for R in 1:D
        println(io, "    boundaries[$R]=$(mfd._boundaries[R])")
    end
    for R in 0:(D - 1)
        if haskey(mfd._isboundary, R)
            println(io, "    isboundary[$R]=$(mfd._isboundary[R])")
        end
    end
    for R in 0:D
        if haskey(mfd._coords, R)
            println(io, "    coords[$R]=$(mfd._coords[R])")
        end
    end
    for R in 0:D
        if haskey(mfd._volumes, R)
            println(io, "    volumes[$R]=$(mfd._volumes[R])")
        end
    end
    println(io, "    weights=$(mfd._weights)")
    for R in 0:D
        if haskey(mfd._dualcoords, R)
            println(io, "    dualcoords[$R]=$(mfd._dualcoords[R])")
        end
    end
    for R in 0:D
        if haskey(mfd._dualvolumes, R)
            println(io, "    dualvolumes[$R]=$(mfd._dualvolumes[R])")
        end
    end
    return print(io, ")")
end

function Defs.invariant(mfd::Manifold{D,C})::Bool where {D,C}
    D ≥ 0 || (@assert false; return false)
    C ≥ D || (@assert false; return false)

    # Check simplices
    Set(keys(mfd._simplices)) == Set(0:D) || (@assert false; return false)
    for R in 0:D
        simplices = get_simplices(mfd, R)
        size(simplices) == (nsimplices(mfd, 0), nsimplices(mfd, R)) ||
            (@assert false; return false)
        for j in axes(simplices, 2)
            sj = sparse_column_rows(simplices, j)
            length(sj) == R + 1 || (@assert false; return false)
        end
    end

    # Check boundaries
    Set(keys(mfd._boundaries)) == Set(1:D) || (@assert false; return false)
    for R in 1:D
        boundaries = get_boundaries(mfd, R)
        size(boundaries) == (nsimplices(mfd, R - 1), nsimplices(mfd, R)) ||
            (@assert false; return false)
        simplicesR = get_simplices(mfd, R)
        simplicesR1 = get_simplices(mfd, R - 1)
        for j in axes(boundaries, 2) # R-simplex
            vj = sparse_column_rows(simplicesR, j)
            sj = sparse_column(boundaries, j)
            for (i, p) in sj    # (R-1)-simplex
                si = sparse_column_rows(simplicesR1, i)
                for k in si     # vertices
                    k ∈ vj || (@assert false; return false)
                end
                abs(p) == 1 || (@assert false; return false)
            end
        end
    end

    # # Check isboundary
    # Set(keys(mfd.isboundary)) == Set(0:(D - 1)) || (@assert false; return false)
    # for R in 0:(D - 1)
    #     isboundary = mfd.isboundary[R]::SparseOp{R,R,One}
    #     size(isboundary) == (nsimplices(mfd, R), nsimplices(mfd, R)) ||
    #         (@assert false; return false)
    # end
    # # TODO: check content as well

    # # Check lookup tables
    # Set(keys(mfd._lookup)) == Set((Ri, Rj) for Ri in 0:D for Rj in 0:D) ||
    #     (@assert false; return false)
    # for Ri in 0:D, Rj in 0:D
    #     lookup = mfd._lookup[(Ri, Rj)]::SparseOp{Ri,Rj,One}
    #     size(lookup) == (nsimplices(mfd, Ri), nsimplices(mfd, Rj)) ||
    #         (@assert false; return false)
    #     if Rj ≥ Ri
    #         for j in 1:size(lookup, 2) # Rj-simplex
    #             vj = sparse_column_rows(mfd.simplices[Rj], j)
    #             length(vj) == Rj + 1 || (@assert false; return false)
    #             sj = sparse_column_rows(lookup, j)
    #             length(sj) == binomial(Rj + 1, Ri + 1) ||
    #                 (@assert false; return false)
    #             for i in sj         # Ri-simplex
    #                 si = sparse_column_rows(mfd.simplices[Ri], i)
    #                 for k in si     # vertices
    #                     k ∈ vj || (@assert false; return false)
    #                 end
    #             end
    #         end
    #         mfd._lookup[(Ri, Rj)] == mfd._lookup[(Rj, Ri)]' ||
    #             (@assert false; return false)
    #     end
    # end

    # Set(keys(mfd._coords)) == Set(0:D) || (@assert false; return false)
    # for R in 0:D
    #     length(mfd._coords[R]) == nsimplices(mfd, R) ||
    #         (@assert false; return false)
    # end
    #
    # Set(keys(mfd._volumes)) == Set(0:D) || (@assert false; return false)
    # for R in 0:D
    #     length(mfd._volumes[R]) == nsimplices(mfd, R) ||
    #         (@assert false; return false)
    # end

    length(mfd._weights) == nsimplices(mfd, 0) || (@assert false; return false)

    # Set(keys(mfd.dualcoords)) == Set(0:D) || (@assert false; return false)
    # for R in 0:D
    #     length(mfd.dualcoords[R]) == nsimplices(mfd, R) ||
    #         (@assert false; return false)
    # end

    # Set(keys(mfd.dualvolumes)) == Set(0:D) || (@assert false; return false)
    # for R in 0:D
    #     length(mfd.dualvolumes[R]) == nsimplices(mfd, R) ||
    #         (@assert false; return false)
    # end

    return true
end

# Comparison

function Base.:(==)(mfd1::Manifold{D,C}, mfd2::Manifold{D,C})::Bool where {D,C}
    mfd1 ≡ mfd2 && return true
    return mfd1.signature == mfd2.signature &&
           mfd1.dualkind == mfd2.dualkind &&
           mfd1.use_weighted_duals == mfd2.use_weighted_duals &&
           mfd1._simplices[D] == mfd2._simplices[D] &&
           mfd1._coords[0] == mfd2._coords[0] &&
           mfd1._weights == mfd2._weights
end
function Base.isequal(mfd1::Manifold{D,C},
                      mfd2::Manifold{D,C})::Bool where {D,C}
    return
    return isequal(mfd1.signature, mfd2.signature) &&
           isequal(mfd1.dualkind, mfd2.dualkind) &&
           isequal(mfd1.use_weighted_duals, mfd2.use_weighted_duals) &&
           isequal(mfd1._simplices[D], mfd2._simplices[D]) &&
           isequal(mfd1._coords[0], mfd2._coords[0]) &&
           isequal(mfd1._weights, mfd2._weights)
end
function Base.hash(mfd::Manifold{D,C}, h::UInt) where {D,C}
    return hash(0x4d34f4ae,
                hash(D,
                     hash(C,
                          hash(mfd.signature,
                               hash(mfd.dualkind,
                                    hash(mfd.use_weighted_duals,
                                         hash(mfd._simplices[D],
                                              hash(mfd._coords[0],
                                                   hash(mfd._weights[0], h)))))))))
end

Base.ndims(::Manifold{D}) where {D} = D

export nsimplices
function nsimplices(mfd::Manifold, ::Val{R}) where {R}
    return size(get_simplices(Val(R), mfd), 2)
end
nsimplices(mfd::Manifold, R::Integer) = size(get_simplices(mfd, R), 2)

export random_point
"""
Return random point in manifold
"""
function random_point(::Val{R}, mfd::Manifold{D,C,S}) where {D,C,R,S}
    @assert 0 ≤ R ≤ D ≤ C
    C == 0 && return SVector{C,S}()
    N = R + 1
    # Choose simplex
    i = ID{R}(rand(1:nsimplices(mfd, R)))
    si = sparse_column_rows(get_simplices(mfd, R), i)
    @assert length(si) == N
    # Choose point in simplex
    λ = abs.(randn(SVector{N,S}))
    λ /= sum(λ)
    x = sum(λ[n] * get_coords(mfd)[si[n]] for n in 1:N)
    return x::SVector{C,S}
end

