[{"article_processing_charge":"No","volume":14465,"date_updated":"2024-02-20T09:13:07Z","oa":1,"arxiv":1,"project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"}],"oa_version":"Preprint","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"János Pach’s Research partially supported by European Research Council (ERC), grant “GeoScape” No. 882971 and by the Hungarian Science Foundation (NKFIH), grant K-131529. Work by Morteza Saghafian is partially supported by the European Research Council (ERC), grant No. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31.","publication_identifier":{"eissn":["16113349"],"isbn":["9783031492716"],"issn":["03029743"]},"_id":"15012","citation":{"mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>31st International Symposium on Graph Drawing and Network Visualization</i>, vol. 14465, Springer Nature, 2024, pp. 339–46, doi:<a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">10.1007/978-3-031-49272-3_23</a>.","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. In: <i>31st International Symposium on Graph Drawing and Network Visualization</i>. Vol 14465. Springer Nature; 2024:339-346. doi:<a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">10.1007/978-3-031-49272-3_23</a>","ista":"Pach J, Saghafian M, Schnider P. 2024. Decomposition of geometric graphs into star-forests. 31st International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LNCS, vol. 14465, 339–346.","short":"J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346.","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” in <i>31st International Symposium on Graph Drawing and Network Visualization</i>, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp. 339–346.","apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2024). Decomposition of geometric graphs into star-forests. In <i>31st International Symposium on Graph Drawing and Network Visualization</i> (Vol. 14465, pp. 339–346). Isola delle Femmine, Palermo, Italy: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">https://doi.org/10.1007/978-3-031-49272-3_23</a>","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” In <i>31st International Symposium on Graph Drawing and Network Visualization</i>, 14465:339–46. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">https://doi.org/10.1007/978-3-031-49272-3_23</a>."},"publication_status":"published","author":[{"id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach","full_name":"Pach, János","first_name":"János"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"},{"first_name":"Patrick","full_name":"Schnider, Patrick","last_name":"Schnider"}],"abstract":[{"lang":"eng","text":"We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n-1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs."}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2306.13201","open_access":"1"}],"alternative_title":["LNCS"],"ec_funded":1,"year":"2024","doi":"10.1007/978-3-031-49272-3_23","title":"Decomposition of geometric graphs into star-forests","external_id":{"arxiv":["2306.13201"]},"page":"339-346","publication":"31st International Symposium on Graph Drawing and Network Visualization","type":"conference","day":"01","status":"public","intvolume":"     14465","department":[{"_id":"HeEd"}],"date_created":"2024-02-18T23:01:03Z","conference":{"start_date":"2023-09-20","end_date":"2023-09-22","location":"Isola delle Femmine, Palermo, Italy","name":"GD: Graph Drawing and Network Visualization"},"date_published":"2024-01-01T00:00:00Z","month":"01","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature"},{"_id":"14345","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"Work by all authors but A. Garber is supported by the European Research Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation.","quality_controlled":"1","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35"}],"oa_version":"Published Version","arxiv":1,"oa":1,"date_updated":"2023-12-13T12:25:06Z","article_processing_charge":"Yes (via OA deal)","abstract":[{"text":"For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2  is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)).","lang":"eng"}],"author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Alexey","last_name":"Garber","full_name":"Garber, Alexey"},{"first_name":"Mohadese","last_name":"Ghafari","full_name":"Ghafari, Mohadese"},{"full_name":"Heiss, Teresa","last_name":"Heiss","orcid":"0000-0002-1780-2689","first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"publication_status":"epub_ahead","citation":{"chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>.","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023.","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2023). On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry (2023).","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2023. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry.","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>","mla":"Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>."},"isi":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00454-023-00566-1"}],"title":"On angles in higher order Brillouin tessellations and related tilings in the plane","external_id":{"isi":["001060727600004"],"arxiv":["2204.01076"]},"year":"2023","doi":"10.1007/s00454-023-00566-1","ec_funded":1,"publication":"Discrete and Computational Geometry","status":"public","day":"07","type":"journal_article","date_created":"2023-09-17T22:01:10Z","department":[{"_id":"HeEd"}],"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"09","date_published":"2023-09-07T00:00:00Z","article_type":"original"},{"ddc":["000"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"Geometric characterization of the persistence of 1D maps","ec_funded":1,"doi":"10.1007/s41468-023-00126-9","year":"2023","oa_version":"Published Version","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"grant_number":"I4887","name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"},{"grant_number":"Z00342","call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","acknowledgement":"Open access funding provided by Austrian Science Fund (FWF). 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. The authors of this paper thank anonymous reviewers for their constructive criticism and Monika Henzinger for detailed comments on an earlier version of this paper.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"_id":"13182","article_processing_charge":"Yes (via OA deal)","oa":1,"date_updated":"2023-10-18T08:13:10Z","author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","orcid":"0000-0002-5372-7890","last_name":"Biswas","full_name":"Biswas, Ranita"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera Di Montesano","full_name":"Cultrera Di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"abstract":[{"lang":"eng","text":"We characterize critical points of 1-dimensional maps paired in persistent homology\r\ngeometrically and this way get elementary proofs of theorems about the symmetry\r\nof persistence diagrams and the variation of such maps. In particular, we identify\r\nbranching points and endpoints of networks as the sole source of asymmetry and\r\nrelate the cycle basis in persistent homology with a version of the stable marriage\r\nproblem. Our analysis provides the foundations of fast algorithms for maintaining a\r\ncollection of sorted lists together with its persistence diagram."}],"citation":{"ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2023. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology (2023).","mla":"Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2023). Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2023."