created projective unsplitter norm and tidelift enterprise referral

.github/FUNDING.yml

@@ -4,6 +4,6 @@ github: [chakravala]
 patreon: dreamscatter
 open_collective: # Replace with a single Open Collective username
 ko_fi: # Replace with a single Ko-fi username
-tidelift: julia/Grassmann
+tidelift: "julia/Grassmann"
 community_bridge: # Replace with a single Community Bridge project-name e.g., cloud-foundry
 custom: https://liberapay.com/chakravala

Project.toml

@@ -14,18 +14,13 @@ LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Reduce = "93e0c654-6965-5f22-aba9-9c1ae6b3c259"
 Requires = "ae029012-a4dd-5104-9daa-d747884805df"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
+SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 
 [compat]
 julia = "1"
 Reduce = "1.2"
 DirectSum = "0.4"
 AbstractTensors = "0.3"
-AbstractLattices = ">=0"
-Combinatorics = ">=0"
-ComputedFieldTypes = ">=0"
-Leibniz = ">=0"
-Requires = ">=0"
-StaticArrays = ">=0"
 
 [extras]
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

README.md

@@ -26,6 +26,7 @@ Additionally, the universal interoperability between different sub-algebras is e
 
   * [Design, code generation](#design-code-generation)
 	 * [Requirements](#requirements)
+	 * [Grassmann for enterprise](#grassmann-for-enterprise)
   * [Direct-sum yields VectorBundle parametric type polymorphism ⨁](#direct-sum-yields-vectorspace-parametric-type-polymorphism-)
 	 * [Interoperability for TensorAlgebra{V}](#interoperability-for-tensoralgebrav)
   * [Generating elements and geometric algebra Λ(V)](#generating-elements-and-geometric-algebra-λv)
@@ -57,7 +58,7 @@ It can be used to work with automatic differentiation and differential geometry,
 
 ```Julia
 using Grassmann, Makie; @basis S"∞+++"
-streamplot(vectorfield(exp((π/4)*(v12+v∞3),V(2,3,4),V(1,2,3)),-1.5..1.5,-1.5..1.5,-1.5..1.5,gridsize=(10,10))
+streamplot(vectorfield(exp((π/4)*(v12+v∞3)),V(2,3,4),V(1,2,3)),-1.5..1.5,-1.5..1.5,-1.5..1.5,gridsize=(10,10))
 ```
 ![paper/img/wave.png](paper/img/wave.png)
 
@@ -89,6 +90,12 @@ The package is compatible via [Requires.jl](https://github.com/MikeInnes/Require
 [GeometryTypes,jl](https://github.com/JuliaGeometry/GeometryTypes.jl),
 [Makie.jl](https://github.com/JuliaPlots/Makie.jl).
 
+## Grassmann for enterprise
+
+Available as part of the Tidelift Subscription
+
+The maintainers of Grassmann and thousands of other packages are working with Tidelift to deliver commercial support and maintenance for the open source dependencies you use to build your applications. Save time, reduce risk, and improve code health, while paying the maintainers of the exact dependencies you use. [Learn more.](https://tidelift.com/subscription/pkg/julia-grassmann?utm_source=julia-grassmann&utm_medium=referral&utm_campaign=enterprise&utm_term=repo)
+
 ## Direct-sum yields `VectorBundle` parametric type polymorphism ⨁
 
 The *DirectSum.jl* package is a work in progress providing the necessary tools to work with an arbitrary `Manifold` specified by an encoding.
@@ -326,7 +333,7 @@ Due to [GeometryTypes,jl](https://github.com/JuliaGeometry/GeometryTypes.jl) `Po
 ```Julia
 using Grassmann, Makie
 basis"2" # Euclidean
-streamplot(vectorfield(v1*exp(π*v12/2)),-1.5..1.5,-1.5..1.5)
+streamplot(vectorfield(exp(π*v12/2)),-1.5..1.5,-1.5..1.5)
 streamplot(vectorfield(exp((π/2)*v12/2)),-1.5..1.5,-1.5..1.5)
 streamplot(vectorfield(exp((π/4)*v12/2)),-1.5..1.5,-1.5..1.5)
 streamplot(vectorfield(v1*exp((π/4)*v12/2)),-1.5..1.5,-1.5..1.5)
@@ -351,7 +358,7 @@ lines(points(f,V(3,4,5)))
 
