@article{8317,
abstract = {When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.},
author = {Aichholzer, Oswin and Akitaya, Hugo A. and Cheung, Kenneth C. and Demaine, Erik D. and Demaine, Martin L. and Fekete, Sándor P. and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane},
issn = {09257721},
journal = {Computational Geometry: Theory and Applications},
publisher = {Elsevier},
title = {{Folding polyominoes with holes into a cube}},
doi = {10.1016/j.comgeo.2020.101700},
volume = {93},
year = {2021},
}
@article{8773,
abstract = {Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.},
author = {Brown, Adam and Romanov, Anna},
issn = {1088-6826},
journal = {Proceedings of the American Mathematical Society},
keywords = {Applied Mathematics, General Mathematics},
number = {1},
pages = {37--52},
publisher = {American Mathematical Society},
title = {{Contravariant forms on Whittaker modules}},
doi = {10.1090/proc/15205},
volume = {149},
year = {2021},
}
@phdthesis{9056,
abstract = {In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations.
The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,
and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics.
We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration
function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss
the implications on persistence for periodic data sets.},
author = {Osang, Georg F},
issn = {2663-337X},
pages = {134},
publisher = {IST Austria},
title = {{Multi-cover persistence and Delaunay mosaics}},
doi = {10.15479/AT:ISTA:9056},
year = {2021},
}
@inproceedings{9253,
abstract = {In March 2020, the Austrian government introduced a widespread lock-down in response to the COVID-19 pandemic. Based on subjective impressions and anecdotal evidence, Austrian public and private life came to a sudden halt. Here we assess the effect of the lock-down quantitatively for all regions in Austria and present an analysis of daily changes of human mobility throughout Austria using near-real-time anonymized mobile phone data. We describe an efficient data aggregation pipeline and analyze the mobility by quantifying mobile-phone traffic at specific point of interests (POIs), analyzing individual trajectories and investigating the cluster structure of the origin-destination graph. We found a reduction of commuters at Viennese metro stations of over 80% and the number of devices with a radius of gyration of less than 500 m almost doubled. The results of studying crowd-movement behavior highlight considerable changes in the structure of mobility networks, revealed by a higher modularity and an increase from 12 to 20 detected communities. We demonstrate the relevance of mobility data for epidemiological studies by showing a significant correlation of the outflow from the town of Ischgl (an early COVID-19 hotspot) and the reported COVID-19 cases with an 8-day time lag. This research indicates that mobile phone usage data permits the moment-by-moment quantification of mobility behavior for a whole country. We emphasize the need to improve the availability of such data in anonymized form to empower rapid response to combat COVID-19 and future pandemics.},
author = {Heiler, Georg and Reisch, Tobias and Hurt, Jan and Forghani, Mohammad and Omani, Aida and Hanbury, Allan and Karimipour, Farid},
booktitle = {2020 IEEE International Conference on Big Data},
isbn = {9781728162515},
location = {Atlanta, GA, United States},
publisher = {IEEE},
title = {{Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic}},
doi = {10.1109/bigdata50022.2020.9378374},
year = {2021},
}
@inproceedings{9296,
abstract = { matching is compatible to two or more labeled point sets of size n with labels {1,…,n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.},
author = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit},
booktitle = {15th International Conference on Algorithms and Computation},
isbn = {9783030682101},
issn = {16113349},
location = {Virtual},
pages = {221--233},
publisher = {Springer Nature},
title = {{On compatible matchings}},
doi = {10.1007/978-3-030-68211-8_18},
volume = {12635},
year = {2021},
}
@article{9317,
abstract = {Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r consists of all points in Rd 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 Rd+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.},
author = {Edelsbrunner, Herbert and Osang, Georg F},
issn = {14320444},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{The multi-cover persistence of Euclidean balls}},
doi = {10.1007/s00454-021-00281-9},
year = {2021},
}
@article{7962,
abstract = {A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.},
author = {Pach, János and Reed, Bruce and Yuditsky, Yelena},
issn = {14320444},
journal = {Discrete and Computational Geometry},
number = {4},
pages = {888--917},
publisher = {Springer Nature},
title = {{Almost all string graphs are intersection graphs of plane convex sets}},
doi = {10.1007/s00454-020-00213-z},
volume = {63},
year = {2020},
}
@inproceedings{8135,
abstract = {Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton and Ölsböck, Katharina and Synak, Peter},
booktitle = {Topological Data Analysis},
isbn = {9783030434076},
issn = {21978549},
pages = {181--218},
publisher = {Springer Nature},
title = {{Radius functions on Poisson–Delaunay mosaics and related complexes experimentally}},
doi = {10.1007/978-3-030-43408-3_8},
volume = {15},
year = {2020},
}
@article{8163,
abstract = {Fejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.},
author = {Vegter, Gert and Wintraecken, Mathijs},
issn = {1588-2896},
journal = {Studia Scientiarum Mathematicarum Hungarica},
number = {2},
pages = {193--199},
publisher = {AKJournals},
title = {{Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes}},
doi = {10.1556/012.2020.57.2.1454},
volume = {57},
year = {2020},
}
@article{8248,
abstract = {We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.},
author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Lieutier, Andre and Wintraecken, Mathijs},
issn = {0179-5376},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{Local conditions for triangulating submanifolds of Euclidean space}},
doi = {10.1007/s00454-020-00233-9},
year = {2020},
}