},"publication_status":"epub_ahead","date_created":"2023-07-02T22:00:44Z","file":[{"access_level":"open_access","date_updated":"2023-07-03T09:41:05Z","file_size":487355,"file_name":"2023_Journal of Applied and Computational Topology_Biswas.pdf","checksum":"697249d5d1c61dea4410b9f021b70fce","date_created":"2023-07-03T09:41:05Z","relation":"main_file","content_type":"application/pdf","creator":"alisjak","file_id":"13185","success":1}],"department":[{"_id":"HeEd"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature","article_type":"original","date_published":"2023-06-17T00:00:00Z","month":"06","file_date_updated":"2023-07-03T09:41:05Z","publication":"Journal of Applied and Computational Topology","status":"public","type":"journal_article","day":"17"},{"ec_funded":1,"doi":"10.1007/s00453-022-01027-6","year":"2023","external_id":{"isi":["000846967100001"]},"title":"A simple algorithm for higher-order Delaunay mosaics and alpha shapes","isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["510"],"publication_status":"published","citation":{"ieee":"H. Edelsbrunner and G. F. Osang, “A simple algorithm for higher-order Delaunay mosaics and alpha shapes,” <i>Algorithmica</i>, vol. 85. Springer Nature, pp. 277–295, 2023.","apa":"Edelsbrunner, H., &#38; Osang, G. F. (2023). A simple algorithm for higher-order Delaunay mosaics and alpha shapes. <i>Algorithmica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00453-022-01027-6\">https://doi.org/10.1007/s00453-022-01027-6</a>","chicago":"Edelsbrunner, Herbert, and Georg F Osang. “A Simple Algorithm for Higher-Order Delaunay Mosaics and Alpha Shapes.” <i>Algorithmica</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00453-022-01027-6\">https://doi.org/10.1007/s00453-022-01027-6</a>.","ama":"Edelsbrunner H, Osang GF. A simple algorithm for higher-order Delaunay mosaics and alpha shapes. <i>Algorithmica</i>. 2023;85:277-295. doi:<a href=\"https://doi.org/10.1007/s00453-022-01027-6\">10.1007/s00453-022-01027-6</a>","mla":"Edelsbrunner, Herbert, and Georg F. Osang. “A Simple Algorithm for Higher-Order Delaunay Mosaics and Alpha Shapes.” <i>Algorithmica</i>, vol. 85, Springer Nature, 2023, pp. 277–95, doi:<a href=\"https://doi.org/10.1007/s00453-022-01027-6\">10.1007/s00453-022-01027-6</a>.","ista":"Edelsbrunner H, Osang GF. 2023. A simple algorithm for higher-order Delaunay mosaics and alpha shapes. Algorithmica. 85, 277–295.","short":"H. Edelsbrunner, G.F. Osang, Algorithmica 85 (2023) 277–295."},"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"id":"464B40D6-F248-11E8-B48F-1D18A9856A87","first_name":"Georg F","last_name":"Osang","full_name":"Osang, Georg F"}],"abstract":[{"lang":"eng","text":"We present a simple algorithm for computing higher-order Delaunay mosaics that works in Euclidean spaces of any finite dimensions. The algorithm selects the vertices of the order-k mosaic from incrementally constructed lower-order mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box to construct the order-k mosaic from its vertices. Beyond this black-box, the algorithm uses only combinatorial operations, thus facilitating easy implementation. We extend this algorithm to compute higher-order α-shapes and provide open-source implementations. We present experimental results for properties of higher-order Delaunay mosaics of random point sets."}],"date_updated":"2023-06-27T12:53:43Z","volume":85,"oa":1,"article_processing_charge":"Yes (via OA deal)","acknowledgement":"Open access funding provided by Austrian Science Fund (FWF). 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.","user_id":"2EBD1598-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35"}],"oa_version":"Published Version","_id":"12086","publication_identifier":{"eissn":["1432-0541"],"issn":["0178-4617"]},"date_published":"2023-01-01T00:00:00Z","article_type":"original","month":"01","language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","department":[{"_id":"HeEd"}],"has_accepted_license":"1","date_created":"2022-09-11T22:01:57Z","file":[{"date_updated":"2023-01-20T10:02:48Z","access_level":"open_access","file_name":"2023_Algorithmica_Edelsbrunner.pdf","file_size":911017,"date_created":"2023-01-20T10:02:48Z","checksum":"71685ca5121f4c837f40c3f8eb50c915","content_type":"application/pdf","relation":"main_file","file_id":"12322","creator":"dernst","success":1}],"type":"journal_article","day":"01","status":"public","intvolume":"        85","page":"277-295","file_date_updated":"2023-01-20T10:02:48Z","publication":"Algorithmica"},{"article_processing_charge":"No","oa":1,"volume":63,"date_updated":"2023-08-16T12:22:07Z","publication_identifier":{"eissn":["1549-960X"],"issn":["1549-9596"]},"pmid":1,"_id":"12544","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF","grant_number":"Z00342"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"oa_version":"Published Version","quality_controlled":"1","acknowledgement":"P.K. acknowledges support from the University of California Multicampus Research Programs and Initiatives (Grant No. M21PR3267) and from the NSF (Grant No.1760485). H.E. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program, 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\nOpen Access is funded by the Austrian Science Fund (FWF).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Koehl, Patrice, et al. “Computing the Volume, Surface Area, Mean, and Gaussian Curvatures of Molecules and Their Derivatives.” <i>Journal of Chemical Information and Modeling</i>, vol. 63, no. 3, American Chemical Society, 2023, pp. 973–85, doi:<a href=\"https://doi.org/10.1021/acs.jcim.2c01346\">10.1021/acs.jcim.2c01346</a>.","ama":"Koehl P, Akopyan A, Edelsbrunner H. Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. <i>Journal of Chemical Information and Modeling</i>. 2023;63(3):973-985. doi:<a href=\"https://doi.org/10.1021/acs.jcim.2c01346\">10.1021/acs.jcim.2c01346</a>","short":"P. Koehl, A. Akopyan, H. Edelsbrunner, Journal of Chemical Information and Modeling 63 (2023) 973–985.","ista":"Koehl P, Akopyan A, Edelsbrunner H. 2023. Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. Journal of Chemical Information and Modeling. 63(3), 973–985.","apa":"Koehl, P., Akopyan, A., &#38; Edelsbrunner, H. (2023). Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. <i>Journal of Chemical Information and Modeling</i>. American Chemical Society. <a href=\"https://doi.org/10.1021/acs.jcim.2c01346\">https://doi.org/10.1021/acs.jcim.2c01346</a>","ieee":"P. Koehl, A. Akopyan, and H. Edelsbrunner, “Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives,” <i>Journal of Chemical Information and Modeling</i>, vol. 63, no. 3. American Chemical Society, pp. 973–985, 2023.","chicago":"Koehl, Patrice, Arseniy Akopyan, and Herbert Edelsbrunner. “Computing the Volume, Surface Area, Mean, and Gaussian Curvatures of Molecules and Their Derivatives.” <i>Journal of Chemical Information and Modeling</i>. American Chemical Society, 2023. <a href=\"https://doi.org/10.1021/acs.jcim.2c01346\">https://doi.org/10.1021/acs.jcim.2c01346</a>."},"publication_status":"published","abstract":[{"lang":"eng","text":"Geometry is crucial in our efforts to comprehend the structures and dynamics of biomolecules. For example, volume, surface area, and integrated mean and Gaussian curvature of the union of balls representing a molecule are used to quantify its interactions with the water surrounding it in the morphometric implicit solvent models. The Alpha Shape theory provides an accurate and reliable method for computing these geometric measures. In this paper, we derive homogeneous formulas for the expressions of these measures and their derivatives with respect to the atomic coordinates, and we provide algorithms that implement them into a new software package, AlphaMol. The only variables in these formulas are the interatomic distances, making them insensitive to translations and rotations. AlphaMol includes a sequential algorithm and a parallel algorithm. In the parallel version, we partition the atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented by a buffer zone to account for atoms whose ball representations may partially cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up compared to an independent serial implementation when using 32 processors. For instance, it takes 31 s to compute the geometric measures and derivatives of each atom in a viral capsid with more than 26 million atoms on 32 Intel processors running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant computations, which ultimately limit the impact of using multiple processors. AlphaMol is available as an OpenSource software."