Grassmann.jl Library

Bivector{V,T} <: TensorGraded{V,2,T}

Graded bivector elements of a Manifold instance V with scalar field T.

GradedVector{V,T} <: TensorGraded{V,1,T}

Graded vector elements of a Manifold instance V with scalar field T.

Manifold{V,T} <: TensorAlgebra{V,T}

Basis parameter locally homeomorphic to V::Submanifold{M} T-module product topology.

Scalar{V,T} <: TensorGraded{V,0,T}

Graded scalar elements of a Manifold instance V with scalar field T.

TensorAlgebra{V,T} <: Number

Universal root tensor type with Manifold instance V with scalar field T.

TensorGraded{V,G,T} <: Manifold{V,T} <: TensorAlgebra

Grade G elements of a Manifold instance V with scalar field T.

TensorMixed{V,T} <: TensorAlgebra{V,T}

Elements of Manifold instance V having non-homogenous grade with scalar field T.

TensorTerm{V,G,T} <: TensorGraded{V,G,T}

Single coefficient for grade G of a Manifold instance V with scalar field T.

Trivector{V,T} <: TensorGraded{V,3,T}

Graded trivector elements of a Manifold instance V with scalar field T.

antisandwich(x::TensorAlgebra,R::TensorAlgebra)

Defined as complementleft(complementright(R)>>>complementright(x)).

bivector(::TensorAlgebra)

Return the bivector (rank 2) part of any TensorAlgebra element.

coabs(t::TensorAlgebra)

Complemented abs defined as complementleft(abs(complementright(t))).

coabs2(t::TensorAlgebra)

Complemented abs2 defined as complementleft(abs2(complementright(t))).

cocbrt(t::TensorAlgebra)

Complemented cbrt defined as complementleft(cbrt(complementright(t))).

cocos(t::TensorAlgebra)

Complemented cos defined as complementleft(cos(complementright(t))).

cocosh(t::TensorAlgebra)

Complemented cosh defined as complementleft(cosh(complementright(t))).

coexp(t::TensorAlgebra)

Complemented exp defined as complementleft(exp(complementright(t))).

coinv(t::TensorAlgebra)

Complemented inv defined as complementleft(inv(complementright(t))).

colog(t::TensorAlgebra)

Complemented log defined as complementleft(log(complementright(t))).

cosandwich(x::TensorAlgebra,R::TensorAlgebra)

Defined as complementleft(sandwich(complementright(x),complementright(R))).

cosin(t::TensorAlgebra)

Complemented sin defined as complementleft(sin(complementright(t))).

cosinh(t::TensorAlgebra)

Complemented sinh defined as complementleft(sinh(complementright(t))).

cosqrt(t::TensorAlgebra)

Complemented sqrt defined as complementleft(sqrt(complementright(t))).

cotan(t::TensorAlgebra)

Complemented tan defined as complementleft(tan(complementright(t))).

cotanh(t::TensorAlgebra)

Complemented tanh defined as complementleft(tanh(complementright(t))).

counit(t::TensorAlgebra)

Pseudo-normalization defined as unitize(t) = t/value(coabs(t)).

gdims(t::TensorGraded{V,G})

Dimensionality of the grade G of V for that TensorAlgebra.

geomabs(t::TensorAlgebra)

Geometric norm defined as geomabs(t) = abs(t) + coabs(t).

isgraded(t) -> Bool

Test whether t is some subtype of TensorGraded.

ismanifold(t) -> Bool

Test whether t is some subtype of Manifold.

ismixed(t) -> Bool

Test whether t is some subtype of TensorMixed.

istensor(t) -> Bool

Test whether t is some subtype of TensorAlgebra.

isterm(t) -> Bool

Test whether t is some subtype of TensorTerm.

mdims(t::TensorAlgebra{V})

Dimensionality of the pseudoscalar V of that TensorAlgebra.

pseudoabs(t::TensorAlgebra)

Complemented abs defined as complementleft(abs(complementright(t))).

pseudoabs2(t::TensorAlgebra)

Complemented abs2 defined as complementleft(abs2(complementright(t))).

pseudocbrt(t::TensorAlgebra)

Complemented cbrt defined as complementleft(cbrt(complementright(t))).

pseudocos(t::TensorAlgebra)

Complemented cos defined as complementleft(cos(complementright(t))).

pseudocosh(t::TensorAlgebra)

Complemented cosh defined as complementleft(cosh(complementright(t))).

pseudoexp(t::TensorAlgebra)

Complemented exp defined as complementleft(exp(complementright(t))).

