[{"oa":1,"publisher":"Springer Nature","quality_controlled":"1","acknowledgement":"This work has been funded by the European Research Council under the European Union’s ERC Grant Agreement Number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). The third author also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. Open access funding provided by the Institute of Science and Technology (IST Austria).","page":"386-434","date_created":"2020-12-12T11:07:02Z","date_published":"2021-07-01T00:00:00Z","doi":"10.1007/s00454-020-00250-8","year":"2021","has_accepted_license":"1","isi":1,"publication":"Discrete & Computational Geometry","day":"01","project":[{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"external_id":{"isi":["000597770300001"]},"article_processing_charge":"Yes (via OA deal)","author":[{"last_name":"Boissonnat","full_name":"Boissonnat, Jean-Daniel","first_name":"Jean-Daniel"},{"full_name":"Kachanovich, Siargey","last_name":"Kachanovich","first_name":"Siargey"},{"orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"title":"Triangulating submanifolds: An elementary and quantified version of Whitney’s method","citation":{"ista":"Boissonnat J-D, Kachanovich S, Wintraecken M. 2021. Triangulating submanifolds: An elementary and quantified version of Whitney’s method. Discrete & Computational Geometry. 66(1), 386–434.","chicago":"Boissonnat, Jean-Daniel, Siargey Kachanovich, and Mathijs Wintraecken. “Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method.” Discrete & Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00250-8.","ieee":"J.-D. Boissonnat, S. Kachanovich, and M. Wintraecken, “Triangulating submanifolds: An elementary and quantified version of Whitney’s method,” Discrete & Computational Geometry, vol. 66, no. 1. Springer Nature, pp. 386–434, 2021.","short":"J.-D. Boissonnat, S. Kachanovich, M. Wintraecken, Discrete & Computational Geometry 66 (2021) 386–434.","apa":"Boissonnat, J.-D., Kachanovich, S., & Wintraecken, M. (2021). Triangulating submanifolds: An elementary and quantified version of Whitney’s method. Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00250-8","ama":"Boissonnat J-D, Kachanovich S, Wintraecken M. Triangulating submanifolds: An elementary and quantified version of Whitney’s method. Discrete & Computational Geometry. 2021;66(1):386-434. doi:10.1007/s00454-020-00250-8","mla":"Boissonnat, Jean-Daniel, et al. “Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method.” Discrete & Computational Geometry, vol. 66, no. 1, Springer Nature, 2021, pp. 386–434, doi:10.1007/s00454-020-00250-8."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","intvolume":" 66","month":"07","abstract":[{"lang":"eng","text":"We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric."}],"oa_version":"Published Version","ec_funded":1,"volume":66,"issue":"1","publication_status":"published","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"language":[{"iso":"eng"}],"file":[{"file_size":983307,"date_updated":"2021-08-06T09:52:29Z","creator":"kschuh","file_name":"2021_DescreteCompGeopmetry_Boissonnat.pdf","date_created":"2021-08-06T09:52:29Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"checksum":"c848986091e56699dc12de85adb1e39c","file_id":"9795"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_type":"original","type":"journal_article","keyword":["Theoretical Computer Science","Computational Theory and Mathematics","Geometry and Topology","Discrete Mathematics and Combinatorics"],"status":"public","_id":"8940","file_date_updated":"2021-08-06T09:52:29Z","department":[{"_id":"HeEd"}],"date_updated":"2023-09-05T15:02:40Z","ddc":["516"]},{"project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"citation":{"ista":"Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 5(1), 99–140.","chicago":"Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic Convergence and Stability of Random Mapper Graphs.” Journal of Applied and Computational Topology. Springer Nature, 2021. https://doi.org/10.1007/s41468-020-00063-x.","ama":"Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 2021;5(1):99-140. doi:10.1007/s41468-020-00063-x","apa":"Brown, A., Bobrowski, O., Munch, E., & Wang, B. (2021). Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-020-00063-x","ieee":"A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence and stability of random mapper graphs,” Journal of Applied and Computational Topology, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.","short":"A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational Topology 5 (2021) 99–140.","mla":"Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper Graphs.” Journal of Applied and Computational Topology, vol. 5, no. 1, Springer Nature, 2021, pp. 99–140, doi:10.1007/s41468-020-00063-x."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","external_id":{"arxiv":["1909.03488"]},"article_processing_charge":"Yes (via OA deal)","author":[{"last_name":"Brown","full_name":"Brown, Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam"},{"first_name":"Omer","full_name":"Bobrowski, Omer","last_name":"Bobrowski"},{"first_name":"Elizabeth","last_name":"Munch","full_name":"Munch, Elizabeth"},{"first_name":"Bei","last_name":"Wang","full_name":"Wang, Bei"}],"title":"Probabilistic convergence and stability of random mapper graphs","acknowledgement":"AB was supported in part by the European Union’s Horizon 2020 research and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No. 754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation, Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like to thank the Institute for Mathematics and its Applications for hosting a workshop titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen Access funding provided by Institute of Science and Technology (IST Austria).","oa":1,"quality_controlled":"1","publisher":"Springer Nature","year":"2021","has_accepted_license":"1","publication":"Journal of Applied and Computational Topology","day":"01","page":"99-140","date_created":"2021-02-11T14:41:02Z","doi":"10.1007/s41468-020-00063-x","date_published":"2021-03-01T00:00:00Z","_id":"9111","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_type":"original","type":"journal_article","status":"public","date_updated":"2023-09-05T15:37:56Z","ddc":["510"],"file_date_updated":"2021-02-11T14:43:59Z","department":[{"_id":"HeEd"}],"abstract":[{"lang":"eng","text":"We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data."}],"oa_version":"Published Version","scopus_import":"1","intvolume":" 5","month":"03","publication_status":"published","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"language":[{"iso":"eng"}],"file":[{"relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"file_id":"9112","checksum":"3f02e9d47c428484733da0f588a3c069","creator":"dernst","file_size":2090265,"date_updated":"2021-02-11T14:43:59Z","file_name":"2020_JourApplCompTopology_Brown.pdf","date_created":"2021-02-11T14:43:59Z"}],"ec_funded":1,"volume":5,"issue":"1"},{"publisher":"Institute of Science and Technology Austria","oa":1,"day":"01","has_accepted_license":"1","year":"2021","date_published":"2021-02-01T00:00:00Z","doi":"10.15479/AT:ISTA:9056","date_created":"2021-02-02T14:11:06Z","page":"134","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"mla":"Osang, Georg F. Multi-Cover Persistence and Delaunay Mosaics. Institute of Science and Technology Austria, 2021, doi:10.15479/AT:ISTA:9056.","short":"G.F. Osang, Multi-Cover Persistence and Delaunay Mosaics, Institute of Science and Technology Austria, 2021.","ieee":"G. F. Osang, “Multi-cover persistence and Delaunay mosaics,” Institute of Science and Technology Austria, Klosterneuburg, 2021.","apa":"Osang, G. F. (2021). Multi-cover persistence and Delaunay mosaics. Institute of Science and Technology Austria, Klosterneuburg. https://doi.org/10.15479/AT:ISTA:9056","ama":"Osang GF. Multi-cover persistence and Delaunay mosaics. 