}],"author":[{"first_name":"Patrice","full_name":"Koehl, Patrice","last_name":"Koehl"},{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy","last_name":"Akopyan","orcid":"0000-0002-2548-617X","first_name":"Arseniy"},{"first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"isi":1,"ddc":["510","540"],"doi":"10.1021/acs.jcim.2c01346","year":"2023","ec_funded":1,"title":"Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives","external_id":{"isi":["000920370700001"],"pmid":["36638318"]},"publication":"Journal of Chemical Information and Modeling","issue":"3","page":"973-985","file_date_updated":"2023-08-16T12:21:13Z","day":"13","type":"journal_article","intvolume":"        63","status":"public","has_accepted_license":"1","department":[{"_id":"HeEd"}],"date_created":"2023-02-12T23:00:59Z","file":[{"access_level":"open_access","date_updated":"2023-08-16T12:21:13Z","checksum":"7d20562269edff1e31b9d6019d4983b0","date_created":"2023-08-16T12:21:13Z","file_size":8069223,"file_name":"2023_JCIM_Koehl.pdf","creator":"dernst","file_id":"14070","relation":"main_file","content_type":"application/pdf","success":1}],"month":"02","date_published":"2023-02-13T00:00:00Z","article_type":"original","scopus_import":"1","publisher":"American Chemical Society","language":[{"iso":"eng"}]},{"has_accepted_license":"1","department":[{"_id":"HeEd"}],"conference":{"start_date":"2022-06-07","name":"SoCG: Symposium on Computational Geometry","end_date":"2022-06-10","location":"Berlin, Germany"},"date_created":"2022-06-01T14:18:04Z","file":[{"success":1,"relation":"main_file","content_type":"application/pdf","creator":"dernst","file_id":"11437","file_size":17580705,"file_name":"2022_LIPICs_Chambers.pdf","checksum":"b25ce40fade4ebc0bcaae176db4f5f1f","date_created":"2022-06-07T07:58:30Z","access_level":"open_access","date_updated":"2022-06-07T07:58:30Z"}],"month":"06","date_published":"2022-06-01T00:00:00Z","scopus_import":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","language":[{"iso":"eng"}],"publication":"38th International Symposium on Computational Geometry","page":"66:1-66:9","file_date_updated":"2022-06-07T07:58:30Z","day":"01","series_title":"LIPIcs","type":"conference","intvolume":"       224","status":"public","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["510"],"doi":"10.4230/LIPIcs.SoCG.2022.66","year":"2022","ec_funded":1,"title":"A cautionary tale: Burning the medial axis is unstable","editor":[{"first_name":"Xavier","full_name":"Goaoc, Xavier","last_name":"Goaoc"},{"first_name":"Michael","full_name":"Kerber, Michael","last_name":"Kerber"}],"article_processing_charge":"No","date_updated":"2023-02-21T09:50:52Z","volume":224,"oa":1,"publication_identifier":{"isbn":["978-3-95977-227-3"],"issn":["1868-8969"]},"_id":"11428","quality_controlled":"1","project":[{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses","grant_number":"M03073"},{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","grant_number":"754411"}],"oa_version":"Published Version","acknowledgement":"Partially supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics” and the European Research Council (ERC), grant no. 788183, “Alpha Shape Theory Extended”. Erin Chambers: Supported in part by the National Science Foundation through grants DBI-1759807, CCF-1907612, and CCF-2106672. Mathijs Wintraecken: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411. The Austrian science fund (FWF) M-3073 Acknowledgements We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie Yan and Tao Ju for sharing code to generate the examples.","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","citation":{"ama":"Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. A cautionary tale: Burning the medial axis is unstable. In: Goaoc X, Kerber M, eds. <i>38th International Symposium on Computational Geometry</i>. Vol 224. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2022:66:1-66:9. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2022.66\">10.4230/LIPIcs.SoCG.2022.66</a>","mla":"Chambers, Erin, et al. “A Cautionary Tale: Burning the Medial Axis Is Unstable.” <i>38th International Symposium on Computational Geometry</i>, edited by Xavier Goaoc and Michael Kerber, vol. 224, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022, p. 66:1-66:9, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2022.66\">10.4230/LIPIcs.SoCG.2022.66</a>.","short":"E. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, in:, X. Goaoc, M. Kerber (Eds.), 38th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022, p. 66:1-66:9.","ista":"Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. 2022. A cautionary tale: Burning the medial axis is unstable. 38th International Symposium on Computational Geometry. SoCG: Symposium on Computational GeometryLIPIcs vol. 224, 66:1-66:9.","ieee":"E. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “A cautionary tale: Burning the medial axis is unstable,” in <i>38th International Symposium on Computational Geometry</i>, Berlin, Germany, 2022, vol. 224, p. 66:1-66:9.","apa":"Chambers, E., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2022). A cautionary tale: Burning the medial axis is unstable. In X. Goaoc &#38; M. Kerber (Eds.), <i>38th International Symposium on Computational Geometry</i> (Vol. 224, p. 66:1-66:9). Berlin, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2022.66\">https://doi.org/10.4230/LIPIcs.SoCG.2022.66</a>","chicago":"Chambers, Erin, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “A Cautionary Tale: Burning the Medial Axis Is Unstable.” In <i>38th International Symposium on Computational Geometry</i>, edited by Xavier Goaoc and Michael Kerber, 224:66:1-66:9. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2022.66\">https://doi.org/10.4230/LIPIcs.SoCG.2022.66</a>."},"publication_status":"published","abstract":[{"text":"The medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible.","lang":"eng"}],"author":[{"last_name":"Chambers","full_name":"Chambers, Erin","first_name":"Erin"},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","last_name":"Fillmore","full_name":"Fillmore, Christopher D","first_name":"Christopher D"},{"id":"2D04F932-F248-11E8-B48F-1D18A9856A87","full_name":"Stephenson, Elizabeth R","last_name":"Stephenson","orcid":"0000-0002-6862-208X","first_name":"Elizabeth R"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220","first_name":"Mathijs"}]},{"status":"public","intvolume":"        30","series_title":"AWMS","type":"book_chapter","day":"27","page":"1-26","publication":"Research in Computational Topology 2","language":[{"iso":"eng"}],"edition":"1","scopus_import":"1","publisher":"Springer Nature","date_published":"2022-01-27T00:00:00Z","month":"01","date_created":"2022-06-07T08:21:11Z","department":[{"_id":"HeEd"}],"author":[{"first_name":"Bea","full_name":"Bleile, Bea","last_name":"Bleile"},{"first_name":"Adélie","full_name":"Garin, Adélie","last_name":"Garin"},{"full_name":"Heiss, Teresa","last_name":"Heiss","orcid":"0000-0002-1780-2689","first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Kelly","full_name":"Maggs, Kelly","last_name":"Maggs"},{"full_name":"Robins, Vanessa","last_name":"Robins","first_name":"Vanessa"}],"abstract":[{"lang":"eng","text":"To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation."}],"citation":{"mla":"Bleile, Bea, et al. “The Persistent Homology of Dual Digital Image Constructions.” <i>Research in Computational Topology 2</i>, edited by Ellen Gasparovic et al., 1st ed., vol. 30, Springer Nature, 2022, pp. 1–26, doi:<a href=\"https://doi.org/10.1007/978-3-030-95519-9_1\">10.1007/978-3-030-95519-9_1</a>.","ama":"Bleile B, Garin A, Heiss T, Maggs K, Robins V. The persistent homology of dual digital image constructions. In: Gasparovic E, Robins V, Turner K, eds. <i>Research in Computational Topology 2</i>. Vol 30. 