From e1fb80d93ab0b2d7a2e6d5956e1327237d088214 Mon Sep 17 00:00:00 2001
From: Michael Reed <18372368+chakravala@users.noreply.github.com>
Date: Fri, 28 Aug 2020 16:05:48 -0400
Subject: [PATCH] improved dyadic product operations

---
 Project.toml        |  2 -
 src/Grassmann.jl    | 42 ++++++++++-----------
 src/algebra.jl      | 46 ++++++++++++++++-------
 src/composite.jl    | 92 ++++++++++++++++++++++-----------------------
 src/forms.jl        |  2 +-
 src/multivectors.jl | 60 ++++++++++++++++-------------
 src/parity.jl       |  4 +-
 7 files changed, 134 insertions(+), 114 deletions(-)

diff --git a/Project.toml b/Project.toml
index fe6ea8f..9d70563 100644
--- a/Project.toml
+++ b/Project.toml
@@ -10,7 +10,6 @@ DirectSum = "22fd7b30-a8c0-5bf2-aabe-97783860d07c"
 Leibniz = "edad4870-8a01-11e9-2d75-8f02e448fc59"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Requires = "ae029012-a4dd-5104-9daa-d747884805df"
-StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
@@ -20,7 +19,6 @@ Leibniz = "0.1"
 DirectSum = "0.7"
 AbstractTensors = "0.5.2"
 ComputedFieldTypes = "0.1"
-StaticArrays = "0"
 Requires = "1"
 
 [extras]
diff --git a/src/Grassmann.jl b/src/Grassmann.jl
index 370db42..f78f4ea 100644
--- a/src/Grassmann.jl
+++ b/src/Grassmann.jl
@@ -5,7 +5,7 @@ module Grassmann
 
 using SparseArrays, ComputedFieldTypes
 using AbstractTensors, Leibniz, DirectSum, Requires
-import AbstractTensors: SVector, MVector, SizedVector, clifford
+import AbstractTensors: Values, Variables, FixedVector, clifford
 
 export ⊕, ℝ, @V_str, @S_str, @D_str, Manifold, SubManifold, Signature, DiagonalForm, value
 export @basis, @basis_str, @dualbasis, @dualbasis_str, @mixedbasis, @mixedbasis_str, Λ
@@ -18,7 +18,7 @@ import Leibniz: Bits, bit2int, indexbits, indices, diffvars, diffmask
 import Leibniz: symmetricmask, indexstring, indexsymbol, combo, digits_fast
 
 import Leibniz: hasconformal, hasinf2origin, hasorigin2inf, g_zero, g_one, bits
-import AbstractTensors: valuetype, scalar, isscalar
+import AbstractTensors: valuetype, scalar, isscalar, ⊗
 import AbstractTensors: vector, isvector, bivector, isbivector, volume, isvolume
 
 ## cache
@@ -62,7 +62,7 @@ end
 ⊘(x::Derivation,y::T) where T<:TensorAlgebra{V} where V = V(x)⊘y
 
 @pure function (V::Signature{N})(d::Leibniz.Derivation{T,O}) where {N,T,O}
-    (O<1||diffvars(V)==0) && (return Chain{V,1,Int}(ones(Int,N)))
+    (O<1||diffvars(V)==0) && (return Chain{V,1,Int}(d.v.λ*ones(Values{N,Int})))
     G,D,C = grade(V),diffvars(V)==1,isdyadic(V)
     G2 = (C ? Int(G/2) : G)-1
     ∇ = sum([getbasis(V,1<<(D ? G : k+G))*getbasis(V,1<<k) for k ∈ 0:G2])
@@ -73,7 +73,7 @@ end
 
 @pure function (M::SubManifold{W,N})(d::Leibniz.Derivation{T,O}) where {W,N,T,O}
     V = isbasis(M) ? W : M
-    (O<1||diffvars(V)==0) && (return Chain{V,1,Int}(ones(Int,N)))
+    (O<1||diffvars(V)==0) && (return Chain{V,1,Int}(d.v.λ*ones(Values{N,Int})))
     G,D,C = grade(V),diffvars(V)==1,isdyadic(V)
     G2 = (C ? Int(G/2) : G)-1
     ∇ = sum([getbasis(V,1<<(D ? G : k+G))*getbasis(V,1<<k) for k ∈ 0:G2])
@@ -93,18 +93,18 @@ function boundary_rank(t,d=gdims(t))
     for k ∈ 2:length(out)-1
         @inbounds out[k] = min(out[k],d[k+1])
     end
-    return SVector(out)
+    return Values(out)
 end
 
 function boundary_null(t)
     d = gdims(t)
     r = boundary_rank(t,d)
     l = length(d)
-    out = zeros(MVector{l,Int})
+    out = zeros(Variables{l,Int})
     for k ∈ 1:l-1
         @inbounds out[k] = d[k+1] - r[k]
     end
-    return SVector(out)
+    return Values(out)
 end
 
 """
@@ -116,11 +116,11 @@ function betti(t::T) where T<:TensorAlgebra
     d = gdims(t)
     r = boundary_rank(t,d)
     l = length(d)-1
-    out = zeros(MVector{l,Int})
+    out = zeros(Variables{l,Int})
     for k ∈ 1:l
         @inbounds out[k] = d[k+1] - r[k] - r[k+1]
     end
-    return SVector(out)
+    return Values(out)
 end
 
 @generated function ↑(ω::T) where T<:TensorAlgebra
@@ -277,12 +277,12 @@ function SparseArrays.sparse(t,cols=columns(t))
     return A
 end
 
-edges(t,cols::SVector) = edges(t,adjacency(t,cols))
+edges(t,cols::Values) = edges(t,adjacency(t,cols))
 function edges(t,adj=adjacency(t))
     ndims(t) == 2 && (return t)
     N = ndims(Manifold(t)); M = points(t)(list(N-1,N)...)
     f = findall(x->!iszero(x),LinearAlgebra.triu(adj))
-    [Chain{M,1}(SVector{2,Int}(f[n].I)) for n ∈ 1:length(f)]
+    [Chain{M,1}(Values{2,Int}(f[n].I)) for n ∈ 1:length(f)]
 end
 
 function facetsinterior(t::Vector{<:Chain{V}}) where V
@@ -323,8 +323,8 @@ function faces(t::Vector{<:Chain{V}},h,::Val{N},g=identity) where {V,N}
     N == 0 && (return [Chain{W,1}(list(1,N))],Int[sum(h)])
     out = Chain{W,1,Int,N}[]
     bnd = Int[]
-    vec = zeros(MVector{ndims(V),Int})
-    val = N+1==ndims(V) ? ∂(Manifold(points(t))(list(1,N+1))(I)) : ones(SVector{binomial(ndims(V),N)})
+    vec = zeros(Variables{ndims(V),Int})
+    val = N+1==ndims(V) ? ∂(Manifold(points(t))(list(1,N+1))(I)) : ones(Values{binomial(ndims(V),N)})
     for i ∈ 1:length(t)
         vec[:] = value(t[i])
         par = DirectSum.indexparity!(vec)
@@ -344,8 +344,8 @@ function faces(t::Vector{<:Chain{V}},h,::Val{N},g=identity) where {V,N}
 end
 
