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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"}],"file":[{"checksum":"dd699303623e96d1478a6ae07210dd05","date_updated":"2020-07-14T12:45:24Z","access_level":"closed","file_name":"IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip","creator":"kschuh","file_size":11827713,"date_created":"2019-02-05T07:43:31Z","content_type":"application/zip","file_id":"5918","relation":"source_file"},{"file_name":"IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf","creator":"kschuh","file_size":4783846,"checksum":"ba163849a190d2b41d66fef0e4983294","date_updated":"2020-07-14T12:45:24Z","access_level":"open_access","content_type":"application/pdf","file_id":"5919","relation":"main_file","date_created":"2019-02-05T07:43:45Z"}],"publication_identifier":{"issn":["2663-337X"]},"date_updated":"2023-09-07T12:25:32Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","year":"2018","oa_version":"Published Version","has_accepted_license":"1","publist_id":"7712","degree_awarded":"PhD","publisher":"Institute of Science and Technology Austria","status":"public","department":[{"_id":"HeEd"}],"page":"171","date_created":"2018-12-11T11:45:10Z","supervisor":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner"}],"month":"06","ddc":["514","516"],"doi":"10.15479/AT:ISTA:th_1026","language":[{"iso":"eng"}],"title":"Multiple covers with balls","citation":{"chicago":"Iglesias Ham, Mabel. “Multiple Covers with Balls.” Institute of Science and Technology Austria, 2018. <a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">https://doi.org/10.15479/AT:ISTA:th_1026</a>.","ista":"Iglesias Ham M. 2018. Multiple covers with balls. Institute of Science and Technology Austria.","ieee":"M. Iglesias Ham, “Multiple covers with balls,” Institute of Science and Technology Austria, 2018.","mla":"Iglesias Ham, Mabel. <i>Multiple Covers with Balls</i>. Institute of Science and Technology Austria, 2018, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">10.15479/AT:ISTA:th_1026</a>.","short":"M. Iglesias Ham, Multiple Covers with Balls, Institute of Science and Technology Austria, 2018.","ama":"Iglesias Ham M. Multiple covers with balls. 2018. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">10.15479/AT:ISTA:th_1026</a>","apa":"Iglesias Ham, M. (2018). <i>Multiple covers with balls</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">https://doi.org/10.15479/AT:ISTA:th_1026</a>"},"alternative_title":["ISTA Thesis"],"author":[{"last_name":"Iglesias Ham","first_name":"Mabel","full_name":"Iglesias Ham, Mabel","id":"41B58C0C-F248-11E8-B48F-1D18A9856A87"}],"type":"dissertation","day":"11"},{"publisher":"Springer","isi":1,"quality_controlled":"1","department":[{"_id":"HeEd"}],"publication":"Geometriae Dedicata","intvolume":"       194","status":"public","page":"55 - 64","month":"06","date_created":"2018-12-11T11:47:57Z","project":[{"grant_number":"291734","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"language":[{"iso":"eng"}],"doi":"10.1007/s10711-017-0265-6","ddc":["510"],"ec_funded":1,"citation":{"mla":"Akopyan, Arseniy. “3-Webs Generated by Confocal Conics and Circles.” <i>Geometriae Dedicata</i>, vol. 194, no. 1, Springer, 2018, pp. 55–64, doi:<a href=\"https://doi.org/10.1007/s10711-017-0265-6\">10.1007/s10711-017-0265-6</a>.","ieee":"A. Akopyan, “3-Webs generated by confocal conics and circles,” <i>Geometriae Dedicata</i>, vol. 194, no. 1. Springer, pp. 55–64, 2018.","ista":"Akopyan A. 2018. 3-Webs generated by confocal conics and circles. Geometriae Dedicata. 194(1), 55–64.","chicago":"Akopyan, Arseniy. “3-Webs Generated by Confocal Conics and Circles.” <i>Geometriae Dedicata</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s10711-017-0265-6\">https://doi.org/10.1007/s10711-017-0265-6</a>.","apa":"Akopyan, A. (2018). 3-Webs generated by confocal conics and circles. <i>Geometriae Dedicata</i>. Springer. <a href=\"https://doi.org/10.1007/s10711-017-0265-6\">https://doi.org/10.1007/s10711-017-0265-6</a>","ama":"Akopyan A. 3-Webs generated by confocal conics and circles. <i>Geometriae Dedicata</i>. 