@article{7567,
abstract = {Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/d−−√) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of non-degeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d2). We conjecture that both bounds are optimal for triangulations in Euclidean space.},
author = {Choudhary, Aruni and Kachanovich, Siargey and Wintraecken, Mathijs},
issn = {1661-8289},
journal = {Mathematics in Computer Science},
pages = {141--176},
publisher = {Springer Nature},
title = {{Coxeter triangulations have good quality}},
doi = {10.1007/s11786-020-00461-5},
volume = {14},
year = {2020},
}
@unpublished{7568,
abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e.manifolds defined as the zero set of some multivariate multivalued functionf:Rd→Rd−n.A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear(PL) approximation based on a triangulationTof the ambient spaceRd. In this paper, we giveconditions under which the PL-approximation of an isomanifold is topologically equivalent to theisomanifold. The conditions can always be met by taking a sufficiently fine triangulationT.},
author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
booktitle = {EUROCG 2020},
pages = {8},
title = {{The topological correctness of the PL-approximation of isomanifolds}},
year = {2020},
}
@article{7666,
abstract = {Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–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 reduced 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.},
author = {Edelsbrunner, Herbert and Ölsböck, Katharina},
issn = {14320444},
journal = {Discrete and Computational Geometry},
pages = {759--775},
publisher = {Springer Nature},
title = {{Tri-partitions and bases of an ordered complex}},
doi = {10.1007/s00454-020-00188-x},
volume = {64},
year = {2020},
}
@article{7791,
abstract = {Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law.},
author = {Akopyan, Arseniy and Karasev, Roman},
issn = {21996768},
journal = {European Journal of Mathematics},
publisher = {Springer Nature},
title = {{When different norms lead to same billiard trajectories?}},
doi = {10.1007/s40879-020-00405-0},
year = {2020},
}
@article{7905,
abstract = {We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.},
author = {Brown, Adam and Wang, Bei},
issn = {0179-5376},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{Sheaf-theoretic stratification learning from geometric and topological perspectives}},
doi = {10.1007/s00454-020-00206-y},
year = {2020},
}