 ∂(t::ChainBundle) = ∂(value(t))
-∂(t::SVector{N,<:Tuple}) where N = ∂.(t)
-∂(t::SVector{N,<:Vector}) where N = ∂.(t)
+∂(t::Values{N,<:Tuple}) where N = ∂.(t)
+∂(t::Values{N,<:Vector}) where N = ∂.(t)
 ∂(t::Tuple{Vector{<:Chain},Vector{Int}}) = ∂(t[1],t[2])
 ∂(t::Vector{<:Chain},u::Vector{Int}) = (f=facets(t,u); f[1][findall(x->!iszero(x),f[2])])
 ∂(t::Vector{<:Chain}) = ndims(t)≠3 ? (f=facetsinterior(t); f[1][setdiff(1:length(f[1]),f[2])]) : edges(t,adjacency(t).%2)
@@ -527,9 +527,9 @@ function __init__()
             c,f = GeometryBasics.coordinates(m),GeometryBasics.faces(m)
             s = size(eltype(c))[1]+1; V = SubManifold(ℝ^s) # s
             n = size(eltype(f))[1]
-            p = ChainBundle([Chain{V,1}(SVector{s,Float64}(1.0,k...)) for k ∈ c])
+            p = ChainBundle([Chain{V,1}(Values{s,Float64}(1.0,k...)) for k ∈ c])
             M = s ≠ n ? p(list(s-n+1,s)) : p
-            t = ChainBundle([Chain{M,1}(SVector{n,Int}(k)) for k ∈ f])
+            t = ChainBundle([Chain{M,1}(Values{n,Int}(k)) for k ∈ f])
             return (p,ChainBundle(∂(t)),t)
         end
         @pure ptype(::GeometryBasics.Point{N,T} where N) where T = T
@@ -692,10 +692,10 @@ function __init__()
         end
         function initmesh(tio::TetGen.TetgenIO, command = "Qp")
             r = TetGen.tetrahedralize(tio, command); V = SubManifold(ℝ^4)
-            p = ChainBundle([Chain{V,1}(SVector{4,Float64}(1.0,k...)) for k ∈ r.points])
-            t = Chain{p,1}.(SVector{4,Int}.(r.tetrahedra))
-            e = Chain{p(2,3,4),1}.(SVector{3,Int}.(r.trifaces))
-            # Chain{p(2,3),1}.(SVector{2,Int}.(r.edges)
+            p = ChainBundle([Chain{V,1}(Values{4,Float64}(1.0,k...)) for k ∈ r.points])
+            t = Chain{p,1}.(Values{4,Int}.(r.tetrahedra))
+            e = Chain{p(2,3,4),1}.(Values{3,Int}.(r.trifaces))
+            # Chain{p(2,3),1}.(Values{2,Int}.(r.edges)
             return p,ChainBundle(e),ChainBundle(t)
         end
         function TetGen.tetrahedralize(mesh::ChainBundle, command = "Qp";
diff --git a/src/algebra.jl b/src/algebra.jl
index 22f5359..10b5104 100644
--- a/src/algebra.jl
+++ b/src/algebra.jl
@@ -9,8 +9,8 @@ import Leibniz: loworder, isnull
 
 ## mutating operations
 
-@pure tvec(N,G,t::Symbol=:Any) = :(SVector{$(binomial(N,G)),$t})
-@pure tvec(N,t::Symbol=:Any) = :(SVector{$(1<<N),$t})
+@pure tvec(N,G,t::Symbol=:Any) = :(Values{$(binomial(N,G)),$t})
+@pure tvec(N,t::Symbol=:Any) = :(Values{$(1<<N),$t})
 @pure tvec(N,μ::Bool) = tvec(N,μ ? :Any : :t)
 
 import Leibniz: g_one, g_zero, Field, ExprField
@@ -353,10 +353,10 @@ Exterior product as defined by the anti-symmetric quotient Λ≡⊗/~
 @inline ∧(a::X,b::Y) where {X<:TensorAlgebra,Y<:TensorAlgebra} = interop(∧,a,b)
 @inline ∧(a::TensorAlgebra{V},b::UniformScaling{T}) where {V,T<:Field} = a∧V(b)
 @inline ∧(a::UniformScaling{T},b::TensorAlgebra{V}) where {V,T<:Field} = V(a)∧b
-@generated ∧(t::T) where T<:SVector{N} where N = wedges([:(t[$i]) for i ∈ 1:N])
-@generated ∧(t::T) where T<:SizedVector{N} where N = wedges([:(t[$i]) for i ∈ 1:N])
-∧(::SVector{0,<:Chain{V}}) where V = one(V) # ∧() = 1
-∧(::SizedVector{0,<:Chain{V}}) where V = one(V)
+@generated ∧(t::T) where T<:Values{N} where N = wedges([:(t[$i]) for i ∈ 1:N])
+@generated ∧(t::T) where T<:FixedVector{N} where N = wedges([:(t[$i]) for i ∈ 1:N])
+∧(::Values{0,<:Chain{V}}) where V = one(V) # ∧() = 1
+∧(::FixedVector{0,<:Chain{V}}) where V = one(V)
 ∧(t::Chain{V,1,<:Chain} where V) = ∧(value(t))
 ∧(a::X,b::Y,c::Z...) where {X<:TensorAlgebra,Y<:TensorAlgebra,Z<:TensorAlgebra} = ∧(a∧b,c...)
 