2018;194(1):55-64. doi:<a href=\"https://doi.org/10.1007/s10711-017-0265-6\">10.1007/s10711-017-0265-6</a>","short":"A. Akopyan, Geometriae Dedicata 194 (2018) 55–64."},"title":"3-Webs generated by confocal conics and circles","day":"01","type":"journal_article","author":[{"orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","full_name":"Akopyan, Arseniy","first_name":"Arseniy"}],"publication_status":"published","license":"https://creativecommons.org/licenses/by/4.0/","oa":1,"file_date_updated":"2020-07-14T12:47:44Z","volume":194,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"issue":"1","article_processing_charge":"Yes (via OA deal)","file":[{"date_created":"2020-01-03T11:35:08Z","content_type":"application/pdf","relation":"main_file","file_id":"7222","checksum":"1febcfc1266486053a069e3425ea3713","date_updated":"2020-07-14T12:47:44Z","access_level":"open_access","file_name":"2018_Springer_Akopyan.pdf","file_size":1140860,"creator":"kschuh"}],"_id":"692","date_published":"2018-06-01T00:00:00Z","abstract":[{"lang":"eng","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."}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2023-09-08T11:40:29Z","external_id":{"isi":["000431418800004"]},"scopus_import":"1","publist_id":"7014","article_type":"original","has_accepted_license":"1","oa_version":"Published Version","year":"2018"},{"publisher":"arXiv","publication_status":"published","main_file_link":[{"url":"https://arxiv.org/abs/1804.03057","open_access":"1"}],"oa":1,"department":[{"_id":"HeEd"},{"_id":"JaMa"}],"status":"public","arxiv":1,"article_processing_charge":"No","article_number":"1804.03057","month":"09","_id":"75","abstract":[{"lang":"eng","text":"We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization."}],"date_created":"2018-12-11T11:44:30Z","date_published":"2018-09-13T00:00:00Z","project":[{"grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"related_material":{"record":[{"status":"public","id":"8156","relation":"dissertation_contains"}]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2023-12-18T10:51:02Z","language":[{"iso":"eng"}],"external_id":{"arxiv":["1804.03057"]},"doi":"10.48550/arXiv.1804.03057","ec_funded":1,"citation":{"apa":"Akopyan, A., Avvakumov, S., &#38; Karasev, R. (2018). Convex fair partitions into arbitrary number of pieces. arXiv. <a href=\"https://doi.org/10.48550/arXiv.1804.03057\">https://doi.org/10.48550/arXiv.1804.03057</a>","ama":"Akopyan A, Avvakumov S, Karasev R. Convex fair partitions into arbitrary number of pieces. 2018. doi:<a href=\"https://doi.org/10.48550/arXiv.1804.03057\">10.48550/arXiv.1804.03057</a>","short":"A. Akopyan, S. Avvakumov, R. Karasev, (2018).","mla":"Akopyan, Arseniy, et al. <i>Convex Fair Partitions into Arbitrary Number of Pieces</i>. 1804.03057, arXiv, 2018, doi:<a href=\"https://doi.org/10.48550/arXiv.1804.03057\">10.48550/arXiv.1804.03057</a>.","ieee":"A. Akopyan, S. Avvakumov, and R. Karasev, “Convex fair partitions into arbitrary number of pieces.” arXiv, 2018.","ista":"Akopyan A, Avvakumov S, Karasev R. 2018. Convex fair partitions into arbitrary number of pieces. 1804.03057.","chicago":"Akopyan, Arseniy, Sergey Avvakumov, and Roman Karasev. “Convex Fair Partitions into Arbitrary Number of Pieces.” arXiv, 2018. <a href=\"https://doi.org/10.48550/arXiv.1804.03057\">https://doi.org/10.48550/arXiv.1804.03057</a>."},"title":"Convex fair partitions into arbitrary number of pieces","day":"13","year":"2018","oa_version":"Preprint","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","last_name":"Akopyan","first_name":"Arseniy","full_name":"Akopyan, Arseniy"},{"id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","last_name":"Avvakumov","full_name":"Avvakumov, Sergey","first_name":"Sergey"},{"last_name":"Karasev","first_name":"Roman","full_name":"Karasev, Roman"}],"type":"preprint"},{"oa_version":"Preprint","year":"2018","publist_id":"7363","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2023-09-11T14:19:12Z","scopus_import":"1","external_id":{"isi":["000423197800019"]},"issue":"4","article_processing_charge":"No","_id":"458","abstract":[{"lang":"eng","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."