################################################################################

# Outer constructor

function Manifold(name::String, simplicesD::SparseOp{0,D,One},
                  coords0::IDVector{0,SVector{C,S}}, weights::IDVector{0,S};
                  signature::Int=+1, dualkind::DualKind=CircumcentricDuals,
                  optimize_mesh::Bool=true,
                  use_weighted_duals::Bool=true) where {D,C,S}
    @assert 0 ≤ D ≤ C

    nvertices, nsimplices = size(simplicesD)
    @assert length(coords0) == nvertices
    @assert length(weights) == nvertices

    if false && D > 0 && nvertices > 0
        # Sort vertices by Morton order
        # (Currently disabled since it doesn't seem to improve speed)
        elmin(x, y) = min.(x, y)
        elmax(x, y) = max.(x, y)
        xmin = reduce(elmin, coords0)
        xmax = reduce(elmax, coords0)
        mvals = [morton(c0, xmin, xmax) for c0 in coords0]
        perm = sortperm(mvals)
        invperm = Array{Int}(undef, nvertices)
        invperm[perm] = 1:nvertices
        simplicesD_array = Array{SVector{D + 1,Int}}(undef, nsimplices)
        for j in axes(simplicesD, 2)
            si0 = sparse_column_rows(simplicesD, j)
            @assert length(si0) == D + 1
            si = SVector{D + 1,Int}(Int(si0[d]) for d in 1:(D + 1))
            simplicesD_array[Int(j)] = invperm[si]
        end
        coords0 = IDVector{0}(coords0.vec[perm])
        weights = IDVector{0}(weights.vec[perm])
        # Sort simplices by first vertex
        sort!(simplicesD_array)
        make_simplicesD = MakeSparse{One}(nvertices, nsimplices)
        for (j, si) in enumerate(simplicesD_array)
            for i in si
                make_simplicesD[i, j] = One()
            end
        end
        simplicesD = SparseOp{0,D,One}(sparse(make_simplicesD))
    end

    # Calculate lower-Dimensional simplices
    simplices = OpDict{Int,One}(D => simplicesD)
    boundaries = OpDict{Int,Int8}()
    for R in D:-1:1
        facesR, boundariesR = calc_faces_boundaries(simplices[R])
        simplices[R - 1] = facesR
        boundaries[R] = boundariesR
    end

    if optimize_mesh && use_weighted_duals
        @assert dualkind == CircumcentricDuals
        # Optimize weights

        # Calculate mask for boundary vertices (which must not be moved)
        if D > 0
            boundary_faces = IDVector{D - 1}(zeros(Int8,
                                                   size(boundaries[D], 1)))
            for j in axes(boundaries[D], 2)
                for (i, s) in sparse_column(boundaries[D], j)
                    boundary_faces[i] += s
                end
            end
            # This might indicate a severe bug; shouldn't faces have
            # opposite orientations when viewed from two neighbouring
            # simplices?
            # @assert all(s -> -1 ≤ s ≤ 1, boundary_faces)
            @assert all(s -> -2 ≤ s ≤ 2, boundary_faces)

            # @assert C == D
            dof = IDVector{0}(ones(SMatrix{C,C,S}, length(coords0)))
            @assert length(boundary_faces) == size(simplices[D - 1], 2)
            for j in axes(simplices[D - 1], 2)
                if isodd(boundary_faces[j])
                    sj = sparse_column_rows(simplices[D - 1], j)
                    sj = SVector{D,ID{0}}(sj[n] for n in 1:D)
                    # initially, no directions are allowed
                    P = zero(SMatrix{C,C,S})
                    for n in 2:D
                        x = coords0[sj[n]] - coords0[sj[1]]
                        # ensure x is orthogonal to the directions in P
                        x -= P * x
                        Q = x * x' / norm2(x)
                        # add one allowed direction
                        P += Q
                    end
                    for i in sparse_column_rows(simplices[D - 1], j)
                        dof[i] = P * dof[i] * P
                        dof[i] = (dof[i] + dof[i]') / 2
                    end
                end
            end

        else
            dof = IDVector{0}(zeros(SMatrix{C,C,S}, length(coords0)))
        end
        dof::IDVector{0,SMatrix{C,C,S}}
        coords0, weights = optimize_mesh1(Val(dualkind), Val(D), simplices,
                                          coords0, dof, weights)
    end

    # TODO: Move this to invariant, maybe add `haskey` for dualcoords
    # if D > 0
    #     check_delaunay(simplices[D], lookup[(D - 1, D)], lookup[(D, D - 1)],
    #                    coords[0], dualcoords[D])
    # end

    # Create D-manifold
    return Manifold(name, signature, dualkind, use_weighted_duals, simplices,
                    boundaries, coords0, weights)
end

################################################################################

export get_simplices
@inline function get_simplices(::Val{R}, mfd::Manifold{D,C,S}) where {R,D,C,S}
    @assert 0 ≤ R ≤ D
    return mfd._simplices[R]::SparseOp{0,R,One}
end
@inline function get_simplices(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    return get_simplices(Val(R), mfd)
end

export get_boundaries
@inline function get_boundaries(::Val{R}, mfd::Manifold{D,C,S}) where {R,D,C,S}
    @assert 0 < R ≤ D
    return mfd._boundaries[R]::SparseOp{R - 1,R,Int8}
end
@inline function get_boundaries(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    return get_boundaries(Val(R), mfd)
end

export get_lookup
@inline function get_lookup(::Val{Ri}, ::Val{Rj},
                            mfd::Manifold{D,C,S}) where {Ri,Rj,D,C,S}
    @assert 0 ≤ Rj ≤ D
    @assert 0 ≤ Ri ≤ D
    !haskey(mfd._lookup, (Ri, Rj)) && calc_lookup!(mfd, Ri, Rj)
    return mfd._lookup[(Ri, Rj)]::SparseOp{Ri,Rj,One}
end
@inline function get_lookup(mfd::Manifold{D,C,S}, Ri::Int,
                            Rj::Int) where {D,C,S}
    return get_lookup(Val(Ri), Val(Rj), mfd)
end

function calc_lookup!(mfd::Manifold{D,C,S}, Ri::Int, Rj::Int) where {D,C,S}
    @assert 0 ≤ Rj ≤ D
    @assert 0 ≤ Ri ≤ D
    @assert !haskey(mfd._lookup, (Ri, Rj))

    # println("[begin calc_lookup!(D=$D, Ri=$Ri, Rj=$Rj)]")

    sizei = nsimplices(mfd, Ri)
    sizej = nsimplices(mfd, Rj)

    if Ri == 0
        # Simplex definitions
        mfd._lookup[(Ri, Rj)] = get_simplices(mfd, Rj)
    elseif Ri == Rj
        # Identity
        mfd._lookup[(Ri, Rj)] = SparseOp{Ri,Rj}(sparse(1:sizei, 1:sizej,
                                                       fill(One(), sizei)))
    elseif Ri == Rj - 1
        # Absolute value of boundaries
        mfd._lookup[(Ri, Rj)] = map(x -> One(x ≠ 0), get_boundaries(mfd, Rj))
    elseif Ri < Rj - 1
        # Chain of two lookup tables
        Rk = Rj - 1
        mfd._lookup[(Ri, Rj)] = map(x -> One(x ≠ 0),
                                    get_lookup(mfd, Ri, Rk) *
                                    get_lookup(mfd, Rk, Rj))
    else
        # Transpose
        mfd._lookup[(Ri, Rj)] = get_lookup(mfd, Rj, Ri)'
    end

