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We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction."}],"title":"The density fingerprint of a periodic point set","article_processing_charge":"No","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa","last_name":"Heiss","full_name":"Heiss, Teresa","orcid":"0000-0002-1780-2689"},{"first_name":"Vitaliy","full_name":" Kurlin , Vitaliy","last_name":" Kurlin "},{"full_name":"Smith, Philip","last_name":"Smith","first_name":"Philip"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220"}],"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","citation":{"chicago":"Edelsbrunner, Herbert, Teresa Heiss, Vitaliy Kurlin , Philip Smith, and Mathijs Wintraecken. “The Density Fingerprint of a Periodic Point Set.” In 37th International Symposium on Computational Geometry (SoCG 2021), 189:32:1-32:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. https://doi.org/10.4230/LIPIcs.SoCG.2021.32.","ista":"Edelsbrunner H, Heiss T, Kurlin V, Smith P, Wintraecken M. 2021. The density fingerprint of a periodic point set. 37th International Symposium on Computational Geometry (SoCG 2021). SoCG: Symposium on Computational Geometry, LIPIcs, vol. 189, 32:1-32:16.","mla":"Edelsbrunner, Herbert, et al. “The Density Fingerprint of a Periodic Point Set.” 37th International Symposium on Computational Geometry (SoCG 2021), vol. 189, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 32:1-32:16, doi:10.4230/LIPIcs.SoCG.2021.32.","short":"H. Edelsbrunner, T. Heiss, V. Kurlin , P. Smith, M. Wintraecken, in:, 37th International Symposium on Computational Geometry (SoCG 2021), Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 32:1-32:16.","ieee":"H. Edelsbrunner, T. Heiss, V. Kurlin , P. Smith, and M. Wintraecken, “The density fingerprint of a periodic point set,” in 37th International Symposium on Computational Geometry (SoCG 2021), Virtual, 2021, vol. 189, p. 32:1-32:16.","apa":"Edelsbrunner, H., Heiss, T., Kurlin , V., Smith, P., & Wintraecken, M. (2021). The density fingerprint of a periodic point set. In 37th International Symposium on Computational Geometry (SoCG 2021) (Vol. 189, p. 32:1-32:16). Virtual: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2021.32","ama":"Edelsbrunner H, Heiss T, Kurlin V, Smith P, Wintraecken M. The density fingerprint of a periodic point set. In: 37th International Symposium on Computational Geometry (SoCG 2021). Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021:32:1-32:16. doi:10.4230/LIPIcs.SoCG.2021.32"},"project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"I4887","name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"},{"_id":"25C5A090-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"The Wittgenstein Prize","grant_number":"Z00312"},{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"date_created":"2021-04-22T08:09:58Z","date_published":"2021-06-02T00:00:00Z","doi":"10.4230/LIPIcs.SoCG.2021.32","page":"32:1-32:16","publication":"37th International Symposium on Computational Geometry (SoCG 2021)","day":"02","year":"2021","has_accepted_license":"1","oa":1,"quality_controlled":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","acknowledgement":"The authors thank Janos Pach for insightful discussions on the topic of thispaper, Morteza Saghafian for finding the one-dimensional counterexample mentioned in Section 5,and Larry Andrews for generously sharing his crystallographic perspective."},{"month":"06","intvolume":" 189","alternative_title":["LIPIcs"],"scopus_import":"1","oa_version":"Published Version","abstract":[{"lang":"eng","text":"Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation."}],"volume":189,"ec_funded":1,"file":[{"access_level":"open_access","relation":"main_file","content_type":"application/pdf","checksum":"22b11a719018b22ecba2471b51f2eb40","file_id":"9611","success":1,"creator":"asandaue","date_updated":"2021-06-28T13:11:39Z","file_size":727817,"date_created":"2021-06-28T13:11:39Z","file_name":"2021_LIPIcs_Biswas.pdf"}],"language":[{"iso":"eng"}],"publication_identifier":{"isbn":["9783959771849"],"issn":["18688969"]},"publication_status":"published","status":"public","type":"conference","conference":{"name":"SoCG: International Symposium on Computational Geometry","end_date":"2021-06-11","location":"Online","start_date":"2021-06-07"},"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)"},"_id":"9604","file_date_updated":"2021-06-28T13:11:39Z","department":[{"_id":"HeEd"}],"ddc":["516"],"date_updated":"2023-02-23T14:02:28Z","quality_controlled":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","oa":1,"doi":"10.4230/LIPIcs.SoCG.2021.16","date_published":"2021-06-02T00:00:00Z","date_created":"2021-06-27T22:01:48Z","day":"02","publication":"Leibniz International Proceedings in Informatics","has_accepted_license":"1","year":"2021","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"The Wittgenstein Prize","grant_number":"Z00342"},{"name":"Discretization in Geometry and Dynamics","grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"}],"article_number":"16","title":"Counting cells of order-k voronoi tessellations in ℝ3 with morse theory","author":[{"first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890","last_name":"Biswas"},{"first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza"}],"article_processing_charge":"No","user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","citation":{"ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2021. Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. Leibniz International Proceedings in Informatics. SoCG: International Symposium on Computational Geometry, LIPIcs, vol. 189, 16.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Counting Cells of Order-k Voronoi Tessellations in ℝ3 with Morse Theory.” In Leibniz International Proceedings in Informatics, Vol. 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. https://doi.org/10.4230/LIPIcs.SoCG.2021.16.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2021). Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. In Leibniz International Proceedings in Informatics (Vol. 189). Online: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2021.16","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. In: Leibniz International Proceedings in Informatics. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021. doi:10.4230/LIPIcs.SoCG.2021.16","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Counting cells of order-k voronoi tessellations in ℝ3 with morse theory,” in Leibniz International Proceedings in Informatics, Online, 2021, vol. 189.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.","mla":"Biswas, Ranita, et al. “Counting Cells of Order-k Voronoi Tessellations in ℝ3 with Morse Theory.” Leibniz International Proceedings in Informatics, vol. 189, 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:10.4230/LIPIcs.SoCG.2021.16."