}],"date_published":"2018-04-01T00:00:00Z","oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1602.04637"}],"publication_status":"published","volume":370,"title":"Incircular nets and confocal conics","ec_funded":1,"citation":{"chicago":"Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2018. <a href=\"https://doi.org/10.1090/tran/7292\">https://doi.org/10.1090/tran/7292</a>.","ieee":"A. Akopyan and A. Bobenko, “Incircular nets and confocal conics,” <i>Transactions of the American Mathematical Society</i>, vol. 370, no. 4. American Mathematical Society, pp. 2825–2854, 2018.","ista":"Akopyan A, Bobenko A. 2018. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 370(4), 2825–2854.","mla":"Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” <i>Transactions of the American Mathematical Society</i>, vol. 370, no. 4, American Mathematical Society, 2018, pp. 2825–54, doi:<a href=\"https://doi.org/10.1090/tran/7292\">10.1090/tran/7292</a>.","short":"A. Akopyan, A. Bobenko, Transactions of the American Mathematical Society 370 (2018) 2825–2854.","ama":"Akopyan A, Bobenko A. Incircular nets and confocal conics. <i>Transactions of the American Mathematical Society</i>. 2018;370(4):2825-2854. doi:<a href=\"https://doi.org/10.1090/tran/7292\">10.1090/tran/7292</a>","apa":"Akopyan, A., &#38; Bobenko, A. (2018). Incircular nets and confocal conics. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/7292\">https://doi.org/10.1090/tran/7292</a>"},"author":[{"last_name":"Akopyan","full_name":"Akopyan, Arseniy","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X"},{"last_name":"Bobenko","first_name":"Alexander","full_name":"Bobenko, Alexander"}],"type":"journal_article","day":"01","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]","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734","name":"International IST Postdoc Fellowship Programme"}],"doi":"10.1090/tran/7292","language":[{"iso":"eng"}],"page":"2825 - 2854","date_created":"2018-12-11T11:46:35Z","month":"04","isi":1,"publisher":"American Mathematical Society","intvolume":"       370","status":"public","department":[{"_id":"HeEd"}],"quality_controlled":"1","publication":"Transactions of the American Mathematical Society"},{"external_id":{"isi":["000415778300010"]},"scopus_import":"1","date_updated":"2023-09-13T08:59:00Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publist_id":"7289","year":"2018","has_accepted_license":"1","oa_version":"Preprint","file_date_updated":"2020-07-14T12:46:38Z","volume":68,"publication_status":"published","oa":1,"file":[{"checksum":"1c8d58cd489a66cd3e2064c1141c8c5e","date_updated":"2020-07-14T12:46:38Z","access_level":"open_access","file_name":"2018_Edelsbrunner.pdf","creator":"dernst","file_size":708357,"date_created":"2019-02-12T06:47:52Z","content_type":"application/pdf","relation":"main_file","file_id":"5953"}],"date_published":"2018-03-01T00:00:00Z","_id":"530","abstract":[{"lang":"eng","text":"Inclusion–exclusion is an effective method for computing the volume of a union of measurable sets. We extend it to multiple coverings, proving short inclusion–exclusion formulas for the subset of Rn covered by at least k balls in a finite set. We implement two of the formulas in dimension n=3 and report on results obtained with our software."}],"article_processing_charge":"No","language":[{"iso":"eng"}],"doi":"10.1016/j.comgeo.2017.06.014","ddc":["000"],"project":[{"name":"Topological Complex Systems","grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"day":"01","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","last_name":"Iglesias Ham","first_name":"Mabel","full_name":"Iglesias Ham, Mabel"}],"type":"journal_article","citation":{"ama":"Edelsbrunner H, Iglesias Ham M. Multiple covers with balls I: Inclusion–exclusion. <i>Computational Geometry: Theory and Applications</i>. 2018;68:119-133. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">10.1016/j.comgeo.2017.06.014</a>","apa":"Edelsbrunner, H., &#38; Iglesias Ham, M. (2018). Multiple covers with balls I: Inclusion–exclusion. <i>Computational Geometry: Theory and Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">https://doi.org/10.1016/j.comgeo.2017.06.014</a>","short":"H. Edelsbrunner, M. Iglesias Ham, Computational Geometry: Theory and Applications 68 (2018) 119–133.","mla":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls I: Inclusion–Exclusion.” <i>Computational Geometry: Theory and Applications</i>, vol. 68, Elsevier, 2018, pp. 119–33, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">10.1016/j.comgeo.2017.06.014</a>.","chicago":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls I: Inclusion–Exclusion.” <i>Computational Geometry: Theory and Applications</i>. Elsevier, 2018. <a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">https://doi.org/10.1016/j.comgeo.2017.06.014</a>.","ieee":"H. Edelsbrunner and M. Iglesias Ham, “Multiple covers with balls I: Inclusion–exclusion,” <i>Computational Geometry: Theory and Applications</i>, vol. 68. Elsevier, pp. 119–133, 2018.","ista":"Edelsbrunner H, Iglesias Ham M. 2018. Multiple covers with balls I: Inclusion–exclusion. Computational Geometry: Theory and Applications. 68, 119–133."},"ec_funded":1,"title":"Multiple covers with balls I: Inclusion–exclusion","publication":"Computational Geometry: Theory and Applications","quality_controlled":"1","department":[{"_id":"HeEd"}],"intvolume":"        68","status":"public","publisher":"Elsevier","isi":1,"month":"03","date_created":"2018-12-11T11:46:59Z","page":"119 - 133"},{"isi":1,"publisher":"Society for Industrial and Applied Mathematics ","intvolume":"        32","status":"public","publication":"SIAM Journal on Discrete Mathematics","quality_controlled":"1","department":[{"_id":"HeEd"}],"page":"2242 - 2257","date_created":"2018-12-11T11:44:24Z","month":"09","project":[{"name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"doi":"10.1137/16M110407X","language":[{"iso":"eng"}],"title":"Counting blanks in polygonal arrangements","citation":{"mla":"Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 32, no. 3, Society for Industrial and Applied Mathematics , 2018, pp. 2242–57, doi:<a href=\"https://doi.org/10.1137/16M110407X\">10.1137/16M110407X</a>.","ieee":"A. Akopyan and E. Segal Halevi, “Counting blanks in polygonal arrangements,” <i>SIAM Journal on Discrete Mathematics</i>, vol. 32, no. 3. Society for Industrial and Applied Mathematics , pp. 2242–2257, 2018.","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.” <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics , 2018. <a href=\"https://doi.org/10.1137/16M110407X\">https://doi.org/10.1137/16M110407X</a>.","apa":"Akopyan, A., &#38; Segal Halevi, E. (2018). Counting blanks in polygonal arrangements. <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics . <a href=\"https://doi.org/10.1137/16M110407X\">https://doi.org/10.1137/16M110407X</a>","ama":"Akopyan A, Segal Halevi E. Counting blanks in polygonal arrangements. <i>SIAM Journal on Discrete Mathematics</i>. 2018;32(3):2242-2257. doi:<a href=\"https://doi.org/10.1137/16M110407X\">10.1137/16M110407X</a>","short":"A. Akopyan, E. Segal Halevi, SIAM Journal on Discrete Mathematics 32 (2018) 2242–2257."},"ec_funded":1,"type":"journal_article","author":[{"full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Segal Halevi, Erel","first_name":"Erel","last_name":"Segal Halevi"}],"day":"06","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1604.00960"}],"oa":1,"publication_status":"published","volume":32,"article_processing_charge":"No","issue":"3","arxiv":1,"date_published":"2018-09-06T00:00:00Z","_id":"58","abstract":[{"lang":"eng","text":"Inside a two-dimensional region (``cake&quot;&quot;), there are m nonoverlapping tiles of a certain kind (``toppings&quot;&quot;). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,&quot;&quot; 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."