@@ -423,10 +423,10 @@ Regressive product as defined by the DeMorgan's law: ∨(ω...) = ⋆⁻¹(∧(
 @inline ∨(a::X,b::Y) where {X<:TensorAlgebra,Y<:TensorAlgebra} = interop(∨,a,b)
 @inline ∨(a::TensorAlgebra{V},b::UniformScaling{T}) where {V,T<:Field} = a∨V(b)
 @inline ∨(a::UniformScaling{T},b::TensorAlgebra{V}) where {V,T<:Field} = V(a)∨b
-@generated ∨(t::T) where T<:SVector = Expr(:call,:∨,[:(t[$k]) for k ∈ 1:length(t)]...)
-@generated ∨(t::T) where T<:SizedVector = Expr(:call,:∨,[:(t[$k]) for k ∈ 1:length(t)]...)
-∨(::SVector{0,<:Chain{V}}) where V = SubManifold(V) # ∨() = I
-∨(::SizedVector{0,<:Chain{V}}) where V = SubManifold(V)
+@generated ∨(t::T) where T<:Values = Expr(:call,:∨,[:(t[$k]) for k ∈ 1:length(t)]...)
+@generated ∨(t::T) where T<:FixedVector = Expr(:call,:∨,[:(t[$k]) for k ∈ 1:length(t)]...)
+∨(::Values{0,<:Chain{V}}) where V = SubManifold(V) # ∨() = I
+∨(::FixedVector{0,<:Chain{V}}) where V = SubManifold(V)
 ∨(t::Chain{V,1,<:Chain} where V) = ∧(value(t))
 ∨(a::X,b::Y,c::Z...) where {X<:TensorAlgebra,Y<:TensorAlgebra,Z<:TensorAlgebra} = ∨(a∨b,c...)
 
@@ -481,9 +481,27 @@ Interior (right) contraction product: ω⋅η = ω∨⋆η
 
 # dyadic products
 
-contraction(a::Chain{W,G,<:Chain},b::Chain{V,1,<:Chain}) where {W,G,V} = Chain{V,1}(a.⋅value(b))
+export outer
+
+outer(a::Leibniz.Derivation,b::Chain{V,1}) where V= outer(V(a),b)
+outer(a::Chain{W},b::Leibniz.Derivation{T,1}) where {W,T} = outer(a,W(b))
+outer(a::Chain{W},b::Chain{V,1}) where {W,V} = Chain{V,1}(a.*value(b))
+
+contraction(a::Chain{W,G},b::Chain{V,1,<:Chain}) where {W,G,V} = Chain{V,1}(column(Ref(a).⋅value(b)))
+contraction(a::Chain{W,G,<:Chain},b::Chain{V,1,<:Chain}) where {W,G,V} = Chain{V,1}(Ref(a).⋅value(b))
 Base.:(:)(a::Chain{V,1,<:Chain},b::Chain{V,1,<:Chain}) where V = sum(value(a).⋅value(b))
 
+# dyadic identity element
+
+@generated Base.:+(g::Chain{V,1,<:Chain{V,1}},t::LinearAlgebra.UniformScaling{Bool}) where V = :(Chain{V,1}(value(g).+$(getalgebra(V).b[Grassmann.list(2,mdims(V)+1)])))
+@generated Base.:+(g::Chain{V,1,<:Chain{V,1}},t::LinearAlgebra.UniformScaling) where V = :(Chain{V,1}(value(g).+t.λ*$(getalgebra(V).b[Grassmann.list(2,mdims(V)+1)])))
+@generated Base.:+(t::LinearAlgebra.UniformScaling{Bool},g::Chain{V,1,<:Chain{V,1}}) where V = :(Chain{V,1}($(getalgebra(V).b[Grassmann.list(2,mdims(V)+1)]).+value(g)))
+@generated Base.:+(t::LinearAlgebra.UniformScaling,g::Chain{V,1,<:Chain{V,1}}) where V = :(Chain{V,1}(t.λ*$(getalgebra(V).b[Grassmann.list(2,mdims(V)+1)]).+value(g)))
+@generated Base.:-(g::Chain{V,1,<:Chain{V,1}},t::LinearAlgebra.UniformScaling{Bool}) where V = :(Chain{V,1}(value(g).-$(getalgebra(V).b[Grassmann.list(2,mdims(V)+1)])))
+@generated Base.:-(g::Chain{V,1,<:Chain{V,1}},t::LinearAlgebra.UniformScaling) where V = :(Chain{V,1}(value(g).-t.λ*$(getalgebra(V).b[Grassmann.list(2,mdims(V)+1)])))
+@generated Base.:-(t::LinearAlgebra.UniformScaling{Bool},g::Chain{V,1,<:Chain{V,1}}) where V = :(Chain{V,1}($(getalgebra(V).b[Grassmann.list(2,mdims(V)+1)]).-value(g)))
+@generated Base.:-(t::LinearAlgebra.UniformScaling,g::Chain{V,1,<:Chain{V,1}}) where V = :(Chain{V,1}(t.λ*$(getalgebra(V).b[Grassmann.list(2,mdims(V)+1)]).-value(g)))
+
 ## cross product
 
 import LinearAlgebra: cross, ×
@@ -794,7 +812,7 @@ for (op,eop) ∈ ((:+,:(+=)),(:-,:(-=)))
                 end
             end
             G = A|B
-            MultiGrade{V,G}(SVector{count_ones(G),TensorGraded{V}}(out))
+            MultiGrade{V,G}(Values{count_ones(G),TensorGraded{V}}(out))
         end
         function $op(a::MultiGrade{V,A},b::T) where T<:TensorGraded{V,B} where {V,A,B}
             N = mdims(V)
@@ -817,9 +835,9 @@ for (op,eop) ∈ ((:+,:(+=)),(:-,:(-=)))
                 end
             end
             G = A|(UInt(1)<<B)
-            MultiGrade{V,G}(SVector{count_ones(G),TensorGraded{V}}(out))
+            MultiGrade{V,G}(Values{count_ones(G),TensorGraded{V}}(out))
         end
-        $op(a::SparseChain{V,A},b::T) where T<:TensorGraded{V,B} where {V,A,B} = MultiGrade{V,(UInt(1)<<A)|(UInt(1)<<B)}(A<B ? SVector(a,b) : SVector(b,a))
+        $op(a::SparseChain{V,A},b::T) where T<:TensorGraded{V,B} where {V,A,B} = MultiGrade{V,(UInt(1)<<A)|(UInt(1)<<B)}(A<B ? Values(a,b) : Values(b,a))
     end
 end
 
diff --git a/src/composite.jl b/src/composite.jl
index bfc9e43..120ddba 100644
--- a/src/composite.jl
+++ b/src/composite.jl
@@ -12,7 +12,7 @@ export exph, log_fast, logh_fast
 function Base.expm1(t::T) where T<:TensorAlgebra
     V = Manifold(t)
     S,term,f = t,(t^2)/2,norm(t)
-    norms = SizedVector{3}(f,norm(term),f)
+    norms = FixedVector{3}(f,norm(term),f)
     k::Int = 3
     @inbounds while norms[2]<norms[1] || norms[2]>1
         S += term
@@ -36,7 +36,7 @@ end
         term = zeros(mvec(N,t))
         S .= B
         out .= value(b^2)/2
-        norms = SizedVector{3}(nb,norm(out),norm(term))
+        norms = FixedVector{3}(nb,norm(out),norm(term))
         k::Int = 3
         @inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
             S += out
@@ -80,7 +80,7 @@ function Base.exp(t::MultiVector,::Val{hint}) where hint
     end
 end
 