2021. doi:10.15479/AT:ISTA:9056","chicago":"Osang, Georg F. “Multi-Cover Persistence and Delaunay Mosaics.” Institute of Science and Technology Austria, 2021. https://doi.org/10.15479/AT:ISTA:9056.","ista":"Osang GF. 2021. Multi-cover persistence and Delaunay mosaics. Klosterneuburg: Institute of Science and Technology Austria."},"title":"Multi-cover persistence and Delaunay mosaics","author":[{"first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116"}],"article_processing_charge":"No","oa_version":"Published Version","abstract":[{"lang":"eng","text":"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,\r\nand 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\r\nfunction 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."}],"place":"Klosterneuburg","month":"02","alternative_title":["ISTA Thesis"],"file":[{"date_created":"2021-02-02T14:09:25Z","file_name":"thesis_source.zip","creator":"patrickd","date_updated":"2021-02-03T10:37:28Z","file_size":13446994,"checksum":"bcf27986147cab0533b6abadd74e7629","file_id":"9063","access_level":"closed","relation":"source_file","content_type":"application/zip"},{"success":1,"file_id":"9064","checksum":"9cc8af266579a464385bbe2aff6af606","relation":"main_file","access_level":"open_access","content_type":"application/pdf","file_name":"thesis_pdfA2b.pdf","date_created":"2021-02-02T14:09:18Z","creator":"patrickd","file_size":5210329,"date_updated":"2021-02-02T14:09:18Z"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["2663-337X"]},"publication_status":"published","degree_awarded":"PhD","related_material":{"record":[{"id":"187","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"8703","status":"public"}]},"_id":"9056","status":"public","type":"dissertation","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["006","514","516"],"supervisor":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"}],"date_updated":"2023-09-07T13:29:01Z","department":[{"_id":"HeEd"},{"_id":"GradSch"}],"file_date_updated":"2021-02-03T10:37:28Z"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"ista":"Osang GF, Edelsbrunner H, Saadatfar M. 2021. Topological signatures and stability of hexagonal close packing and Barlow stackings. Soft Matter. 17(40), 9107–9115.","chicago":"Osang, Georg F, Herbert Edelsbrunner, and Mohammad Saadatfar. “Topological Signatures and Stability of Hexagonal Close Packing and Barlow Stackings.” Soft Matter. Royal Society of Chemistry , 2021. https://doi.org/10.1039/d1sm00774b.","apa":"Osang, G. F., Edelsbrunner, H., & Saadatfar, M. (2021). Topological signatures and stability of hexagonal close packing and Barlow stackings. Soft Matter. Royal Society of Chemistry . https://doi.org/10.1039/d1sm00774b","ama":"Osang GF, Edelsbrunner H, Saadatfar M. Topological signatures and stability of hexagonal close packing and Barlow stackings. Soft Matter. 2021;17(40):9107-9115. doi:10.1039/d1sm00774b","short":"G.F. Osang, H. Edelsbrunner, M. Saadatfar, Soft Matter 17 (2021) 9107–9115.","ieee":"G. F. Osang, H. Edelsbrunner, and M. Saadatfar, “Topological signatures and stability of hexagonal close packing and Barlow stackings,” Soft Matter, vol. 17, no. 40. Royal Society of Chemistry , pp. 9107–9115, 2021.","mla":"Osang, Georg F., et al. “Topological Signatures and Stability of Hexagonal Close Packing and Barlow Stackings.” Soft Matter, vol. 17, no. 40, Royal Society of Chemistry , 2021, pp. 9107–15, doi:10.1039/d1sm00774b."},"title":"Topological signatures and stability of hexagonal close packing and Barlow stackings","article_processing_charge":"No","external_id":{"isi":["000700090000001"],"pmid":["34569592"]},"author":[{"orcid":"0000-0002-8882-5116","full_name":"Osang, Georg F","last_name":"Osang","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","first_name":"Georg F"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Mohammad","full_name":"Saadatfar, Mohammad","last_name":"Saadatfar"}],"project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"name":"The Wittgenstein Prize","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"publication":"Soft Matter","day":"20","year":"2021","has_accepted_license":"1","isi":1,"date_created":"2021-10-31T23:01:30Z","doi":"10.1039/d1sm00774b","date_published":"2021-10-20T00:00:00Z","page":"9107-9115","acknowledgement":"MS acknowledges the support by Australian Research Council funding through the ARC Training Centre for M3D Innovation (IC180100008). MS thanks M. Hanifpour and N. Francois for their input and valuable discussions. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, grant no. 788183 and from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","oa":1,"quality_controlled":"1","publisher":"Royal Society of Chemistry ","ddc":["540"],"date_updated":"2023-10-03T09:24:27Z","file_date_updated":"2023-10-03T09:21:42Z","department":[{"_id":"HeEd"}],"_id":"10204","status":"public","article_type":"original","type":"journal_article","language":[{"iso":"eng"}],"file":[{"creator":"dernst","file_size":4678788,"date_updated":"2023-10-03T09:21:42Z","file_name":"2021_SoftMatter_acceptedversion_Osang.pdf","date_created":"2023-10-03T09:21:42Z","relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"checksum":"b4da0c420530295e61b153960f6cb350","file_id":"14385"}],"publication_status":"published","publication_identifier":{"eissn":["1744-6848"],"issn":["1744-683X"]},"ec_funded":1,"volume":17,"issue":"40","pmid":1,"oa_version":"Submitted Version","abstract":[{"text":"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.","lang":"eng"}],"intvolume":" 17","month":"10","scopus_import":"1"},{"_id":"9605","type":"conference","conference":{"location":"Online","end_date":"2021-06-11","start_date":"2021-06-07","name":"SoCG: International Symposium on Computational Geometry"},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"status":"public","date_updated":"2023-10-04T12:03:39Z","ddc":["516"],"file_date_updated":"2021-06-28T12:40:47Z","department":[{"_id":"HeEd"}],"abstract":[{"lang":"eng","text":"Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness. "}],"oa_version":"Published Version","scopus_import":"1","alternative_title":["LIPIcs"],"month":"06","intvolume":" 189","publication_identifier":{"issn":["18688969"],"isbn":["9783959771849"]},"publication_status":"published","file":[{"date_created":"2021-06-28T12:40:47Z","file_name":"2021_LIPIcs_Corbet.pdf","creator":"cziletti","date_updated":"2021-06-28T12:40:47Z","file_size":"1367983","checksum":"0de217501e7ba8b267d58deed0d51761","file_id":"9610","success":1,"access_level":"open_access","relation":"main_file","content_type":"application/pdf"}],"language":[{"iso":"eng"}],"related_material":{"link":[{"url":"https://arxiv.org/abs/2103.07823","relation":"extended_version"}],"record":[{"status":"public","id":"12709","relation":"later_version"}]},"volume":189,"article_number":"27","citation":{"ista":"Corbet R, Kerber M, Lesnick M, Osang GF. 2021. Computing the multicover bifiltration. Leibniz International Proceedings in Informatics. SoCG: International Symposium on Computational Geometry, LIPIcs, vol. 189, 27.","chicago":"Corbet, René, Michael Kerber, Michael Lesnick, and Georg F Osang. “Computing the Multicover Bifiltration.” In Leibniz International Proceedings in Informatics, Vol. 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. https://doi.org/10.4230/LIPIcs.SoCG.2021.27.","ieee":"R. Corbet, M. Kerber, M. Lesnick, and G. F. Osang, “Computing the multicover bifiltration,” in Leibniz International Proceedings in Informatics, Online, 2021, vol. 189.","short":"R. Corbet, M. Kerber, M. Lesnick, G.F. Osang, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.","ama":"Corbet R, Kerber M, Lesnick M, Osang GF. Computing the multicover bifiltration. In: Leibniz International Proceedings in Informatics. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021. doi:10.4230/LIPIcs.SoCG.2021.27","apa":"Corbet, R., Kerber, M., Lesnick, M., & Osang, G. F. (2021). Computing the multicover bifiltration. In Leibniz International Proceedings in Informatics (Vol. 189). Online: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2021.27","mla":"Corbet, René, et al. “Computing the Multicover Bifiltration.” Leibniz International Proceedings in Informatics, vol. 189, 27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:10.4230/LIPIcs.SoCG.2021.27."},"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","author":[{"last_name":"Corbet","full_name":"Corbet, René","first_name":"René"},{"last_name":"Kerber","full_name":"Kerber, Michael","first_name":"Michael"},{"last_name":"Lesnick","full_name":"Lesnick, Michael","first_name":"Michael"},{"first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8882-5116","full_name":"Osang, Georg F","last_name":"Osang"}],"article_processing_charge":"No","external_id":{"arxiv":["2103.07823"]},"title":"Computing the multicover bifiltration","acknowledgement":"The authors want to thank the reviewers for many helpful comments and suggestions.","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","oa":1,"has_accepted_license":"1","year":"2021","day":"02","publication":"Leibniz International Proceedings in Informatics","doi":"10.4230/LIPIcs.SoCG.2021.27","date_published":"2021-06-02T00:00:00Z","date_created":"2021-06-27T22:01:49Z"},{"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","citation":{"ista":"Boissonnat J-D, Kachanovich S, Wintraecken M. 2021. Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations. 37th International Symposium on Computational Geometry (SoCG 2021). SoCG: Symposium on Computational GeometryLeibniz International Proceedings in Informatics (LIPIcs), LIPIcs, vol. 189, 17:1-17:16.","chicago":"Boissonnat, Jean-Daniel, Siargey Kachanovich, and Mathijs Wintraecken. “Tracing Isomanifolds in Rd in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations.” In 37th International Symposium on Computational Geometry (SoCG 2021), 189:17:1-17:16. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. https://doi.org/10.4230/LIPIcs.SoCG.2021.17.","ama":"Boissonnat J-D, Kachanovich S, Wintraecken M. Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations. In: 37th International Symposium on Computational Geometry (SoCG 2021). Vol 189. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021:17:1-17:16. doi:10.4230/LIPIcs.SoCG.2021.17","apa":"Boissonnat, J.-D., Kachanovich, S., & Wintraecken, M. (2021). Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations. In 37th International Symposium on Computational Geometry (SoCG 2021) (Vol. 189, p. 17:1-17:16). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2021.17","ieee":"J.-D. Boissonnat, S. Kachanovich, and M. Wintraecken, “Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations,” in 37th International Symposium on Computational Geometry (SoCG 2021), Virtual, 2021, vol. 189, p. 17:1-17:16.","short":"J.-D. Boissonnat, S. Kachanovich, M. Wintraecken, in:, 37th International Symposium on Computational Geometry (SoCG 2021), Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2021, p. 17:1-17:16.","mla":"Boissonnat, Jean-Daniel, et al. “Tracing Isomanifolds in Rd in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations.” 37th International Symposium on Computational Geometry (SoCG 2021), vol. 189, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 17:1-17:16, doi:10.4230/LIPIcs.SoCG.2021.17."},"title":"Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations","author":[{"last_name":"Boissonnat","full_name":"Boissonnat, Jean-Daniel","first_name":"Jean-Daniel"},{"first_name":"Siargey","full_name":"Kachanovich, Siargey","last_name":"Kachanovich"},{"orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs"}],"article_processing_charge":"No","project":[{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"day":"02","publication":"37th International Symposium on Computational Geometry (SoCG 2021)","has_accepted_license":"1","year":"2021","date_published":"2021-06-02T00:00:00Z","doi":"10.4230/LIPIcs.SoCG.2021.17","date_created":"2021-06-02T10:10:55Z","page":"17:1-17:16","acknowledgement":"We thank Dominique Attali, Guilherme de Fonseca, Arijit Ghosh, Vincent Pilaud and Aurélien Alvarez for their comments and suggestions. We also acknowledge the reviewers.","quality_controlled":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","oa":1,"ddc":["005","516","514"],"date_updated":"2023-10-10T07:34:34Z","department":[{"_id":"HeEd"}],"file_date_updated":"2021-06-02T10:22:33Z","_id":"9441","series_title":"Leibniz International Proceedings in Informatics (LIPIcs)","status":"public","type":"conference","conference":{"start_date":"2021-06-07","end_date":"2021-06-11","location":"Virtual","name":"SoCG: Symposium on Computational Geometry"},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"file":[{"checksum":"c322aa48d5d35a35877896cc565705b6","file_id":"9442","success":1,"access_level":"open_access","relation":"main_file","content_type":"application/pdf","date_created":"2021-06-02T10:22:33Z","file_name":"LIPIcs-SoCG-2021-17.pdf","creator":"mwintrae","date_updated":"2021-06-02T10:22:33Z","file_size":1972902}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["1868-8969"],"isbn":["978-3-95977-184-9"]},"publication_status":"published","related_material":{"record":[{"status":"public","id":"12960","relation":"later_version"}]},"volume":189,"ec_funded":1,"oa_version":"Published Version","abstract":[{"lang":"eng","text":"Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. "}],"place":"Dagstuhl, Germany","month":"06","intvolume":" 189","alternative_title":["LIPIcs"]},{"page":"938-976","doi":"10.1007/s00454-020-00240-w","date_published":"2021-10-01T00:00:00Z","date_created":"2020-09-06T22:01:13Z","isi":1,"year":"2021","day":"01","publication":"Discrete and Computational Geometry","publisher":"Springer Nature","quality_controlled":"1","oa":1,"acknowledgement":"This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. W.K.S. was also supported by the Australian Research Council (DP1401000851). A.V.A. was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha).","author":[{"last_name":"Akopyan","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy"},{"first_name":"Alexander I.","last_name":"Bobenko","full_name":"Bobenko, Alexander I."},{"full_name":"Schief, Wolfgang K.","last_name":"Schief","first_name":"Wolfgang K."},{"first_name":"Jan","last_name":"Techter","full_name":"Techter, Jan"}],"article_processing_charge":"No","external_id":{"arxiv":["1908.00856"],"isi":["000564488500002"]},"title":"On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs","citation":{"mla":"Akopyan, Arseniy, et al. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” Discrete and Computational Geometry, vol. 66, Springer Nature, 2021, pp. 938–76, doi:10.1007/s00454-020-00240-w.","ieee":"A. Akopyan, A. I. Bobenko, W. K. Schief, and J. Techter, “On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs,” Discrete and Computational Geometry, vol. 66. Springer Nature, pp. 938–976, 2021.","short":"A. Akopyan, A.I. Bobenko, W.K. Schief, J. Techter, Discrete and Computational Geometry 66 (2021) 938–976.","apa":"Akopyan, A., Bobenko, A. I., Schief, W. K., & Techter, J. (2021). On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00240-w","ama":"Akopyan A, Bobenko AI, Schief WK, Techter J. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. 2021;66:938-976. doi:10.1007/s00454-020-00240-w","chicago":"Akopyan, Arseniy, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” Discrete and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00240-w.","ista":"Akopyan A, Bobenko AI, Schief WK, Techter J. 2021. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. 66, 938–976."