    # println("[end calc_lookup!(D=$D, Ri=$Ri, Rj=$Rj)]")
    return nothing
end

export get_isboundary
@inline function get_isboundary(::Val{R}, mfd::Manifold{D,C,S}) where {R,D,C,S}
    @assert 0 ≤ R < D
    !haskey(mfd._isboundary, R) && calc_isboundary!(mfd, R)
    return mfd._isboundary[R]::SparseOp{R,R,One}
end
@inline function get_isboundary(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    return get_isboundary(Val(R), mfd)
end

function calc_isboundary!(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    @assert 0 ≤ R < D
    @assert !haskey(mfd._isboundary, R)

    # println("[begin calc_isboundary!(D=$D, R=$R)]")

    I = Int[]
    J = Int[]
    V = One[]
    nelts = nsimplices(mfd, R)
    if R == D - 1
        boundariesD = get_boundaries(mfd, D)
        isbndface = IDVector{D - 1}(zeros(Bool, size(boundariesD, 1)))
        for j in axes(boundariesD, 2)
            for i in sparse_column_rows(boundariesD, j)
                isbndface[i] = !isbndface[i]
            end
        end
        for (i, b) in enumerate(isbndface)
            if b
                push!(I, Int(i))
                push!(J, Int(i))
                push!(V, One())
            end
        end
    else
        lookup = get_lookup(mfd, R, D - 1)
        for j0 in findnz(get_isboundary(mfd, D - 1).op)[1]
            j = ID{D - 1}(j0)
            for i in sparse_column_rows(lookup, j)
                push!(I, Int(i))
                push!(J, Int(i))
                push!(V, One())
            end
        end
    end
    mfd._isboundary[R] = SparseOp{R,R}(sparse(I, J, V, nelts, nelts, max))

    # println("[end calc_isboundary!(D=$D, R=$R)]")
    return nothing
end

export get_coords
@inline get_coords(mfd::Manifold) = get_coords(Val(0), mfd)
@inline function get_coords(::Val{R}, mfd::Manifold{D,C,S}) where {R,D,C,S}
    @assert 0 ≤ R ≤ D
    !haskey(mfd._coords, R) && calc_coords!(mfd, R)
    return mfd._coords[R]::IDVector{R,SVector{C,S}}
end
@inline function get_coords(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    return get_coords(Val(R), mfd)
end

function calc_coords!(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    @assert 0 ≤ R ≤ D
    @assert !haskey(mfd._coords, R)
    # println("[begin calc_coords!(D=$D, R=$R)]")
    mfd._coords[R] = calc_coords(get_simplices(mfd, R), get_coords(mfd))
    # println("[end calc_coords!(D=$D, R=$R)]")
    return nothing
end

export get_volumes
@inline function get_volumes(::Val{R}, mfd::Manifold{D,C,S}) where {R,D,C,S}
    @assert 0 ≤ R ≤ D
    !haskey(mfd._volumes, R) && calc_volumes!(mfd, R)
    return mfd._volumes[R]::IDVector{R,S}
end
@inline function get_volumes(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    return get_volumes(Val(R), mfd)
end

function calc_volumes!(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    @assert 0 ≤ R ≤ D
    @assert !haskey(mfd._volumes, R)
    # println("[begin calc_volumes!(D=$D, R=$R)]")
    mfd._volumes[R] = calc_volumes(get_simplices(mfd, R), get_coords(mfd),
                                   mfd.signature)

    # Ensure that all volumes are strictly positive
    allpositive = all(>(0), mfd._volumes[R])
    if !allpositive
        @warn "Not all  $R-volumes are strictly positive"
        @show count(≤(0), mfd._volumes[R])
        @show findmin(mfd._volumes[R])
        @show get_coords(mfd, R)[ID{R}(findmin(mfd._volumes[R])[2])]
    end

    # println("[end calc_volumes!(D=$D, R=$R)]")
    return nothing
end

export get_weights
@inline function get_weights(mfd::Manifold{D,C,S}) where {D,C,S}
    return mfd._weights::IDVector{0,S}
end

export get_dualcoords
@inline function get_dualcoords(::Val{R}, mfd::Manifold{D,C,S}) where {R,D,C,S}
    @assert 0 ≤ R ≤ D
    !haskey(mfd._dualcoords, R) && calc_dualcoords!(mfd, R)
    return mfd._dualcoords[R]::IDVector{R,SVector{C,S}}
end
@inline function get_dualcoords(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    return get_dualcoords(Val(R), mfd)
end

function calc_dualcoords!(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    @assert 0 ≤ R ≤ D
    @assert !haskey(mfd._dualcoords, R)
    # println("[begin calc_dualcoords!(D=$D, R=$R)]")
    if mfd.use_weighted_duals
        mfd._dualcoords[R] = calc_dualcoords(Val(mfd.dualkind),
                                             get_simplices(mfd, R),
                                             get_coords(mfd), get_weights(mfd))
    else
        mfd._dualcoords[R] = calc_dualcoords(Val(mfd.dualkind),
                                             get_simplices(mfd, R),
                                             get_coords(mfd))
    end
    # println("[end calc_dualcoords!(D=$D, R=$R)]")
    return nothing
end

export get_dualvolumes
@inline function get_dualvolumes(::Val{R}, mfd::Manifold{D,C,S}) where {R,D,C,S}
    @assert 0 ≤ R ≤ D
    !haskey(mfd._dualvolumes, R) && calc_dualvolumes!(mfd, R)
    return mfd._dualvolumes[R]::IDVector{R,S}
end
@inline function get_dualvolumes(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    return get_dualvolumes(Val(R), mfd)
end

function calc_dualvolumes!(mfd::Manifold{D,C,S}, R::Int) where {D,C,S}
    @assert 0 ≤ R ≤ D
    @assert !haskey(mfd._dualvolumes, R)
    # println("[begin calc_dualvolumes!(D=$D, R=$R)]")
    if mfd.dualkind == BarycentricDuals
        mfd._dualvolumes[R] = calc_dualvolumes(Val(mfd.dualkind), Val(D),
                                               Val(R), get_simplices(mfd, R),
                                               mfd.lookup, get_coords(mfd),
                                               get_volumes(mfd, D))
        @assert all(x -> x != 0 && isfinite(x), mfd._dualvolumes[R])
    elseif mfd.dualkind == CircumcentricDuals
        if R == D
            mfd._dualvolumes[D] = IDVector{D}(fill(S(1), nsimplices(mfd, D)))
        else
            mfd._dualvolumes[R] = calc_dualvolumes(Val(mfd.dualkind), Val(D),
                                                   get_simplices(mfd, R),
                                                   get_simplices(mfd, R + 1),
                                                   get_lookup(mfd, R + 1, R),
                                                   get_coords(mfd),
                                                   get_dualcoords(mfd, R),
                                                   get_dualcoords(mfd, R + 1),
                                                   get_dualvolumes(mfd, R + 1))
        end

        # # TODO: These should be equivalent
        # dualvolsR = calc_dualvolumes(Val(BarycentricDuals), Val(R), mfd)
        # if !(mfd._dualvolumes[R] ≈ dualvolsR)
        #     @show D R mfd._dualvolumes[R] dualvolsR
        # end
        # @assert mfd._dualvolumes[R] ≈ dualvolsR

    else
        @assert false
    end

    # Ensure that all dual volumes are strictly positive
    allpositive = all(>(0), mfd._dualvolumes[R])
    if !allpositive
        @warn "Not all dual $R-volumes are strictly positive"
        @show count(≤(0), mfd._dualvolumes[R])
        @show findmin(mfd._dualvolumes[R])
        @show get_dualcoords(mfd, R)[ID{R}(findmin(mfd._dualvolumes[R])[2])]
    end