1st ed. AWMS. Cham: Springer Nature; 2022:1-26. doi:<a href=\"https://doi.org/10.1007/978-3-030-95519-9_1\">10.1007/978-3-030-95519-9_1</a>","short":"B. Bleile, A. Garin, T. Heiss, K. Maggs, V. Robins, in:, E. Gasparovic, V. Robins, K. Turner (Eds.), Research in Computational Topology 2, 1st ed., Springer Nature, Cham, 2022, pp. 1–26.","ista":"Bleile B, Garin A, Heiss T, Maggs K, Robins V. 2022.The persistent homology of dual digital image constructions. In: Research in Computational Topology 2. Association for Women in Mathematics Series, vol. 30, 1–26.","apa":"Bleile, B., Garin, A., Heiss, T., Maggs, K., &#38; Robins, V. (2022). The persistent homology of dual digital image constructions. In E. Gasparovic, V. Robins, &#38; K. Turner (Eds.), <i>Research in Computational Topology 2</i> (1st ed., Vol. 30, pp. 1–26). Cham: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-95519-9_1\">https://doi.org/10.1007/978-3-030-95519-9_1</a>","ieee":"B. Bleile, A. Garin, T. Heiss, K. Maggs, and V. Robins, “The persistent homology of dual digital image constructions,” in <i>Research in Computational Topology 2</i>, 1st ed., vol. 30, E. Gasparovic, V. Robins, and K. Turner, Eds. Cham: Springer Nature, 2022, pp. 1–26.","chicago":"Bleile, Bea, Adélie Garin, Teresa Heiss, Kelly Maggs, and Vanessa Robins. “The Persistent Homology of Dual Digital Image Constructions.” In <i>Research in Computational Topology 2</i>, edited by Ellen Gasparovic, Vanessa Robins, and Katharine Turner, 1st ed., 30:1–26. AWMS. Cham: Springer Nature, 2022. <a href=\"https://doi.org/10.1007/978-3-030-95519-9_1\">https://doi.org/10.1007/978-3-030-95519-9_1</a>."},"publication_status":"published","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"}],"quality_controlled":"1","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This project started during the Women in Computational Topology workshop held in Canberra in July of 2019. All authors are very grateful for its organisation and the financial support for the workshop from the Mathematical Sciences Institute at ANU, the US National Science Foundation through the award CCF-1841455, the Australian Mathematical Sciences Institute and the Association for Women in Mathematics. AG is supported by the Swiss National Science Foundation grant CRSII5_177237. TH is supported by the European Research Council (ERC) Horizon 2020 project “Alpha Shape Theory Extended” No. 788183. KM is supported by the ERC Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 859860. VR was supported by Australian Research Council Future Fellowship FT140100604 during the early stages of this project.","publication_identifier":{"eisbn":["9783030955199"],"isbn":["9783030955182"]},"_id":"11440","article_processing_charge":"No","date_updated":"2022-06-07T08:32:42Z","volume":30,"oa":1,"editor":[{"full_name":"Gasparovic, Ellen","last_name":"Gasparovic","first_name":"Ellen"},{"first_name":"Vanessa","last_name":"Robins","full_name":"Robins, Vanessa"},{"first_name":"Katharine","full_name":"Turner, Katharine","last_name":"Turner"}],"arxiv":1,"external_id":{"arxiv":["2102.11397"]},"title":"The persistent homology of dual digital image constructions","ec_funded":1,"year":"2022","doi":"10.1007/978-3-030-95519-9_1","place":"Cham","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2102.11397"}],"alternative_title":["Association for Women in Mathematics Series"]},{"status":"public","type":"journal_article","day":"27","file_date_updated":"2022-07-27T09:25:53Z","publication":"Leibniz International Proceedings on Mathematics","language":[{"iso":"eng"}],"publisher":"Schloss Dagstuhl - Leibniz Zentrum für Informatik","date_published":"2022-07-27T00:00:00Z","month":"07","date_created":"2022-07-27T09:27:34Z","file":[{"date_updated":"2022-07-27T09:25:53Z","access_level":"open_access","date_created":"2022-07-27T09:25:53Z","checksum":"b2f511e8b1cae5f1892b0cdec341acac","file_name":"D-S-E.pdf","file_size":639266,"creator":"scultrer","file_id":"11659","content_type":"application/pdf","relation":"main_file"}],"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"has_accepted_license":"1","author":[{"first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano"},{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"abstract":[{"lang":"eng","text":"The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements."}],"citation":{"chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz Zentrum für Informatik, n.d.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz Zentrum für Informatik.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz Zentrum für Informatik.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Leibniz International Proceedings on Mathematics (n.d.).","mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Leibniz International Proceedings on Mathematics</i>, Schloss Dagstuhl - Leibniz Zentrum für Informatik.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Leibniz International Proceedings on Mathematics</i>."},"publication_status":"submitted","oa_version":"Submitted Version","project":[{"grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35"}],"quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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.","_id":"11658","article_processing_charge":"No","oa":1,"date_updated":"2022-07-28T07:57:48Z","title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","ec_funded":1,"year":"2022","ddc":["510"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"}},{"date_created":"2022-07-27T09:31:15Z","file":[{"content_type":"application/pdf","relation":"main_file","creator":"scultrer","file_id":"11661","file_name":"window 1.pdf","file_size":564836,"date_created":"2022-07-27T09:30:30Z","checksum":"95903f9d1649e8e437a967b6f2f64730","date_updated":"2022-07-27T09:30:30Z","access_level":"open_access"}],"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","date_published":"2022-07-25T00:00:00Z","month":"07","file_date_updated":"2022-07-27T09:30:30Z","publication":"LIPIcs","status":"public","type":"journal_article","day":"25","ddc":["510"],"alternative_title":["LIPIcs"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs","ec_funded":1,"year":"2022","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. ","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35"}],"quality_controlled":"1","oa_version":"Submitted Version","_id":"11660","oa":1,"date_updated":"2022-07-28T08:05:34Z","article_processing_charge":"No","author":[{"first_name":"Ranita","orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita","last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza"}],"abstract":[{"text":"We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. ","lang":"eng"}],"publication_status":"submitted","citation":{"apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs,” <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, n.d.","mla":"Biswas, Ranita, et al. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” <i>LIPIcs</i>, Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. <i>LIPIcs</i>.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, LIPIcs (n.d.)."}},{"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["510"],"doi":"10.1007/s40879-020-00405-0","year":"2022","ec_funded":1,"external_id":{"arxiv":["1912.12685"]},"title":"When different norms lead to same billiard trajectories?","arxiv":1,"date_updated":"2024-02-22T15:58:42Z","volume":8,"oa":1,"article_processing_charge":"Yes (via OA deal)","_id":"7791","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"acknowledgement":"AA was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). RK was supported by the Federal professorship program Grant 1.456.2016/1.4 and the Russian Foundation for Basic Research Grants 18-01-00036 and 19-01-00169. Open access funding provided by Institute of Science and Technology (IST Austria). The authors thank Alexey Balitskiy, Milena Radnović, and Serge Tabachnikov for useful discussions.","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","oa_version":"Published Version","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"publication_status":"published","citation":{"ama":"Akopyan A, Karasev R. When different norms lead to same billiard trajectories? <i>European Journal of Mathematics</i>. 2022;8(4):1309-1312. doi:<a href=\"https://doi.org/10.1007/s40879-020-00405-0\">10.1007/s40879-020-00405-0</a>","mla":"Akopyan, Arseniy, and Roman Karasev. “When Different Norms Lead to Same Billiard Trajectories?” <i>European Journal of Mathematics</i>, vol. 8, no. 4, Springer Nature, 2022, pp. 1309–12, doi:<a href=\"https://doi.org/10.1007/s40879-020-00405-0\">10.1007/s40879-020-00405-0</a>.","ista":"Akopyan A, Karasev R. 2022. When different norms lead to same billiard trajectories? European Journal of Mathematics. 