Showing papers by "Herbert Edelsbrunner published in 2021"
TL;DR: In this article, a method based on k-fold covers was proposed to distinguish between two common representations of close packings of identical spheres consisting of hexagonal layers, called Barlow stackings, appear abundantly in minerals and metals.
Abstract: Two common representations of close packings of identical spheres consisting of hexagonal layers, called Barlow stackings, appear abundantly in minerals and metals. These motifs, however, occupy an identical portion of space and bear identical first-order topological signatures as measured by persistent homology. Here we present a novel method based on k-fold covers that unambiguously distinguishes between these patterns. Moreover, our approach provides topological evidence that the FCC motif is the more stable of the two in the context of evolving experimental sphere packings during the transition from disordered to an ordered state. We conclude that our approach can be generalised to distinguish between various Barlow stackings manifested in minerals and metals.
3 citations
TL;DR: In this paper, a combinatorial analog of the classic Helmholz-Hodge decomposition of a smooth vector field is presented, which can be computed by a matrix reduction algorithm.
Abstract: Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholz-Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, $K$, and every dimension, $p$, there is a partition of the set of $p$-cells into a maximal $p$-tree, a maximal $p$-cotree, and a collection of $p$-cells whose cardinality is the $p$-th Betti number of $K$. Given an ordering of the $p$-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.
3 citations
TL;DR: In this paper, the authors model a crystal as a periodic point set and present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties, and prove invariance under isometries, continuity and completeness in the generic case.
Abstract: Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.
3 citations
TL;DR: In this paper, the persistence diagrams of the k-fold cover of X and r are computed in terms of the number of points of X within distance of r. While standard methods suffice for the filtration in scale, they need novel geometric and topological concepts for the filtering in depth, such as a rhomboid tiling and a zigzag module of Delaunay mosaics.
Abstract: Given a locally finite $$X \subseteq {{{\mathbb {R}}}}^d$$
and a radius $$r \ge 0$$
, the k-fold cover of X and r consists of all points in $${{{\mathbb {R}}}}^d$$
that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in
$${{{\mathbb {R}}}}^{d+1}$$
whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.
2 citations
07 Jun 2021
TL;DR: In this paper, the authors model a crystal as a periodic point set and present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties, and prove invariance under isometries, continuity and completeness in the generic case.
Abstract: Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.
2 citations
TL;DR: In this article, the authors considered the dual of the order-k Voronoi tessellation and proved that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps.
Abstract: Given a locally finite set $$X \subseteq {{\mathbb {R}}}^d$$
and an integer $$k \ge 0$$
, we consider the function $${\mathbf{w}_{k}^{}} :{\mathrm{Del}_{k}{({X})}} \rightarrow {{\mathbb {R}}}$$
on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.
1 citations
TL;DR: In this article, the authors consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process, and study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point.
Abstract: Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.