},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"volume":66,"ec_funded":1,"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"publication_status":"published","language":[{"iso":"eng"}],"scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1908.00856"}],"month":"10","intvolume":" 66","abstract":[{"text":"Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.","lang":"eng"}],"oa_version":"Preprint","department":[{"_id":"HeEd"}],"date_updated":"2024-03-07T14:51:11Z","article_type":"original","type":"journal_article","status":"public","_id":"8338"},{"abstract":[{"text":"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.","lang":"eng"}],"oa_version":"Published Version","scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00454-020-00233-9"}],"month":"09","intvolume":" 66","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"publication_status":"published","language":[{"iso":"eng"}],"volume":66,"ec_funded":1,"_id":"8248","type":"journal_article","article_type":"original","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"status":"public","date_updated":"2024-03-07T14:54:59Z","ddc":["510"],"department":[{"_id":"HeEd"}],"acknowledgement":"Open access funding provided by the Institute of Science and Technology (IST Austria). Arijit Ghosh is supported by the Ramanujan Fellowship (No. SB/S2/RJN-064/2015), India.\r\nThis work has been funded by the European Research Council under the European Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). The third author is supported by Ramanujan Fellowship (No. SB/S2/RJN-064/2015), India. The fifth author also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411.","publisher":"Springer Nature","quality_controlled":"1","oa":1,"has_accepted_license":"1","isi":1,"year":"2021","day":"01","publication":"Discrete and Computational Geometry","page":"666-686","doi":"10.1007/s00454-020-00233-9","date_published":"2021-09-01T00:00:00Z","date_created":"2020-08-11T07:11:51Z","project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411"}],"citation":{"mla":"Boissonnat, Jean-Daniel, et al. “Local Conditions for Triangulating Submanifolds of Euclidean Space.” Discrete and Computational Geometry, vol. 66, Springer Nature, 2021, pp. 666–86, doi:10.1007/s00454-020-00233-9.","apa":"Boissonnat, J.-D., Dyer, R., Ghosh, A., Lieutier, A., & Wintraecken, M. (2021). Local conditions for triangulating submanifolds of Euclidean space. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00233-9","ama":"Boissonnat J-D, Dyer R, Ghosh A, Lieutier A, Wintraecken M. Local conditions for triangulating submanifolds of Euclidean space. Discrete and Computational Geometry. 2021;66:666-686. doi:10.1007/s00454-020-00233-9","ieee":"J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, and M. Wintraecken, “Local conditions for triangulating submanifolds of Euclidean space,” Discrete and Computational Geometry, vol. 66. Springer Nature, pp. 666–686, 2021.","short":"J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, M. Wintraecken, Discrete and Computational Geometry 66 (2021) 666–686.","chicago":"Boissonnat, Jean-Daniel, Ramsay Dyer, Arijit Ghosh, Andre Lieutier, and Mathijs Wintraecken. “Local Conditions for Triangulating Submanifolds of Euclidean Space.” Discrete and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00233-9.","ista":"Boissonnat J-D, Dyer R, Ghosh A, Lieutier A, Wintraecken M. 2021. Local conditions for triangulating submanifolds of Euclidean space. Discrete and Computational Geometry. 66, 666–686."},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Jean-Daniel","full_name":"Boissonnat, Jean-Daniel","last_name":"Boissonnat"},{"first_name":"Ramsay","last_name":"Dyer","full_name":"Dyer, Ramsay"},{"first_name":"Arijit","last_name":"Ghosh","full_name":"Ghosh, Arijit"},{"full_name":"Lieutier, Andre","last_name":"Lieutier","first_name":"Andre"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs"}],"external_id":{"isi":["000558119300001"]},"article_processing_charge":"Yes (via OA deal)","title":"Local conditions for triangulating submanifolds of Euclidean space"},{"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"publication_status":"published","file":[{"file_size":1013730,"date_updated":"2020-11-25T09:06:41Z","creator":"dernst","file_name":"2020_DiscreteCompGeometry_Brown.pdf","date_created":"2020-11-25T09:06:41Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"file_id":"8803","checksum":"487a84ea5841b75f04f66d7ebd71b67e"}],"language":[{"iso":"eng"}],"volume":65,"abstract":[{"lang":"eng","text":"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."}],"oa_version":"Published Version","scopus_import":"1","month":"06","intvolume":" 65","date_updated":"2024-03-07T15:01:58Z","ddc":["510"],"file_date_updated":"2020-11-25T09:06:41Z","department":[{"_id":"HeEd"}],"_id":"7905","article_type":"original","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"status":"public","isi":1,"has_accepted_license":"1","year":"2021","day":"01","publication":"Discrete and Computational Geometry","page":"1166-1198","doi":"10.1007/s00454-020-00206-y","date_published":"2021-06-01T00:00:00Z","date_created":"2020-05-30T10:26:04Z","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). This work was partially supported by NSF IIS-1513616 and NSF ABI-1661375. The authors would like to thank the anonymous referees for their insightful comments.","publisher":"Springer Nature","quality_controlled":"1","oa":1,"citation":{"ista":"Brown A, Wang B. 2021. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. 65, 1166–1198.","chicago":"Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” Discrete and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00206-y.","short":"A. Brown, B. Wang, Discrete and Computational Geometry 65 (2021) 1166–1198.","ieee":"A. Brown and B. Wang, “Sheaf-theoretic stratification learning from geometric and topological perspectives,” Discrete and Computational Geometry, vol. 65. Springer Nature, pp. 1166–1198, 2021.","apa":"Brown, A., & Wang, B. (2021). Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00206-y","ama":"Brown A, Wang B. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. 2021;65:1166-1198. doi:10.1007/s00454-020-00206-y","mla":"Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” Discrete and Computational Geometry, vol. 65, Springer Nature, 2021, pp. 1166–98, doi:10.1007/s00454-020-00206-y."},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","author":[{"full_name":"Brown, Adam","last_name":"Brown","first_name":"Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425"},{"first_name":"Bei","last_name":"Wang","full_name":"Wang, Bei"}],"external_id":{"isi":["000536324700001"],"arxiv":["1712.07734"]},"article_processing_charge":"Yes (via OA deal)","title":"Sheaf-theoretic stratification learning from geometric and topological perspectives","project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}]},{"project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411"}],"article_processing_charge":"Yes (via OA deal)","author":[{"last_name":"Choudhary","full_name":"Choudhary, Aruni","first_name":"Aruni"},{"last_name":"Kachanovich","full_name":"Kachanovich, Siargey","first_name":"Siargey"},{"last_name":"Wintraecken","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs"}],"title":"Coxeter triangulations have good quality","citation":{"ista":"Choudhary A, Kachanovich S, Wintraecken M. 2020. Coxeter triangulations have good quality. Mathematics in Computer Science. 14, 141–176.","chicago":"Choudhary, Aruni, Siargey Kachanovich, and Mathijs Wintraecken. “Coxeter Triangulations Have Good Quality.” Mathematics in Computer Science. Springer Nature, 2020. https://doi.org/10.1007/s11786-020-00461-5.","ama":"Choudhary A, Kachanovich S, Wintraecken M. Coxeter triangulations have good quality. Mathematics in Computer Science. 