[{"citation":{"chicago":"Iglesias Ham, Mabel. “Multiple Covers with Balls.” Institute of Science and Technology Austria, 2018. https://doi.org/10.15479/AT:ISTA:th_1026.","ista":"Iglesias Ham M. 2018. Multiple covers with balls. Institute of Science and Technology Austria.","mla":"Iglesias Ham, Mabel. Multiple Covers with Balls. Institute of Science and Technology Austria, 2018, doi:10.15479/AT:ISTA:th_1026.","ieee":"M. Iglesias Ham, “Multiple covers with balls,” Institute of Science and Technology Austria, 2018.","short":"M. Iglesias Ham, Multiple Covers with Balls, Institute of Science and Technology Austria, 2018.","apa":"Iglesias Ham, M. (2018). Multiple covers with balls. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th_1026","ama":"Iglesias Ham M. Multiple covers with balls. 2018. doi:10.15479/AT:ISTA:th_1026"},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publist_id":"7712","author":[{"full_name":"Iglesias Ham, Mabel","last_name":"Iglesias Ham","first_name":"Mabel","id":"41B58C0C-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"No","title":"Multiple covers with balls","has_accepted_license":"1","year":"2018","day":"11","page":"171","doi":"10.15479/AT:ISTA:th_1026","date_published":"2018-06-11T00:00:00Z","date_created":"2018-12-11T11:45:10Z","publisher":"Institute of Science and Technology Austria","oa":1,"supervisor":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"date_updated":"2023-09-07T12:25:32Z","ddc":["514","516"],"file_date_updated":"2020-07-14T12:45:24Z","department":[{"_id":"HeEd"}],"_id":"201","type":"dissertation","status":"public","pubrep_id":"1026","publication_identifier":{"issn":["2663-337X"]},"degree_awarded":"PhD","publication_status":"published","file":[{"file_size":11827713,"date_updated":"2020-07-14T12:45:24Z","creator":"kschuh","file_name":"IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip","date_created":"2019-02-05T07:43:31Z","content_type":"application/zip","relation":"source_file","access_level":"closed","file_id":"5918","checksum":"dd699303623e96d1478a6ae07210dd05"},{"content_type":"application/pdf","relation":"main_file","access_level":"open_access","checksum":"ba163849a190d2b41d66fef0e4983294","file_id":"5919","file_size":4783846,"date_updated":"2020-07-14T12:45:24Z","creator":"kschuh","file_name":"IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf","date_created":"2019-02-05T07:43:45Z"}],"language":[{"iso":"eng"}],"abstract":[{"text":"We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications.","lang":"eng"}],"oa_version":"Published Version","alternative_title":["ISTA Thesis"],"month":"06"},{"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116"}],"publist_id":"7732","title":"The multi-cover persistence of Euclidean balls","citation":{"chicago":"Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence of Euclidean Balls,” Vol. 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. https://doi.org/10.4230/LIPIcs.SoCG.2018.34.","ista":"Edelsbrunner H, Osang GF. 2018. The multi-cover persistence of Euclidean balls. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 99, 34.","mla":"Edelsbrunner, Herbert, and Georg F. Osang. The Multi-Cover Persistence of Euclidean Balls. Vol. 99, 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, doi:10.4230/LIPIcs.SoCG.2018.34.","ieee":"H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean balls,” presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary, 2018, vol. 99.","short":"H. Edelsbrunner, G.F. Osang, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.","apa":"Edelsbrunner, H., & Osang, G. F. (2018). The multi-cover persistence of Euclidean balls (Vol. 99). Presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2018.34","ama":"Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. In: Vol 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2018. doi:10.4230/LIPIcs.SoCG.2018.34"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"article_number":"34","date_created":"2018-12-11T11:45:05Z","doi":"10.4230/LIPIcs.SoCG.2018.34","date_published":"2018-06-11T00:00:00Z","year":"2018","has_accepted_license":"1","day":"11","oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","acknowledgement":"This work is 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).","file_date_updated":"2020-07-14T12:45:19Z","department":[{"_id":"HeEd"}],"date_updated":"2023-09-07T13:29:00Z","ddc":["516"],"conference":{"start_date":"2018-06-11","location":"Budapest, Hungary","end_date":"2018-06-14","name":"SoCG: Symposium on Computational Geometry"},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"type":"conference","status":"public","_id":"187","volume":99,"related_material":{"record":[{"id":"9317","status":"public","relation":"later_version"},{"relation":"dissertation_contains","id":"9056","status":"public"}]},"publication_status":"published","language":[{"iso":"eng"}],"file":[{"file_name":"2018_LIPIcs_Edelsbrunner_Osang.pdf","date_created":"2018-12-18T09:27:22Z","file_size":528018,"date_updated":"2020-07-14T12:45:19Z","creator":"dernst","checksum":"d8c0533ad0018eb4ed1077475eb8fc18","file_id":"5738","content_type":"application/pdf","relation":"main_file","access_level":"open_access"}],"scopus_import":1,"alternative_title":["LIPIcs"],"intvolume":" 99","month":"06","abstract":[{"lang":"eng","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 from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers. "}],"oa_version":"Published Version"},{"type":"journal_article","article_type":"original","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"status":"public","_id":"692","department":[{"_id":"HeEd"}],"file_date_updated":"2020-07-14T12:47:44Z","date_updated":"2023-09-08T11:40:29Z","ddc":["510"],"scopus_import":"1","month":"06","intvolume":" 194","abstract":[{"text":"We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.","lang":"eng"}],"oa_version":"Published Version","volume":194,"issue":"1","ec_funded":1,"publication_status":"published","file":[{"file_size":1140860,"date_updated":"2020-07-14T12:47:44Z","creator":"kschuh","file_name":"2018_Springer_Akopyan.pdf","date_created":"2020-01-03T11:35:08Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","checksum":"1febcfc1266486053a069e3425ea3713","file_id":"7222"}],"language":[{"iso":"eng"}],"project":[{"call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme","grant_number":"291734"}],"publist_id":"7014","author":[{"first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan"}],"external_id":{"isi":["000431418800004"]},"article_processing_charge":"Yes (via OA deal)","title":"3-Webs generated by confocal conics and circles","citation":{"chicago":"Akopyan, Arseniy. “3-Webs Generated by Confocal Conics and Circles.” Geometriae Dedicata. Springer, 2018. https://doi.org/10.1007/s10711-017-0265-6.","ista":"Akopyan A. 2018. 3-Webs generated by confocal conics and circles. Geometriae Dedicata. 194(1), 55–64.","mla":"Akopyan, Arseniy. “3-Webs Generated by Confocal Conics and Circles.” Geometriae Dedicata, vol. 194, no. 1, Springer, 2018, pp. 55–64, doi:10.1007/s10711-017-0265-6.","ama":"Akopyan A. 3-Webs generated by confocal conics and circles. Geometriae Dedicata. 2018;194(1):55-64. doi:10.1007/s10711-017-0265-6","apa":"Akopyan, A. (2018). 3-Webs generated by confocal conics and circles. Geometriae Dedicata. Springer. https://doi.org/10.1007/s10711-017-0265-6","short":"A. Akopyan, Geometriae Dedicata 194 (2018) 55–64.","ieee":"A. Akopyan, “3-Webs generated by confocal conics and circles,” Geometriae Dedicata, vol. 194, no. 1. Springer, pp. 55–64, 2018."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publisher":"Springer","quality_controlled":"1","oa":1,"page":"55 - 64","date_published":"2018-06-01T00:00:00Z","doi":"10.1007/s10711-017-0265-6","date_created":"2018-12-11T11:47:57Z","has_accepted_license":"1","isi":1,"year":"2018","day":"01","publication":"Geometriae Dedicata"},{"ec_funded":1,"issue":"3","volume":32,"language":[{"iso":"eng"}],"publication_status":"published","intvolume":" 32","month":"09","main_file_link":[{"url":"https://arxiv.org/abs/1604.00960","open_access":"1"}],"scopus_import":"1","oa_version":"Preprint","abstract":[{"lang":"eng","text":"Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks."}],"department":[{"_id":"HeEd"}],"date_updated":"2023-09-11T12:48:39Z","status":"public","type":"journal_article","_id":"58","date_created":"2018-12-11T11:44:24Z","date_published":"2018-09-06T00:00:00Z","doi":"10.1137/16M110407X","page":"2242 - 2257","publication":"SIAM Journal on Discrete Mathematics","day":"06","year":"2018","isi":1,"oa":1,"publisher":"Society for Industrial and Applied Mathematics ","quality_controlled":"1","title":"Counting blanks in polygonal arrangements","article_processing_charge":"No","external_id":{"arxiv":["1604.00960"],"isi":["000450810500036"]},"publist_id":"7996","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","last_name":"Akopyan","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X"},{"last_name":"Segal Halevi","full_name":"Segal Halevi, Erel","first_name":"Erel"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"ista":"Akopyan A, Segal Halevi E. 2018. Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. 32(3), 2242–2257.","chicago":"Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics , 2018. https://doi.org/10.1137/16M110407X.","ieee":"A. Akopyan and E. Segal Halevi, “Counting blanks in polygonal arrangements,” SIAM Journal on Discrete Mathematics, vol. 32, no. 3. Society for Industrial and Applied Mathematics , pp. 2242–2257, 2018.","short":"A. Akopyan, E. Segal Halevi, SIAM Journal on Discrete Mathematics 32 (2018) 2242–2257.","apa":"Akopyan, A., & Segal Halevi, E. (2018). Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics . https://doi.org/10.1137/16M110407X","ama":"Akopyan A, Segal Halevi E. Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. 2018;32(3):2242-2257. doi:10.1137/16M110407X","mla":"Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” SIAM Journal on Discrete Mathematics, vol. 32, no. 3, Society for Industrial and Applied Mathematics , 2018, pp. 2242–57, doi:10.1137/16M110407X."},"project":[{"call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme","grant_number":"291734"}]},{"date_created":"2018-12-11T11:46:35Z","doi":"10.1090/tran/7292","date_published":"2018-04-01T00:00:00Z","page":"2825 - 2854","publication":"Transactions of the American Mathematical Society","day":"01","year":"2018","isi":1,"oa":1,"quality_controlled":"1","publisher":"American Mathematical Society","acknowledgement":"DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”; People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) REA grant agreement n◦[291734]","title":"Incircular nets and confocal conics","article_processing_charge":"No","external_id":{"isi":["000423197800019"]},"publist_id":"7363","author":[{"orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy"},{"last_name":"Bobenko","full_name":"Bobenko, Alexander","first_name":"Alexander"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"ista":"Akopyan A, Bobenko A. 2018. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 370(4), 2825–2854.","chicago":"Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” Transactions of the American Mathematical Society. American Mathematical Society, 2018. https://doi.org/10.1090/tran/7292.","ama":"Akopyan A, Bobenko A. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 2018;370(4):2825-2854. doi:10.1090/tran/7292","apa":"Akopyan, A., & Bobenko, A. (2018). Incircular nets and confocal conics. Transactions of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/tran/7292","ieee":"A. Akopyan and A. Bobenko, “Incircular nets and confocal conics,” Transactions of the American Mathematical Society, vol. 370, no. 4. American Mathematical Society, pp. 2825–2854, 2018.","short":"A. Akopyan, A. Bobenko, Transactions of the American Mathematical Society 370 (2018) 2825–2854.","mla":"Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” Transactions of the American Mathematical Society, vol. 370, no. 4, American Mathematical Society, 2018, pp. 2825–54, doi:10.1090/tran/7292."},"project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734","name":"International IST Postdoc Fellowship Programme"}],"ec_funded":1,"volume":370,"issue":"4","language":[{"iso":"eng"}],"publication_status":"published","intvolume":" 370","month":"04","main_file_link":[{"url":"https://arxiv.org/abs/1602.04637","open_access":"1"}],"scopus_import":"1","oa_version":"Preprint","abstract":[{"text":"We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the plane. These are congruences of straight lines in the plane with the combinatorics of the square grid, combinatorially colored as a checkerboard, such that all black coordinate quadrilaterals possess inscribed circles. We show how this larger class of IC-nets appears quite naturally in Laguerre geometry of oriented planes and spheres and leads to new remarkable incidence theorems. Most of our results are valid in hyperbolic and spherical geometries as well. We present also generalizations in spaces of higher dimension, called checkerboard IS-nets. The construction of these nets is based on a new 9 inspheres incidence theorem.","lang":"eng"}],"department":[{"_id":"HeEd"}],"date_updated":"2023-09-11T14:19:12Z","status":"public","type":"journal_article","_id":"458"}]