8(4), 1309–1312.","short":"A. Akopyan, R. Karasev, European Journal of Mathematics 8 (2022) 1309–1312.","ieee":"A. Akopyan and R. Karasev, “When different norms lead to same billiard trajectories?,” <i>European Journal of Mathematics</i>, vol. 8, no. 4. Springer Nature, pp. 1309–1312, 2022.","apa":"Akopyan, A., &#38; Karasev, R. (2022). When different norms lead to same billiard trajectories? <i>European Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40879-020-00405-0\">https://doi.org/10.1007/s40879-020-00405-0</a>","chicago":"Akopyan, Arseniy, and Roman Karasev. “When Different Norms Lead to Same Billiard Trajectories?” <i>European Journal of Mathematics</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s40879-020-00405-0\">https://doi.org/10.1007/s40879-020-00405-0</a>."},"abstract":[{"lang":"eng","text":"Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law."}],"author":[{"orcid":"0000-0002-2548-617X","last_name":"Akopyan","full_name":"Akopyan, Arseniy","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Karasev","full_name":"Karasev, Roman","first_name":"Roman"}],"has_accepted_license":"1","department":[{"_id":"HeEd"}],"date_created":"2020-05-03T22:00:48Z","file":[{"checksum":"f53e71fd03744075adcd0b8fc1b8423d","date_created":"2020-05-04T10:33:42Z","file_size":263926,"file_name":"2020_EuropMathematics_Akopyan.pdf","access_level":"open_access","date_updated":"2020-07-14T12:48:03Z","file_id":"7796","creator":"dernst","relation":"main_file","content_type":"application/pdf"}],"month":"12","date_published":"2022-12-01T00:00:00Z","article_type":"original","publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"issue":"4","publication":"European Journal of Mathematics","file_date_updated":"2020-07-14T12:48:03Z","page":"1309 - 1312","day":"01","type":"journal_article","intvolume":"         8","status":"public"},{"language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","date_published":"2021-10-01T00:00:00Z","article_type":"original","month":"10","date_created":"2020-09-06T22:01:13Z","department":[{"_id":"HeEd"}],"status":"public","intvolume":"        66","type":"journal_article","day":"01","page":"938-976","publication":"Discrete and Computational Geometry","title":"On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs","external_id":{"isi":["000564488500002"],"arxiv":["1908.00856"]},"ec_funded":1,"year":"2021","doi":"10.1007/s00454-020-00240-w","isi":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1908.00856"}],"author":[{"last_name":"Akopyan","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","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.","last_name":"Schief","full_name":"Schief, Wolfgang K."},{"full_name":"Techter, Jan","last_name":"Techter","first_name":"Jan"}],"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"}],"publication_status":"published","citation":{"ieee":"A. Akopyan, A. I. Bobenko, W. K. Schief, and J. Techter, “On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs,” <i>Discrete and Computational Geometry</i>, vol. 66. Springer Nature, pp. 938–976, 2021.","apa":"Akopyan, A., Bobenko, A. I., Schief, W. K., &#38; Techter, J. (2021). On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00240-w\">https://doi.org/10.1007/s00454-020-00240-w</a>","chicago":"Akopyan, Arseniy, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00240-w\">https://doi.org/10.1007/s00454-020-00240-w</a>.","mla":"Akopyan, Arseniy, et al. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” <i>Discrete and Computational Geometry</i>, vol. 66, Springer Nature, 2021, pp. 938–76, doi:<a href=\"https://doi.org/10.1007/s00454-020-00240-w\">10.1007/s00454-020-00240-w</a>.","ama":"Akopyan A, Bobenko AI, Schief WK, Techter J. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. <i>Discrete and Computational Geometry</i>. 2021;66:938-976. doi:<a href=\"https://doi.org/10.1007/s00454-020-00240-w\">10.1007/s00454-020-00240-w</a>","short":"A. Akopyan, A.I. Bobenko, W.K. Schief, J. Techter, Discrete and Computational Geometry 66 (2021) 938–976.","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."},"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).","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"}],"quality_controlled":"1","_id":"8338","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"volume":66,"date_updated":"2024-03-07T14:51:11Z","oa":1,"article_processing_charge":"No","arxiv":1},{"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"isi":1,"related_material":{"record":[{"status":"public","id":"187","relation":"earlier_version"}]},"ddc":["516"],"doi":"10.1007/s00454-021-00281-9","year":"2021","ec_funded":1,"title":"The multi-cover persistence of Euclidean balls","external_id":{"isi":["000635460400001"]},"volume":65,"oa":1,"date_updated":"2023-08-07T14:35:44Z","article_processing_charge":"Yes (via OA deal)","_id":"9317","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"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).","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa_version":"Published Version","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"quality_controlled":"1","publication_status":"published","citation":{"ieee":"H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean balls,” <i>Discrete and Computational Geometry</i>, vol. 65. Springer Nature, pp. 1296–1313, 2021.","apa":"Edelsbrunner, H., &#38; Osang, G. F. (2021). The multi-cover persistence of Euclidean balls. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-021-00281-9\">https://doi.org/10.1007/s00454-021-00281-9</a>","chicago":"Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence of Euclidean Balls.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-021-00281-9\">https://doi.org/10.1007/s00454-021-00281-9</a>.","ama":"Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. <i>Discrete and Computational Geometry</i>. 2021;65:1296–1313. doi:<a href=\"https://doi.org/10.1007/s00454-021-00281-9\">10.1007/s00454-021-00281-9</a>","mla":"Edelsbrunner, Herbert, and Georg F. Osang. “The Multi-Cover Persistence of Euclidean Balls.” <i>Discrete and Computational Geometry</i>, vol. 65, Springer Nature, 2021, pp. 1296–1313, doi:<a href=\"https://doi.org/10.1007/s00454-021-00281-9\">10.1007/s00454-021-00281-9</a>.","short":"H. Edelsbrunner, G.F. Osang, Discrete and Computational Geometry 65 (2021) 1296–1313.","ista":"Edelsbrunner H, Osang GF. 2021. The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. 65, 1296–1313."},"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"}],"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","full_name":"Osang, Georg F","first_name":"Georg F"}],"has_accepted_license":"1","department":[{"_id":"HeEd"}],"date_created":"2021-04-11T22:01:15Z","file":[{"success":1,"file_id":"10394","creator":"cchlebak","relation":"main_file","content_type":"application/pdf","checksum":"59b4e1e827e494209bcb4aae22e1d347","date_created":"2021-12-01T10:56:53Z","file_size":677704,"file_name":"2021_DisCompGeo_Edelsbrunner_Osang.pdf","access_level":"open_access","date_updated":"2021-12-01T10:56:53Z"}],"month":"03","date_published":"2021-03-31T00:00:00Z","article_type":"original","publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"publication":"Discrete and Computational Geometry","file_date_updated":"2021-12-01T10:56:53Z","page":"1296–1313","day":"31","type":"journal_article","intvolume":"        65","status":"public"},{"file_date_updated":"2021-04-22T08:08:14Z","page":"32:1-32:16","publication":"37th International Symposium on Computational Geometry (SoCG 2021)","status":"public","intvolume":"       189","type":"conference","day":"02","date_created":"2021-04-22T08:09:58Z","file":[{"date_updated":"2021-04-22T08:08:14Z","access_level":"open_access","date_created":"2021-04-22T08:08:14Z","checksum":"1787baef1523d6d93753b90d0c109a6d","file_name":"df_socg_final_version.pdf","file_size":3117435,"file_id":"9346","creator":"mwintrae","content_type":"application/pdf","relation":"main_file","success":1}],"conference":{"name":"SoCG: Symposium on Computational Geometry","location":"Virtual","end_date":"2021-06-11","start_date":"2021-06-07"},"department":[{"_id":"HeEd"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","date_published":"2021-06-02T00:00:00Z","month":"06","user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","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.","oa_version":"Published Version","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics","grant_number":"I4887"},{"call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"25C5A090-B435-11E9-9278-68D0E5697425","grant_number":"Z00312"},{"call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"}],"quality_controlled":"1","_id":"9345","publication_identifier":{"issn":["1868-8969"]},"oa":1,"date_updated":"2023-02-23T13:55:40Z","volume":189,"article_processing_charge":"No","author":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-1780-2689","full_name":"Heiss, Teresa","last_name":"Heiss","first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Vitaliy","last_name":" Kurlin ","full_name":" Kurlin , Vitaliy"},{"first_name":"Philip","full_name":"Smith, Philip","last_name":"Smith"},{"first_name":"Mathijs","orcid":"0000-0002-7472-2220","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"text":"Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. 