}},{"status":"public","type":"conference","conference":{"location":"Uppsala, Sweden","end_date":"2021-05-27","start_date":"2021-05-24","name":"DGMM: International Conference on Discrete Geometry and Mathematical Morphology"},"_id":"9824","department":[{"_id":"HeEd"}],"date_updated":"2022-05-31T06:58:21Z","month":"05","intvolume":" 12708","alternative_title":["LNCS"],"scopus_import":"1","oa_version":"None","abstract":[{"text":"We define a new compact coordinate system in which each integer triplet addresses a voxel in the BCC grid, and we investigate some of its properties. We propose a characterization of 3D discrete analytical planes with their topological features (in the Cartesian and in the new coordinate system) such as the interrelation between the thickness of the plane and the separability constraint we aim to obtain.","lang":"eng"}],"volume":12708,"ec_funded":1,"language":[{"iso":"eng"}],"publication_identifier":{"issn":["03029743"],"isbn":["9783030766566"],"eissn":["16113349"]},"publication_status":"published","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"title":"Body centered cubic grid - coordinate system and discrete analytical plane definition","author":[{"first_name":"Lidija","full_name":"Čomić, Lidija","last_name":"Čomić"},{"first_name":"Rita","full_name":"Zrour, Rita","last_name":"Zrour"},{"last_name":"Largeteau-Skapin","full_name":"Largeteau-Skapin, Gaëlle","first_name":"Gaëlle"},{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","last_name":"Biswas","orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita"},{"full_name":"Andres, Eric","last_name":"Andres","first_name":"Eric"}],"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"short":"L. Čomić, R. Zrour, G. Largeteau-Skapin, R. Biswas, E. Andres, in:, Discrete Geometry and Mathematical Morphology, Springer Nature, 2021, pp. 152–163.","ieee":"L. Čomić, R. Zrour, G. Largeteau-Skapin, R. Biswas, and E. Andres, “Body centered cubic grid - coordinate system and discrete analytical plane definition,” in Discrete Geometry and Mathematical Morphology, Uppsala, Sweden, 2021, vol. 12708, pp. 152–163.","apa":"Čomić, L., Zrour, R., Largeteau-Skapin, G., Biswas, R., & Andres, E. (2021). Body centered cubic grid - coordinate system and discrete analytical plane definition. In Discrete Geometry and Mathematical Morphology (Vol. 12708, pp. 152–163). Uppsala, Sweden: Springer Nature. https://doi.org/10.1007/978-3-030-76657-3_10","ama":"Čomić L, Zrour R, Largeteau-Skapin G, Biswas R, Andres E. Body centered cubic grid - coordinate system and discrete analytical plane definition. In: Discrete Geometry and Mathematical Morphology. Vol 12708. Springer Nature; 2021:152-163. doi:10.1007/978-3-030-76657-3_10","mla":"Čomić, Lidija, et al. “Body Centered Cubic Grid - Coordinate System and Discrete Analytical Plane Definition.” Discrete Geometry and Mathematical Morphology, vol. 12708, Springer Nature, 2021, pp. 152–63, doi:10.1007/978-3-030-76657-3_10.","ista":"Čomić L, Zrour R, Largeteau-Skapin G, Biswas R, Andres E. 2021. Body centered cubic grid - coordinate system and discrete analytical plane definition. Discrete Geometry and Mathematical Morphology. DGMM: International Conference on Discrete Geometry and Mathematical Morphology, LNCS, vol. 12708, 152–163.","chicago":"Čomić, Lidija, Rita Zrour, Gaëlle Largeteau-Skapin, Ranita Biswas, and Eric Andres. “Body Centered Cubic Grid - Coordinate System and Discrete Analytical Plane Definition.” In Discrete Geometry and Mathematical Morphology, 12708:152–63. Springer Nature, 2021. https://doi.org/10.1007/978-3-030-76657-3_10."},"quality_controlled":"1","publisher":"Springer Nature","acknowledgement":"This work has been partially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project no. 451-03-68/2020-14/200156: “Innovative scientific and artistic research from the FTS (activity) domain” (LČ), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183 (RB), and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35 (RB).","date_published":"2021-05-16T00:00:00Z","doi":"10.1007/978-3-030-76657-3_10","date_created":"2021-08-08T22:01:29Z","page":"152-163","day":"16","publication":"Discrete Geometry and Mathematical Morphology","year":"2021"},{"oa_version":"Preprint","abstract":[{"text":"When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.","lang":"eng"}],"month":"02","intvolume":" 93","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/1910.09917v3","open_access":"1"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["09257721"]},"publication_status":"published","related_material":{"record":[{"status":"public","id":"6989","relation":"shorter_version"}]},"volume":93,"_id":"8317","status":"public","article_type":"original","type":"journal_article","date_updated":"2023-08-04T10:57:42Z","department":[{"_id":"HeEd"}],"acknowledgement":"This research was performed in part at the 33rd Bellairs Winter Workshop on Computational Geometry. We thank all other participants for a fruitful atmosphere. H. Akitaya was supported by NSF CCF-1422311 & 1423615. Z. Masárová was partially funded by Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","publisher":"Elsevier","quality_controlled":"1","oa":1,"day":"01","publication":"Computational Geometry: Theory and Applications","isi":1,"year":"2021","date_published":"2021-02-01T00:00:00Z","doi":"10.1016/j.comgeo.2020.101700","date_created":"2020-08-30T22:01:09Z","article_number":"101700","project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"The Wittgenstein Prize"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ista":"Aichholzer O, Akitaya HA, Cheung KC, Demaine ED, Demaine ML, Fekete SP, Kleist L, Kostitsyna I, Löffler M, Masárová Z, Mundilova K, Schmidt C. 2021. Folding polyominoes with holes into a cube. Computational Geometry: Theory and Applications. 93, 101700.","chicago":"Aichholzer, Oswin, Hugo A. Akitaya, Kenneth C. Cheung, Erik D. Demaine, Martin L. Demaine, Sándor P. Fekete, Linda Kleist, et al. “Folding Polyominoes with Holes into a Cube.” Computational Geometry: Theory and Applications. Elsevier, 2021. https://doi.org/10.1016/j.comgeo.2020.101700.","apa":"Aichholzer, O., Akitaya, H. A., Cheung, K. C., Demaine, E. D., Demaine, M. L., Fekete, S. P., … Schmidt, C. (2021). Folding polyominoes with holes into a cube. Computational Geometry: Theory and Applications. Elsevier. https://doi.org/10.1016/j.comgeo.2020.101700","ama":"Aichholzer O, Akitaya HA, Cheung KC, et al. Folding polyominoes with holes into a cube. Computational Geometry: Theory and Applications. 2021;93. doi:10.1016/j.comgeo.2020.101700","short":"O. Aichholzer, H.A. Akitaya, K.C. Cheung, E.D. Demaine, M.L. Demaine, S.P. Fekete, L. Kleist, I. Kostitsyna, M. Löffler, Z. Masárová, K. Mundilova, C. Schmidt, Computational Geometry: Theory and Applications 93 (2021).","ieee":"O. Aichholzer et al., “Folding polyominoes with holes into a cube,” Computational Geometry: Theory and Applications, vol. 93. Elsevier, 2021.","mla":"Aichholzer, Oswin, et al. “Folding Polyominoes with Holes into a Cube.” Computational Geometry: Theory and Applications, vol. 93, 101700, Elsevier, 2021, doi:10.1016/j.comgeo.2020.101700."},"title":"Folding polyominoes with holes into a cube","author":[{"last_name":"Aichholzer","full_name":"Aichholzer, Oswin","first_name":"Oswin"},{"first_name":"Hugo A.","full_name":"Akitaya, Hugo A.","last_name":"Akitaya"},{"first_name":"Kenneth C.","last_name":"Cheung","full_name":"Cheung, Kenneth C."},{"first_name":"Erik D.","full_name":"Demaine, Erik D.","last_name":"Demaine"},{"last_name":"Demaine","full_name":"Demaine, Martin L.","first_name":"Martin L."