-@pure isR301(V::DiagonalForm) = DirectSum.diagonalform(V) == SVector(1,1,1,0)
+@pure isR301(V::DiagonalForm) = DirectSum.diagonalform(V) == Values(1,1,1,0)
 @pure isR301(::SubManifold{V}) where V = isR301(V)
 @pure isR301(V) = false
 
@@ -129,7 +129,7 @@ function qlog(w::T,x::Int=10000) where T<:TensorAlgebra
     w2,f = w^2,norm(w)
     prod = w*w2
     S,term = w,prod/3
-    norms = SizedVector{3}(f,norm(term),f)
+    norms = FixedVector{3}(f,norm(term),f)
     k::Int = 5
     @inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ x
         S += term
@@ -156,7 +156,7 @@ end # http://www.netlib.org/cephes/qlibdoc.html#qlog
         S .= value(b)
         out .= value(b*w2)
         term .= out/3
-        norms = SizedVector{3}(f,norm(term),f)
+        norms = FixedVector{3}(f,norm(term),f)
         k::Int = 5
         @inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ x
             S += term
@@ -197,7 +197,7 @@ function Base.cosh(t::T) where T<:TensorAlgebra
     τ = t^2
     S,term = τ/2,(τ^2)/24
     f = norm(S)
-    norms = SizedVector{3}(f,norm(term),f)
+    norms = FixedVector{3}(f,norm(term),f)
     k::Int = 6
     @inbounds while norms[2]<norms[1] || norms[2]>1
         S += term
@@ -222,7 +222,7 @@ end
         term = zeros(mvec(N,t))
         S .= value(τ)/2
         out .= value((τ^2))/24
-        norms = SizedVector{3}(norm(S),norm(out),norm(term))
+        norms = FixedVector{3}(norm(S),norm(out),norm(term))
         k::Int = 6
         @inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
             S += out
@@ -246,7 +246,7 @@ function Base.sinh(t::T) where T<:TensorAlgebra
     V = Manifold(t)
     τ,f = t^2,norm(t)
     S,term = t,(t*τ)/6
-    norms = SizedVector{3}(f,norm(term),f)
+    norms = FixedVector{3}(f,norm(term),f)
     k::Int = 5
     @inbounds while norms[2]<norms[1] || norms[2]>1
         S += term
@@ -271,7 +271,7 @@ end
         term = zeros(mvec(N,t))
         S .= value(b)
         out .= value(b*τ)/6
-        norms = SizedVector{3}(norm(S),norm(out),norm(term))
+        norms = FixedVector{3}(norm(S),norm(out),norm(term))
         k::Int = 5
         @inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
             S += out
@@ -294,7 +294,7 @@ for (logfast,expf) ∈ ((:log_fast,:exp),(:logh_fast,:exph))
     @eval function $logfast(t::T) where T<:TensorAlgebra
         V = Manifold(t)
         term = zero(V)
-        norm = SizedVector{2}(0.,0.)
+        norm = FixedVector{2}(0.,0.)
         while true
             en = $expf(term)
             term -= 2(en-t)/(en+t)
@@ -337,12 +337,12 @@ end=#
 
 function Cramer(N::Int,j=0)
     t = j ≠ 0 ? :T : :t
-    x,y = SVector{N}([Symbol(:x,i) for i ∈ 1:N]),SVector{N}([Symbol(:y,i) for i ∈ 1:N])
+    x,y = Values{N}([Symbol(:x,i) for i ∈ 1:N]),Values{N}([Symbol(:y,i) for i ∈ 1:N])
     xy = [:(($(x[1+i]),$(y[1+i])) = ($(x[i])∧$t[$(1+i-j)],$t[end-$i]∧$(y[i]))) for i ∈ 1:N-1-j]
     return x,y,xy
 end
 
-@generated function Base.:\(t::SVector{M,<:Chain{V,1}},v::Chain{V,1}) where {M,V}
+@generated function Base.:\(t::Values{M,<:Chain{V,1}},v::Chain{V,1}) where {M,V}
     W = M≠mdims(V) ? SubManifold(M) : V; N = M-1
     if M == 1 && (V === ℝ1 || V == 1)
         return :(Chain{V,1}(Values(v[1]/t[1][1])))
@@ -377,16 +377,16 @@ end
         :(Chain{$W,1}(column($(Expr(:call,:.⋅,out,:(Ref(detx))))./abs2(detx)))))
 end
 
