Year: 2008
Type: supplementary-materials
Abstract: Here we present a brief exposition of some material from algebraic topology that we use in our methods. We include it for completeness, as it may not be familiar for many readers. In particular, we define simplicial complexes, simplicial homology groups, and state the theorem cited in the Results section. See [Bott and Tu, 1982, Ewald, 1996, Hatcher, 2002] for more details. 1 Preliminary definitions Convexity. A set C ⊂ R n is called convex if, for all x, y ∈ C, the straight line segment between x and y is contained in C: {tx + (1 − t)y | 0 ≤ t ≤ 1} ⊆ C. Convex hull. The convex hull of a set of points x1,..., xr ∈ R n is the set of points conv({x1,..., xr}): = { r∑ tixi | ∑ ti = 1 and ti ≥ 0 for all i}. i=1
Source: Lincoln (University of Nebraska)
Authors: Carina Curto, Vladimir Itskov
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Stanford Bio-X
OpenAlex ID: W2100789261