 ```Julia
 using Grassmann, Makie; @basis S"∞+++"
-streamplot(vectorfield(exp((π/4)*(v12+v∞3),V(2,3,4)),-1.5..1.5,-1.5..1.5,-1.5..1.5,gridsize=(10,10))
+streamplot(vectorfield(exp((π/4)*(v12+v∞3)),V(2,3,4)),-1.5..1.5,-1.5..1.5,-1.5..1.5,gridsize=(10,10))
 ```
 ![paper/img/orb.png](paper/img/orb.png)
 

paper/paper.tex

@@ -376,11 +376,11 @@ \section{Leibniz operators and Grassmann's Hodge-DeRahm theory}
 	\end{align*}
 \end{theorem}
 \begin{proof}
-	$\partial\omega = \omega\cdot\nabla = \star(\star\omega\wedge\star^2\nabla) = (-1)^n(-1)^{nk}\star d\star\omega$.
+	$\partial\omega = \omega\cdot\nabla = \star\inv(\star\omega\wedge\star^2\nabla) = (-1)^n(-1)^{nk}\star d\star\omega$.
 
 
 	Then  substitute this into $\int_M \omega\wedge(-1)^{mk+m+1}\star\star d\star\eta = (-1)^{km+m+1}(-1)^{(m-k+1)(k-1)}\int_M\omega\wedge d\star\eta$,
-	with identity $(-1)^{km+m+1}(-1)^{(m-k+1)(k-1)}=(-1)^k$ and
+	apply the identity $(-1)^{km+m+1}(-1)^{(m-k+1)(k-1)}=(-1)^k$ and
 	$ (-1)^k\int_M\omega\wedge d\star\eta = \int_M d(\omega\wedge\star\eta) - (-1)^{k-1}\omega\wedge d\star\eta = \int_M d\omega\wedge\star\eta$.
 	Stokes identity can be proved by relying on a variant of the \textit{common factor theorem} by Browne \cite{browne}.
 \end{proof}
@@ -457,7 +457,7 @@ \section{Leibniz operators and Grassmann's Hodge-DeRahm theory}
 \begin{lstlisting}[language = Julia]
 using Grassmann, Makie
 basis"2" # Euclidean
-streamplot(vectorfield(v1*exp(π*v12/2)),-1.5..1.5,-1.5..1.5)
+streamplot(vectorfield(exp(π*v12/2)),-1.5..1.5,-1.5..1.5)
 streamplot(vectorfield(exp((π/2)*v12/2)),-1.5..1.5,-1.5..1.5)
 streamplot(vectorfield(exp((π/4)*v12/2)),-1.5..1.5,-1.5..1.5)
 streamplot(vectorfield(v1*exp((π/4)*v12/2)),-1.5..1.5,-1.5..1.5)
@@ -511,7 +511,7 @@ \section{Leibniz operators and Grassmann's Hodge-DeRahm theory}
 \end{align*}
 having dimensional equivalence brought by the Grassmann-Poincare-Hodge complement duality,
 \begin{align*}
-	\mc H^{n-p}M &\cong \frac{\ker(d\Omega^{n-p}M)}{\im{d\Omega^{n-p-1}M}}, & \dim\mc H^pM &= \dim\frac{\ker(\partial\Omega^pM)}{\im{\partial\Omega^{p+1}M}}
+	\mc H^{n-p}M &\cong \frac{\ker(d\Omega^{n-p}M)}{\im{d\Omega^{n-p+1}M}}, & \dim\mc H^pM &= \dim\frac{\ker(\partial\Omega^pM)}{\im{\partial\Omega^{p+1}M}}
 \end{align*}
 The rank of the grade $p$ boundary incidence operator is
 $$ \text{rank}\gen{\partial\gen{M}_{p+1}}_p = \min\set{\dim\gen{\partial\gen{M}_{p+1}}_p,\dim\gen{M}_{p+1}} $$
@@ -527,7 +527,7 @@ \section{Leibniz operators and Grassmann's Hodge-DeRahm theory}
 
 In Fig. 4, different possible discrete bivector topologies in a projective Riemann sphere setting are examined. 
 The figures are based on the product topology of two rotation bivectors.
-When the Euclidean $\mb R^4$ basis is combined with projective geometric algebra, resulting one parameter Lie groups can be visualized in the form of a torus in $\mb R^3$. 
+When the Euclidean $\mb R^4$ basis is combined with projective geometric algebra, resulting one parameter Lie groups can be visualized as a fibration of a torus in $\mb R^3$. 