pseudoinv(t::TensorAlgebra)

Complemented inv defined as complementleft(inv(complementright(t))).

pseudolog(t::TensorAlgebra)

Complemented log defined as complementleft(log(complementright(t))).

pseudoscalar(::TensorAlgebra), volume(::TensorAlgebra)

Return the pseudoscalar (full rank) part of any TensorAlgebra element.

pseudosin(t::TensorAlgebra)

Complemented sin defined as complementleft(sin(complementright(t))).

pseudosinh(t::TensorAlgebra)

Complemented sinh defined as complementleft(sinh(complementright(t))).

pseudosqrt(t::TensorAlgebra)

Complemented sqrt defined as complementleft(sqrt(complementright(t))).

pseudotan(t::TensorAlgebra)

Complemented tan defined as complementleft(tan(complementright(t))).

pseudotanh(t::TensorAlgebra)

Complemented tanh defined as complementleft(tanh(complementright(t))).

scalar(::TensorAlgebra)

Return the scalar (rank 0) part of any TensorAlgebra element.

tdims(t::TensorAlgebra{V})

Dimensionality of the superalgebra of V for that TensorAlgebra.

trivector(::TensorAlgebra)

Return the trivector (rank 3) part of any TensorAlgebra element.

unit(t::Number)

Normalization defined as unit(t) = t/abs(t).

unitnorm(t::TensorAlgebra)

Geometric normalization defined as unitnorm(t) = t/norm(geomabs(t)).

value(::TensorAlgebra)

Returns the internal Values representation of a TensorAlgebra element.

valuetype(t::TensorAlgebra{V,T}) where {V,T} = T

Returns type of a TensorAlgebra element value's internal representation.

vector(::TensorAlgebra)

Return the vector (rank 1) part of any TensorAlgebra element.

rank(::Manifold{n})

Dimensionality n of the Manifold subspace representation.

@co fun(args...)

Use the macro @co to make a pseudoscalar complement variant of any functions:

julia> @co myfun(x)
comyfun (generic function with 1 method)

Now comyfun(x) = complementleft(myfun(complementright(x))) is defined.

julia> @co myproduct(a,b)
comyproduct (generic function with 1 method)

Now comyproduct(a,b) = complementleft(myproduct(!a,!b)) is defined.

@pseudo fun(args...)

Use the macro @pseudo to make a pseudoscalar complement variant of any functions:

julia> @pseudo myfun(x)
pseudomyfun (generic function with 1 method)

Now pseudomyfun(x) = complementleft(myfun(complementright(x))) is defined.

julia> @pseudo myproduct(a,b)
pseudomyproduct (generic function with 1 method)

Now pseudomyproduct(a,b) = complementleft(myproduct(!a,!b)) is defined.

DirectSum.Basis{V} <: SubAlgebra{V} <: TensorAlgebra{V}

Grassmann basis container with cache of Submanifold elements and their Symbol names.

DirectSum.ExtendedBasis{V} <: SubAlgebra{V} <: TensorAlgebra{V}

Grassmann basis container without a dedicated Submanifold cache (only lazy caching).

Infinity{V} <: TensorGraded{V,0} <: TensorAlgebra{V}

Infinite quantity Infinity of the Grassmann algebra over V.

One{V} <: TensorGraded{V,0} <: TensorAlgebra{V}

Unit quantity One of the Grassmann algebra over V.

Single{V,G,B,T} <: TensorTerm{V,G,T} <: TensorGraded{V,G,T}

Single type with pseudoscalar V::Manifold, grade/rank G::Int, B::Submanifold{V,G}, field T::Type.

DirectSum.SparseBasis{V} <: SubAlgebra{V} <: TensorAlgebra{V}

Grassmann basis with sparse cache of Submanifold{G,V} elements and their Symbol names.

Submanifold{V,G,B} <: TensorGraded{V,G} <: Manifold{G}

Basis type with pseudoscalar V::Manifold, grade/rank G::Int, bits B::UInt64.

TensorBundle{n,ℙ,g,ν,μ} <: Manifold{n}

A manifold representing the space of tensors with a given metric and basis for projection.

Let n be the rank of a Manifold{n}. The type TensorBundle{n,ℙ,g,ν,μ} uses byte-encoded data available at pre-compilation, where ℙ specifies the basis for up and down projection, g is a bilinear form that specifies the metric of the space, and μ is an integer specifying the order of the tangent bundle (i.e. multiplicity limit of Leibniz-Taylor monomials). Lastly, ν is the number of tangent variables.