2020;14:141-176. doi:10.1007/s11786-020-00461-5","apa":"Choudhary, A., Kachanovich, S., & Wintraecken, M. (2020). Coxeter triangulations have good quality. Mathematics in Computer Science. Springer Nature. https://doi.org/10.1007/s11786-020-00461-5","short":"A. Choudhary, S. Kachanovich, M. Wintraecken, Mathematics in Computer Science 14 (2020) 141–176.","ieee":"A. Choudhary, S. Kachanovich, and M. Wintraecken, “Coxeter triangulations have good quality,” Mathematics in Computer Science, vol. 14. Springer Nature, pp. 141–176, 2020.","mla":"Choudhary, Aruni, et al. “Coxeter Triangulations Have Good Quality.” Mathematics in Computer Science, vol. 14, Springer Nature, 2020, pp. 141–76, doi:10.1007/s11786-020-00461-5."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"publisher":"Springer Nature","quality_controlled":"1","page":"141-176","date_created":"2020-03-05T13:30:18Z","date_published":"2020-03-01T00:00:00Z","doi":"10.1007/s11786-020-00461-5","year":"2020","has_accepted_license":"1","publication":"Mathematics in Computer Science","day":"01","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"type":"journal_article","article_type":"original","status":"public","_id":"7567","file_date_updated":"2020-11-20T10:18:02Z","department":[{"_id":"HeEd"}],"date_updated":"2021-01-12T08:14:13Z","ddc":["510"],"scopus_import":"1","intvolume":" 14","month":"03","abstract":[{"text":"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.","lang":"eng"}],"oa_version":"Published Version","ec_funded":1,"volume":14,"publication_status":"published","publication_identifier":{"eissn":["1661-8289"],"issn":["1661-8270"]},"language":[{"iso":"eng"}],"file":[{"creator":"dernst","file_size":872275,"date_updated":"2020-11-20T10:18:02Z","file_name":"2020_MathCompScie_Choudhary.pdf","date_created":"2020-11-20T10:18:02Z","relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"file_id":"8783","checksum":"1d145f3ab50ccee735983cb89236e609"}]},{"doi":"10.1007/978-3-030-43408-3_8","date_published":"2020-06-22T00:00:00Z","date_created":"2020-07-19T22:00:59Z","page":"181-218","day":"22","publication":"Topological Data Analysis","has_accepted_license":"1","year":"2020","publisher":"Springer Nature","quality_controlled":"1","oa":1,"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","title":"Radius functions on Poisson–Delaunay mosaics and related complexes experimentally","author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","last_name":"Nikitenko","full_name":"Nikitenko, Anton"},{"last_name":"Ölsböck","full_name":"Ölsböck, Katharina","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina"},{"full_name":"Synak, Peter","last_name":"Synak","first_name":"Peter","id":"331776E2-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"chicago":"Edelsbrunner, Herbert, Anton Nikitenko, Katharina Ölsböck, and Peter Synak. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” In Topological Data Analysis, 15:181–218. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-43408-3_8.","ista":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. 2020. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. Topological Data Analysis. , Abel Symposia, vol. 15, 181–218.","mla":"Edelsbrunner, Herbert, et al. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” Topological Data Analysis, vol. 15, Springer Nature, 2020, pp. 181–218, doi:10.1007/978-3-030-43408-3_8.","ieee":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions on Poisson–Delaunay mosaics and related complexes experimentally,” in Topological Data Analysis, 2020, vol. 15, pp. 181–218.","short":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, P. Synak, in:, Topological Data Analysis, Springer Nature, 2020, pp. 181–218.","apa":"Edelsbrunner, H., Nikitenko, A., Ölsböck, K., & Synak, P. (2020). Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In Topological Data Analysis (Vol. 15, pp. 181–218). Springer Nature. https://doi.org/10.1007/978-3-030-43408-3_8","ama":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In: Topological Data Analysis. Vol 15. Springer Nature; 2020:181-218. doi:10.1007/978-3-030-43408-3_8"},"project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"638176","name":"Efficient Simulation of Natural Phenomena at Extremely Large Scales","_id":"2533E772-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"volume":15,"ec_funded":1,"file":[{"date_created":"2020-10-08T08:56:14Z","file_name":"2020-B-01-PoissonExperimentalSurvey.pdf","creator":"dernst","date_updated":"2020-10-08T08:56:14Z","file_size":2207071,"file_id":"8628","checksum":"7b5e0de10675d787a2ddb2091370b8d8","success":1,"access_level":"open_access","relation":"main_file","content_type":"application/pdf"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["21932808"],"isbn":["9783030434076"],"eissn":["21978549"]},"publication_status":"published","month":"06","intvolume":" 15","alternative_title":["Abel Symposia"],"scopus_import":"1","oa_version":"Submitted Version","abstract":[{"lang":"eng","text":"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."}],"department":[{"_id":"HeEd"}],"file_date_updated":"2020-10-08T08:56:14Z","ddc":["510"],"date_updated":"2021-01-12T08:17:06Z","status":"public","type":"conference","_id":"8135"},{"page":"143-158","date_published":"2020-11-17T00:00:00Z","doi":"10.1515/mathm-2020-0106","date_created":"2021-03-16T08:55:19Z","has_accepted_license":"1","year":"2020","day":"17","publication":"Mathematical Morphology - Theory and Applications","publisher":"De Gruyter","quality_controlled":"1","oa":1,"acknowledgement":"This work has been partially supported by the European Research Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation programme, grant no. 788183, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. ","author":[{"last_name":"Biswas","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita"},{"last_name":"Largeteau-Skapin","full_name":"Largeteau-Skapin, Gaëlle","first_name":"Gaëlle"},{"full_name":"Zrour, Rita","last_name":"Zrour","first_name":"Rita"},{"full_name":"Andres, Eric","last_name":"Andres","first_name":"Eric"}],"article_processing_charge":"No","title":"Digital objects in rhombic dodecahedron grid","citation":{"mla":"Biswas, Ranita, et al. “Digital Objects in Rhombic Dodecahedron Grid.” Mathematical Morphology - Theory and Applications, vol. 4, no. 1, De Gruyter, 2020, pp. 143–58, doi:10.1515/mathm-2020-0106.","ieee":"R. Biswas, G. Largeteau-Skapin, R. Zrour, and E. Andres, “Digital objects in rhombic dodecahedron grid,” Mathematical Morphology - Theory and Applications, vol. 4, no. 1. De Gruyter, pp. 143–158, 2020.","short":"R. Biswas, G. Largeteau-Skapin, R. Zrour, E. Andres, Mathematical Morphology - Theory and Applications 4 (2020) 143–158.","ama":"Biswas R, Largeteau-Skapin G, Zrour R, Andres E. Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. 2020;4(1):143-158. doi:10.1515/mathm-2020-0106","apa":"Biswas, R., Largeteau-Skapin, G., Zrour, R., & Andres, E. (2020). Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. De Gruyter. https://doi.org/10.1515/mathm-2020-0106","chicago":"Biswas, Ranita, Gaëlle Largeteau-Skapin, Rita Zrour, and Eric Andres. “Digital Objects in Rhombic Dodecahedron Grid.” Mathematical Morphology - Theory and Applications. De Gruyter, 2020. https://doi.org/10.1515/mathm-2020-0106.","ista":"Biswas R, Largeteau-Skapin G, Zrour R, Andres E. 2020. Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. 4(1), 143–158."