}],"external_id":{"isi":["000450810500036"],"arxiv":["1604.00960"]},"scopus_import":"1","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2023-09-11T12:48:39Z","oa_version":"Preprint","year":"2018","publist_id":"7996"},{"volume":6,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"file_date_updated":"2020-07-14T12:47:28Z","oa":1,"publication_status":"published","_id":"6355","date_published":"2018-05-31T00:00:00Z","abstract":[{"lang":"eng","text":"We  prove  that  any  cyclic  quadrilateral  can  be  inscribed  in  any  closed  convex C1-curve.  The smoothness condition is not required if the quadrilateral is a rectangle."}],"file":[{"content_type":"application/pdf","relation":"main_file","file_id":"6356","date_created":"2019-04-30T06:14:58Z","file_name":"2018_ForumMahtematics_Akopyan.pdf","file_size":249246,"creator":"dernst","date_updated":"2020-07-14T12:47:28Z","access_level":"open_access","checksum":"5a71b24ba712a3eb2e46165a38fbc30a"}],"article_number":"e7","article_processing_charge":"No","arxiv":1,"publication_identifier":{"issn":["2050-5094"]},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2023-09-19T14:50:12Z","external_id":{"isi":["000433915500001"],"arxiv":["1712.10205"]},"year":"2018","has_accepted_license":"1","oa_version":"Published Version","status":"public","intvolume":"         6","department":[{"_id":"UlWa"},{"_id":"HeEd"},{"_id":"JaMa"}],"quality_controlled":"1","publication":"Forum of Mathematics, Sigma","isi":1,"publisher":"Cambridge University Press","date_created":"2019-04-30T06:09:57Z","month":"05","doi":"10.1017/fms.2018.7","ddc":["510"],"language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"8156"}]},"project":[{"name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425"}],"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","last_name":"Akopyan","full_name":"Akopyan, Arseniy","first_name":"Arseniy"},{"last_name":"Avvakumov","first_name":"Sergey","full_name":"Avvakumov, Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87"}],"type":"journal_article","day":"31","title":"Any cyclic quadrilateral can be inscribed in any closed convex smooth curve","ec_funded":1,"citation":{"chicago":"Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be Inscribed in Any Closed Convex Smooth Curve.” <i>Forum of Mathematics, Sigma</i>. Cambridge University Press, 2018. <a href=\"https://doi.org/10.1017/fms.2018.7\">https://doi.org/10.1017/fms.2018.7</a>.","ista":"Akopyan A, Avvakumov S. 2018. Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. Forum of Mathematics, Sigma. 6, e7.","ieee":"A. Akopyan and S. Avvakumov, “Any cyclic quadrilateral can be inscribed in any closed convex smooth curve,” <i>Forum of Mathematics, Sigma</i>, vol. 6. Cambridge University Press, 2018.","mla":"Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be Inscribed in Any Closed Convex Smooth Curve.” <i>Forum of Mathematics, Sigma</i>, vol. 6, e7, Cambridge University Press, 2018, doi:<a href=\"https://doi.org/10.1017/fms.2018.7\">10.1017/fms.2018.7</a>.","short":"A. Akopyan, S. Avvakumov, Forum of Mathematics, Sigma 6 (2018).","ama":"Akopyan A, Avvakumov S. Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. <i>Forum of Mathematics, Sigma</i>. 2018;6. doi:<a href=\"https://doi.org/10.1017/fms.2018.7\">10.1017/fms.2018.7</a>","apa":"Akopyan, A., &#38; Avvakumov, S. (2018). Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. <i>Forum of Mathematics, Sigma</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fms.2018.7\">https://doi.org/10.1017/fms.2018.7</a>"}},{"year":"2018","oa_version":"Preprint","publist_id":"7948","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2023-09-13T08:49:16Z","external_id":{"isi":["000444141200005"],"arxiv":["1702.05172"]},"scopus_import":"1","abstract":[{"lang":"eng","text":"The goal of this article is to introduce the reader to the theory of intrinsic geometry of convex surfaces. We illustrate the power of the tools by proving a theorem on convex surfaces containing an arbitrarily long closed simple geodesic. Let us remind ourselves that a curve in a surface is called geodesic if every sufficiently short arc of the