| --- |
| pipeline_tag: text-classification |
| widget: |
| - text: "Dijkstra's algorithm is an algorithm for finding the paths between nodes in a weighted graph" |
| example_title: "Djikstra" |
| - text: "Proposition 1. The sum of the differential weight d on the Dowker complex D(X, Y, R) is the number of elements of Y ." |
| example_title: "Proposition" |
| - text: "Proof. This follows from showing that t is order-reversing in the following way: if σ ⊆ τ, then t(σ) ≥ t(τ)." |
| example_title: "Proof 1" |
| - text: "Theorem 1. Given the Dowker complex D(X, Y, R) and differential weight d, one can reconstruct R up to a bijection on Y ." |
| example_title: "Theorem" |
| - text: "The Dowker complex D(X,Y,R) is a functor between an appropriately con- structed category of relations and the category of abstract simplicial complexes. We prove this fact in Theorem 3 along with a few other observations." |
| example_title: "Basic" |
| - text: "Lemma 1. Let f : X → Y be a simplicial map. For every pair of simplices σ, τ of X satisfying σ ⊆ τ, their images in Y satisfy f(σ) ⊆ f(τ)." |
| example_title: "Theorem" |
| - text: "Since f is a simplicial map, then f(σ) is a simplex of Y and so is f(τ). If σ ⊆ τ, then every vertex v of σ is also a vertex of τ. By the definition of simplicial maps, f(v) is a vertex of both f(σ) and f(τ). Conversely, every vertex of f(σ) is the image of some vertex w of σ. □" |
| example_title: "Proof without keyword" |
| - text: "The Dowker complex defined in Definition 2 is a covariant functor D : Rel → Asc." |
| example_title: "th without keyword" |
| --- |