    # println("[end calc_dualvolumes!(D=$D, R=$R)]")
    return nothing
end

export get_simplex_tree
@inline function get_simplex_tree(mfd::Manifold{D,C,S}) where {D,C,S}
    mfd._simplex_tree === nothing && calc_simplex_tree!(mfd)
    return mfd._simplex_tree::KDTree{SVector{C,S},Euclidean,S}
end

function calc_simplex_tree!(mfd::Manifold{D,C,S}) where {D,C,S}
    @assert mfd._simplex_tree === nothing
    # println("[begin calc_simplex_tree!(D=$D)]")
    mfd._simplex_tree = KDTree(get_coords(mfd, D).vec)
    # println("[end calc_simplex_tree!(D=$D)]")
    return nothing
end

################################################################################

struct Face{D,N}
    vertices::SVector{N,ID{0}}
    parent::ID{D}
    parity::Int8
end

"""
Calculate faces and boundaries
"""
function calc_faces_boundaries(simplices::SparseOp{0,D,One}) where {D}
    D::Int
    @assert 0 < D

    nvertices, nsimplices = size(simplices)

    # See arXiv:1103.3076v2 [cs.NA], section 7
    facelist = Face{D,D}[]
    for j in axes(simplices, 2)
        ks = SVector{D + 1,ID{0}}(sparse_column_rows(simplices, j)[n]
                                  for n in 1:(D + 1))
        for n in 1:(D + 1)
            # Leave out vertex n
            ls = deleteat(ks, n)
            p = bitsign(isodd(n - 1))
            push!(facelist, Face{D,D}(ls, j, p))
        end
    end
    sort!(facelist; by=f -> f.vertices)

    # Convert facelist into sparse matrix and create boundary operator
    fI = Int[]
    fJ = Int[]
    fV = One[]
    bI = Int[]
    bJ = Int[]
    bV = Int8[]

    nfaces = 0
    oldvertices = zero(SVector{D,Int})
    for f in facelist
        if f.vertices ≠ oldvertices
            oldvertices = f.vertices
            # We found a new face
            nfaces += 1
            for i in f.vertices
                push!(fI, Int(i))
                push!(fJ, nfaces)
                push!(fV, One())
            end
        end
        # Add boundary face
        push!(bI, nfaces)
        push!(bJ, f.parent)
        push!(bV, f.parity)
    end

    faces = SparseOp{0,D - 1}(sparse(fI, fJ, fV, nvertices, nfaces))
    boundaries = SparseOp{D - 1,D}(sparse(bI, bJ, bV, nfaces, nsimplices))

    return faces, boundaries
end

################################################################################

function optimize_mesh1(::Val{CircumcentricDuals}, ::Val{D},
                        simplices::OpDict{Int,One},
                        coords::IDVector{0,SVector{C,S}},
                        dof::IDVector{0,SMatrix{C,C,S}},
                        weights::IDVector{0,S}) where {D,C,S}
    D::Int
    C::Int
    @assert 0 ≤ D ≤ C

    for D′ in D:D
        α = S(1) / 2D′
        for iter in 1:100
            new_coords = copy(coords)
            new_weights = copy(weights)
            for R in 1:D′
                shift_cs, shift_ws = improve_mesh(simplices[R], coords, weights)
                for i in axes(new_coords, 1)
                    # nnzev = count(isapprox(1), eigvals(dof[i]))
                    # if nnzev ≤ D′
                    # new_coords[i] += α * dof[i] * shift_cs[i]
                    new_weights[i] += α * shift_ws[i]
                    # end
                end
            end
            avg_weight = sum(new_weights) / length(new_weights)
            for i in axes(new_weights, 1)
                new_weights[i] -= avg_weight
            end
            if !(sum(new_weights) + 1 ≈ 1)
                @show maximum(abs.(new_weights)) sum(new_weights)
            end
            @assert sum(new_weights) + 1 ≈ 1
            coords = new_coords
            weights = new_weights
        end
    end

    return coords, weights
end

################################################################################

"""
Calculate coordinates
"""
function calc_coords(simplices::SparseOp{0,D,One},
                     coords0::IDVector{0,SVector{C,S}}) where {D,C,S}
    nvertices, nsimplices = size(simplices)
    D == 0 && return coords0
    coords = IDVector{D}(Array{SVector{C,S}}(undef, nsimplices))
    @inbounds for i in axes(coords, 1)
        si0 = sparse_column_rows(simplices, i)
        si = SVector{D + 1}(si0[n] for n in 1:(D + 1))
        xs = SVector{D + 1}(coords0[i] for i in si)
        coords[i] = barycentre(xs)
    end
    return coords
end

"""
Calculate volumes
"""
function calc_volumes(simplices::SparseOp{0,D,One},
                      coords::IDVector{0,SVector{C,S}},
                      signature::Int=1) where {D,C,S}
    nvertices, nsimplices = size(simplices)
    D == 0 && return IDVector{D}(fill(one(S), nvertices))
    volumes = IDVector{D}(Array{S}(undef, nsimplices))
    @inbounds for i in axes(volumes, 1)
        si0 = sparse_column_rows(simplices, i)
        si = SVector{D + 1}(si0[n] for n in 1:(D + 1))
        xs = SVector{D + 1}(Form{C,1}(coords[i]) for i in si)
        # volumes[i] = volume(xs)
        vol = volume(xs)
        svol = signed_volume(xs)
        svols = signed_volume(xs, signature)
        if !(abs(svol) ≈ vol)
            @show D C i vol svol
        end
        @assert abs(svol) ≈ vol
        volumes[i] = svols * sign(svol)
    end
    return volumes
end

"""
Calculate barycentric dual coordinates
"""
function calc_dualcoords(::Val{BarycentricDuals}, simplices::SparseOp{0,D,One},
                         coords::IDVector{0,SVector{C,S}}) where {D,C,S}
    D == 0 && return coords
    nsimplices = size(simplices, 2)
    dualcoords = IDVector{D}(Array{SVector{C,S}}(undef, nsimplices))
    @inbounds for i in axes(dualcoords, 1)
        si0 = sparse_column_rows(simplices, i)
        si = SVector{D + 1}(si0[n] for n in 1:(D + 1))
        xs = SVector{D + 1}(Form{C,1}(coords[i]) for i in si)
        dualcoords[i] = barycentre(xs)
    end
    return dualcoords
end

"""
Calculate circumcentric dual coordinates
"""
function calc_dualcoords(::Val{CircumcentricDuals},
                         simplices::SparseOp{0,D,One},
                         coords::IDVector{0,SVector{C,S}}) where {D,C,S}
    nvertices, nsimplices = size(simplices)
    @assert length(coords) == nvertices
    D == 0 && return coords
    dualcoords = IDVector{D}(Array{SVector{C,S}}(undef, nsimplices))
    @inbounds for i in axes(dualcoords, 1)
        si0 = sparse_column_rows(simplices, i)
        si = SVector{D + 1}(si0[n] for n in 1:(D + 1))
        xs = SVector{D + 1}(Form{C,1}(coords[i]) for i in si)
        dualcoords[i] = circumcentre(xs)
    end
    return dualcoords
end

"""
Calculate weighted circumcentric dual coordinates
"""
function calc_dualcoords(::Val{CircumcentricDuals},
                         simplices::SparseOp{0,D,One},
                         coords::IDVector{0,SVector{C,S}},
                         weights::IDVector{0,S}) where {D,C,S}
    nvertices, nsimplices = size(simplices)
    @assert length(coords) == nvertices
    @assert length(weights) == nvertices
    D == 0 && return coords
    dualcoords = IDVector{D}(Array{SVector{C,S}}(undef, nsimplices))
    @inbounds for i in axes(dualcoords, 1)
        si0 = sparse_column_rows(simplices, i)
        si = SVector{D + 1}(si0[n] for n in 1:(D + 1))
        xs = SVector{D + 1}(Form{C,1}(coords[i]) for i in si)
        ws = SVector{D + 1}(Form{C,0}((weights[i],)) for i in si)
        dualcoords[i] = circumcentre(xs, ws)
    end
    return dualcoords
end

