Grassmann.jl Library

Manifold{V} <: TensorAlgebra{V}

Basis parametrization locally homeomorphic to ℝ^n product topology.

TV[ elements ]
TV{T}[ elements ]

Create Values literals using array construction syntax. The element type is inferred by promoting elements to a common type or set to T when T is provided explicitly.

Examples:

• TV[1.0, 2.0] creates a length-2 Values of Float64 elements.
• TV{Float32}[1, 2] creates a length-2 Values of Float32 elements.

A couple of helpful type aliases are also provided:

• TV_F64[1, 2] creates a lenght-2 Values of Float64 elements
• TV_F32[1, 2] creates a lenght-2 Values of Float32 elements

TensorAlgebra{V} <: Number

Universal root tensor type with Manifold instance parameter V.

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

Graded elements of a TensorAlgebra in a Manifold topology.

TensorMixed{V} <: TensorAlgebra{V}

Elements of TensorAlgebra having non-homogenous grade.

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

Terms of a TensorAlgebra having a single coefficient.

_InitialValue

A singleton type for representing "universal" initial value (identity element).

The idea is that, given op for mapfoldl, virtually, we define an "extended" version of it by

op′(::_InitialValue, x) = x
op′(acc, x) = op(acc, x)

This is just a conceptually useful model to have in mind and we don't actually define op′ here (yet?). But see Base.BottomRF for how it might work in action.

(It is related to that you can always turn a semigroup without an identity into a monoid by "adjoining" an element that acts as the identity.)

Return either the statically known Val() or runtime length()

bivector(::TensorAlgebra)

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

mdims(t::TensorAlgebra{V})

Dimensionality of the pseudoscalar V of that TensorAlgebra.

Returns the common Val of the inputs (or else throws a DimensionMismatch)

scalar(::TensorAlgebra)

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

trivector(::TensorAlgebra)

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

values(::TensorAlgebra)

Returns the internal Values representation of a TensorAlgebra element.

valuetype(t::TensorAlgebra)

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

vector(::TensorAlgebra)

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

pseudoscalar(::TensorAlgebra)

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

rank(::Manifold{n})

Dimensionality n of the Manifold subspace representation.

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).

Simplex{V,G,B,𝕂} <: TensorTerm{V,G} <: TensorGraded{V,G}

Simplex type with pseudoscalar V::Manifold, grade/rank G::Int, B::SubManifold{V,G}, field 𝕂::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}

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.

basis(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.

@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.

Chain{V,G,𝕂} <: TensorGraded{V,G}

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

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

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

MultiVector{V,𝕂} <: TensorMixed{V} <: TensorAlgebra{V}

Chain type with pseudoscalar V::Manifold and scalar field 𝕂::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 = involute(η)\ω⊖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))

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

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

∨(ω::TensorAlgebra,η::TensorAlgebra)

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

*(ω::TensorAlgebra,η::TensorAlgebra)

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

Sandwich product: ω>>>η = ω⊖η⊖(~ω)

betti(::TensorAlgebra)

Compute the Betti numbers.

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

Cross product: ω×η = ⋆(ω∧η)

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

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

clifford(ω::TensorAlgebra)

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

complementleft(::TensorAlgebra)

Non-metric variant Grassmann-Poincare left complement.

complementlefthodge(ω::TensorAlgebra)

Grassmann-Poincare left complement: ⋆'ω = I∗'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.

complementright(::TensorAlgebra)

Non-metric variant of Grassmann-Poincare-Hodge complement.

complementrighthodge(ω::TensorAlgebra)

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

getbasis(V::Manifold,v)

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

χ(::TensorAlgebra)

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