-@generated function Base.in(v::Chain{V,1},t::SVector{N,<:Chain{V,1}}) where {V,N}
+@generated function Base.in(v::Chain{V,1},t::Values{N,<:Chain{V,1}}) where {V,N}
     if N == mdims(V)
         x,y,xy = Grassmann.Cramer(N-1)
         mid = [:(s==signbit(($(x[i])∧v∧$(y[end-i]))[1])) for i ∈ 1:N-2]
-        out = SVector(:(s==signbit((v∧$(y[end]))[1])),mid...,:(s==signbit(($(x[end])∧v)[1])))
+        out = Values(:(s==signbit((v∧$(y[end]))[1])),mid...,:(s==signbit(($(x[end])∧v)[1])))
         return Expr(:block,:((x1,y1)=(t[1],t[end])),xy...,:(s=signbit((t[1]∧$(y[end]))[1])),ands(out))
     else
         x,y,xy = Grassmann.Cramer(N-1,1)
         mid = [:(signscalar(($(x[i])∧(v-x1)∧$(y[end-i]))/d)) for i ∈ 1:N-2]
-        out = SVector(:(signscalar((v∧∧(vectors(t,v)))/d)),mid...,:(signscalar(($(x[end])∧(v-x1))/d)))
+        out = Values(:(signscalar((v∧∧(vectors(t,v)))/d)),mid...,:(signscalar(($(x[end])∧(v-x1))/d)))
         return Expr(:block,:(T=vectors(t)),:((x1,y1)=(t[1],T[end])),xy...,
             :($(x[end])=$(x[end-1])∧T[end-1];d=$(x[end])∧T[end]),ands(out))
     end
@@ -398,11 +398,11 @@ end
     M > mdims(V) && (return :(tt = _transpose(t,$W); tt⋅inv(Chain{$W,1}(t)⋅tt)))
     x,y,xy = Grassmann.Cramer(N)
     val = if iseven(N)
-        Expr(:call,:SVector,y[end],[:($(y[end-i])∧$(x[i])) for i ∈ 1:N-1]...,x[end])
+        Expr(:call,:Values,y[end],[:($(y[end-i])∧$(x[i])) for i ∈ 1:N-1]...,x[end])
     elseif M≠mdims(V)
-        Expr(:call,:SVector,y[end],[:($(iseven(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,:(-$(x[end])))
+        Expr(:call,:Values,y[end],[:($(iseven(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,:(-$(x[end])))
     else
-        Expr(:call,:SVector,:(-$(y[end])),[:($(isodd(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,x[end])
+        Expr(:call,:Values,:(-$(y[end])),[:($(isodd(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,x[end])
     end
     out = if M≠mdims(V)
         :(vector.($(Expr(:call,:./,val,:((t[1]∧$(y[end])))))))
@@ -412,16 +412,16 @@ end
     return Expr(:block,:((x1,y1)=(t[1],t[end])),xy...,:(_transpose($out,$W)))
 end
 
-@generated function grad(T::SVector{M,<:Chain{V,1}}) where {M,V}
+@generated function grad(T::Values{M,<:Chain{V,1}}) where {M,V}
     W = M≠mdims(V) ? SubManifold(M) : V; N = mdims(V)-1
     M < mdims(V) && (return :(ct = Chain{$W,1}(T); map(↓(V),ct⋅inv(_transpose(T,$W)⋅ct))))
     x,y,xy = Grassmann.Cramer(N)
     val = if iseven(N)
-        Expr(:call,:SVector,[:($(y[end-i])∧$(x[i])) for i ∈ 1:N-1]...,x[end])
+        Expr(:call,:Values,[:($(y[end-i])∧$(x[i])) for i ∈ 1:N-1]...,x[end])
     elseif M≠mdims(V)
-        Expr(:call,:SVector,y[end],[:($(iseven(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,:(-$(x[end])))
+        Expr(:call,:Values,y[end],[:($(iseven(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,:(-$(x[end])))
     else
-        Expr(:call,:SVector,[:($(isodd(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,x[end])
+        Expr(:call,:Values,[:($(isodd(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,x[end])
     end
     out = if M≠mdims(V)
         :(vector.($(Expr(:call,:./,val,:((t[1]∧$(y[end])))))))
@@ -431,7 +431,7 @@ end
     return Expr(:block,:(t=_transpose(T,$W)),:((x1,y1)=(t[1],t[end])),xy...,:(_transpose($out,↓(V))))
 end
 
-@generated function Base.:\(t::SVector{N,<:Chain{M,1}},v::Chain{V,1}) where {N,M,V}
+@generated function Base.:\(t::Values{N,<:Chain{M,1}},v::Chain{V,1}) where {N,M,V}
     W = M≠mdims(V) ? SubManifold(N) : V
     if mdims(M) > mdims(V)
         :(ct=Chain{$W,1}(t); ct⋅(inv(_transpose(t,$W)⋅ct)⋅v))
@@ -453,7 +453,7 @@ INV(m::Chain{V,1,<:Chain{V,1}}) where V = Chain{V,1,Chain{V,1}}(inv(SMatrix(m)))
 export vandermonde
 
 @generated approx(x,y::Chain{V}) where V = :(polynom(x,$(Val(mdims(V))))⋅y)
-approx(x,y::SVector{N}) where N = value(polynom(x,Val(N)))⋅y
+approx(x,y::Values{N}) where N = value(polynom(x,Val(N)))⋅y
 approx(x,y::AbstractVector) = [x^i for i ∈ 0:length(y)-1]⋅y
 
 vandermonde(x::Array,y::Array,N::Int) = vandermonde(x,N)\y[:] # compute ((inv(X'*X))*X')*y
@@ -468,8 +468,8 @@ end
 vandermonde(x,y,V) = (length(x)≠mdims(V) ? _vandermonde(x,V) : vandermonde(x,V))\y
 vandermonde(x,V) = transpose(_vandermonde(x,V))
 _vandermonde(x::Chain,V) = _vandermonde(value(x),V)
-@generated _vandermonde(x::SVector{N},V) where N = :(Chain{$(SubManifold(N)),1}(polynom.(x,$(Val(mdims(V))))))
-@generated polynom(x,::Val{N}) where N = Expr(:call,:(Chain{$(SubManifold(N)),1}),Expr(:call,:SVector,[:(x^$i) for i ∈ 0:N-1]...))
+@generated _vandermonde(x::Values{N},V) where N = :(Chain{$(SubManifold(N)),1}(polynom.(x,$(Val(mdims(V))))))
+@generated polynom(x,::Val{N}) where N = Expr(:call,:(Chain{$(SubManifold(N)),1}),Expr(:call,:Values,[:(x^$i) for i ∈ 0:N-1]...))
 
 function vandermondeinterp(x,y,V,grid) # grid=384
     coef = vandermonde(x,y,V) # Vandermonde ((inv(X'*X))*X')*y
@@ -486,11 +486,11 @@ end
     quote
         p = points(t)
         M,A = ↓(Manifold(p)),p[c[1]]
-        Chain{M,1}.($(Expr(:.,:SVector,v)))
+        Chain{M,1}.($(Expr(:.,:Values,v)))
     end
 end
-@pure list(a::Int,b::Int) = SVector{max(0,b-a+1),Int}(a:b...)
-vectors(x::SVector{N,<:Chain{V}},y=x[1]) where {N,V} = ↓(V).(x[list(2,N)].-y)
+@pure list(a::Int,b::Int) = Values{max(0,b-a+1),Int}(a:b...)
+vectors(x::Values{N,<:Chain{V}},y=x[1]) where {N,V} = ↓(V).(x[list(2,N)].-y)
 vectors(x::Chain{V,1},y=x[1]) where V = vectors(value(x),y)
 #point(x,y=x[1]) = y∧∧(vectors(x))
 