Zero{V} <: TensorGraded{V,0} <: TensorAlgebra{V}

Null quantity Zero of the Grassmann algebra over V.

clifford(ω::TensorAlgebra)

Clifford conjugate of an element: clifford(ω) = involute(reverse(ω))

involute(ω::TensorAlgebra)

Involute of an element: ~ω = (-1)^grade(ω)*ω

reverse(ω::TensorAlgebra)

Reverse of an element: ~ω = (-1)^(grade(ω)(grade(ω)-1)/2)ω

~(ω::TensorAlgebra)

Reverse of an element: ~ω = (-1)^(grade(ω)(grade(ω)-1)/2)ω

imag(ω::TensorAlgebra)

The imag part (ω-(~ω))/2 is defined by abs2(imag(ω)) == -(imag(ω)^2).

real(ω::TensorAlgebra)

The real part (ω+(~ω))/2 is defined by abs2(real(ω)) == real(ω)^2.

alloc(V::Manifold,:V,"v","w","∂","ϵ")

Generates Basis declaration having Manifold specified by V. The first argument provides pseudoscalar specifications, the second argument is the variable name for the Manifold, and the third and fourth argument are variable prefixes of the Submanifold vector names (and covector basis names).

getbasis(V::Manifold,v)

Fetch a specific Submanifold{G,V} element from an optimal SubAlgebra{V} selection.

nameindex(V::Int) -> Int

Returns a default name index for an integer input.

nameindex(a::NTuple{4, String}) -> Int

Get a unique index for a set of name options, based on a cached result.

nameindex(V::T) -> Int

Returns the name index for V when V is a TensorBundle or Manifold.

namelist(V) -> String

Get the cached name list for the options specified in V.

pseudoclifford(ω::TensorAlgebra)

Pseudo-clifford conjugate element: pseudoclifford(ω) = pseudoinvolute(pseudoreverse(ω))

pseudoinvolute(ω::TensorAlgebra)

Anti-involute of an element: ~ω = (-1)^pseudograde(ω)*ω

pseudoreverse(ω::TensorAlgebra)

Anti-reverse of an element: ~ω = (-1)^(pseudograde(ω)(pseudograde(ω)-1)/2)ω

tensorhash(d, o; c=0, C=0) -> Int

Compute a unique integer representation for a set of options.

Arguments:

• d: Number of tangent variables (corresponds to ν).
• o: Order of Leibniz-Taylor monomials multiplicity limit (corresponds to μ).
• c: Coefficient used to specify the metric (corresponds to g). Defaults to 0.
• C: Additional coefficient. Defaults to 0.

Returns

A unique integer representation that can be used as a cache key.

@basis

Generates Submanifold elements having Manifold specified by V. As a result of this macro, all of the Submanifold{V,G} elements generated by that TensorBundle become available in the local workspace with the specified naming. The first argument provides pseudoscalar specifications, the second argument is the variable name for the Manifold, and the third and fourth argument are variable prefixes of the Submanifold vector names (and covector basis names). Default for @basis M is @basis M V v w ∂ ϵ.

@dualbasis

Generates Submanifold elements having Manifold specified by V'. As a result of this macro, all of the Submanifold{V',G} elements generated by that TensorBundle become available in the local workspace with the specified naming. The first argument provides pseudoscalar specifications, the second argument is the variable name for the dual Manifold, and the third and fourth argument are variable prefixes of the Submanifold covector names (and tensor field basis names). Default for @dualbasis M is @dualbasis M VV w ϵ.

@mixedbasis

Generates Submanifold elements having Manifold specified by V⊕V'. As a result of this macro, all of the Submanifold{V⊕V',G} elements generated by that TensorBundle become available in the local workspace with the specified naming. The first argument provides pseudoscalar specifications, the second argument is the variable name for the Manifold, and the third and fourth argument are variable prefixes of the Submanifold vector names (and covector basis names). Default for @mixedbasis M is @mixedbasis M V v w ∂ ϵ.

AbstractSpinor{V,T} <: TensorMixed{V,T} <: TensorAlgebra{V,T}

Elements of TensorAlgebra having non-homogenous grade being a spinor in the abstract.

Chain{V,G,T} <: TensorGraded{V,G,T} <: TensorAlgebra{V,T}

Chain type with pseudoscalar V::Manifold, grade/rank G::Int, scalar field T::Type.

ChainBundle{V,G,T,P} <: Manifold{V,T} <: TensorAlgebra{V,T}

Subsets of a bundle cross-section over a Manifold topology.