},{"first_name":"Sándor P.","full_name":"Fekete, Sándor P.","last_name":"Fekete"},{"first_name":"Linda","full_name":"Kleist, Linda","last_name":"Kleist"},{"full_name":"Kostitsyna, Irina","last_name":"Kostitsyna","first_name":"Irina"},{"full_name":"Löffler, Maarten","last_name":"Löffler","first_name":"Maarten"},{"last_name":"Masárová","orcid":"0000-0002-6660-1322","full_name":"Masárová, Zuzana","first_name":"Zuzana","id":"45CFE238-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Klara","last_name":"Mundilova","full_name":"Mundilova, Klara"},{"full_name":"Schmidt, Christiane","last_name":"Schmidt","first_name":"Christiane"}],"article_processing_charge":"No","external_id":{"isi":["000579185100004"],"arxiv":["1910.09917"]}},{"_id":"8773","keyword":["Applied Mathematics","General Mathematics"],"status":"public","type":"journal_article","article_type":"original","date_updated":"2023-08-04T11:11:47Z","department":[{"_id":"HeEd"}],"oa_version":"Preprint","abstract":[{"text":"Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.","lang":"eng"}],"intvolume":" 149","month":"01","main_file_link":[{"url":"https://arxiv.org/abs/1910.08286","open_access":"1"}],"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"eissn":["1088-6826"],"issn":["0002-9939"]},"ec_funded":1,"volume":149,"issue":"1","project":[{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"chicago":"Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.” Proceedings of the American Mathematical Society. American Mathematical Society, 2021. https://doi.org/10.1090/proc/15205.","ista":"Brown A, Romanov A. 2021. Contravariant forms on Whittaker modules. Proceedings of the American Mathematical Society. 149(1), 37–52.","mla":"Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.” Proceedings of the American Mathematical Society, vol. 149, no. 1, American Mathematical Society, 2021, pp. 37–52, doi:10.1090/proc/15205.","ieee":"A. Brown and A. Romanov, “Contravariant forms on Whittaker modules,” Proceedings of the American Mathematical Society, vol. 149, no. 1. American Mathematical Society, pp. 37–52, 2021.","short":"A. Brown, A. Romanov, Proceedings of the American Mathematical Society 149 (2021) 37–52.","apa":"Brown, A., & Romanov, A. (2021). Contravariant forms on Whittaker modules. Proceedings of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/proc/15205","ama":"Brown A, Romanov A. Contravariant forms on Whittaker modules. Proceedings of the American Mathematical Society. 2021;149(1):37-52. doi:10.1090/proc/15205"},"title":"Contravariant forms on Whittaker modules","article_processing_charge":"No","external_id":{"arxiv":["1910.08286"],"isi":["000600416300004"]},"author":[{"id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam","full_name":"Brown, Adam","last_name":"Brown"},{"full_name":"Romanov, Anna","last_name":"Romanov","first_name":"Anna"}],"acknowledgement":"We would like to thank Peter Trapa for useful discussions, and Dragan Milicic and Arun Ram for valuable feedback on the structure of the paper. The first author acknowledges the support of the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie Grant Agreement No. 754411. The second author is\r\nsupported by the National Science Foundation Award No. 1803059.","oa":1,"publisher":"American Mathematical Society","quality_controlled":"1","publication":"Proceedings of the American Mathematical Society","day":"01","year":"2021","isi":1,"date_created":"2020-11-19T10:17:40Z","date_published":"2021-01-01T00:00:00Z","doi":"10.1090/proc/15205","page":"37-52"},{"publication_status":"published","publication_identifier":{"isbn":["9781728162515"]},"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"In March 2020, the Austrian government introduced a widespread lock-down in response to the COVID-19 pandemic. Based on subjective impressions and anecdotal evidence, Austrian public and private life came to a sudden halt. Here we assess the effect of the lock-down quantitatively for all regions in Austria and present an analysis of daily changes of human mobility throughout Austria using near-real-time anonymized mobile phone data. We describe an efficient data aggregation pipeline and analyze the mobility by quantifying mobile-phone traffic at specific point of interests (POIs), analyzing individual trajectories and investigating the cluster structure of the origin-destination graph. We found a reduction of commuters at Viennese metro stations of over 80% and the number of devices with a radius of gyration of less than 500 m almost doubled. The results of studying crowd-movement behavior highlight considerable changes in the structure of mobility networks, revealed by a higher modularity and an increase from 12 to 20 detected communities. We demonstrate the relevance of mobility data for epidemiological studies by showing a significant correlation of the outflow from the town of Ischgl (an early COVID-19 hotspot) and the reported COVID-19 cases with an 8-day time lag. This research indicates that mobile phone usage data permits the moment-by-moment quantification of mobility behavior for a whole country. We emphasize the need to improve the availability of such data in anonymized form to empower rapid response to combat COVID-19 and future pandemics."}],"oa_version":"Preprint","main_file_link":[{"url":"https://arxiv.org/abs/2008.10064","open_access":"1"}],"scopus_import":"1","month":"03","date_updated":"2023-08-07T14:00:13Z","department":[{"_id":"HeEd"}],"_id":"9253","conference":{"name":"Big Data: International Conference on Big Data","start_date":"2020-12-10","location":"Atlanta, GA, United States","end_date":"2020-12-13"},"type":"conference","status":"public","year":"2021","isi":1,"publication":"2020 IEEE International Conference on Big Data","day":"19","page":"3123-3132","date_created":"2021-03-21T11:34:07Z","date_published":"2021-03-19T00:00:00Z","doi":"10.1109/bigdata50022.2020.9378374","oa":1,"publisher":"IEEE","quality_controlled":"1","citation":{"ieee":"G. Heiler et al., “Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic,” in 2020 IEEE International Conference on Big Data, Atlanta, GA, United States, 2021, pp. 3123–3132.","short":"G. Heiler, T. Reisch, J. Hurt, M. Forghani, A. Omani, A. Hanbury, F. Karimipour, in:, 2020 IEEE International Conference on Big Data, IEEE, 2021, pp. 3123–3132.","ama":"Heiler G, Reisch T, Hurt J, et al. Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. In: 2020 IEEE International Conference on Big Data. IEEE; 2021:3123-3132. doi:10.1109/bigdata50022.2020.9378374","apa":"Heiler, G., Reisch, T., Hurt, J., Forghani, M., Omani, A., Hanbury, A., & Karimipour, F. (2021). Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. In 2020 IEEE International Conference on Big Data (pp. 3123–3132). Atlanta, GA, United States: IEEE. https://doi.org/10.1109/bigdata50022.2020.9378374","mla":"Heiler, Georg, et al. “Country-Wide Mobility Changes Observed Using Mobile Phone Data during COVID-19 Pandemic.” 2020 IEEE International Conference on Big Data, IEEE, 2021, pp. 3123–32, doi:10.1109/bigdata50022.2020.9378374.","ista":"Heiler G, Reisch T, Hurt J, Forghani M, Omani A, Hanbury A, Karimipour F. 2021. Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. 