"""
Calculate barycentric dual volumes
"""
function calc_dualvolumes(::Val{BarycentricDuals}, ::Val{R},
                          manifold::Manifold{D,C,S}) where {R,D,C,S}
    R::Int
    D::Int
    C::Int
    @assert 0 ≤ R ≤ D ≤ C

    simplices_R = get_simplices(manifold, R)

    dualvolumes = IDVector{R}(Array{S}(undef, size(simplices_R, 2)))

    if R == D
        @inbounds for i in axes(dualvolumes, 1)
            dualvolumes[i] = 1
        end

    elseif false && R == 0
        lookup_D_R = get_lookup(manifold, D, R)

        volumes_D = get_volumes(manifold, D)
        for i in axes(dualvolumes, 1)
            vol = S(0)
            for j in sparse_column_rows(lookup_D_R, i)
                vol += volumes_D[j]
            end
            dualvolumes[i] = vol / (D + 1)
        end

    elseif R == D - 1
        lookup_D_R = get_lookup(manifold, D, R)
        dualcoords_R = get_dualcoords(manifold, R)
        dualcoords_D = get_dualcoords(manifold, D)

        @inbounds for i in axes(dualvolumes, 1)
            bci = dualcoords_R[i]

            vol = S(0)
            for j in sparse_column_rows(lookup_D_R, i)
                bcj = dualcoords_D[j]

                vol += volume(SVector(bci, bcj))
            end
            dualvolumes[i] = vol
        end

    elseif R == D - 2
        lookup_D_R = get_lookup(manifold, D, R)
        lookup_D1_D = get_lookup(manifold, D - 1, D)
        lookup_R_D1 = get_lookup(manifold, R, D - 1)
        dualcoords_R = get_dualcoords(manifold, R)
        dualcoords_D = get_dualcoords(manifold, D)
        dualcoords_D1 = get_dualcoords(manifold, D - 1)

        # Loop over all `R`-forms
        @inbounds for i in axes(dualvolumes, 1)
            bci = dualcoords_R[i]

            vol = S(0)
            # Loop over all neighbouring `D`-forms
            for j in sparse_column_rows(lookup_D_R, i)
                bcj = dualcoords_D[j]

                # Loop over all contained `D-1`-forms that contain `i`
                for k in sparse_column_rows(lookup_D1_D, j)
                    if i ∈ sparse_column_rows(lookup_R_D1, k)
                        bck = dualcoords_D1[k]

                        vol += volume(SVector(bci, bcj, bck))
                    end
                end
            end
            dualvolumes[i] = vol
        end

    elseif R == D - 3
        lookup_D_R = get_lookup(manifold, D, R)
        lookup_D1_D = get_lookup(manifold, D - 1, D)
        lookup_R_D1 = get_lookup(manifold, R, D - 1)
        lookup_D2_D1 = get_lookup(manifold, D - 2, D - 1)
        lookup_R_D2 = get_lookup(manifold, R, D - 2)
        dualcoords_R = get_dualcoords(manifold, R)
        dualcoords_D = get_dualcoords(manifold, D)
        dualcoords_D1 = get_dualcoords(manifold, D - 1)
        dualcoords_D2 = get_dualcoords(manifold, D - 2)

        # Loop over all `R`-forms
        @inbounds for i in axes(dualvolumes, 1)
            bci = dualcoords_R[i]

            vol = S(0)
            # Loop over all neighbouring `D`-forms
            for j in sparse_column_rows(lookup_D_R, i)
                bcj = dualcoords_D[j]

                # Loop over all contained `D-1`-forms that contain `i`
                for k in sparse_column_rows(lookup_D1_D, j)
                    if i ∈ sparse_column_rows(lookup_R_D1, k)
                        bck = dualcoords_D1[k]

                        # Loop over all contained `D-2`-forms that contain `i`
                        for l in sparse_column_rows(lookup_D2_D1, k)
                            if i ∈ sparse_column_rows(lookup_R_D2, l)
                                bcl = dualcoords_D2[l]

                                vol += volume(SVector(bci, bcj, bck, bcl))
                            end
                        end
                    end
                end
            end
            dualvolumes[i] = vol
        end

    elseif R == D - 4
        lookup_D_R = get_lookup(manifold, D, R)
        lookup_D1_D = get_lookup(manifold, D - 1, D)
        lookup_R_D1 = get_lookup(manifold, R, D - 1)
        lookup_D2_D1 = get_lookup(manifold, D - 2, D - 1)
        lookup_R_D2 = get_lookup(manifold, R, D - 2)
        lookup_D3_D2 = get_lookup(manifold, D - 3, D - 2)
        lookup_R_D3 = get_lookup(manifold, R, D - 3)
        dualcoords_R = get_dualcoords(manifold, R)
        dualcoords_D = get_dualcoords(manifold, D)
        dualcoords_D1 = get_dualcoords(manifold, D - 1)
        dualcoords_D2 = get_dualcoords(manifold, D - 2)
        dualcoords_D3 = get_dualcoords(manifold, D - 3)

        # Loop over all `R`-forms
        @inbounds for i in axes(dualvolumes, 1)
            bci = dualcoords_R[i]

            vol = S(0)
            # Loop over all neighbouring `D`-forms
            for j in sparse_column_rows(lookup_D_R, i)
                bcj = dualcoords_D[j]

                # Loop over all contained `D-1`-forms that contain `i`
                for k in sparse_column_rows(lookup_D1_D, j)
                    if i ∈ sparse_column_rows(lookup_R_D1, k)
                        bck = dualcoords_D1[k]

                        # Loop over all contained `D-2`-forms that contain `i`
                        for l in sparse_column_rows(lookup_D2_D1, k)
                            if i ∈ sparse_column_rows(lookup_R_D2, l)
                                bcl = dualcoords_D2[l]

                                # Loop over all contained `D-3`-forms
                                # that contain `i`
                                for m in sparse_column_rows(lookup_D3_D2, l)
                                    if i ∈ sparse_column_rows(lookup_R_D3, m)
                                        bcm = dualcoords_D3[m]

                                        vol += volume(SVector(bci, bcj, bck,
                                                              bcl, bcm))
                                    end
                                end
                            end
                        end
                    end
                end
            end
            dualvolumes[i] = vol
        end

    elseif R == D - 5
        lookup_D_R = get_lookup(manifold, D, R)
        lookup_D1_D = get_lookup(manifold, D - 1, D)
        lookup_R_D1 = get_lookup(manifold, R, D - 1)
        lookup_D2_D1 = get_lookup(manifold, D - 2, D - 1)
        lookup_R_D2 = get_lookup(manifold, R, D - 2)
        lookup_D3_D2 = get_lookup(manifold, D - 3, D - 2)
        lookup_R_D3 = get_lookup(manifold, R, D - 3)
        lookup_D4_D3 = get_lookup(manifold, D - 4, D - 3)
        lookup_R_D4 = get_lookup(manifold, R, D - 4)
        dualcoords_R = get_dualcoords(manifold, R)
        dualcoords_D = get_dualcoords(manifold, D)
        dualcoords_D1 = get_dualcoords(manifold, D - 1)
        dualcoords_D2 = get_dualcoords(manifold, D - 2)
        dualcoords_D3 = get_dualcoords(manifold, D - 3)
        dualcoords_D4 = get_dualcoords(manifold, D - 4)