CoSpinor{V,T} <: AbstractSpinor{V,T} <: TensorAlgebra{V,T}

PsuedoSpinor (odd grade) type with pseudoscalar V::Manifold and scalar T::Type.

Couple{V,B,T} <: AbstractSpinor{V,T} <: TensorAlgebra{V,T}

Pair of values with V::Manifold, basis B::Submanifold, scalar field of T::Type.

Multivector{V,T} <: TensorMixed{V,T} <: TensorAlgebra{V,T}

Chain type with pseudoscalar V::Manifold and scalar field T::Type.

Phasor{V,B,T} <: AbstractSpinor{V,T} <: TensorAlgebra{V,T}

Magnitude/phase angle with V::Manifold, frequency B::Type, and amplitude T::Type.

PseudoCouple{V,B,T} <: AbstractSpinor{V,T} <: TensorAlgebra{V,T}

Pair of values with V::Manifold, basis B::Submanifold, pseudoscalar of T::Type.

Spinor{V,T} <: AbstractSpinor{V,T} <: TensorAlgebra{V,T}

Spinor (even grade) type with pseudoscalar V::Manifold and scalar field T::Type.

∧(ω::TensorAlgebra,η::TensorAlgebra)

Exterior product as defined by the anti-symmetric quotient Λ≡⊗/~

∨(ω::TensorAlgebra,η::TensorAlgebra)

Regressive product as defined by the DeMorgan's law: ∨(ω...) = ⋆⁻¹(∧(⋆.(ω)...))

∗(ω::TensorAlgebra,η::TensorAlgebra)

Reversed geometric product: ω∗η = (~ω)*η

⊘(ω::TensorAlgebra,η::TensorAlgebra)

General sandwich product: ω⊘η = reverse(η)⊖ω⊖involute(η)

For normalized even grade η it is ω⊘η = (~η)⊖ω⊖η

⊙(ω::TensorAlgebra,η::TensorAlgebra)

Symmetrization projection: ⊙(ω...) = ∑(∏(σ.(ω)...))/factorial(length(ω))

⊠(ω::TensorAlgebra,η::TensorAlgebra)

Anti-symmetrization projection: ⊠(ω...) = ∑(∏(πσ.(ω)...))/factorial(length(ω))

*(ω::TensorAlgebra,η::TensorAlgebra)

Geometric algebraic product: ω⊖η = (-1)ᵖdet(ω∩η)⊗(Λ(ω⊖η)∪L(ω⊕η))

contraction(ω::TensorAlgebra,η::TensorAlgebra)

Interior (right) contraction product: ω⋅η = ω∨⋆η

even(t)

Selects the even part (t+involute(t))/2 and is defined by even grade.

odd(t)

Selects the odd part (t-involute(t))/2 and is defined by odd grade.

∨(ω::TensorAlgebra,η::TensorAlgebra)

Regressive product as defined by the DeMorgan's law: ∨(ω...) = ⋆⁻¹(∧(⋆.(ω)...))

>>>(ω::TensorAlgebra,η::TensorAlgebra)

Traditional sandwich product: ω>>>η = ω⊖η⊖clifford(ω)

For normalized even grade η it is ω>>>η = ω⊖η⊖(~ω)

↑(ω::TensorAlgebra{V}) where V # project

Canonical up-project operation from the space V, based on either Euclidean projective geometry, or the Riemann sphere, or conformal geometric algebra, or potentially other future canonical specifications. Optional arguments expose lower-level building blocks: ↑(ω,b) — use an explicit projective basis element b, or in the CGA context ↑(ω,p,m) — parameterise the conformal split with point-like part p and Minkowski part m. See also ↓ for the inverse down-reject operation.

↓(ω::TensorAlgebra{V}) where V # reject

Canonical down-reject operation from the space V, based on either Euclidean projective geometry, or the Riemann sphere, or conformal geometric algebra, or potentially other future canonical specifications. Optional arguments expose lower-level building blocks: ↓(ω,b) — use an explicit projective basis element b, or in the CGA context ↓(ω,p,m) — parameterise the conformal split with point-like part p and Minkowski part m. See also ↑ for the inverse up-project operation.

betti(::TensorAlgebra)

Compute combinatoric Betti numbers based on the count_gdims and boundary_rank methods.

boundary_null(t)

Dimension of the nullspace (kernel) of the combinatorial boundary operator on t, which can be an element of the TensorAlgebra. For each grade k the function lists Values with entries dₖ₊₁ - rₖ, where the computation follows from d = count_gdims(t) and r = boundary_rank(t) utility methods. Intuitively, this measures how many k‑chains are boundaries of (k+1)‑chains and therefore vanish under the ∂ operation.