2020 IEEE International Conference on Big Data. Big Data: International Conference on Big Data, 3123–3132.","chicago":"Heiler, Georg, Tobias Reisch, Jan Hurt, Mohammad Forghani, Aida Omani, Allan Hanbury, and Farid Karimipour. “Country-Wide Mobility Changes Observed Using Mobile Phone Data during COVID-19 Pandemic.” In 2020 IEEE International Conference on Big Data, 3123–32. IEEE, 2021. https://doi.org/10.1109/bigdata50022.2020.9378374."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","external_id":{"isi":["000662554703032"],"arxiv":["2008.10064"]},"article_processing_charge":"No","author":[{"last_name":"Heiler","full_name":"Heiler, Georg","first_name":"Georg"},{"first_name":"Tobias","full_name":"Reisch, Tobias","last_name":"Reisch"},{"first_name":"Jan","last_name":"Hurt","full_name":"Hurt, Jan"},{"first_name":"Mohammad","last_name":"Forghani","full_name":"Forghani, Mohammad"},{"last_name":"Omani","full_name":"Omani, Aida","first_name":"Aida"},{"last_name":"Hanbury","full_name":"Hanbury, Allan","first_name":"Allan"},{"id":"2A2BCDC4-CF62-11E9-BE5E-3B1EE6697425","first_name":"Farid","full_name":"Karimipour, Farid","orcid":"0000-0001-6746-4174","last_name":"Karimipour"}],"title":"Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic"},{"date_updated":"2023-08-07T14:35:44Z","ddc":["516"],"file_date_updated":"2021-12-01T10:56:53Z","department":[{"_id":"HeEd"}],"_id":"9317","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","publication_status":"published","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"language":[{"iso":"eng"}],"file":[{"success":1,"file_id":"10394","checksum":"59b4e1e827e494209bcb4aae22e1d347","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_name":"2021_DisCompGeo_Edelsbrunner_Osang.pdf","date_created":"2021-12-01T10:56:53Z","file_size":677704,"date_updated":"2021-12-01T10:56:53Z","creator":"cchlebak"}],"ec_funded":1,"volume":65,"related_material":{"record":[{"status":"public","id":"187","relation":"earlier_version"}]},"abstract":[{"text":"Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r consists of all points in Rd that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in Rd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.","lang":"eng"}],"oa_version":"Published Version","scopus_import":"1","intvolume":" 65","month":"03","citation":{"ama":"Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. 2021;65:1296–1313. doi:10.1007/s00454-021-00281-9","apa":"Edelsbrunner, H., & Osang, G. F. (2021). The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-021-00281-9","ieee":"H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean balls,” Discrete and Computational Geometry, vol. 65. Springer Nature, pp. 1296–1313, 2021.","short":"H. Edelsbrunner, G.F. Osang, Discrete and Computational Geometry 65 (2021) 1296–1313.","mla":"Edelsbrunner, Herbert, and Georg F. Osang. “The Multi-Cover Persistence of Euclidean Balls.” Discrete and Computational Geometry, vol. 65, Springer Nature, 2021, pp. 1296–1313, doi:10.1007/s00454-021-00281-9.","ista":"Edelsbrunner H, Osang GF. 2021. The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. 65, 1296–1313.","chicago":"Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence of Euclidean Balls.” Discrete and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-021-00281-9."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"Yes (via OA deal)","external_id":{"isi":["000635460400001"]},"author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"last_name":"Osang","full_name":"Osang, Georg F","first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"}],"title":"The multi-cover persistence of Euclidean balls","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"year":"2021","isi":1,"has_accepted_license":"1","publication":"Discrete and Computational Geometry","day":"31","page":"1296–1313","date_created":"2021-04-11T22:01:15Z","date_published":"2021-03-31T00:00:00Z","doi":"10.1007/s00454-021-00281-9","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha), and by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF)\r\nOpen Access funding provided by the Institute of Science and Technology (IST Austria).","oa":1,"publisher":"Springer Nature","quality_controlled":"1"},{"acknowledgement":"We would like to thank the anonymous referees for their useful comments and suggestions. János Pach is partially supported by Austrian Science Fund (FWF) grant Z 342-N31 and by ERC Advanced grant “GeoScape.” István Tomon is partially supported by Swiss National Science Foundation grant no. 200021_196965, and thanks the support of MIPT Moscow. Both authors are partially supported by The Russian Government in the framework of MegaGrant no. 075-15-2019-1926.","oa":1,"publisher":"Elsevier","quality_controlled":"1","year":"2021","isi":1,"has_accepted_license":"1","publication":"Journal of Combinatorial Theory. Series B","day":"09","page":"21-37","date_created":"2021-06-27T22:01:47Z","doi":"10.1016/j.jctb.2021.05.004","date_published":"2021-06-09T00:00:00Z","project":[{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"The Wittgenstein Prize"}],"citation":{"ista":"Pach J, Tomon I. 2021. Erdős-Hajnal-type results for monotone paths. Journal of Combinatorial Theory. Series B. 151, 21–37.","chicago":"Pach, János, and István Tomon. “Erdős-Hajnal-Type Results for Monotone Paths.” Journal of Combinatorial Theory. Series B. Elsevier, 2021. https://doi.org/10.1016/j.jctb.2021.05.004.","ama":"Pach J, Tomon I. Erdős-Hajnal-type results for monotone paths. Journal of Combinatorial Theory Series B. 2021;151:21-37. doi:10.1016/j.jctb.2021.05.004","apa":"Pach, J., & Tomon, I. (2021). Erdős-Hajnal-type results for monotone paths. Journal of Combinatorial Theory. Series B. Elsevier. https://doi.org/10.1016/j.jctb.2021.05.004","ieee":"J. Pach and I. Tomon, “Erdős-Hajnal-type results for monotone paths,” Journal of Combinatorial Theory. Series B, vol. 151. Elsevier, pp. 21–37, 2021.","short":"J. Pach, I. Tomon, Journal of Combinatorial Theory. Series B 151 (2021) 21–37.","mla":"Pach, János, and István Tomon. “Erdős-Hajnal-Type Results for Monotone Paths.” Journal of Combinatorial Theory. Series B, vol. 151, Elsevier, 2021, pp. 21–37, doi:10.1016/j.jctb.2021.05.004."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","external_id":{"isi":["000702280800002"]},"author":[{"first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach","full_name":"Pach, János"},{"first_name":"István","last_name":"Tomon","full_name":"Tomon, István"}],"title":"Erdős-Hajnal-type results for monotone paths","abstract":[{"lang":"eng","text":"An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck > 0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least n^ck . This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs.\r\nA key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k) > 0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k ≥ 3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees."}],"oa_version":"Published Version","scopus_import":"1","intvolume":" 151","month":"06","publication_status":"published","publication_identifier":{"issn":["0095-8956"]},"language":[{"iso":"eng"}],"file":[{"relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"checksum":"15fbc9064cd9d1c777ac0043b78c8f12","file_id":"9612","creator":"asandaue","file_size":418168,"date_updated":"2021-06-28T13:33:23Z","file_name":"2021_JournalOfCombinatorialTheory_Pach.pdf","date_created":"2021-06-28T13:33:23Z"}],"volume":151,"_id":"9602","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":"2023-08-10T13:38:00Z","ddc":["510"],"file_date_updated":"2021-06-28T13:33:23Z","department":[{"_id":"HeEd"}]},{"publication_identifier":{"eissn":["19326203"]},"publication_status":"published","file":[{"creator":"asandaue","file_size":2706919,"date_updated":"2021-08-09T09:25:41Z","file_name":"2021_PLoSONE_Graff.pdf","date_created":"2021-08-09T09:25:41Z","relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"checksum":"0277aa155d5db1febd2cb384768bba5f","file_id":"9832"}],"language":[{"iso":"eng"}],"issue":"7","volume":16,"abstract":[{"lang":"eng","text":"Heart rate variability (hrv) is a physiological phenomenon of the variation in the length of the time interval between consecutive heartbeats. In many cases it could be an indicator of the development of pathological states. The classical approach to the analysis of hrv includes time domain methods and frequency domain methods. However, attempts are still being made to define new and more effective hrv assessment tools. Persistent homology is a novel data analysis tool developed in the recent decades that is rooted at algebraic topology. The Topological Data Analysis (TDA) approach focuses on examining the shape of the data in terms of connectedness and holes, and has recently proved to be very effective in various fields of research. In this paper we propose the use of persistent homology to the hrv analysis. We recall selected topological descriptors used in the literature and we introduce some new topological descriptors that reflect the specificity of hrv, and we discuss their relation to the standard hrv measures. In particular, we show that this novel approach provides a collection of indices that might be at least as useful as the classical parameters in differentiating between series of beat-to-beat intervals (RR-intervals) in healthy subjects and patients suffering from a stroke episode."}],"pmid":1,"oa_version":"Published Version","scopus_import":"1","month":"07","intvolume":" 16","date_updated":"2023-08-10T14:21:42Z","ddc":["006"],"file_date_updated":"2021-08-09T09:25:41Z","department":[{"_id":"HeEd"}],"_id":"9821","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","isi":1,"has_accepted_license":"1","year":"2021","day":"01","publication":"PLoS ONE","doi":"10.1371/journal.pone.0253851","date_published":"2021-07-01T00:00:00Z","date_created":"2021-08-08T22:01:28Z","acknowledgement":"We express our gratitude to the anonymous referees who provided constructive comments that helped us improve the quality of the paper.","publisher":"Public Library of Science","quality_controlled":"1","oa":1,"citation":{"chicago":"Graff, Grzegorz, Beata Graff, Pawel Pilarczyk, Grzegorz Jablonski, Dariusz Gąsecki, and Krzysztof Narkiewicz. “Persistent Homology as a New Method of the Assessment of Heart Rate Variability.” PLoS ONE. Public Library of Science, 2021. https://doi.org/10.1371/journal.pone.0253851.","ista":"Graff G, Graff B, Pilarczyk P, Jablonski G, Gąsecki D, Narkiewicz K. 2021. Persistent homology as a new method of the assessment of heart rate variability. PLoS ONE. 16(7), e0253851.","mla":"Graff, Grzegorz, et al. “Persistent Homology as a New Method of the Assessment of Heart Rate Variability.” PLoS ONE, vol. 16, no. 7, e0253851, Public Library of Science, 2021, doi:10.1371/journal.pone.0253851.","short":"G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, K. Narkiewicz, PLoS ONE 16 (2021).","ieee":"G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, and K. Narkiewicz, “Persistent homology as a new method of the assessment of heart rate variability,” PLoS ONE, vol. 16, no. 7. Public Library of Science, 2021.","ama":"Graff G, Graff B, Pilarczyk P, Jablonski G, Gąsecki D, Narkiewicz K. Persistent homology as a new method of the assessment of heart rate variability. PLoS ONE. 2021;16(7). doi:10.1371/journal.pone.0253851","apa":"Graff, G., Graff, B., Pilarczyk, P., Jablonski, G., Gąsecki, D., & Narkiewicz, K. (2021). Persistent homology as a new method of the assessment of heart rate variability. PLoS ONE. Public Library of Science. https://doi.org/10.1371/journal.pone.0253851"},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","author":[{"first_name":"Grzegorz","full_name":"Graff, Grzegorz","last_name":"Graff"},{"first_name":"Beata","full_name":"Graff, Beata","last_name":"Graff"},{"id":"3768D56A-F248-11E8-B48F-1D18A9856A87","first_name":"Pawel","last_name":"Pilarczyk","full_name":"Pilarczyk, Pawel"},{"orcid":"0000-0002-3536-9866","full_name":"Jablonski, Grzegorz","last_name":"Jablonski","first_name":"Grzegorz","id":"4483EF78-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Gąsecki","full_name":"Gąsecki, Dariusz","first_name":"Dariusz"},{"first_name":"Krzysztof","full_name":"Narkiewicz, Krzysztof","last_name":"Narkiewicz"}],"external_id":{"pmid":["34292957"],"isi":["000678124900050"]},"article_processing_charge":"Yes","title":"Persistent homology as a new method of the assessment of heart rate variability","article_number":"e0253851"},{"ec_funded":1,"file":[{"date_created":"2023-08-14T11:55:10Z","file_name":"2023_ExperimentalMath_Akopyan.pdf","creator":"dernst","date_updated":"2023-08-14T11:55:10Z","file_size":1966019,"checksum":"3514382e3a1eb87fa6c61ad622874415","file_id":"14053","success":1,"access_level":"open_access","relation":"main_file","content_type":"application/pdf"}],"language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1944-950X"],"issn":["1058-6458"]},"publication_status":"published","month":"10","scopus_import":"1","oa_version":"Published Version","abstract":[{"lang":"eng","text":"Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density."}],"department":[{"_id":"HeEd"}],"file_date_updated":"2023-08-14T11:55:10Z","ddc":["510"],"date_updated":"2023-08-14T11:57:07Z","status":"public","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)"},"_id":"10222","doi":"10.1080/10586458.2021.1980459","date_published":"2021-10-25T00:00:00Z","date_created":"2021-11-07T23:01:25Z","page":"1-15","day":"25","publication":"Experimental Mathematics","isi":1,"has_accepted_license":"1","year":"2021","publisher":"Taylor and Francis","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 no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.\r\nWe are grateful to Dmitry Zaporozhets and Christoph Thäle for valuable comments and for directing us to relevant references. We also thank to Anton Mellit for a useful discussion on Bessel functions.","title":"The beauty of random polytopes inscribed in the 2-sphere","author":[{"first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan"},{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201","last_name":"Nikitenko"}],"external_id":{"isi":["000710893500001"],"arxiv":["2007.07783"]},"article_processing_charge":"Yes (via OA deal)","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Akopyan, Arseniy, et al. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” Experimental Mathematics, Taylor and Francis, 2021, pp. 1–15, doi:10.1080/10586458.2021.1980459.","apa":"Akopyan, A., Edelsbrunner, H., & Nikitenko, A. (2021). The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics. Taylor and Francis. https://doi.org/10.1080/10586458.2021.1980459","ama":"Akopyan A, Edelsbrunner H, Nikitenko A. The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics. 2021:1-15. doi:10.1080/10586458.2021.1980459","ieee":"A. Akopyan, H. Edelsbrunner, and A. Nikitenko, “The beauty of random polytopes inscribed in the 2-sphere,” Experimental Mathematics. Taylor and Francis, pp. 1–15, 2021.","short":"A. Akopyan, H. Edelsbrunner, A. Nikitenko, Experimental Mathematics (2021) 1–15.","chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, and Anton Nikitenko. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” Experimental Mathematics. Taylor and Francis, 2021. https://doi.org/10.1080/10586458.2021.1980459.","ista":"Akopyan A, Edelsbrunner H, Nikitenko A. 2021. The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics., 1–15."},"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"Z00342","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics","grant_number":"I4887"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}]},{"_id":"8940","status":"public","keyword":["Theoretical Computer Science","Computational Theory and Mathematics","Geometry and Topology","Discrete Mathematics and Combinatorics"],"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)"},"ddc":["516"],"date_updated":"2023-09-05T15:02:40Z","department":[{"_id":"HeEd"}],"file_date_updated":"2021-08-06T09:52:29Z","oa_version":"Published Version","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."}],"month":"07","intvolume":" 66","file":[{"success":1,"checksum":"c848986091e56699dc12de85adb1e39c","file_id":"9795","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_name":"2021_DescreteCompGeopmetry_Boissonnat.pdf","date_created":"2021-08-06T09:52:29Z","file_size":983307,"date_updated":"2021-08-06T09:52:29Z","creator":"kschuh"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"publication_status":"published","volume":66,"issue":"1","ec_funded":1,"project":[{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"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.","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","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.","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."},"title":"Triangulating submanifolds: An elementary and quantified version of Whitney’s method","author":[{"full_name":"Boissonnat, Jean-Daniel","last_name":"Boissonnat","first_name":"Jean-Daniel"},{"full_name":"Kachanovich, Siargey","last_name":"Kachanovich","first_name":"Siargey"},{"last_name":"Wintraecken","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"Yes (via OA deal)","external_id":{"isi":["000597770300001"]},"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).","publisher":"Springer Nature","quality_controlled":"1","oa":1,"day":"01","publication":"Discrete & Computational Geometry","isi":1,"has_accepted_license":"1","year":"2021","date_published":"2021-07-01T00:00:00Z","doi":"10.1007/s00454-020-00250-8","date_created":"2020-12-12T11:07:02Z","page":"386-434"},{"date_published":"2021-03-01T00:00:00Z","doi":"10.1007/s41468-020-00063-x","date_created":"2021-02-11T14:41:02Z","page":"99-140","day":"01","publication":"Journal of Applied and Computational Topology","has_accepted_license":"1","year":"2021","publisher":"Springer Nature","quality_controlled":"1","oa":1,"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).","title":"Probabilistic convergence and stability of random mapper graphs","author":[{"id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam","last_name":"Brown","full_name":"Brown, Adam"},{"last_name":"Bobrowski","full_name":"Bobrowski, Omer","first_name":"Omer"},{"full_name":"Munch, Elizabeth","last_name":"Munch","first_name":"Elizabeth"},{"full_name":"Wang, Bei","last_name":"Wang","first_name":"Bei"}],"article_processing_charge":"Yes (via OA deal)","external_id":{"arxiv":["1909.03488"]},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"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.","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.","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.","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."},"project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"issue":"1","volume":5,"ec_funded":1,"file":[{"file_id":"9112","checksum":"3f02e9d47c428484733da0f588a3c069","success":1,"content_type":"application/pdf","access_level":"open_access","relation":"main_file","date_created":"2021-02-11T14:43:59Z","file_name":"2020_JourApplCompTopology_Brown.pdf","date_updated":"2021-02-11T14:43:59Z","file_size":2090265,"creator":"dernst"}],"language":[{"iso":"eng"}],"publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"publication_status":"published","month":"03","intvolume":" 5","scopus_import":"1","oa_version":"Published Version","abstract":[{"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.","lang":"eng"}],"file_date_updated":"2021-02-11T14:43:59Z","department":[{"_id":"HeEd"}],"ddc":["510"],"date_updated":"2023-09-05T15:37:56Z","status":"public","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)"},"_id":"9111"},{"page":"134","doi":"10.15479/AT:ISTA:9056","date_published":"2021-02-01T00:00:00Z","date_created":"2021-02-02T14:11:06Z","has_accepted_license":"1","year":"2021","day":"01","publisher":"Institute of Science and Technology Austria","oa":1,"author":[{"last_name":"Osang","full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116","first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"No","title":"Multi-cover persistence and Delaunay mosaics","citation":{"mla":"Osang, Georg F. Multi-Cover Persistence and Delaunay Mosaics. Institute of Science and Technology Austria, 2021, doi:10.15479/AT:ISTA:9056.","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","ieee":"G. F. Osang, “Multi-cover persistence and Delaunay mosaics,” Institute of Science and Technology Austria, Klosterneuburg, 2021.","short":"G.F. Osang, Multi-Cover Persistence and Delaunay Mosaics, Institute of Science and Technology Austria, 2021.","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."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","related_material":{"record":[{"relation":"part_of_dissertation","id":"187","status":"public"},{"relation":"part_of_dissertation","id":"8703","status":"public"}]},"publication_identifier":{"issn":["2663-337X"]},"publication_status":"published","degree_awarded":"PhD","file":[{"file_id":"9063","checksum":"bcf27986147cab0533b6abadd74e7629","access_level":"closed","relation":"source_file","content_type":"application/zip","date_created":"2021-02-02T14:09:25Z","file_name":"thesis_source.zip","creator":"patrickd","date_updated":"2021-02-03T10:37:28Z","file_size":13446994},{"creator":"patrickd","file_size":5210329,"date_updated":"2021-02-02T14:09:18Z","file_name":"thesis_pdfA2b.pdf","date_created":"2021-02-02T14:09:18Z","relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"checksum":"9cc8af266579a464385bbe2aff6af606","file_id":"9064"}],"language":[{"iso":"eng"}],"alternative_title":["ISTA Thesis"],"month":"02","place":"Klosterneuburg","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."