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"issue":"1","volume":4,"ec_funded":1,"publication_identifier":{"issn":["2353-3390"]},"publication_status":"published","file":[{"content_type":"application/pdf","access_level":"open_access","relation":"main_file","checksum":"4a1043fa0548a725d464017fe2483ce0","file_id":"9272","success":1,"date_updated":"2021-03-22T08:56:37Z","file_size":3668725,"creator":"dernst","date_created":"2021-03-22T08:56:37Z","file_name":"2020_MathMorpholTheoryAppl_Biswas.pdf"}],"language":[{"iso":"eng"}],"month":"11","intvolume":" 4","abstract":[{"text":"Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system.","lang":"eng"}],"oa_version":"Published Version","file_date_updated":"2021-03-22T08:56:37Z","department":[{"_id":"HeEd"}],"date_updated":"2021-03-22T09:01:50Z","ddc":["510"],"article_type":"original","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"status":"public","_id":"9249"},{"language":[{"iso":"eng"}],"publication_identifier":{"isbn":["9783030687656"],"eissn":["1611-3349"],"issn":["0302-9743"]},"publication_status":"published","volume":12590,"oa_version":"Preprint","abstract":[{"text":"We call a multigraph non-homotopic if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on n>1 vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with n vertices and m>4n edges is larger than cm2n for some constant c>0 , and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as n is fixed and m⟶∞ .","lang":"eng"}],"month":"09","intvolume":" 12590","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/2006.14908","open_access":"1"}],"date_updated":"2021-04-06T11:32:32Z","department":[{"_id":"HeEd"}],"series_title":"LNCS","_id":"9299","status":"public","type":"conference","conference":{"location":"Virtual, Online","end_date":"2020-09-18","start_date":"2020-09-16","name":"GD: Graph Drawing and Network Visualization"},"day":"20","publication":"28th International Symposium on Graph Drawing and Network Visualization","year":"2020","date_published":"2020-09-20T00:00:00Z","doi":"10.1007/978-3-030-68766-3_28","date_created":"2021-03-28T22:01:44Z","page":"359-371","acknowledgement":"Supported by the National Research, Development and Innovation Office, NKFIH, KKP-133864, K-131529, K-116769, K-132696, by the Higher Educational Institutional Excellence Program 2019 NKFIH-1158-6/2019, the Austrian Science Fund (FWF), grant Z 342-N31, by the Ministry of Education and Science of the Russian Federation MegaGrant No. 075-15-2019-1926, and by the ERC Synergy Grant “Dynasnet” No. 810115. A full version can be found at https://arxiv.org/abs/2006.14908.","publisher":"Springer Nature","quality_controlled":"1","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Pach, János, et al. “Crossings between Non-Homotopic Edges.” 28th International Symposium on Graph Drawing and Network Visualization, vol. 12590, Springer Nature, 2020, pp. 359–71, doi:10.1007/978-3-030-68766-3_28.","short":"J. Pach, G. Tardos, G. Tóth, in:, 28th International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2020, pp. 359–371.","ieee":"J. Pach, G. Tardos, and G. Tóth, “Crossings between non-homotopic edges,” in 28th International Symposium on Graph Drawing and Network Visualization, Virtual, Online, 2020, vol. 12590, pp. 359–371.","apa":"Pach, J., Tardos, G., & Tóth, G. (2020). Crossings between non-homotopic edges. In 28th International Symposium on Graph Drawing and Network Visualization (Vol. 12590, pp. 359–371). Virtual, Online: Springer Nature. https://doi.org/10.1007/978-3-030-68766-3_28","ama":"Pach J, Tardos G, Tóth G. Crossings between non-homotopic edges. In: 28th International Symposium on Graph Drawing and Network Visualization. Vol 12590. LNCS. Springer Nature; 2020:359-371. doi:10.1007/978-3-030-68766-3_28","chicago":"Pach, János, Gábor Tardos, and Géza Tóth. “Crossings between Non-Homotopic Edges.” In 28th International Symposium on Graph Drawing and Network Visualization, 12590:359–71. LNCS. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-68766-3_28.","ista":"Pach J, Tardos G, Tóth G. 2020. Crossings between non-homotopic edges. 28th International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network VisualizationLNCS vol. 12590, 359–371."},"title":"Crossings between non-homotopic edges","author":[{"full_name":"Pach, János","last_name":"Pach","first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4"},{"first_name":"Gábor","full_name":"Tardos, Gábor","last_name":"Tardos"},{"full_name":"Tóth, Géza","last_name":"Tóth","first_name":"Géza"}],"article_processing_charge":"No","external_id":{"arxiv":["2006.14908"]},"project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"The Wittgenstein Prize","grant_number":"Z00342"}]},{"oa":1,"publisher":"Carleton University","quality_controlled":"1","acknowledgement":"This research is partially supported by the Office of Naval Research, through grant no. N62909-18-1-2038, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","date_created":"2021-07-04T22:01:26Z","date_published":"2020-12-14T00:00:00Z","doi":"10.20382/jocg.v11i2a7","page":"162-182","publication":"Journal of Computational Geometry","day":"14","year":"2020","has_accepted_license":"1","project":[{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics","grant_number":"I4887"}],"title":"Topological data analysis in information space","article_processing_charge":"Yes","author":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"id":"2E36B656-F248-11E8-B48F-1D18A9856A87","first_name":"Ziga","full_name":"Virk, Ziga","last_name":"Virk"},{"first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Hubert","last_name":"Wagner"}],"user_id":"6785fbc1-c503-11eb-8a32-93094b40e1cf","citation":{"ista":"Edelsbrunner H, Virk Z, Wagner H. 2020. Topological data analysis in information space. Journal of Computational Geometry. 11(2), 162–182.","chicago":"Edelsbrunner, Herbert, Ziga Virk, and Hubert Wagner. “Topological Data Analysis in Information Space.” Journal of Computational Geometry. Carleton University, 2020. https://doi.org/10.20382/jocg.v11i2a7.","apa":"Edelsbrunner, H., Virk, Z., & Wagner, H. (2020). Topological data analysis in information space. Journal of Computational Geometry. Carleton University. https://doi.org/10.20382/jocg.v11i2a7","ama":"Edelsbrunner H, Virk Z, Wagner H. Topological data analysis in information space. Journal of Computational Geometry. 2020;11(2):162-182. doi:10.20382/jocg.v11i2a7","short":"H. Edelsbrunner, Z. Virk, H. Wagner, Journal of Computational Geometry 11 (2020) 162–182.","ieee":"H. Edelsbrunner, Z. Virk, and H. Wagner, “Topological data analysis in information space,” Journal of Computational Geometry, vol. 11, no. 2. Carleton University, pp. 162–182, 2020.","mla":"Edelsbrunner, Herbert, et al. “Topological Data Analysis in Information Space.” Journal of Computational Geometry, vol. 11, no. 2, Carleton University, 2020, pp. 162–82, doi:10.20382/jocg.v11i2a7."},"intvolume":" 11","month":"12","scopus_import":"1","oa_version":"Published Version","abstract":[{"text":"Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.","lang":"eng"}],"license":"https://creativecommons.org/licenses/by/3.0/","issue":"2","volume":11,"language":[{"iso":"eng"}],"file":[{"relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"file_id":"9882","checksum":"f02d0b2b3838e7891a6c417fc34ffdcd","creator":"asandaue","file_size":1449234,"date_updated":"2021-08-11T11:55:11Z","file_name":"2020_JournalOfComputationalGeometry_Edelsbrunner.pdf","date_created":"2021-08-11T11:55:11Z"}],"publication_status":"published","publication_identifier":{"eissn":["1920180X"]},"status":"public","tmp":{"short":"CC BY (3.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/3.0/legalcode","name":"Creative Commons Attribution 3.0 Unported (CC BY 3.0)"},"type":"journal_article","article_type":"original","_id":"9630","department":[{"_id":"HeEd"}],"file_date_updated":"2021-08-11T11:55:11Z","ddc":["510","000"],"date_updated":"2021-08-11T12:26:34Z"},{"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","citation":{"chicago":"Akopyan, Arseniy, Richard Schwartz, and Serge Tabachnikov. “Billiards in Ellipses Revisited.” European Journal of Mathematics. Springer Nature, 2020. https://doi.org/10.1007/s40879-020-00426-9.","ista":"Akopyan A, Schwartz R, Tabachnikov S. 2020. Billiards in ellipses revisited. European Journal of Mathematics.","mla":"Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” European Journal of Mathematics, Springer Nature, 2020, doi:10.1007/s40879-020-00426-9.","apa":"Akopyan, A., Schwartz, R., & Tabachnikov, S. (2020). Billiards in ellipses revisited. European Journal of Mathematics. Springer Nature. https://doi.org/10.1007/s40879-020-00426-9","ama":"Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. European Journal of Mathematics. 