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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.","lang":"eng"}],"volume":15,"ec_funded":1,"file":[{"file_size":2207071,"date_updated":"2020-10-08T08:56:14Z","creator":"dernst","file_name":"2020-B-01-PoissonExperimentalSurvey.pdf","date_created":"2020-10-08T08:56:14Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"file_id":"8628","checksum":"7b5e0de10675d787a2ddb2091370b8d8"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["21932808"],"isbn":["9783030434076"],"eissn":["21978549"]},"publication_status":"published","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"call_identifier":"H2020","_id":"2533E772-B435-11E9-9278-68D0E5697425","grant_number":"638176","name":"Efficient Simulation of Natural Phenomena at Extremely Large Scales"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"title":"Radius functions on Poisson–Delaunay mosaics and related complexes experimentally","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Nikitenko, Anton","last_name":"Nikitenko","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Katharina","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","full_name":"Ölsböck, Katharina","last_name":"Ölsböck"},{"id":"331776E2-F248-11E8-B48F-1D18A9856A87","first_name":"Peter","full_name":"Synak, Peter","last_name":"Synak"}],"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.","short":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, P. Synak, in:, Topological Data Analysis, Springer Nature, 2020, pp. 181–218.","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.","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"},"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).","date_published":"2020-06-22T00:00:00Z","doi":"10.1007/978-3-030-43408-3_8","date_created":"2020-07-19T22:00:59Z","page":"181-218","day":"22","publication":"Topological Data Analysis","has_accepted_license":"1","year":"2020"},{"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. ","oa":1,"publisher":"De Gruyter","quality_controlled":"1","publication":"Mathematical Morphology - Theory and Applications","day":"17","year":"2020","has_accepted_license":"1","date_created":"2021-03-16T08:55:19Z","date_published":"2020-11-17T00:00:00Z","doi":"10.1515/mathm-2020-0106","page":"143-158","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"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","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","short":"R. Biswas, G. Largeteau-Skapin, R. Zrour, E. Andres, Mathematical Morphology - Theory and Applications 4 (2020) 143–158.","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.","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.","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.","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."},"title":"Digital objects in rhombic dodecahedron grid","article_processing_charge":"No","author":[{"full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890","last_name":"Biswas","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Largeteau-Skapin, Gaëlle","last_name":"Largeteau-Skapin","first_name":"Gaëlle"},{"first_name":"Rita","full_name":"Zrour, Rita","last_name":"Zrour"},{"first_name":"Eric","full_name":"Andres, Eric","last_name":"Andres"}],"oa_version":"Published Version","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"}],"intvolume":" 4","month":"11","language":[{"iso":"eng"}],"file":[{"relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"checksum":"4a1043fa0548a725d464017fe2483ce0","file_id":"9272","creator":"dernst","file_size":3668725,"date_updated":"2021-03-22T08:56:37Z","file_name":"2020_MathMorpholTheoryAppl_Biswas.pdf","date_created":"2021-03-22T08:56:37Z"}],"publication_status":"published","publication_identifier":{"issn":["2353-3390"]},"ec_funded":1,"issue":"1","volume":4,"_id":"9249","status":"public","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","ddc":["510"],"date_updated":"2021-03-22T09:01:50Z","department":[{"_id":"HeEd"}],"file_date_updated":"2021-03-22T08:56:37Z"},{"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.","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","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","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","external_id":{"arxiv":["2006.14908"]},"article_processing_charge":"No","author":[{"full_name":"Pach, János","last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János"},{"first_name":"Gábor","full_name":"Tardos, Gábor","last_name":"Tardos"},{"first_name":"Géza","last_name":"Tóth","full_name":"Tóth, Géza"}],"project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"The Wittgenstein Prize","grant_number":"Z00342"}],"publication":"28th International Symposium on Graph Drawing and Network Visualization","day":"20","year":"2020","date_created":"2021-03-28T22:01:44Z","doi":"10.1007/978-3-030-68766-3_28","date_published":"2020-09-20T00:00:00Z","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.","oa":1,"publisher":"Springer Nature","quality_controlled":"1","date_updated":"2021-04-06T11:32:32Z","department":[{"_id":"HeEd"}],"series_title":"LNCS","_id":"9299","status":"public","conference":{"end_date":"2020-09-18","location":"Virtual, Online","start_date":"2020-09-16","name":"GD: Graph Drawing and Network Visualization"},"type":"conference","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"isbn":["9783030687656"],"eissn":["1611-3349"],"issn":["0302-9743"]},"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"}],"intvolume":" 12590","month":"09","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2006.14908"}],"scopus_import":"1"},{"oa":1,"quality_controlled":"1","publisher":"Carleton University","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":[{"name":"Discretization in Geometry and Dynamics","grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"}],"title":"Topological data analysis in information space","article_processing_charge":"Yes","author":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Virk, Ziga","last_name":"Virk","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","first_name":"Ziga"},{"last_name":"Wagner","full_name":"Wagner, Hubert","first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"user_id":"6785fbc1-c503-11eb-8a32-93094b40e1cf","citation":{"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.","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.","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","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","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.","ista":"Edelsbrunner H, Virk Z, Wagner H. 2020. Topological data analysis in information space. Journal of Computational Geometry. 11(2), 162–182."},"intvolume":" 11","month":"12","scopus_import":"1","oa_version":"Published Version","abstract":[{"lang":"eng","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."}],"license":"https://creativecommons.org/licenses/by/3.0/","volume":11,"issue":"2","language":[{"iso":"eng"}],"file":[{"access_level":"open_access","relation":"main_file","content_type":"application/pdf","file_id":"9882","checksum":"f02d0b2b3838e7891a6c417fc34ffdcd","success":1,"creator":"asandaue","date_updated":"2021-08-11T11:55:11Z","file_size":1449234,"date_created":"2021-08-11T11:55:11Z","file_name":"2020_JournalOfComputationalGeometry_Edelsbrunner.pdf"}],"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)"},"article_type":"original","type":"journal_article","_id":"9630","file_date_updated":"2021-08-11T11:55:11Z","department":[{"_id":"HeEd"}],"ddc":["510","000"],"date_updated":"2021-08-11T12:26:34Z"},{"date_published":"2020-09-09T00:00:00Z","doi":"10.1007/s40879-020-00426-9","date_created":"2020-09-20T22:01:38Z","year":"2020","day":"09","publication":"European Journal of Mathematics","quality_controlled":"1","publisher":"Springer Nature","oa":1,"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.","author":[{"first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan"},{"last_name":"Schwartz","full_name":"Schwartz, Richard","first_name":"Richard"},{"full_name":"Tabachnikov, Serge","last_name":"Tabachnikov","first_name":"Serge"}],"article_processing_charge":"No","external_id":{"arxiv":["2001.02934"]},"title":"Billiards in ellipses revisited","citation":{"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.","ama":"Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. European Journal of Mathematics. 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","mla":"Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” European Journal of Mathematics, Springer Nature, 2020, doi:10.1007/s40879-020-00426-9.","ista":"Akopyan A, Schwartz R, Tabachnikov S. 2020. Billiards in ellipses revisited. European Journal of Mathematics.","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."},"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"ec_funded":1,"publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"publication_status":"published","language":[{"iso":"eng"}],"scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/2001.02934","open_access":"1"}],"month":"09","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."}],"oa_version":"Preprint","department":[{"_id":"HeEd"}],"date_updated":"2021-12-02T15:10:17Z","article_type":"original","type":"journal_article","status":"public","_id":"8538"}]