        # Loop over all `R`-forms
        @inbounds for i in axes(dualvolumes, 1)
            bci = dualcoords_R[i]

            vol = S(0)
            # Loop over all neighbouring `D`-forms
            for j in sparse_column_rows(lookup_D_R, i)
                bcj = dualcoords_D[j]

                # Loop over all contained `D-1`-forms that contain `i`
                for k in sparse_column_rows(lookup_D1_D, j)
                    if i ∈ sparse_column_rows(lookup_R_D1, k)
                        bck = dualcoords_D1[k]

                        # Loop over all contained `D-2`-forms that contain `i`
                        for l in sparse_column_rows(lookup_D2_D1, k)
                            if i ∈ sparse_column_rows(lookup_R_D2, l)
                                bcl = dualcoords_D2[l]

                                # Loop over all contained `D-3`-forms
                                # that contain `i`
                                for m in sparse_column_rows(lookup_D3_D2, l)
                                    if i ∈ sparse_column_rows(lookup_R_D3, m)
                                        bcm = dualcoords_D3[m]

                                        # Loop over all contained
                                        # `D-4`-forms that contain `i`
                                        for n in
                                            sparse_column_rows(lookup_D4_D3, m)
                                            if i ∈
                                               sparse_column_rows(lookup_R_D4,
                                                                  n)
                                                bcn = dualcoords_D3[n]

                                                vol += volume(SVector(bci, bcj,
                                                                      bck, bcl,
                                                                      bcm, bcn))
                                            end
                                        end
                                    end
                                end
                            end
                        end
                    end
                end
            end
            dualvolumes[i] = vol
        end

    else
        @assert false
    end

    return dualvolumes
end

"""
Calculate circumcentric dual volumes

See [1198555.1198667, section 6.2.1]
"""
function calc_dualvolumes(::Val{CircumcentricDuals}, ::Val{D},
                          simplices::SparseOp{0,R,One},
                          simplices1::SparseOp{0,R1,One},
                          parents::SparseOp{R1,R,One},
                          coords::IDVector{0,SVector{C,S}},
                          dualcoords::IDVector{R,SVector{C,S}},
                          dualcoords1::IDVector{R1,SVector{C,S}},
                          dualvolumes1::IDVector{R1,S},
                          signature::Int=1) where {D,R,R1,C,S}
    R::Int
    R1::Int
    C::Int
    @assert 0 ≤ R ≤ R1 ≤ D
    @assert R1 == R + 1
    nvertices, nsimplices = size(simplices)
    dualvolumes = IDVector{R}(Array{S}(undef, nsimplices))
    # Loop over all `R`-simplices
    @inbounds for i in axes(dualvolumes, 1)
        si0 = sparse_column_rows(simplices, i)
        @assert length(si0) == R + 1
        si = SVector{R + 1,ID{0}}(si0[n] for n in 1:(R + 1))
        xsi = SVector{R + 1}(Form{C,1,S}(coords[i]) for i in si)
        # bci = barycentre(xsi)::Form{C,1,S}
        cci = Form{C,1,S}(dualcoords[i])

        voli = zero(S)
        # Loop over all neighbouring `R+1`-simplices
        for j in sparse_column_rows(parents, i)
            sj0 = sparse_column_rows(simplices1, j)
            @assert length(sj0) == R + 2
            sj = SVector{R + 2,ID{0}}(sj0[n] for n in 1:(R + 2))

            xsj = SVector{R + 2}(Form{C,1,S}(coords[j]) for j in sj)
            bcj = barycentre(xsj)::Form{C,1,S}
            ccj = Form{C,1,S}(dualcoords1[j])

            ysi = map(y -> y - xsi[1], deleteat(xsi, 1))   # R
            ysj = map(y -> y - xsj[1], deleteat(xsj, 1))   # R+1
            @assert !isempty(ysj)
            ni = ∧(ysj) ⋅ ∧(ysi)   # 1
            ni::Form{C,1,S}
            qsi = map(y -> norm(ni ⋅ y) < 10 * eps(S), ysi)
            @assert all(qsi)
            lni = norm(ni)
            @assert lni > 0
            nni = ni / lni

            s0 = (bcj - xsi[1]) ⋅ nni
            s0::Form{C,0,S}
            s = bitsign(signbit(s0[]))
            h0 = (ccj - cci) ⋅ nni
            h0::Form{C,0,S}
            h = s * h0[]

            b = dualvolumes1[j]
            volj = b * h

            voli += volj
        end

        dualvolumes[i] = voli / (D - R)
    end
    return dualvolumes
end

end


text:
using Random, IterTools, Combinatorics

function SetGameTM(basic = true)
    drawsize = basic ? 9 : 12
    setsneeded = div(drawsize, 2)
    setsof3 = Vector{Vector{NTuple{4, String}}}()
    draw = Vector{NTuple{4, String}}()
    deck = collect(Iterators.product(["red", "green", "purple"], ["one", "two", "three"],
        ["oval", "squiggle", "diamond"], ["solid", "open", "striped"]))

    while length(setsof3) != setsneeded
        empty!(draw)
        empty!(setsof3)
        map(x -> push!(draw, x), shuffle(deck)[1:drawsize])
        for threecards in combinations(draw, 3)
            canuse = true
            for i in 1:4
                u = length(unique(map(x->x[i], threecards)))
                if u != 3 && u != 1
                    canuse = false
                end
            end
            if canuse
                push!(setsof3, threecards)
            end
        end
    end

    println("Dealt $drawsize cards:")
    for card in draw
        println("    $card")
    end
    println("\nFormed these cards into $setsneeded sets:")
    for set in setsof3
        for card in set
            println("    $card")
        end
        println()
    end
end

SetGameTM()
SetGameTM(false)

ext: jl

max_stars_repo_name: ethansaxenian/RosettaDecode

max_issues_repo_name: ethansaxenian/RosettaDecode

max_forks_repo_name: ethansaxenian/RosettaDecode


text:
### A Pluto.jl notebook ###
# v0.17.5

using Markdown
using InteractiveUtils

# ╔═╡ 0c66d310-bf10-11ec-1be7-5d0878ec4a0b
begin
	using Markdown
	using Unitful
	using UnitfulRecipes
	using Plots
	using LsqFit
	using PlutoUI
	plotly()
end

# ╔═╡ ec84079f-cb49-4b60-a5b7-6ea1cf6a0240
md"""
### Fotoefekta un Komptona nobīdes šķērsgriezumi
Šodien apskatīsimies kādā veidā fotona enerģija un mērķa atomu skaitlis $Z$ ietekmē fotoefektu un komptona nobīdes varbūtību.

#### Šķērsgriezums
Domājot tēlaini, kāda objekta šķērsgriezums ir lauka vienība, ko prezentē kādam virzienam. Šķērsgriezums mēra $\mathrm{cm}^2$, kā lauka vienība, bet nav obligāti saistīts ar reālo virsmu. Piemēram, elektrona šķērsgriezumu var attēlot kā aplis ar virsmas lauku $\sigma$ ap elektronu. Bet šis šķērsgriezums var mainīties ar enerģiju, tāpēc nav saistīts ar kaut ko fizikālo.

Pieņemsim, ka kāda fotona plūsma $I$ ar enerģiju $E$ nonāk uz kāda materiāla slāni ar biezumu $t$  un $n$ atomiem uz lauka vienību. Atomu šķērsgriezums ir $\sigma$. Reakciju skaits sekundē $N_{reakcijas}$ var noteikt šādi:

$N_{reakcijas} = \sigma I n $

Parametru $n$ var iegūt šādi:

$n=\frac{\rho t N_A}{M_A}$,

kur $N_A$ ir Avogrado skaitlis un $M_A$ ir molāra masa.