2020. doi:10.1007/s40879-020-00426-9","short":"A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics (2020).","ieee":"A. Akopyan, R. Schwartz, and S. Tabachnikov, “Billiards in ellipses revisited,” European Journal of Mathematics. Springer Nature, 2020."},"title":"Billiards in ellipses revisited","external_id":{"arxiv":["2001.02934"]},"article_processing_charge":"No","author":[{"orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy"},{"full_name":"Schwartz, Richard","last_name":"Schwartz","first_name":"Richard"},{"full_name":"Tabachnikov, Serge","last_name":"Tabachnikov","first_name":"Serge"}],"acknowledgement":" This paper would not be written if not for Dan Reznik’s curiosity and persistence; we are very grateful to him. We also thank R. Garcia and J. Koiller for interesting discussions. It is a pleasure to thank the Mathematical Institute of the University of Heidelberg for its stimulating atmosphere. ST thanks M. Bialy for interesting discussions and the Tel Aviv\r\nUniversity for its invariable hospitality. AA was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 78818 Alpha). RS is supported by NSF Grant DMS-1807320. ST was supported by NSF grant DMS-1510055 and SFB/TRR 191.","oa":1,"quality_controlled":"1","publisher":"Springer Nature","publication":"European Journal of Mathematics","day":"09","year":"2020","date_created":"2020-09-20T22:01:38Z","date_published":"2020-09-09T00:00:00Z","doi":"10.1007/s40879-020-00426-9","_id":"8538","status":"public","type":"journal_article","article_type":"original","date_updated":"2021-12-02T15:10:17Z","department":[{"_id":"HeEd"}],"oa_version":"Preprint","abstract":[{"lang":"eng","text":"We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods."}],"month":"09","main_file_link":[{"url":"https://arxiv.org/abs/2001.02934","open_access":"1"}],"scopus_import":"1","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"ec_funded":1},{"department":[{"_id":"HeEd"}],"file_date_updated":"2020-07-14T12:48:06Z","date_updated":"2023-08-02T06:49:16Z","ddc":["510"],"conference":{"name":"SoCG: Symposium on Computational Geometry","location":"Zürich, Switzerland","end_date":"2020-06-26","start_date":"2020-06-22"},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"type":"conference","status":"public","_id":"7952","ec_funded":1,"volume":164,"related_material":{"record":[{"id":"9649","status":"public","relation":"later_version"}]},"publication_status":"published","publication_identifier":{"issn":["1868-8969"],"isbn":["978-3-95977-143-6"]},"language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","access_level":"open_access","relation":"main_file","checksum":"38cbfa4f5d484d267a35d44d210df044","file_id":"7969","date_updated":"2020-07-14T12:48:06Z","file_size":1009739,"creator":"dernst","date_created":"2020-06-17T10:13:34Z","file_name":"2020_LIPIcsSoCG_Boissonnat.pdf"}],"scopus_import":"1","alternative_title":["LIPIcs"],"intvolume":" 164","month":"06","abstract":[{"text":"Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary. ","lang":"eng"}],"oa_version":"Published Version","article_processing_charge":"No","author":[{"last_name":"Boissonnat","full_name":"Boissonnat, Jean-Daniel","first_name":"Jean-Daniel"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220"}],"title":"The topological correctness of PL-approximations of isomanifolds","citation":{"ista":"Boissonnat J-D, Wintraecken M. 2020. The topological correctness of PL-approximations of isomanifolds. 36th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 164, 20:1-20:18.","chicago":"Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL-Approximations of Isomanifolds.” In 36th International Symposium on Computational Geometry, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. https://doi.org/10.4230/LIPIcs.SoCG.2020.20.","apa":"Boissonnat, J.-D., & Wintraecken, M. (2020). The topological correctness of PL-approximations of isomanifolds. In 36th International Symposium on Computational Geometry (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2020.20","ama":"Boissonnat J-D, Wintraecken M. The topological correctness of PL-approximations of isomanifolds. In: 36th International Symposium on Computational Geometry. Vol 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:10.4230/LIPIcs.SoCG.2020.20","ieee":"J.-D. Boissonnat and M. Wintraecken, “The topological correctness of PL-approximations of isomanifolds,” in 36th International Symposium on Computational Geometry, Zürich, Switzerland, 2020, vol. 164.","short":"J.-D. Boissonnat, M. Wintraecken, in:, 36th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.","mla":"Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL-Approximations of Isomanifolds.” 36th International Symposium on Computational Geometry, vol. 164, 20:1-20:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:10.4230/LIPIcs.SoCG.2020.20."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"article_number":"20:1-20:18","date_created":"2020-06-09T07:24:11Z","doi":"10.4230/LIPIcs.SoCG.2020.20","date_published":"2020-06-01T00:00:00Z","year":"2020","has_accepted_license":"1","publication":"36th International Symposium on Computational Geometry","day":"01","oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1"},{"oa":1,"quality_controlled":"1","publisher":"Springer Nature","date_created":"2018-12-11T11:44:29Z","doi":"10.1007/978-3-030-36020-7_1","date_published":"2020-06-21T00:00:00Z","page":"1-27","publication":"Geometric Aspects of Functional Analysis","day":"21","year":"2020","isi":1,"project":[{"call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117"}],"title":"Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures","editor":[{"full_name":"Klartag, Bo'az","last_name":"Klartag","first_name":"Bo'az"},{"full_name":"Milman, Emanuel","last_name":"Milman","first_name":"Emanuel"}],"article_processing_charge":"No","external_id":{"isi":["000557689300003"],"arxiv":["1808.07350"]},"author":[{"first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan"},{"first_name":"Roman","last_name":"Karasev","full_name":"Karasev, Roman"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ista":"Akopyan A, Karasev R. 2020.Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In: Geometric Aspects of Functional Analysis. vol. 2256, 1–27.","chicago":"Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian Measures and Radial Non-Gaussian Measures.” In Geometric Aspects of Functional Analysis, edited by Bo’az Klartag and Emanuel Milman, 2256:1–27. LNM. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-36020-7_1.","ieee":"A. Akopyan and R. Karasev, “Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures,” in Geometric Aspects of Functional Analysis, vol. 2256, B. Klartag and E. Milman, Eds. Springer Nature, 2020, pp. 1–27.","short":"A. Akopyan, R. Karasev, in:, B. Klartag, E. Milman (Eds.), Geometric Aspects of Functional Analysis, Springer Nature, 2020, pp. 1–27.","ama":"Akopyan A, Karasev R. Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In: Klartag B, Milman E, eds. Geometric Aspects of Functional Analysis. Vol 2256. LNM. Springer Nature; 2020:1-27. doi:10.1007/978-3-030-36020-7_1","apa":"Akopyan, A., & Karasev, R. (2020). Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In B. Klartag & E. Milman (Eds.), Geometric Aspects of Functional Analysis (Vol. 2256, pp. 1–27). Springer Nature. https://doi.org/10.1007/978-3-030-36020-7_1","mla":"Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian Measures and Radial Non-Gaussian Measures.” Geometric Aspects of Functional Analysis, edited by Bo’az Klartag and Emanuel Milman, vol. 2256, Springer Nature, 2020, pp. 1–27, doi:10.1007/978-3-030-36020-7_1."