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.","lang":"eng"}],"publication_status":"published","citation":{"chicago":"Edelsbrunner, Herbert, Teresa Heiss, Vitaliy  Kurlin , Philip Smith, and Mathijs Wintraecken. “The Density Fingerprint of a Periodic Point Set.” In <i>37th International Symposium on Computational Geometry (SoCG 2021)</i>, 189:32:1-32:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">https://doi.org/10.4230/LIPIcs.SoCG.2021.32</a>.","apa":"Edelsbrunner, H., Heiss, T.,  Kurlin , V., Smith, P., &#38; Wintraecken, M. (2021). The density fingerprint of a periodic point set. In <i>37th International Symposium on Computational Geometry (SoCG 2021)</i> (Vol. 189, p. 32:1-32:16). Virtual: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">https://doi.org/10.4230/LIPIcs.SoCG.2021.32</a>","ieee":"H. Edelsbrunner, T. Heiss, V.  Kurlin , P. Smith, and M. Wintraecken, “The density fingerprint of a periodic point set,” in <i>37th International Symposium on Computational Geometry (SoCG 2021)</i>, Virtual, 2021, vol. 189, p. 32:1-32:16.","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.","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.","ama":"Edelsbrunner H, Heiss T,  Kurlin  V, Smith P, Wintraecken M. The density fingerprint of a periodic point set. In: <i>37th International Symposium on Computational Geometry (SoCG 2021)</i>. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021:32:1-32:16. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">10.4230/LIPIcs.SoCG.2021.32</a>","mla":"Edelsbrunner, Herbert, et al. “The Density Fingerprint of a Periodic Point Set.” <i>37th International Symposium on Computational Geometry (SoCG 2021)</i>, vol. 189, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 32:1-32:16, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">10.4230/LIPIcs.SoCG.2021.32</a>."},"ddc":["004","516"],"alternative_title":["LIPIcs"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"The density fingerprint of a periodic point set","ec_funded":1,"year":"2021","doi":"10.4230/LIPIcs.SoCG.2021.32"},{"abstract":[{"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.","lang":"eng"}],"author":[{"orcid":"0000-0002-5372-7890","last_name":"Biswas","full_name":"Biswas, Ranita","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"citation":{"mla":"Biswas, Ranita, et al. “Counting Cells of Order-k Voronoi Tessellations in ℝ<sup>3</sup> with Morse Theory.” <i>Leibniz International Proceedings in Informatics</i>, vol. 189, 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">10.4230/LIPIcs.SoCG.2021.16</a>.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. In: <i>Leibniz International Proceedings in Informatics</i>. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">10.4230/LIPIcs.SoCG.2021.16</a>","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2021. Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. Leibniz International Proceedings in Informatics. SoCG: International Symposium on Computational Geometry, LIPIcs, vol. 189, 16.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory,” in <i>Leibniz International Proceedings in Informatics</i>, Online, 2021, vol. 189.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2021). Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. In <i>Leibniz International Proceedings in Informatics</i> (Vol. 189). Online: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">https://doi.org/10.4230/LIPIcs.SoCG.2021.16</a>","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Counting Cells of Order-k Voronoi Tessellations in ℝ<sup>3</sup> with Morse Theory.” In <i>Leibniz International Proceedings in Informatics</i>, Vol. 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">https://doi.org/10.4230/LIPIcs.SoCG.2021.16</a>."},"publication_status":"published","publication_identifier":{"issn":["18688969"],"isbn":["9783959771849"]},"_id":"9604","quality_controlled":"1","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF"},{"grant_number":"I4887","name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"}],"oa_version":"Published Version","user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","article_processing_charge":"No","volume":189,"date_updated":"2023-02-23T14:02:28Z","oa":1,"title":"Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory","doi":"10.4230/LIPIcs.SoCG.2021.16","year":"2021","ec_funded":1,"ddc":["516"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"alternative_title":["LIPIcs"],"article_number":"16","intvolume":"       189","status":"public","day":"02","type":"conference","publication":"Leibniz International Proceedings in Informatics","file_date_updated":"2021-06-28T13:11:39Z","scopus_import":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","language":[{"iso":"eng"}],"month":"06","date_published":"2021-06-02T00:00:00Z","conference":{"start_date":"2021-06-07","location":"Online","end_date":"2021-06-11","name":"SoCG: International Symposium on Computational Geometry"},"file":[{"relation":"main_file","content_type":"application/pdf","file_id":"9611","creator":"asandaue","success":1,"access_level":"open_access","date_updated":"2021-06-28T13:11:39Z","file_size":727817,"file_name":"2021_LIPIcs_Biswas.pdf","checksum":"22b11a719018b22ecba2471b51f2eb40","date_created":"2021-06-28T13:11:39Z"}],"date_created":"2021-06-27T22:01:48Z","has_accepted_license":"1","department":[{"_id":"HeEd"}]},{"publisher":"Royal Society of Chemistry ","scopus_import":"1","language":[{"iso":"eng"}],"month":"10","date_published":"2021-10-20T00:00:00Z","article_type":"original","file":[{"date_updated":"2023-10-03T09:21:42Z","access_level":"open_access","file_name":"2021_SoftMatter_acceptedversion_Osang.pdf","file_size":4678788,"date_created":"2023-10-03T09:21:42Z","checksum":"b4da0c420530295e61b153960f6cb350","content_type":"application/pdf","relation":"main_file","file_id":"14385","creator":"dernst","success":1}],"date_created":"2021-10-31T23:01:30Z","has_accepted_license":"1","department":[{"_id":"HeEd"}],"intvolume":"        17","status":"public","day":"20","type":"journal_article","issue":"40","publication":"Soft Matter","page":"9107-9115","file_date_updated":"2023-10-03T09:21:42Z","title":"Topological signatures and stability of hexagonal close packing and Barlow stackings","external_id":{"isi":["000700090000001"],"pmid":["34569592"]},"doi":"10.1039/d1sm00774b","year":"2021","ec_funded":1,"ddc":["540"],"isi":1,"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."}],"author":[{"first_name":"Georg F","last_name":"Osang","full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Saadatfar","full_name":"Saadatfar, Mohammad","first_name":"Mohammad"}],"publication_status":"published","citation":{"ama":"Osang GF, Edelsbrunner H, Saadatfar M. Topological signatures and stability of hexagonal close packing and Barlow stackings. <i>Soft Matter</i>. 2021;17(40):9107-9115. doi:<a href=\"https://doi.org/10.1039/d1sm00774b\">10.1039/d1sm00774b</a>","mla":"Osang, Georg F., et al. “Topological Signatures and Stability of Hexagonal Close Packing and Barlow Stackings.” <i>Soft Matter</i>, vol. 17, no. 40, Royal Society of Chemistry , 2021, pp. 9107–15, doi:<a href=\"https://doi.org/10.1039/d1sm00774b\">10.1039/d1sm00774b</a>.","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.","short":"G.F. Osang, H. Edelsbrunner, M. Saadatfar, Soft Matter 17 (2021) 9107–9115.","apa":"Osang, G. F., Edelsbrunner, H., &#38; Saadatfar, M. (2021). Topological signatures and stability of hexagonal close packing and Barlow stackings. <i>Soft Matter</i>. Royal Society of Chemistry . <a href=\"https://doi.org/10.1039/d1sm00774b\">https://doi.org/10.1039/d1sm00774b</a>","ieee":"G. F. Osang, H. Edelsbrunner, and M. Saadatfar, “Topological signatures and stability of hexagonal close packing and Barlow stackings,” <i>Soft Matter</i>, vol. 17, no. 40. Royal Society of Chemistry , pp. 9107–9115, 2021.","chicago":"Osang, Georg F, Herbert Edelsbrunner, and Mohammad Saadatfar. “Topological Signatures and Stability of Hexagonal Close Packing and Barlow Stackings.” <i>Soft Matter</i>. Royal Society of Chemistry , 2021. <a href=\"https://doi.org/10.1039/d1sm00774b\">https://doi.org/10.1039/d1sm00774b</a>."