@@ -538,12 +538,13 @@ volumes(m) = mdims(Manifold(m))≠2 ? volumes(m,detsimplex(m)) : edgelength.(val
 detsimplex(m::Vector{<:Chain{V}}) where V = ∧(m)/factorial(mdims(V)-1)
 detsimplex(m::ChainBundle) = detsimplex(value(m))
 mean(m::T) where T<:AbstractVector{<:Chain} = sum(m)/length(m)
-mean(m::T) where T<:SVector = sum(m)/length(m)
+mean(m::T) where T<:Values = sum(m)/length(m)
 mean(m::Chain{V,1,<:Chain} where V) = mean(value(m))
-barycenter(m::SVector{N,<:Chain}) where N = (s=sum(m);s/s[1])
+barycenter(m::Values{N,<:Chain}) where N = (s=sum(m);s/s[1])
 barycenter(m::Vector{<:Chain}) = (s=sum(m);s/s[1])
 barycenter(m::Chain{V,1,<:Chain} where V) = barycenter(value(m))
-curl(m::SVector{N,<:Chain{V}} where N) where V = curl(Chain{V,1}(m))
+curl(m::FixedVector{N,<:Chain{V}} where N) where V = curl(Chain{V,1}(m))
+curl(m::Values{N,<:Chain{V}} where N) where V = curl(Chain{V,1}(m))
 curl(m::T) where T<:TensorAlgebra = Manifold(m)(∇)×m
 LinearAlgebra.det(t::Chain{V,1,<:Chain} where V) = ∧(t)
 LinearAlgebra.det(m::Vector{<:Chain{V}}) where V = ∧(m)
@@ -581,12 +582,12 @@ function refinemesh!(::R,p::ChainBundle{W},e,t,η,_=nothing) where {W,R<:Abstrac
     p = points(t)
     x,T,V = value(p),value(t),Manifold(p)
     for i ∈ η
-        push!(x,Chain{V,1}(SVector(1,(x[i+1][2]+x[i][2])/2)))
+        push!(x,Chain{V,1}(Values(1,(x[i+1][2]+x[i][2])/2)))
     end
     sort!(x,by=x->x[2]); submesh!(p)
-    e[end] = Chain{p(2),1}(SVector(length(x)))
+    e[end] = Chain{p(2),1}(Values(length(x)))
     for i ∈ length(t)+2:length(x)
-        push!(T,Chain{p,1}(SVector{2,Int}(i-1,i)))
+        push!(T,Chain{p,1}(Values{2,Int}(i-1,i)))
     end
 end
 
@@ -649,25 +650,20 @@ if VERSION >= v"1.4"
         (T = promote_type(TA, Chain{V,G,𝕂,X}); mul!(similar(B, T, (size(transA, 1), size(B, 2))), transA, B, 1, 0))
 end
 
-#=@generated function StaticArrays._diff(::Size{S}, a::SVector{Q,<:Chain}, ::Val{D}) where {S,D,Q}
-    N = length(S)
-    Snew = ([n==D ? S[n]-1 : S[n] for n = 1:N]...,)
-
+@generated function AbstractTensors._diff(::Val{N}, a::Values{Q,<:Chain}, ::Val{1}) where {N,Q}
+    Snew = N-1
     exprs = Array{Expr}(undef, Snew)
-    itr = [1:n for n = Snew]
-
-    for i1 = Base.product(itr...)
+    for i1 = Base.product(1:Snew)
         i2 = copy([i1...])
-        i2[D] = i1[D] + 1
+        i2[1] = i1[1] + 1
         exprs[i1...] = :(a[$(i2...)] - a[$(i1...)])
     end
-
     return quote
         Base.@_inline_meta
-        T = eltype(a)
-        @inbounds return similar_type(a, T, Size($Snew))(tuple($(exprs...)))
+        elements = tuple($(exprs...))
+        @inbounds return AbstractTensors.similar_type(a, eltype(a), Val($Snew))(elements)
     end
-end=#
+end
 
 Base.map(fn, x::MultiVector{V}) where V = MultiVector{V}(map(fn, value(x)))
 Base.map(fn, x::Chain{V,G}) where {V,G} = Chain{V,G}(map(fn,value(x)))
@@ -715,7 +711,7 @@ Base.iterate(t::Orthogrid,state) = (s=state+1; s≤length(t) ? (getindex(t,s),s)
 @pure Base.lastindex(t::Orthogrid) = length(t)
 @pure Base.lastindex(t::Orthogrid,i::Int) = size(t)[i]
 @pure Base.getindex(t::Orthogrid,i::CartesianIndex) = getindex(t,i.I...)
-@pure Base.getindex(t::Orthogrid{V},i::Vararg{Int}) where V = Chain{V,1}(value(t.x.min)+(SVector(i)-1).*step(t))
+@pure Base.getindex(t::Orthogrid{V},i::Vararg{Int}) where V = Chain{V,1}(value(t.x.min)+(Values(i)-1).*step(t))
 
 Base.IndexStyle(::Orthogrid) = IndexCartesian()
 function Base.getindex(A::Orthogrid, I::Int)
diff --git a/src/forms.jl b/src/forms.jl
index 2ac14cc..3e0b9ff 100644
--- a/src/forms.jl
+++ b/src/forms.jl
@@ -40,7 +40,7 @@
         return :b
     elseif W⊆V
         if G == 1
-            ind = SVector{mdims(W),Int}(indices(bits(W),mdims(V)))
+            ind = Values{mdims(W),Int}(indices(bits(W),mdims(V)))
             :(@inbounds Chain{w,1,T}(b.v[$ind]))
         else quote
             out,N = zeros(choicevec(M,G,valuetype(b))),mdims(V)
diff --git a/src/multivectors.jl b/src/multivectors.jl
index e885dbd..daf9598 100644
--- a/src/multivectors.jl
+++ b/src/multivectors.jl
@@ -21,7 +21,7 @@ export UniformScaling, I, points
 ## Chain{V,G,𝕂}
 
 @computed struct Chain{V,G,𝕂} <: TensorGraded{V,G}
-    v::SVector{binomial(mdims(V),G),𝕂}
+    v::Values{binomial(mdims(V),G),𝕂}
 end
 