 When the fourth $v_\infty$ basis direction is rotated into the Minkowski plane, the double rotation becomes a helix with translational single rotation. In examples (b)$\sim$(c) the bivector is modulated.
 
 \begin{figure}[t]

src/Grassmann.jl

@@ -3,9 +3,9 @@ module Grassmann
 #   This file is part of Grassmann.jl. It is licensed under the GPL license
 #   Grassmann Copyright (C) 2019 Michael Reed
 
-using Combinatorics, StaticArrays, Requires
+using Combinatorics, StaticArrays, SparseArrays
 using ComputedFieldTypes, AbstractLattices
-using DirectSum, AbstractTensors
+using DirectSum, AbstractTensors, Requires
 
 export vectorspace, ⊕, ℝ, @V_str, @S_str, @D_str, Signature,DiagonalForm,SubManifold, value
 import DirectSum: hasinf, hasorigin, mixedmode, dual, value, vectorspace, V0, ⊕, pre, vsn

src/multivectors.jl

@@ -828,6 +828,61 @@ Base.copysign(x::Simplex{V,G,B,T},y::Simplex{V,G,B,T}) where {V,G,B,T} = Simplex
     return A
 end
 
-# genfun
+# Euclidean norm (unsplitter)
+
+unsplitstart(g) = 1|((UInt(1)<<(g-1)-1)<<2)
+unsplitend(g) = (UInt(1)<<g-1)<<2
+
+const unsplitter_cache = SparseMatrixCSC{Float64,Int64}[]
+@pure unsplitter_calc(n) = (n2=Int(n/2);sparse(1:n2,1:n2,1,n,n)+sparse(1:n2,(n2+1):n,-1/2,n,n)+sparse((n2+1):n,(n2+1):n,1/2,n,n)+sparse((n2+1):n,1:n2,1,n,n))
+@pure function unsplitter(n::Int)
+    n2 = Int(n/2)
+    for k ∈ length(unsplitter_cache)+1:n2
+        push!(unsplitter_cache,unsplitter_calc(2k))
+    end
+    @inbounds unsplitter_cache[n2]
+end
+@pure unsplitter(n,g) = unsplitter(bladeindex(n,unsplitend(g))-bladeindex(n,unsplitstart(g)))
+
+for implex ∈ (MSB...,Basis)
+    @eval begin
+        #norm(t::$implex) = norm(unsplitval(t))
+        function unsplitvalue(a::$implex{V,G}) where {V,G}
+            !(hasinf(V) && hasorigin(V)) && (return value(a))
+            #T = valuetype(a)
+            #$(insert_expr((:N,:t,:out),:mvec,:T,:(typeof((one(T)/(2one(T))))))...)
+            #out = copy(value(a,t))
+            return unsplitvalue(MChain(a))
+        end
+    end
+end
 
+for Chain ∈ MSC
+    @eval begin
+        #norm(t::$Chain) = norm(unsplitval(t))
+        function unsplitvalue(a::$Chain{T,V,G}) where {T,V,G}
+            !(hasinf(V) && hasorigin(V)) && (return value(a))
+            $(insert_expr((:N,:t,:out),:mvec,:T,:(typeof((one(T)/(2one(T))))))...)
+            out = copy(value(a,mvec(N,G,t)))
+            bi = bladeindex(N,unsplitstart(G)):bladeindex(N,unsplitend(G))-1
+            out[bi] = unsplitter(N,G)*out[bi]
+            return out
+        end
+    end
+end
 
+@eval begin
+    #norm(t::MultiVector) = norm(unsplitval(t))
+    function unsplitvalue(a::MultiVector{T,V}) where {T,V}
+        !(hasinf(V) && hasorigin(V)) && (return value(a))
+        $(insert_expr((:N,:t,:out),:mvec,:T,:(typeof((one(T)/(2one(T))))))...)
+        out = copy(value(a,mvec(N,t)))
+        for G ∈ 1:N-1
+            bi = basisindex(N,unsplitstart(G)):basisindex(N,unsplitend(G))-1
+            out[bi] = unsplitter(N,G)*out[bi]
+        end
+        return out
+    end
+end
+
+# genfun