},"intvolume":" 2256","month":"06","main_file_link":[{"url":"https://arxiv.org/abs/1808.07350","open_access":"1"}],"scopus_import":"1","oa_version":"Preprint","abstract":[{"lang":"eng","text":"We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class\r\nof compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2.\r\nWe use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian\r\nmeasures."}],"ec_funded":1,"volume":2256,"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"eisbn":["9783030360207"],"issn":["00758434"],"isbn":["9783030360191"],"eissn":["16179692"]},"status":"public","type":"book_chapter","_id":"74","series_title":"LNM","department":[{"_id":"HeEd"},{"_id":"JaMa"}],"date_updated":"2023-08-17T13:48:31Z"},{"oa":1,"quality_controlled":"1","publisher":"SIAM","date_created":"2020-03-01T23:00:39Z","doi":"10.1137/S0040585X97T989726","date_published":"2020-02-13T00:00:00Z","page":"595-614","publication":"Theory of Probability and its Applications","day":"13","year":"2020","isi":1,"project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"title":"Weighted Poisson–Delaunay mosaics","external_id":{"isi":["000551393100007"],"arxiv":["1705.08735"]},"article_processing_charge":"No","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201","last_name":"Nikitenko"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ista":"Edelsbrunner H, Nikitenko A. 2020. Weighted Poisson–Delaunay mosaics. Theory of Probability and its Applications. 64(4), 595–614.","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” Theory of Probability and Its Applications. SIAM, 2020. https://doi.org/10.1137/S0040585X97T989726.","ieee":"H. Edelsbrunner and A. Nikitenko, “Weighted Poisson–Delaunay mosaics,” Theory of Probability and its Applications, vol. 64, no. 4. SIAM, pp. 595–614, 2020.","short":"H. Edelsbrunner, A. Nikitenko, Theory of Probability and Its Applications 64 (2020) 595–614.","ama":"Edelsbrunner H, Nikitenko A. Weighted Poisson–Delaunay mosaics. Theory of Probability and its Applications. 2020;64(4):595-614. doi:10.1137/S0040585X97T989726","apa":"Edelsbrunner, H., & Nikitenko, A. (2020). Weighted Poisson–Delaunay mosaics. Theory of Probability and Its Applications. SIAM. https://doi.org/10.1137/S0040585X97T989726","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” Theory of Probability and Its Applications, vol. 64, no. 4, SIAM, 2020, pp. 595–614, doi:10.1137/S0040585X97T989726."},"intvolume":" 64","month":"02","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1705.08735"}],"scopus_import":"1","oa_version":"Preprint","abstract":[{"text":"Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$.","lang":"eng"}],"ec_funded":1,"issue":"4","volume":64,"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"eissn":["10957219"],"issn":["0040585X"]},"status":"public","article_type":"original","type":"journal_article","_id":"7554","department":[{"_id":"HeEd"}],"date_updated":"2023-08-18T06:45:48Z"},{"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_type":"original","type":"journal_article","status":"public","_id":"7666","file_date_updated":"2020-11-20T13:22:21Z","department":[{"_id":"HeEd"}],"date_updated":"2023-08-21T06:13:48Z","ddc":["510"],"scopus_import":"1","intvolume":" 64","month":"03","abstract":[{"lang":"eng","text":"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."}],"oa_version":"Published Version","ec_funded":1,"volume":64,"publication_status":"published","publication_identifier":{"issn":["01795376"],"eissn":["14320444"]},"language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"file_id":"8786","checksum":"f8cc96e497f00c38340b5dafe0cb91d7","file_size":701673,"date_updated":"2020-11-20T13:22:21Z","creator":"dernst","file_name":"2020_DiscreteCompGeo_Edelsbrunner.pdf","date_created":"2020-11-20T13:22:21Z"}],"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"article_processing_charge":"Yes (via OA deal)","external_id":{"isi":["000520918800001"]},"author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina","full_name":"Ölsböck, Katharina","orcid":"0000-0002-4672-8297","last_name":"Ölsböck"}],"title":"Tri-partitions and bases of an ordered complex","citation":{"ista":"Edelsbrunner H, Ölsböck K. 2020. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 64, 759–775.","chicago":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00188-x.","short":"H. Edelsbrunner, K. Ölsböck, Discrete and Computational Geometry 64 (2020) 759–775.","ieee":"H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,” Discrete and Computational Geometry, vol. 64. Springer Nature, pp. 759–775, 2020.","ama":"Edelsbrunner H, Ölsböck K. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 2020;64:759-775. doi:10.1007/s00454-020-00188-x","apa":"Edelsbrunner, H., & Ölsböck, K. (2020). Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00188-x","mla":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry, vol. 64, Springer Nature, 2020, pp. 759–75, doi:10.1007/s00454-020-00188-x."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1,"publisher":"Springer Nature","quality_controlled":"1","acknowledgement":"This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF).","page":"759-775","date_created":"2020-04-19T22:00:56Z","date_published":"2020-03-20T00:00:00Z","doi":"10.1007/s00454-020-00188-x","year":"2020","has_accepted_license":"1","isi":1,"publication":"Discrete and Computational Geometry","day":"20"},{"project":[{"name":"The Wittgenstein Prize","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"title":"Almost all string graphs are intersection graphs of plane convex sets","article_processing_charge":"No","external_id":{"isi":["000538229000001"],"arxiv":["1803.06710"]},"author":[{"id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János","last_name":"Pach","full_name":"Pach, János"},{"first_name":"Bruce","full_name":"Reed, Bruce","last_name":"Reed"},{"first_name":"Yelena","full_name":"Yuditsky, Yelena","last_name":"Yuditsky"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ama":"Pach J, Reed B, Yuditsky Y. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 2020;63(4):888-917. doi:10.1007/s00454-020-00213-z","apa":"Pach, J., Reed, B., & Yuditsky, Y. (2020). Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00213-z","ieee":"J. Pach, B. Reed, and Y. Yuditsky, “Almost all string graphs are intersection graphs of plane convex sets,” Discrete and Computational Geometry, vol. 63, no. 4. Springer Nature, pp. 888–917, 2020.","short":"J. Pach, B. Reed, Y. Yuditsky, Discrete and Computational Geometry 63 (2020) 888–917.","mla":"Pach, János, et al. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.” Discrete and Computational Geometry, vol. 63, no. 4, Springer Nature, 2020, pp. 888–917, doi:10.1007/s00454-020-00213-z.","ista":"Pach J, Reed B, Yuditsky Y. 2020. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 63(4), 888–917.","chicago":"Pach, János, Bruce Reed, and Yelena Yuditsky. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00213-z."},"oa":1,"quality_controlled":"1","publisher":"Springer Nature","date_created":"2020-06-14T22:00:51Z","doi":"10.1007/s00454-020-00213-z","date_published":"2020-06-05T00:00:00Z","page":"888-917","publication":"Discrete and Computational Geometry","day":"05","year":"2020","isi":1,"status":"public","article_type":"original","type":"journal_article","_id":"7962","department":[{"_id":"HeEd"}],"date_updated":"2023-08-21T08:49:18Z","intvolume":" 63","month":"06","main_file_link":[{"url":"https://arxiv.org/abs/1803.06710","open_access":"1"}],"scopus_import":"1","oa_version":"Preprint","abstract":[{"lang":"eng","text":"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."}],"volume":63,"issue":"4","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"eissn":["14320444"],"issn":["01795376"]}}]