#### Dati
Mēs varam atrast dati par šķērsgriezumu atkārība no enerģijas [XCOM datubāzē](https://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html) no NIST mājas lapas.
Izvelēsim _Element_ un _Enter additional energies by hand_ un kliķšinām `Submit information`.

Nākamājā logā, rakstīsim vai nu _Atomic number_ $Z$ (piemēram "56") vai _Symbol_ (piemēram "Fe"). Zem _Graph options_, kliķšinām
* Total Attenuation with Coherent Scattering
* Incoherent Scattering (Komptona izkliede)
* Photoelectric Absorption
Izvēlēsim vēlamo enerģijas diapazonu (pietiek līdz 10 MeV)
un kļiķšinām `Submit Information`. (Var arī citus, bet mēs pagaidām gribam zināt tikai par fotoefektu un Komptona izkliedi.)

Parādās grafiks ar šķersgriezumu $\tilde{\sigma}$ [$\mathrm{g}\, \mathrm{cm}^{-2}$] y-asī pret fotona enerģiju x-asī.

#### Citi noderīgi dati
[Mendelejeva tabula](https://pubchem.ncbi.nlm.nih.gov/periodic-table/)
Spiežot uz katru elementu var iegūt papildus informāciju ($\rho$, $M_A$, utt. )

"""

# ╔═╡ e735388c-6631-4a83-84a3-458931dd63cf
md"""
### Uzdevums 1.
#### Dažādi veidi, kā skatīties uz šķērsgriezumu.
a) Kā mēs varētu dabūt atoma šķērsgriezumu $\sigma$ no $\tilde{\sigma}$?
"""

# ╔═╡ 701363c4-3989-4f79-84cd-0972de60506c
md"""
b) Kāds ir fotoefekta šķērsgriezums pie $E=1$ MeV vienam dzelzs atomam?
"""

# ╔═╡ af8d9681-e622-4b2c-b7cb-7069835dc88e
md"""
c) Cik biezs dzelzs slānis ir vajadzīgs, lai samazinātu fotonu plūsmu uz $1 \%$ pie $E=10$ keV? Pie $E=1$ keV? Kas notiek, ja slānis ir no svīna?
"""

# ╔═╡ d69d9cf6-278f-4cdf-8afe-215d9028cf65
md"""
### Uzdevums 2.
#### Komptona izkliedes šķērsgriezums atkarībā no $Z$.
a) Izvirziet hipotēzi par to, kādā veidā Komptona izkliedes šķērsgriezums ir atkarīgs no $Z$.
"""

# ╔═╡ 04ce5b6e-f60b-4c73-b98a-548d5e9ca36a
md"""
b) Jūsu modelī, vai labāk izmantot $\sigma$ [$\mathrm{cm}^2$] vai $\tilde{\sigma}$ $\mathrm{cm}^2 \, \mathrm{g}^{-1}$]
"""

# ╔═╡ 71933852-57e2-4e24-8ca2-7dc91577d061
md"""
c) Izmantojiet [XCOM datubāzi](https://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html), lai iegūtu šķērsgriezumu vērtības pie vairākām $Z$ vērtībām pie kādas konkrētas enerģijas.
"""

# ╔═╡ af0029ec-251c-4451-9ec3-4b217e739a90
md"""
d) Pārbaudiet Jūsu modeli ar iegūtajiem datiem un komentējiet iespējamas atšķirības. _Julia_ valodā modeli var rakstīt šādi

```@. model(x,p) =p[1]*f1(x)+p[2]*f2(x)+p[3]*f3(x)+...```

Tad, definējiet minējumu Jūsu parametriem

``` p0=[a,b,c]```

Beidzot, ja X=[$X_1,X_2,...,X_n$] ir X-dati un Y=[$Y_1,Y_2,...,Y_n$] ir Y-dati, izmantojiet pavēles

```
fit=curve_fit(model,xdata,ydata,p0)
println("p = ",fit.param)
println("stderr = ", stderror(fit))
plot(X,Y,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red)
plot!(X,model(X,fit.param))
```
"""

# ╔═╡ adadb085-e762-41b9-a2a9-9982035fb8af
md"""
### Uzdevums 3.
#### Fotoefekta šķērsgriezums atkarībā no $Z$.

"""

# ╔═╡ e7540bb4-2a31-452e-94b3-127715d0f69b
md"""
a) Izmantojiet XCOM datubāzē lai iegūtu fotoefekta šķērsgriezumu vērtības pie vairākām  vērtībām pie kādas konkrētas enerģijas. Pie kuras enerģijas vērtības ņemsiet Jūsu mērījumus? Kāpēc?
"""

# ╔═╡ a52de69b-3fda-4616-88b1-236aa601704c
md"""
b) Zīmējiet Jūsu "datus" ar $Z$ vērtību x-asī un šķērsgriezumu uz $y$-asi. Ja X=[$X_1,X_2,...,X_n$] un Y=[$Y_1,Y_2,...,Y_n$], izmantojiet pavēli ```plot(X,Y)```
"""

# ╔═╡ 95f0e1eb-9461-4db6-a98e-6803ad9c67d8
md"""
c) Izvirziet modeli, kas varētu aprakstīt datus. (Skatieties logaritmiskajā skalā ar pavēli ```plot(X,Y,yaxis=:log, xaxis=:log)```. Ko tas nozīmē?)
"""

# ╔═╡ 419e92ea-47b6-4365-8136-fd775813f042
md"""
d) Pārbaudiet Jūsu modeli un nosakiet parametrus.
"""

# ╔═╡ 65c443af-c62c-469f-9c63-559542868652
md"""
### Uzdevums 4.
#### Fotoefekta šķērsgriezums atkarībā no $E$.
a) Izvirziet hipotēzi par pēkšņiem šķērsgriezuma palielinājumu pie dažādām enerģijām. (Atcerēties fotoefekta būtību un Einšteina aprakstu.)
"""

# ╔═╡ 905896c6-aa45-4957-9ed9-e18288016b46
md"""
b) Izvelēties kādu elementu. No tabulas, ko veido XCOM programma, rakstiet šķērsgriezumu pie dažādām enerģijām. Kādu energijas diapazonu Jūs izvēlējāt? Kāpēc?
"""

# ╔═╡ b4fe3ae0-7f91-4a49-9b39-8035200ecfdd
md"""
c) Līdzīgi iepriekšējam uzdevumam, veidojiet un pārbaudiet modeli, kas apraksta fotoefekta šķērsgriezumu atkarībā no enerģijas.
"""

# ╔═╡ 7921514e-6fde-45a9-b275-0efeb588e286
md"""
### Uzdevums 5.
#### Radiācijas ekranēšana.
a) Kāpēc svīns tiek izmantots, lai ekranētu rentgena starus?
"""

# ╔═╡ 3ab2cf7b-f2d1-41c2-8530-3e419980a5dd
md"""
b) Pēc kuras funkcijas samazināsies fotonu plūsma, kas šķērso vielas slāni, atkarībā no slāņa biezumu. Pamatojiet Jūsu atbildi ar teorētiskiem vai skaitliskiem aprēķiniem vai argumentiem.)
"""

# ╔═╡ 5ec2c1f9-983e-4236-aa8a-b7f76f059b2d
md"""
## Piezīmes par fitēšanu
Fitēšanas rūtīnas ir tikai programmas. Tādēļ viņas var nonākt strupceļā, un ir vajadīga cilvēka iesaistīšanas, lai nonāktu pie risinājuma.

### Tolerance
Piemēram, kad mums ir ļoti mazi skaitli, ļoti vienkāršs fits ir neveiksmīgs. Izskatās, ka funkcija `curve_fit` no pakates LsqFit.jl nestrādā labi ar mazām vērtībām.

