https://github.com/chakravala/Leibniz.jl

Tensor algebra utility library
https://github.com/chakravala/Leibniz.jl

calculus derivation laplacian math tensor-algebra

Last synced: about 2 months ago
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Tensor algebra utility library

• Host: GitHub
• URL: https://github.com/chakravala/Leibniz.jl
• Owner: chakravala
• License: agpl-3.0
• Created: 2019-06-07T14:56:07.000Z (over 5 years ago)
• Default Branch: master
• Last Pushed: 2024-04-26T03:41:17.000Z (5 months ago)
• Last Synced: 2024-07-04T20:16:33.123Z (3 months ago)
• Topics: calculus, derivation, laplacian, math, tensor-algebra
• Language: Julia
• Homepage: https://grassmann.crucialflow.com
• Size: 110 KB
• Stars: 19
• Watchers: 4
• Forks: 5
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • Funding: .github/FUNDING.yml
  • License: LICENSE

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• awesome-sciml - chakravala/Leibniz.jl: Tensor algebra utility library

README

        
# Leibniz.jl

*Bit entanglements for tensor algebra derivations and hypergraphs*

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Although intended for compatibility use with the [Grassmann.jl](https://github.com/chakravala/Grassmann.jl) package for multivariable differential operators and tensor field operations, `Leibniz` can be used independently.

### Extended dual index printing with full alphanumeric characters #62'

To help provide a commonly shared and readable indexing to the user, some print methods are provided:

```julia

julia> Leibniz.printindices(stdout,Leibniz.indices(UInt(2^62-1)),false,"v")

v₁₂₃₄₅₆₇₈₉₀abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ

julia> Leibniz.printindices(stdout,Leibniz.indices(UInt(2^62-1)),false,"w")

w¹²³⁴⁵⁶⁷⁸⁹⁰ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz

```

An application of this is in `Grassmann` and `DirectSum`, where dual indexing is used.

# Derivation

Generates the tensor algebra of multivariable symmetric Leibniz differentials and interfaces `using Reduce, Grassmann` to provide the `∇,Δ` vector field operators, enabling  mixed-symmetry tensors with arbitrary multivariate `Grassmann` manifolds.

```Julia

julia> using Leibniz, Grassmann

julia> V = tangent(ℝ^3,4,3)

⟨+++⟩

julia> V(∇)

∂₁v₁ + ∂₂v₂ + ∂₃v₃

julia> V(∇^2)

0 + 1∂₁∂₁ + 1∂₂∂₂ + 1∂₃∂₃

julia> V(∇^3)

0 + 1∂₁∂₁∂₁v₁ + 1∂₂∂₂∂₂v₂ + 1∂₃∂₃∂₃v₃ + 1∂₂∂₁₂v₁ + 1∂₃∂₁₃v₁ + 1∂₁∂₁₂v₂ + 1∂₃∂₂₃v₂ + 1∂₁∂₁₃v₃ + 1∂₂∂₂₃v₃

julia> V(∇^4)

0.0 + 1∂₁∂₁∂₁∂₁ + 1∂₂∂₂∂₂∂₂ + 1∂₃∂₃∂₃∂₃ + 2∂₁₂∂₁₂ + 2∂₁₃∂₁₃ + 2∂₂₃∂₂₃

julia> ∇^2 == Δ

true

julia> ∇, Δ

(∂ₖvₖ, ∂ₖ²v)

```

This is an initial undocumented pre-release registration for testing with other packages.