},"pmid":1,"_id":"10204","publication_identifier":{"eissn":["1744-6848"],"issn":["1744-683X"]},"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.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize"}],"oa_version":"Submitted Version","volume":17,"date_updated":"2023-10-03T09:24:27Z","oa":1,"article_processing_charge":"No"},{"department":[{"_id":"HeEd"}],"has_accepted_license":"1","file":[{"file_id":"14053","creator":"dernst","relation":"main_file","content_type":"application/pdf","success":1,"access_level":"open_access","date_updated":"2023-08-14T11:55:10Z","checksum":"3514382e3a1eb87fa6c61ad622874415","date_created":"2023-08-14T11:55:10Z","file_size":1966019,"file_name":"2023_ExperimentalMath_Akopyan.pdf"}],"date_created":"2021-11-07T23:01:25Z","article_type":"original","date_published":"2021-10-25T00:00:00Z","month":"10","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Taylor and Francis","file_date_updated":"2023-08-14T11:55:10Z","page":"1-15","publication":"Experimental Mathematics","type":"journal_article","day":"25","status":"public","isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["510"],"ec_funded":1,"doi":"10.1080/10586458.2021.1980459","year":"2021","external_id":{"isi":["000710893500001"],"arxiv":["2007.07783"]},"title":"The beauty of random polytopes inscribed in the 2-sphere","article_processing_charge":"Yes (via OA deal)","date_updated":"2023-08-14T11:57:07Z","oa":1,"arxiv":1,"oa_version":"Published Version","quality_controlled":"1","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics","grant_number":"I4887"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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.","publication_identifier":{"issn":["1058-6458"],"eissn":["1944-950X"]},"_id":"10222","citation":{"ista":"Akopyan A, Edelsbrunner H, Nikitenko A. 2021. The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics., 1–15.","short":"A. Akopyan, H. Edelsbrunner, A. Nikitenko, Experimental Mathematics (2021) 1–15.","mla":"Akopyan, Arseniy, et al. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>, Taylor and Francis, 2021, pp. 1–15, doi:<a href=\"https://doi.org/10.1080/10586458.2021.1980459\">10.1080/10586458.2021.1980459</a>.","ama":"Akopyan A, Edelsbrunner H, Nikitenko A. The beauty of random polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. 2021:1-15. doi:<a href=\"https://doi.org/10.1080/10586458.2021.1980459\">10.1080/10586458.2021.1980459</a>","chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, and Anton Nikitenko. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>. Taylor and Francis, 2021. <a href=\"https://doi.org/10.1080/10586458.2021.1980459\">https://doi.org/10.1080/10586458.2021.1980459</a>.","apa":"Akopyan, A., Edelsbrunner, H., &#38; Nikitenko, A. (2021). The beauty of random polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. Taylor and Francis. <a href=\"https://doi.org/10.1080/10586458.2021.1980459\">https://doi.org/10.1080/10586458.2021.1980459</a>","ieee":"A. Akopyan, H. Edelsbrunner, and A. Nikitenko, “The beauty of random polytopes inscribed in the 2-sphere,” <i>Experimental Mathematics</i>. Taylor and Francis, pp. 1–15, 2021."},"publication_status":"published","author":[{"last_name":"Akopyan","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","last_name":"Nikitenko","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201"}],"abstract":[{"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.","lang":"eng"}]},{"year":"2021","doi":"10.1007/978-3-030-76657-3_10","ec_funded":1,"title":"Body centered cubic grid - coordinate system and discrete analytical plane definition","alternative_title":["LNCS"],"citation":{"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 <i>Discrete Geometry and Mathematical Morphology</i>, Uppsala, Sweden, 2021, vol. 12708, pp. 152–163.","apa":"Čomić, L., Zrour, R., Largeteau-Skapin, G., Biswas, R., &#38; Andres, E. (2021). Body centered cubic grid - coordinate system and discrete analytical plane definition. In <i>Discrete Geometry and Mathematical Morphology</i> (Vol. 12708, pp. 152–163). Uppsala, Sweden: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-76657-3_10\">https://doi.org/10.1007/978-3-030-76657-3_10</a>","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 <i>Discrete Geometry and Mathematical Morphology</i>, 12708:152–63. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/978-3-030-76657-3_10\">https://doi.org/10.1007/978-3-030-76657-3_10</a>.","mla":"Čomić, Lidija, et al. “Body Centered Cubic Grid - Coordinate System and Discrete Analytical Plane Definition.” <i>Discrete Geometry and Mathematical Morphology</i>, vol. 12708, Springer Nature, 2021, pp. 152–63, doi:<a href=\"https://doi.org/10.1007/978-3-030-76657-3_10\">10.1007/978-3-030-76657-3_10</a>.","ama":"Čomić L, Zrour R, Largeteau-Skapin G, Biswas R, Andres E. Body centered cubic grid - coordinate system and discrete analytical plane definition. In: <i>Discrete Geometry and Mathematical Morphology</i>. Vol 12708. Springer Nature; 2021:152-163. doi:<a href=\"https://doi.org/10.1007/978-3-030-76657-3_10\">10.1007/978-3-030-76657-3_10</a>","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.","short":"L. Čomić, R. Zrour, G. Largeteau-Skapin, R. Biswas, E. Andres, in:, Discrete Geometry and Mathematical Morphology, Springer Nature, 2021, pp. 152–163."},"publication_status":"published","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"}],"author":[{"full_name":"Čomić, Lidija","last_name":"Čomić","first_name":"Lidija"},{"first_name":"Rita","full_name":"Zrour, Rita","last_name":"Zrour"},{"full_name":"Largeteau-Skapin, Gaëlle","last_name":"Largeteau-Skapin","first_name":"Gaëlle"},{"full_name":"Biswas, Ranita","last_name":"Biswas","orcid":"0000-0002-5372-7890","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Eric","last_name":"Andres","full_name":"Andres, Eric"}],"article_processing_charge":"No","date_updated":"2022-05-31T06:58:21Z","volume":12708,"publication_identifier":{"issn":["03029743"],"isbn":["9783030766566"],"eissn":["16113349"]},"_id":"9824","quality_controlled":"1","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"oa_version":"None","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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).","month":"05","date_published":"2021-05-16T00:00:00Z","scopus_import":"1","publisher":"Springer Nature","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"conference":{"start_date":"2021-05-24","name":"DGMM: International Conference on Discrete Geometry and Mathematical Morphology","end_date":"2021-05-27","location":"Uppsala, Sweden"},"date_created":"2021-08-08T22:01:29Z","day":"16","type":"conference","intvolume":"     12708","status":"public","publication":"Discrete Geometry and Mathematical Morphology","page":"152-163"},{"alternative_title":["Abel Symposia"],"ddc":["510"],"ec_funded":1,"year":"2020","doi":"10.1007/978-3-030-43408-3_8","title":"Radius functions on Poisson–Delaunay mosaics and related complexes experimentally","article_processing_charge":"No","oa":1,"volume":15,"date_updated":"2021-01-12T08:17:06Z","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"_id":"2533E772-B435-11E9-9278-68D0E5697425","name":"Efficient Simulation of Natural Phenomena at Extremely Large Scales","call_identifier":"H2020","grant_number":"638176"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35"}],"quality_controlled":"1","oa_version":"Submitted Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","publication_identifier":{"eissn":["21978549"],"issn":["21932808"],"isbn":["9783030434076"]},"_id":"8135","citation":{"chicago":"Edelsbrunner, Herbert, Anton Nikitenko, Katharina Ölsböck, and Peter Synak. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” In <i>Topological Data Analysis</i>, 15:181–218. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">https://doi.org/10.1007/978-3-030-43408-3_8</a>.","ieee":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions on Poisson–Delaunay mosaics and related complexes experimentally,” in <i>Topological Data Analysis</i>, 2020, vol. 15, pp. 181–218.","apa":"Edelsbrunner, H., Nikitenko, A., Ölsböck, K., &#38; Synak, P. (2020). Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In <i>Topological Data Analysis</i> (Vol. 15, pp. 181–218). Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">https://doi.org/10.1007/978-3-030-43408-3_8</a>","short":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, P. Synak, in:, Topological Data Analysis, Springer Nature, 2020, pp. 181–218.","ista":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. 2020. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. Topological Data Analysis. , Abel Symposia, vol. 15, 181–218.","ama":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In: <i>Topological Data Analysis</i>. Vol 15. Springer Nature; 2020:181-218. doi:<a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">10.1007/978-3-030-43408-3_8</a>","mla":"Edelsbrunner, Herbert, et al. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” <i>Topological Data Analysis</i>, vol. 15, Springer Nature, 2020, pp. 181–218, doi:<a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">10.1007/978-3-030-43408-3_8</a>."},"publication_status":"published","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"full_name":"Nikitenko, Anton","last_name":"Nikitenko","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"},{"id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","full_name":"Ölsböck, Katharina","last_name":"Ölsböck","first_name":"Katharina"},{"first_name":"Peter","full_name":"Synak, Peter","last_name":"Synak","id":"331776E2-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"text":"Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.","lang":"eng"}],"department":[{"_id":"HeEd"}],"has_accepted_license":"1","file":[{"checksum":"7b5e0de10675d787a2ddb2091370b8d8","date_created":"2020-10-08T08:56:14Z","file_size":2207071,"file_name":"2020-B-01-PoissonExperimentalSurvey.pdf","access_level":"open_access","date_updated":"2020-10-08T08:56:14Z","success":1,"creator":"dernst","file_id":"8628","relation":"main_file","content_type":"application/pdf"}],"date_created":"2020-07-19T22:00:59Z","date_published":"2020-06-22T00:00:00Z","month":"06","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature","file_date_updated":"2020-10-08T08:56:14Z","page":"181-218","publication":"Topological Data Analysis","type":"conference","day":"22","status":"public","intvolume":"        15"},{"publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"_id":"8538","oa_version":"Preprint","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"}],"quality_controlled":"1","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","acknowledgement":" This paper would not be written if not for Dan Reznik’s curiosity and persistence; we are very grateful to him. We also thank R. Garcia and J. Koiller for interesting discussions. It is a pleasure to thank the Mathematical Institute of the University of Heidelberg for its stimulating atmosphere. ST thanks M. Bialy for interesting discussions and the Tel Aviv\r\nUniversity for its invariable hospitality. AA was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 78818 Alpha). RS is supported by NSF Grant DMS-1807320. ST was supported by NSF grant DMS-1510055 and SFB/TRR 191.","arxiv":1,"article_processing_charge":"No","oa":1,"date_updated":"2021-12-02T15:10:17Z","abstract":[{"text":"We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.","lang":"eng"}],"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","last_name":"Akopyan","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X"},{"first_name":"Richard","full_name":"Schwartz, Richard","last_name":"Schwartz"},{"first_name":"Serge","full_name":"Tabachnikov, Serge","last_name":"Tabachnikov"}],"citation":{"chicago":"Akopyan, Arseniy, Richard Schwartz, and Serge Tabachnikov. “Billiards in Ellipses Revisited.” <i>European Journal of Mathematics</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s40879-020-00426-9\">https://doi.org/10.1007/s40879-020-00426-9</a>.","ieee":"A. Akopyan, R. Schwartz, and S. Tabachnikov, “Billiards in ellipses revisited,” <i>European Journal of Mathematics</i>. Springer Nature, 2020.","apa":"Akopyan, A., Schwartz, R., &#38; Tabachnikov, S. (2020). Billiards in ellipses revisited. <i>European Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40879-020-00426-9\">https://doi.org/10.1007/s40879-020-00426-9</a>","short":"A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics (2020).","ista":"Akopyan A, Schwartz R, Tabachnikov S. 2020. Billiards in ellipses revisited. European Journal of Mathematics.","mla":"Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” <i>European Journal of Mathematics</i>, Springer Nature, 2020, doi:<a href=\"https://doi.org/10.1007/s40879-020-00426-9\">10.1007/s40879-020-00426-9</a>.","ama":"Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. <i>European Journal of Mathematics</i>. 2020. doi:<a href=\"https://doi.org/10.1007/s40879-020-00426-9\">10.1007/s40879-020-00426-9</a>"},"publication_status":"published","main_file_link":[{"url":"https://arxiv.org/abs/2001.02934","open_access":"1"}],"title":"Billiards in ellipses revisited","external_id":{"arxiv":["2001.02934"]},"doi":"10.1007/s40879-020-00426-9","year":"2020","ec_funded":1,"publication":"European Journal of Mathematics","status":"public","day":"09","type":"journal_article","date_created":"2020-09-20T22:01:38Z","department":[{"_id":"HeEd"}],"scopus_import":"1","publisher":"Springer Nature","language":[{"iso":"eng"}],"month":"09","date_published":"2020-09-09T00:00:00Z","article_type":"original"},{"citation":{"ama":"Osang GF, Rouxel-Labbé M, Teillaud M. Generalizing CGAL periodic Delaunay triangulations. In: <i>28th Annual European Symposium on Algorithms</i>. Vol 173. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:<a href=\"https://doi.org/10.4230/LIPIcs.ESA.2020.75\">10.4230/LIPIcs.ESA.2020.75</a>","mla":"Osang, Georg F., et al. “Generalizing CGAL Periodic Delaunay Triangulations.” <i>28th Annual European Symposium on Algorithms</i>, vol. 173, 75, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:<a href=\"https://doi.org/10.4230/LIPIcs.ESA.2020.75\">10.4230/LIPIcs.ESA.2020.75</a>.","short":"G.F. Osang, M. Rouxel-Labbé, M. Teillaud, in:, 28th Annual European Symposium on Algorithms, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.","ista":"Osang GF, Rouxel-Labbé M, Teillaud M. 2020. Generalizing CGAL periodic Delaunay triangulations. 28th Annual European Symposium on Algorithms. ESA: Annual European Symposium on Algorithms, LIPIcs, vol. 173, 75.","ieee":"G. F. Osang, M. Rouxel-Labbé, and M. Teillaud, “Generalizing CGAL periodic Delaunay triangulations,” in <i>28th Annual European Symposium on Algorithms</i>, Virtual, Online; Pisa, Italy, 2020, vol. 173.","apa":"Osang, G. F., Rouxel-Labbé, M., &#38; Teillaud, M. (2020). Generalizing CGAL periodic Delaunay triangulations. In <i>28th Annual European Symposium on Algorithms</i> (Vol. 173). Virtual, Online; Pisa, Italy: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.ESA.2020.75\">https://doi.org/10.4230/LIPIcs.ESA.2020.75</a>","chicago":"Osang, Georg F, Mael Rouxel-Labbé, and Monique Teillaud. “Generalizing CGAL Periodic Delaunay Triangulations.” In <i>28th Annual European Symposium on Algorithms</i>, Vol. 173. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. <a href=\"https://doi.org/10.4230/LIPIcs.ESA.2020.75\">https://doi.org/10.4230/LIPIcs.ESA.2020.75</a>."},"publication_status":"published","author":[{"id":"464B40D6-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8882-5116","full_name":"Osang, Georg F","last_name":"Osang","first_name":"Georg F"},{"first_name":"Mael","full_name":"Rouxel-Labbé, Mael","last_name":"Rouxel-Labbé"},{"full_name":"Teillaud, Monique","last_name":"Teillaud","first_name":"Monique"}],"abstract":[{"text":"Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity. 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