 """
@@ -30,27 +30,29 @@ end
 Chain type with pseudoscalar `V::Manifold`, grade/rank `G::Int`, scalar field `𝕂::Type`.
 """
 Chain{V,G}(val::S) where {V,G,S<:AbstractVector{𝕂}} where 𝕂 = Chain{V,G,𝕂}(val)
-#Chain{V,G}(args::𝕂...) where {V,G,𝕂} = Chain{V,G}(SVector{binomial(mdims(V),G)}(args...))
+Chain{V}(val::S) where {V,S<:TupleVector{N,𝕂}} where {N,𝕂} = Chain{V,1,𝕂}(val)
+Chain(val::S) where S<:TupleVector{N,𝕂} where {N,𝕂} = Chain{SubManifold(N),1,𝕂}(val)
+#Chain{V,G}(args::𝕂...) where {V,G,𝕂} = Chain{V,G}(Values{binomial(mdims(V),G)}(args...))
 @generated function Chain{V,G}(args::𝕂...) where {V,G,𝕂}
     bg = binomial(mdims(V),G)
-    ref = SVector{bg}([:(args[$i]) for i ∈ 1:bg])
-    :(Chain{V,G}($(Expr(:call,:(SVector{$bg,𝕂}),ref...))))
+    ref = Values{bg}([:(args[$i]) for i ∈ 1:bg])
+    :(Chain{V,G}($(Expr(:call,:(Values{$bg,𝕂}),ref...))))
 end
 
 @generated function Chain{V}(args::𝕂...) where {V,𝕂}
-    bg = mdims(V); ref = SVector{bg}([:(args[$i]) for i ∈ 1:bg])
-    :(Chain{V,1}($(Expr(:call,:(SVector{$bg,𝕂}),ref...))))
+    bg = mdims(V); ref = Values{bg}([:(args[$i]) for i ∈ 1:bg])
+    :(Chain{V,1}($(Expr(:call,:(Values{$bg,𝕂}),ref...))))
 end
 
 @generated function Chain(args::𝕂...) where 𝕂
     N = length(args)
     V = SubManifold(N)
-    ref = SVector{N}([:(args[$i]) for i ∈ 1:N])
-    :(Chain{$V,1}($(Expr(:call,:(SVector{$N,𝕂}),ref...))))
+    ref = Values{N}([:(args[$i]) for i ∈ 1:N])
+    :(Chain{$V,1}($(Expr(:call,:(Values{$N,𝕂}),ref...))))
 end
 
-Chain(v::Chain{V,G,𝕂}) where {V,G,𝕂} = Chain{V,G}(SVector{binomial(mdims(V),G),𝕂}(v.v))
-Chain{𝕂}(v::Chain{V,G}) where {V,G,𝕂} = Chain{V,G}(SVector{binomial(mdims(V),G),𝕂}(v.v))
+Chain(v::Chain{V,G,𝕂}) where {V,G,𝕂} = Chain{V,G}(Values{binomial(mdims(V),G),𝕂}(v.v))
+Chain{𝕂}(v::Chain{V,G}) where {V,G,𝕂} = Chain{V,G}(Values{binomial(mdims(V),G),𝕂}(v.v))
 
 export Chain
 getindex(m::Chain,i::Int) = m.v[i]
@@ -64,7 +66,7 @@ Base.firstindex(m::Chain) = 1
 Base.zero(::Type{<:Chain{V,G,T}}) where {V,G,T} = Chain{V,G}(zeros(svec(mdims(V),G,T)))
 Base.zero(::Chain{V,G,T}) where {V,G,T} = Chain{V,G}(zeros(svec(mdims(V),G,T)))
 
-transpose_row(t::SVector{N,<:Chain{V}},i,W=V) where {N,V} = Chain{W,1}(getindex.(t,i))
+transpose_row(t::Values{N,<:Chain{V}},i,W=V) where {N,V} = Chain{W,1}(getindex.(t,i))
 transpose_row(t::FixedVector{N,<:Chain{V}},i,W=V) where {N,V} = Chain{W,1}(getindex.(t,i))
 transpose_row(t::Chain{V,1,<:Chain},i) where V = transpose_row(value(t),i,V)
 @generated _transpose(t::Values{N,<:Chain{V,1}},W=V) where {N,V} = :(Chain{V,1}(transpose_row.(Ref(t),$(Values{mdims(V)}(1:mdims(V))),W)))
@@ -134,6 +136,12 @@ end
 ==(a::T,b::Chain{V}) where T<:TensorTerm{V} where V = b==a
 ==(a::Chain{V},b::T) where T<:TensorTerm{V} where V = prod(0==value(b).==value(a))
 
+Base.ones(::Type{Chain{V,G,T,X}}) where {V,G,T,X} = Chain{V,G,T}(ones(Values{X,T}))
+Base.ones(::Type{Chain{V,G,T,X}}) where {V,G,T<:Chain,X} = Chain{V,G,T}(ones.(ntuple(n->T,mdims(V))))
+⊗(a::Type{<:Chain{V}},b::Type{<:Chain{W}}) where {V,W} = Chain{V,1,Chain{W,1,Float64,mdims(W)},mdims(V)}
+⊗(a::Type{<:Chain{V,1}},b::Type{<:Chain{W,1}}) where {V,W} = Chain{V,1,Chain{W,1,Float64,mdims(W)},mdims(V)}
+⊗(a::Type{<:Chain{V,1}},b::Type{<:Chain{W,1,T}}) where {V,W,T} = Chain{V,1,Chain{W,1,T,mdims(W)},mdims(V)}
+
 """
     ChainBundle{V,G,P} <: Manifold{V} <: TensorAlgebra{V}
 
@@ -156,7 +164,7 @@ end
 @pure bundle(::ChainBundle{V,G,T,P} where {V,G,T}) where P = P
 @pure deletebundle!(V) = deletebundle!(bundle(V))
 @pure function deletebundle!(P::Int)
-    bundle_cache[P] = [Chain{ℝ^0,0,Int}(SVector(0))]
+    bundle_cache[P] = [Chain{ℝ^0,0,Int}(Values(0))]
 end
 @pure isbundle(::ChainBundle) = true
 @pure isbundle(t) = false
@@ -204,7 +212,7 @@ Base.show(io::IO,m::ChainBundle) = print(io,showbundle(m),length(m))
 ## MultiVector{V,𝕂}
 
 @computed struct MultiVector{V,T} <: TensorMixed{V}
-    v::SVector{1<<mdims(V),T}
+    v::Values{1<<mdims(V),T}
 end
 
 """
@@ -324,7 +332,7 @@ end
 
 SparseChain{V,G}(v::SparseVector{T,Int}) where {V,G,T} = SparseChain{V,G,T}(v)
 