boundary_rank(t)

Returns the rank of the combinatorial boundary operator evaluated on t, which can be an element of the TensorAlgebra. Internally the function counts how many graded dimensions of t remain after applying the boundary operator ∂ on it, minimized to not exceed the original graded space. This results in a list of Values whose entries rₖ satisfy 0 ≤ rₖ ≤ dₖ computed with d = count_gdims(t) as utility methods.

cayley(V,op=*)

Compute the cayley table with op(a,b) for each Submanifold basis of V.

compactio(::Bool)

Toggles compact numbers for extended Grassmann elements (enabled by default)

↑(ω::TensorAlgebra{V}) where V # project

Canonical up-project operation from the space V, based on either Euclidean projective geometry, or the Riemann sphere, or conformal geometric algebra, or potentially other future canonical specifications. Optional arguments expose lower-level building blocks: ↑(ω,b) — use an explicit projective basis element b, or in the CGA context ↑(ω,p,m) — parameterise the conformal split with point-like part p and Minkowski part m. See also ↓ for the inverse down-reject operation.

↓(ω::TensorAlgebra{V}) where V # reject

Canonical down-reject operation from the space V, based on either Euclidean projective geometry, or the Riemann sphere, or conformal geometric algebra, or potentially other future canonical specifications. Optional arguments expose lower-level building blocks: ↓(ω,b) — use an explicit projective basis element b, or in the CGA context ↓(ω,p,m) — parameterise the conformal split with point-like part p and Minkowski part m. See also ↑ for the inverse up-project operation.

cross(ω::TensorAlgebra,η::TensorAlgebra)

Cross product: ω×η = ⋆(ω∧η)

dot(ω::TensorAlgebra,η::TensorAlgebra)

Interior (right) contraction product: ω⋅η = ω∨⋆η

Δ = ∇^2 = ∂ₖ²v # laplacian::Laplacian

Abstract laplacian as second-order Derivation{Bool,2} is Laplacian operator dispatch.

∇ = ∂ₖvₖ # nabla::Nabla

Abstract nabla as first-order Derivation{Bool,1} is Nabla operator dispatch.

Δ = ∇^2 = ∂ₖ²v # laplacian::Laplacian

Abstract laplacian as second-order Derivation{Bool,2} is Laplacian operator dispatch.

∇ = ∂ₖvₖ # nabla::Nabla

Abstract nabla as first-order Derivation{Bool,1} is Nabla operator dispatch.

Δ = ∇^2 = ∂ₖ²v # laplacian::Laplacian

Abstract laplacian as second-order Derivation{Bool,2} is Laplacian operator dispatch.

∇ = ∂ₖvₖ # nabla::Nabla

Abstract nabla as first-order Derivation{Bool,1} is Nabla operator dispatch.

complementleft(::TensorAlgebra)

Euclidean metric variant Grassmann left complement.

complementlefthodge(ω::TensorAlgebra)

Grassmann-Hodge left complement: ⋆'ω = I∗'ω

complementright(::TensorAlgebra)

Euclidean metric variant of Grassmann right complement.

complementrighthodge(ω::TensorAlgebra)

Grassmann-Hodge right complement: ⋆ω = ω∗I

bit2int(b::BitVector) -> UInt

Takes a BitVector and converts it to a corresponding unsigned integer representation.

digitsfast(b,N::Int) -> Values{N+1,Int}

Calculates the digits of a binary number b up to length N.

gdimsall(N) -> Values{N+1,Int}

Returns Values{N+1,Int} representing binomial coefficients from 0 to N.

gdimseven(N) -> Values

Returns Values representing binomial coefficients for even values from 0 to N.

gdimsodd(N) -> Values

Returns Values representing binomial coefficients for odd values from 1 to N.

getbasis(V::Manifold,v)

Fetch a specific SubManifold{G,V} element from an optimal SubAlgebra{V} selection.

indexbits(N,indices) -> BitVector

Converts a list of indices into a BitVector of length N, used to create an array where only the specified indices are set to true.

indices(b::UInt) -> Vector
indices(b::UInt,N::UInt) -> Vector

Computes the indices at which a binary number b has bits equal to 1. The N argument (optional) specifies the length of the binary representation.

intlog(M::Integer) -> Int

This function computes the integer 2-logarithm of an integer.

χ(::TensorAlgebra)

Compute the Euler characteristic χ = ∑ₚ(-1)ᵖbₚ.