"""

# ╔═╡ 70a934e4-b76c-40a6-84ec-dbeb43869722
begin
	xdata=[1,2,3,4,5]
	ydata=2.0e-24 .* xdata
	@. myModel1(x,p)=p[1]+p[2]*x
	p0=[0,3e-24]
	fit=curve_fit(myModel1,xdata,ydata,p0)
	println("p = ",fit.param)
	println("stderr = ", stderror(fit))
	plot(xdata,ydata,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red,ylims=(0,2e-23))
	plot!(xdata,myModel1(xdata,fit.param))
end

# ╔═╡ a4fd6ed0-87da-4e84-a9ae-4aaad315b377
	with_terminal() do
		println("p = ",fit.param)
		println("stderr = ", stderror(fit))
	end

# ╔═╡ 44459c45-38f9-453c-b2a5-726ca1610330
md"""
Vienkāršs risinājums būtu reizināt mūsu $y$ vērtības ar $10^{24}$, un pēc tam pievienot apgriezto reizinātāju ($10^{-24}$) fita rezultātam. Tomēr šī metode nav ļoti apmierinoša. Vai tā strādās, ja mums vienkārši ir liels diapazons?

Cits risinājums ir izmantot "keyword arguments" (`kwargs`) `x_tol` un `g_tol`. Funkcija `curve_fit` meklē fita parametrus iteratīvi. Šie parametri nosaka, kad funkcia `curve_fit` būs "apmierināta" ar savu rezultātu, t.i., kad būs pēdēja iterācija. Parametrs `x_tol` ir saistīts ar relatīvu parametru vērtību izmaiņu starp iterācijām. Kad relatīva parametru izmaiņa ir mazākā par `x_tol`, tad `curve_fit` beidzās. Savukārt parametrs `g_tol` ir saistīts ar relatīvā $\Xi^2$ funkcijas gradientu. Atkaļ, kad tas gradients būs mazāk par `g_tol`, iterācijas beidzās.

Parametriem `x_tol` un `g_tol` ir _default_ vērtības, bet izskatās, ka tie nav adekvāti. Varam mēģināt citus.

`fit=curve_fit(myModel1,xdata,ydata,p0,x_tol=1e-24,g_tol=1e-24)`
Precīzas vērtības ir jāmeklē. Ja izvēlēsimies parāk lielas vērtības, fits varētu būt nepareizs. Izvēlojoties pārāk mazāk vērtības, fits varētu prasīt ļoti daudz laika, vai vispār nekonverģēt.
"""

# ╔═╡ 844b8483-bcc9-4a91-9be5-4784f3002d7a
begin
	fit2=curve_fit(myModel1,xdata,ydata,p0,x_tol=1e-24,g_tol=1e-24)
	plot(xdata,ydata,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red,ylims=(0,2e-23))
	plot!(xdata,myModel1(xdata,fit2.param))
end

# ╔═╡ c4fd2179-f382-4d35-8799-2a3efef9f321
	with_terminal() do
		println("p = ",fit2.param)
		println("stderr = ", stderror(fit2))
	end

# ╔═╡ dbbab02a-8a4f-4b44-937c-d1ef38ab966a
md"""
### Eksponentiālas funkcijas
Pieņemsim, ka vēlāmies pielīdzināt modeli $y=ax^n$ kādiem datiem. Mēs zinām, ka tādu vienādojumu varam pārvērst pa lineāru vienādojumu šādi: $\log y =\log a+n \log x$.
Mēģināsim abus variantus:
"""

# ╔═╡ aa9d574c-b58d-42ea-a861-125807674b28
begin
	X=[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0]
	Y=2.5 .* X.^(3)
	Y=Y .+ sqrt.(Y).*randn(length(Y))
	plot(X,Y,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red) #,ylims=(0,2e-23)))
	@. myExpModel1(x,p)=p[1]*x^p[2]
	P0=[1.0,1.0]
	fit3=curve_fit(myExpModel1,X,Y,P0)
	plot!(X,myExpModel1(X,fit3.param))
end

# ╔═╡ 84d1d9a0-054b-4e2a-a90a-d27380276521
with_terminal() do
	println("p = ",fit3.param)
	println("stderr = ", stderror(fit3))
end

# ╔═╡ 4565a5f8-7bb9-47ec-a656-1e73edd391e1
begin
	lX=log.(X)
	lY=log.(Y)
	plot(lX,lY,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red)
	@. myLogModel1(x,p)=p[1]+p[2]*x
	Pl0=[1.0,1.0]
	fit4=curve_fit(myLogModel1,lX,lY,Pl0)
	plot!(lX,myLogModel1(lX,fit4.param))
end

# ╔═╡ b8d74e23-653d-4d37-9707-854d64cbc3c1
with_terminal() do
	println("p = ",fit4.param)
	println("stderr = ", stderror(fit4))
end

# ╔═╡ 51dec9ac-ab2d-430c-b277-38bcfaed52f3
md"""
Izskatās, ka pirmas fits ($y=ax^n$) atbilst kļūdu robežas. Otrais fits ($\log y = \log a + n\log x$) ir nedaudz sliktāks.
Kāpēc? Lai pielīdzinātu parametrus, tiek minimizēti attiecīgi lielumi $\sqrt ( \sum_i (f(x,p)-y_i)^2 )$ vai $\sqrt ( \sum_i (\log (f(x,p)) - \log y_i)^2 )$, kur f(x,p) ir modelis un $y_i=y_i^0+y_i^{err}$, bet kļūdas to ietekmē savādāk zem logaritma funkcijas.
Varam mēģināt uzlabot izmantojot svarus (_weights_):
```
weights=Y
fit5=curve_fit(myExpModel1,X,Y,weights,PP0)
```
Tāds svara faktors ņēm vērā, ka kļūdas tika definētas, ka $\propto \sqrt(y)$, līdz ar to, relatīva kļūda samazinās, jo lielāka $y$ vērtība. Tāpēc, vēlāmies, lai lielākas $y$ vērtības ietekmē vairāk.
Skatīsimies, kas notiek:
"""

# ╔═╡ 94c7c9b7-2de8-4a93-8b42-56127012fedb
begin
	weights=Y
	plot(X,Y,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red) #,ylims=(0,2e-23)))
	PP0=[1.0,1.0]
	fit5=curve_fit(myExpModel1,X,Y,weights,PP0)
	plot!(X,myExpModel1(X,fit5.param))
end

# ╔═╡ d04bbf04-ab8b-4905-b14c-05578426ed55
with_terminal() do
	println("p = ",fit5.param)
	println("stderr = ", stderror(fit5))
end

# ╔═╡ f48e07bf-0d5e-462e-9013-15a07100d5f3
md"""
Kā var redzēt, fits funkcijai $y=ax^n$ ir ļoti tuvu patiesībai un novērtēta kļūda ir maza.
"""

# ╔═╡ 456093b4-f9ef-412d-8950-34ec661f02fd
begin
	plot(lX,lY,linestyle=:auto,linewidth=0,markershape=:circle, markercolor=:red)
	PPl0=[1.0,1.0]
	fit6=curve_fit(myLogModel1,lX,lY,weights,PPl0)
	plot!(lX,myLogModel1(lX,fit6.param))
end

# ╔═╡ 492fddf6-d464-47bc-bc9b-0d41be2564b5
with_terminal() do
	println("p = ",fit6.param)
	println("stderr = ", stderror(fit6))
end

# ╔═╡ e013b711-339b-44ef-87df-d476b2e231d1
md"""
Fits funkcijai $\log y = \log a + n\log x$ arī uzlabojās, bet tomēr nav tik labs.
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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UnitfulRecipes = "42071c24-d89e-48dd-8a24-8a12d9b8861f"

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[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
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