Reference

Simplicial Complexes

Construction

Accessing

Hodge.verticesFunction
vertices(sc)

Return an array containg the vertices of the SimplicialComplex sc.

See also simplices. Notice that this function returns an array of integers while simplices(sc, 0) returns an array of singleton arrays containing the vertices.

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Hodge.simplicesFunction
simplices(sc[, k])

Return all simplices of the SimplicialComplex sc whose dimension equals k.

If the parameter k is not given, return all simplices of sc including the empty face.

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Hodge.numsimplicesFunction
numsimplices(sc[, k])

Return the number of k-dimensional simplices of sc.

If the parameter k is not given, return the total number of simplices including the empty face.

This function is a more efficient implementation of length ∘ simplices.

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Topological Operators

Cochains

Construction

Hodge.CochainType
Cochain{R, n}

Represent the n-th group $C^n(K; R)$ of cochains over the ring R whose basespace is the simplicial complex $K$.

For constructing Cochains, see also the methods zero_cochain and indicator_cochain.

The elements of this type may be seem as functions from the n-simplices of K to R or as skew-symmetric n-tensors over the vertices of K. This second perspective follows the ideas from the paper:

  • Jiang, X., Lim, L., Yao, Y. et al. Statistical ranking and combinatorial Hodge theory. Math. Program. 127, 203–244 (2011). https://doi.org/10.1007/s10107-010-0419-x
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Hodge.zero_cochainFunction
zero_cochain(R, K, n)

Construct an identically zero n-cochain over R and whose base space is K.

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Hodge.indicator_cochainFunction
indicator_cochain(R, K, σ)

Return the indicator function f of the n-simplex σ as a Cochain. That is, a cochain such that f(σ) = 1 and f(τ) = 0 for all other n-simplices of K.

Notice that, nevertheless, f is still skew-symmetric over permutations of σs indices.

If Κ does not contain σ, the returned cochain is identically zero.

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Operators

Hodge.basespaceFunction
basespace(ω::Cochain)

Return the simplicial complex that ω is associated with.

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Hodge.normFunction
norm(ω[, p])

Calculate the p-norm of the Cochain ω.

By default, p=2.

Return a floating point, no matter the base ring of ω.

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Hodge.innerFunction
inner(ω, ξ)

Usual inner product between Cochains.

Warning

For complex Cochains, the conjugation is taken on the first entry.

This inner product sees a n-cochain as a free vector space over the (non-oriented) n-simplices of their base space. Formally,

\[\sum_{\sigma \in \mathrm{simplices}(K,n)} \overline{f(σ)} g(σ).\]

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Hodge.coboundaryFunction
coboundary(ω)

The coboundary or discrete exterior derivative of a Cochain.

The coboundary of $ω$ applied to a simplex $σ$ equals the alternating sum of $ω$ applied to the faces of $σ$.

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Hodge.hodgeFunction
hodge(ω)

Hodge decomposition of a Cochain using inner as the inner product.

Return a tuple (α,β,γ) such that

ω == coboundary(α) + coboundary_adj(β) + γ
laplacian(γ) == 0
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