-for Vec ∈ (:(SVector{L,T}),:(SubArray{T,1,SVector{L,T}}))
+for Vec ∈ (:(Values{L,T}),:(SubArray{T,1,Values{L,T}}))
     @eval function chainvalues(V::Manifold{N},m::$Vec,::Val{G}) where {N,G,L,T}
         bng = binomial(N,G)
         G∉(0,N) && sum(m .== 0)/bng < fill_limit && (return Chain{V,G,T}(m))
@@ -375,7 +383,7 @@ end
 ## MultiGrade{V,G}
 
 @computed struct MultiGrade{V,G} <: TensorMixed{V}
-    v::SVector{count_ones(G),TensorGraded{V}}
+    v::Values{count_ones(G),TensorGraded{V}}
 end
 
 @doc """
@@ -387,7 +395,7 @@ Sparse multivector type with pseudoscalar `V::Manifold` and grade encoding `G::U
 terms(v::MultiGrade) = v.v
 value(v::MultiGrade) = reduce(vcat,value.(terms(v)))
 
-MultiGrade{V}(v::Vector{T}) where T<:TensorGraded{V} where V = MultiGrade{V,|(UInt(1).<<rank.(v)...)}(SVector(v...))
+MultiGrade{V}(v::Vector{T}) where T<:TensorGraded{V} where V = MultiGrade{V,|(UInt(1).<<rank.(v)...)}(Values(v...))
 MultiGrade(v::Vector{T}) where T<:TensorGraded{V} where V = MultiGrade{V}(v)
 MultiGrade(m::T) where T<:TensorAlgebra = m
 MultiGrade(m::Chain{T,V,G}) where {T,V,G} = chainvalues(V,value(m),Val{G}())
@@ -395,13 +403,13 @@ MultiGrade(m::Chain{T,V,G}) where {T,V,G} = chainvalues(V,value(m),Val{G}())
 function MultiGrade(m::MultiVector{V,T}) where {V,T}
     N = mdims(V)
     sum(m.v .== 0)/(1<<N) < fill_limit && (return m)
-    out = zeros(SizedArray{Tuple{N+1},TensorGraded{V},1,1})
+    out = zeros(FixedVector{N,TensorGraded{V}})
     G = zero(UInt)
     for i ∈ 0:N
         @inbounds !prod(m[i].==0) && (G|=UInt(1)<<i;out[i+1]=chainvalues(V,m[i],Val{i}()))
     end
     cG = count_ones(G)
-    return cG≠1 ? MultiGrade{V,G}(SVector{cG,T}(out[indices(G,N+1)]...)) : out[1]
+    return cG≠1 ? MultiGrade{V,G}(Values{cG,T}(out[indices(G,N+1)]...)) : out[1]
 end
 
 function show(io::IO, m::MultiGrade{V,G}) where {V,G}
@@ -474,19 +482,19 @@ Base.isapprox(a::S,b::T) where {S<:MultiVector,T<:MultiVector} = Manifold(a)==Ma
 
 Leibniz.gdims(t::Tuple{Vector{<:Chain},Vector{Int}}) = gdims(t[1][findall(x->!iszero(x),t[2])])
 function Leibniz.gdims(t::Vector{<:Chain})
-    out = zeros(MVector{mdims(Manifold(points(t)))+1,Int})
+    out = zeros(Variables{mdims(Manifold(points(t)))+1,Int})
     out[mdims(Manifold(t))+1] = length(t)
     return out
 end
-function Leibniz.gdims(t::SVector{N,<:Vector}) where N
-    out = zeros(MVector{mdims(points(t[1]))+1,Int})
+function Leibniz.gdims(t::Values{N,<:Vector}) where N
+    out = zeros(Variables{mdims(points(t[1]))+1,Int})
     for i ∈ 1:N
         out[mdims(Manifold(t[i]))+1] = length(t[i])
     end
     return out
 end
-function Leibniz.gdims(t::SVector{N,<:Tuple}) where N
-    out = zeros(MVector{mdims(points(t[1][1]))+1,Int})
+function Leibniz.gdims(t::Values{N,<:Tuple}) where N
+    out = zeros(Variables{mdims(points(t[1][1]))+1,Int})
     for i ∈ 1:N
         out[mdims(Manifold(t[i][1]))+1] = length(t[i][1])
     end
@@ -494,7 +502,7 @@ function Leibniz.gdims(t::SVector{N,<:Tuple}) where N
 end
 function Leibniz.gdims(t::MultiVector{V}) where V
     N = mdims(V)
-    out = zeros(MVector{N+1,Int})
+    out = zeros(Variables{N+1,Int})
     bs = binomsum_set(N)
     for G ∈ 0:N
         ib = indexbasis(N,G)
@@ -505,8 +513,8 @@ function Leibniz.gdims(t::MultiVector{V}) where V
     return out
 end
 
-Leibniz.χ(t::SVector{N,<:Vector}) where N = (B=gdims(t);sum([B[t]*(-1)^t for t ∈ 1:length(B)]))
-Leibniz.χ(t::SVector{N,<:Tuple}) where N = (B=gdims(t);sum([B[t]*(-1)^t for t ∈ 1:length(B)]))
+Leibniz.χ(t::Values{N,<:Vector}) where N = (B=gdims(t);sum([B[t]*(-1)^t for t ∈ 1:length(B)]))
+Leibniz.χ(t::Values{N,<:Tuple}) where N = (B=gdims(t);sum([B[t]*(-1)^t for t ∈ 1:length(B)]))
 
 ## Adjoint
 
diff --git a/src/parity.jl b/src/parity.jl
index 1a2b341..4d0ef28 100644
--- a/src/parity.jl
+++ b/src/parity.jl
@@ -205,8 +205,8 @@ export odd, even, angular, radial, ₊, ₋, ǂ
 
 @pure signbit(V::T) where T<:Manifold{N} where N = (ib=indexbasis(N); parity.(Ref(V),ib,ib))
 @pure signbit(V::T,G) where T<:Manifold{N} where N = (ib=indexbasis(N,G); parity.(Ref(V),ib,ib))
-@pure angular(V::T) where T<:Manifold = SVector(findall(signbit(V))...)
-@pure radial(V::T) where T<:Manifold = SVector(findall(.!signbit(V))...)
+@pure angular(V::T) where T<:Manifold = Values(findall(signbit(V))...)
+@pure radial(V::T) where T<:Manifold = Values(findall(.!signbit(V))...)
 @pure angular(V::T,G) where T<:Manifold = findall(signbit(V,G))
 @pure radial(V::T,G) where T<:Manifold = findall(.!signbit(V,G))