}],"oa_version":"Published Version","file_date_updated":"2021-02-03T10:37:28Z","department":[{"_id":"HeEd"},{"_id":"GradSch"}],"supervisor":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"}],"date_updated":"2023-09-07T13:29:01Z","ddc":["006","514","516"],"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)"},"status":"public","_id":"9056"},{"project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"Z00342","name":"The Wittgenstein Prize","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"external_id":{"isi":["000700090000001"],"pmid":["34569592"]},"article_processing_charge":"No","author":[{"first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","orcid":"0000-0002-8882-5116","full_name":"Osang, Georg F"},{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"last_name":"Saadatfar","full_name":"Saadatfar, Mohammad","first_name":"Mohammad"}],"title":"Topological signatures and stability of hexagonal close packing and Barlow stackings","citation":{"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.","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.","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.","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","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","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."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"quality_controlled":"1","publisher":"Royal Society of Chemistry ","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.","page":"9107-9115","date_created":"2021-10-31T23:01:30Z","date_published":"2021-10-20T00:00:00Z","doi":"10.1039/d1sm00774b","year":"2021","has_accepted_license":"1","isi":1,"publication":"Soft Matter","day":"20","article_type":"original","type":"journal_article","status":"public","_id":"10204","file_date_updated":"2023-10-03T09:21:42Z","department":[{"_id":"HeEd"}],"date_updated":"2023-10-03T09:24:27Z","ddc":["540"],"scopus_import":"1","intvolume":" 17","month":"10","abstract":[{"lang":"eng","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."}],"oa_version":"Submitted Version","pmid":1,"ec_funded":1,"volume":17,"issue":"40","publication_status":"published","publication_identifier":{"issn":["1744-683X"],"eissn":["1744-6848"]},"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,"file_id":"14385","checksum":"b4da0c420530295e61b153960f6cb350"}]},{"volume":189,"related_material":{"link":[{"url":"https://arxiv.org/abs/2103.07823","relation":"extended_version"}],"record":[{"id":"12709","status":"public","relation":"later_version"}]},"language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","access_level":"open_access","relation":"main_file","checksum":"0de217501e7ba8b267d58deed0d51761","file_id":"9610","success":1,"date_updated":"2021-06-28T12:40:47Z","file_size":"1367983","creator":"cziletti","date_created":"2021-06-28T12:40:47Z","file_name":"2021_LIPIcs_Corbet.pdf"}],"publication_status":"published","publication_identifier":{"isbn":["9783959771849"],"issn":["18688969"]},"intvolume":" 189","month":"06","scopus_import":"1","alternative_title":["LIPIcs"],"oa_version":"Published Version","abstract":[{"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. ","lang":"eng"}],"department":[{"_id":"HeEd"}],"file_date_updated":"2021-06-28T12:40:47Z","ddc":["516"],"date_updated":"2023-10-04T12:03:39Z","status":"public","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)"},"type":"conference","_id":"9605","date_created":"2021-06-27T22:01:49Z","date_published":"2021-06-02T00:00:00Z","doi":"10.4230/LIPIcs.SoCG.2021.27","publication":"Leibniz International Proceedings in Informatics","day":"02","year":"2021","has_accepted_license":"1","oa":1,"quality_controlled":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","acknowledgement":"The authors want to thank the reviewers for many helpful comments and suggestions.","title":"Computing the multicover bifiltration","article_processing_charge":"No","external_id":{"arxiv":["2103.07823"]},"author":[{"last_name":"Corbet","full_name":"Corbet, René","first_name":"René"},{"first_name":"Michael","last_name":"Kerber","full_name":"Kerber, Michael"},{"last_name":"Lesnick","full_name":"Lesnick, Michael","first_name":"Michael"},{"full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116","last_name":"Osang","first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"}],"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","citation":{"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.","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","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","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.","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.","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."},"article_number":"27"},{"author":[{"first_name":"Jean-Daniel","last_name":"Boissonnat","full_name":"Boissonnat, Jean-Daniel"},{"first_name":"Siargey","last_name":"Kachanovich","full_name":"Kachanovich, Siargey"},{"orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"No","title":"Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations","citation":{"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.","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","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","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.","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."},"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","project":[{"_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"page":"17:1-17:16","doi":"10.4230/LIPIcs.SoCG.2021.17","date_published":"2021-06-02T00:00:00Z","date_created":"2021-06-02T10:10:55Z","has_accepted_license":"1","year":"2021","day":"02","publication":"37th International Symposium on Computational Geometry (SoCG 2021)","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","oa":1,"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.","file_date_updated":"2021-06-02T10:22:33Z","department":[{"_id":"HeEd"}],"date_updated":"2023-10-10T07:34:34Z","ddc":["005","516","514"],"type":"conference","conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2021-06-11","location":"Virtual","start_date":"2021-06-07"},"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":"9441","series_title":"Leibniz International Proceedings in Informatics (LIPIcs)","related_material":{"record":[{"status":"public","id":"12960","relation":"later_version"}]},"volume":189,"ec_funded":1,"publication_identifier":{"issn":["1868-8969"],"isbn":["978-3-95977-184-9"]},"publication_status":"published","file":[{"file_name":"LIPIcs-SoCG-2021-17.pdf","date_created":"2021-06-02T10:22:33Z","file_size":1972902,"date_updated":"2021-06-02T10:22:33Z","creator":"mwintrae","success":1,"file_id":"9442","checksum":"c322aa48d5d35a35877896cc565705b6","content_type":"application/pdf","relation":"main_file","access_level":"open_access"}],"language":[{"iso":"eng"}],"alternative_title":["LIPIcs"],"place":"Dagstuhl, Germany","month":"06","intvolume":" 189","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. "}],"oa_version":"Published Version"},{"oa":1,"publisher":"Springer Nature","quality_controlled":"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).","page":"938-976","date_created":"2020-09-06T22:01:13Z","doi":"10.1007/s00454-020-00240-w","date_published":"2021-10-01T00:00:00Z","year":"2021","isi":1,"publication":"Discrete and Computational Geometry","day":"01","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"}],"external_id":{"arxiv":["1908.00856"],"isi":["000564488500002"]},"article_processing_charge":"No","author":[{"full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Bobenko","full_name":"Bobenko, Alexander I.","first_name":"Alexander I."},{"first_name":"Wolfgang K.","full_name":"Schief, Wolfgang K.","last_name":"Schief"},{"first_name":"Jan","last_name":"Techter","full_name":"Techter, Jan"}],"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.","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","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.","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","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1908.00856"}],"scopus_import":"1","intvolume":" 66","month":"10","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","ec_funded":1,"volume":66,"publication_status":"published","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"language":[{"iso":"eng"}],"article_type":"original","type":"journal_article","status":"public","_id":"8338","department":[{"_id":"HeEd"}],"date_updated":"2024-03-07T14:51:11Z"},{"project":[{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425"}],"external_id":{"isi":["000558119300001"]},"article_processing_charge":"Yes (via OA deal)","author":[{"full_name":"Boissonnat, Jean-Daniel","last_name":"Boissonnat","first_name":"Jean-Daniel"},{"first_name":"Ramsay","full_name":"Dyer, Ramsay","last_name":"Dyer"},{"first_name":"Arijit","full_name":"Ghosh, Arijit","last_name":"Ghosh"},{"full_name":"Lieutier, Andre","last_name":"Lieutier","first_name":"Andre"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken"}],"title":"Local conditions for triangulating submanifolds of Euclidean space","citation":{"short":"J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, M. Wintraecken, Discrete and Computational Geometry 66 (2021) 666–686.","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.","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","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.","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.","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."},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa":1,"publisher":"Springer Nature","quality_controlled":"1","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.","page":"666-686","date_created":"2020-08-11T07:11:51Z","doi":"10.1007/s00454-020-00233-9","date_published":"2021-09-01T00:00:00Z","year":"2021","has_accepted_license":"1","isi":1,"publication":"Discrete and Computational Geometry","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)"},"article_type":"original","type":"journal_article","status":"public","_id":"8248","department":[{"_id":"HeEd"}],"date_updated":"2024-03-07T14:54:59Z","ddc":["510"],"main_file_link":[{"url":"https://doi.org/10.1007/s00454-020-00233-9","open_access":"1"}],"scopus_import":"1","intvolume":" 66","month":"09","abstract":[{"lang":"eng","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."}],"oa_version":"Published Version","ec_funded":1,"volume":66,"publication_status":"published","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"language":[{"iso":"eng"}]},{"title":"Sheaf-theoretic stratification learning from geometric and topological perspectives","author":[{"full_name":"Brown, Adam","last_name":"Brown","id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam"},{"full_name":"Wang, Bei","last_name":"Wang","first_name":"Bei"}],"external_id":{"arxiv":["1712.07734"],"isi":["000536324700001"]},"article_processing_charge":"Yes (via OA deal)","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","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.","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","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","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."},"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"doi":"10.1007/s00454-020-00206-y","date_published":"2021-06-01T00:00:00Z","date_created":"2020-05-30T10:26:04Z","page":"1166-1198","day":"01","publication":"Discrete and Computational Geometry","has_accepted_license":"1","isi":1,"year":"2021","publisher":"Springer Nature","quality_controlled":"1","oa":1,"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.","department":[{"_id":"HeEd"}],"file_date_updated":"2020-11-25T09:06:41Z","ddc":["510"],"date_updated":"2024-03-07T15:01:58Z","status":"public","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)"},"_id":"7905","volume":65,"file":[{"success":1,"checksum":"487a84ea5841b75f04f66d7ebd71b67e","file_id":"8803","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_name":"2020_DiscreteCompGeometry_Brown.pdf","date_created":"2020-11-25T09:06:41Z","file_size":1013730,"date_updated":"2020-11-25T09:06:41Z","creator":"dernst"}],"language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"publication_status":"published","month":"06","intvolume":" 65","scopus_import":"1","oa_version":"Published Version","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."}]},{"department":[{"_id":"HeEd"}],"file_date_updated":"2020-11-20T10:18:02Z","ddc":["510"],"date_updated":"2021-01-12T08:14:13Z","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","_id":"7567","ec_funded":1,"volume":14,"language":[{"iso":"eng"}],"file":[{"success":1,"file_id":"8783","checksum":"1d145f3ab50ccee735983cb89236e609","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_name":"2020_MathCompScie_Choudhary.pdf","date_created":"2020-11-20T10:18:02Z","file_size":872275,"date_updated":"2020-11-20T10:18:02Z","creator":"dernst"}],"publication_status":"published","publication_identifier":{"eissn":["1661-8289"],"issn":["1661-8270"]},"intvolume":" 14","month":"03","scopus_import":"1","oa_version":"Published Version","abstract":[{"lang":"eng","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."}],"title":"Coxeter triangulations have good quality","article_processing_charge":"Yes (via OA deal)","author":[{"first_name":"Aruni","last_name":"Choudhary","full_name":"Choudhary, Aruni"},{"full_name":"Kachanovich, Siargey","last_name":"Kachanovich","first_name":"Siargey"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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.","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","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","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.","short":"A. Choudhary, S. Kachanovich, M. Wintraecken, Mathematics in Computer Science 14 (2020) 141–176.","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."},"project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"},{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411"}],"date_created":"2020-03-05T13:30:18Z","doi":"10.1007/s11786-020-00461-5","date_published":"2020-03-01T00:00:00Z","page":"141-176","publication":"Mathematics in Computer Science","day":"01","year":"2020","has_accepted_license":"1","oa":1,"quality_controlled":"1","publisher":"Springer Nature"}]