curve is length minimizing; if, in addition, it has no self-intersections, we call it simple geodesic. A tetrahedron with equal opposite edges is called isosceles. The axiomatic method of Alexandrov geometry allows us to work with the metrics of convex surfaces directly, without approximating it first by a smooth or polyhedral metric. Such approximations destroy the closed geodesics on the surface; therefore it is difficult (if at all possible) to apply approximations in the proof of our theorem. On the other hand, a proof in the smooth or polyhedral case usually admits a translation into Alexandrov’s language; such translation makes the result more general. In fact, our proof resembles a translation of the proof given by Protasov. Note that the main theorem implies in particular that a smooth convex surface does not have arbitrarily long simple closed geodesics. However we do not know a proof of this corollary that is essentially simpler than the one presented below."}],"_id":"106","date_published":"2018-09-01T00:00:00Z","issue":"3","article_processing_charge":"No","arxiv":1,"volume":40,"oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1702.05172"}],"publication_status":"published","author":[{"orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","full_name":"Akopyan, Arseniy","first_name":"Arseniy"},{"first_name":"Anton","full_name":"Petrunin, Anton","last_name":"Petrunin"}],"type":"journal_article","day":"01","title":"Long geodesics on convex surfaces","citation":{"apa":"Akopyan, A., &#38; Petrunin, A. (2018). Long geodesics on convex surfaces. <i>Mathematical Intelligencer</i>. Springer. <a href=\"https://doi.org/10.1007/s00283-018-9795-5\">https://doi.org/10.1007/s00283-018-9795-5</a>","ama":"Akopyan A, Petrunin A. Long geodesics on convex surfaces. <i>Mathematical Intelligencer</i>. 2018;40(3):26-31. doi:<a href=\"https://doi.org/10.1007/s00283-018-9795-5\">10.1007/s00283-018-9795-5</a>","short":"A. Akopyan, A. Petrunin, Mathematical Intelligencer 40 (2018) 26–31.","mla":"Akopyan, Arseniy, and Anton Petrunin. “Long Geodesics on Convex Surfaces.” <i>Mathematical Intelligencer</i>, vol. 40, no. 3, Springer, 2018, pp. 26–31, doi:<a href=\"https://doi.org/10.1007/s00283-018-9795-5\">10.1007/s00283-018-9795-5</a>.","ieee":"A. Akopyan and A. Petrunin, “Long geodesics on convex surfaces,” <i>Mathematical Intelligencer</i>, vol. 40, no. 3. Springer, pp. 26–31, 2018.","ista":"Akopyan A, Petrunin A. 2018. Long geodesics on convex surfaces. Mathematical Intelligencer. 40(3), 26–31.","chicago":"Akopyan, Arseniy, and Anton Petrunin. “Long Geodesics on Convex Surfaces.” <i>Mathematical Intelligencer</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s00283-018-9795-5\">https://doi.org/10.1007/s00283-018-9795-5</a>."},"doi":"10.1007/s00283-018-9795-5","language":[{"iso":"eng"}],"date_created":"2018-12-11T11:44:40Z","month":"09","page":"26 - 31","intvolume":"        40","status":"public","quality_controlled":"1","department":[{"_id":"HeEd"}],"publication":"Mathematical Intelligencer","isi":1,"publisher":"Springer"},{"day":"01","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan"},{"full_name":"Balitskiy, Alexey","first_name":"Alexey","last_name":"Balitskiy"},{"full_name":"Grigorev, Mikhail","first_name":"Mikhail","last_name":"Grigorev"}],"type":"journal_article","citation":{"mla":"Akopyan, Arseniy, et al. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” <i>Discrete &#38; Computational Geometry</i>, vol. 59, no. 4, Springer, 2018, pp. 1001–09, doi:<a href=\"https://doi.org/10.1007/s00454-017-9883-x\">10.1007/s00454-017-9883-x</a>.","ista":"Akopyan A, Balitskiy A, Grigorev M. 2018. On the circle covering theorem by A.W. Goodman and R.E. Goodman. Discrete &#38; Computational Geometry. 59(4), 1001–1009.","ieee":"A. Akopyan, A. Balitskiy, and M. Grigorev, “On the circle covering theorem by A.W. Goodman and R.E. Goodman,” <i>Discrete &#38; Computational Geometry</i>, vol. 59, no. 4. Springer, pp. 1001–1009, 2018.","chicago":"Akopyan, Arseniy, Alexey Balitskiy, and Mikhail Grigorev. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” <i>Discrete &#38; Computational Geometry</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s00454-017-9883-x\">https://doi.org/10.1007/s00454-017-9883-x</a>.","apa":"Akopyan, A., Balitskiy, A., &#38; Grigorev, M. (2018). On the circle covering theorem by A.W. Goodman and R.E. Goodman. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s00454-017-9883-x\">https://doi.org/10.1007/s00454-017-9883-x</a>","ama":"Akopyan A, Balitskiy A, Grigorev M. On the circle covering theorem by A.W. Goodman and R.E. Goodman. <i>Discrete &#38; Computational Geometry</i>. 2018;59(4):1001-1009. doi:<a href=\"https://doi.org/10.1007/s00454-017-9883-x\">10.1007/s00454-017-9883-x</a>","short":"A. Akopyan, A. Balitskiy, M. Grigorev, Discrete &#38; Computational Geometry 59 (2018) 1001–1009."},"ec_funded":1,"title":"On the circle covering theorem by A.W. Goodman and R.E. Goodman","language":[{"iso":"eng"}],"ddc":["516","000"],"doi":"10.1007/s00454-017-9883-x","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","grant_number":"291734"}],"month":"06","date_created":"2018-12-11T11:49:57Z","page":"1001-1009","publication":"Discrete & Computational Geometry","quality_controlled":"1","department":[{"_id":"HeEd"}],"status":"public","intvolume":"        59","publisher":"Springer","isi":1,"article_type":"original","publist_id":"6324","has_accepted_license":"1","year":"2018","oa_version":"Published Version","scopus_import":"1","external_id":{"isi":["000432205500011"]},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2023-09-20T12:08:51Z","publication_identifier":{"issn":["01795376"],"eissn":["14320444"]},"file":[{"date_created":"2019-01-18T09:27:36Z","success":1,"content_type":"application/pdf","relation":"main_file","file_id":"5844","date_updated":"2019-01-18T09:27:36Z","access_level":"open_access","file_name":"2018_DiscreteComp_Akopyan.pdf","file_size":482518,"creator":"dernst"}],"abstract":[{"lang":"eng","text":"In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, … , rn in the plane, it is always possible to cover them by a disk of radius R= ∑ ri, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K⊂ Rd with homothety coefficients τ1, … , τn> 0 , it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets."}],"_id":"1064","date_published":"2018-06-01T00:00:00Z","article_processing_charge":"Yes (via OA deal)","issue":"4","file_date_updated":"2019-01-18T09:27:36Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"volume":59,"publication_status":"published","oa":1},{"publication_identifier":{"issn":["1631073X"]},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2023-09-13T09:34:12Z","scopus_import":"1","external_id":{"isi":["000430402700009"],"arxiv":["1805.01652"]},"oa_version":"Preprint","year":"2018","publist_id":"7420","article_type":"original","volume":356,"oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1805.01652"}],"publication_status":"published","_id":"409","date_published":"2018-04-01T00:00:00Z","abstract":[{"text":"We give a simple proof of T. Stehling's result [4], whereby in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except a finite number are hexagons.","lang":"eng"}],"issue":"4","article_processing_charge":"No","arxiv":1,"doi":"10.1016/j.crma.2018.03.005","language":[{"iso":"eng"}],"type":"journal_article","author":[{"full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"}],"day":"01","title":"On the number of non-hexagons in a planar tiling","citation":{"short":"A. Akopyan, Comptes Rendus Mathematique 356 (2018) 412–414.","ama":"Akopyan A. On the number of non-hexagons in a planar tiling. <i>Comptes Rendus Mathematique</i>. 2018;356(4):412-414. doi:<a href=\"https://doi.org/10.1016/j.crma.2018.03.005\">10.1016/j.crma.2018.03.005</a>","apa":"Akopyan, A. (2018). On the number of non-hexagons in a planar tiling. <i>Comptes Rendus Mathematique</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.crma.2018.03.005\">https://doi.org/10.1016/j.crma.2018.03.005</a>","chicago":"Akopyan, Arseniy. “On the Number of Non-Hexagons in a Planar Tiling.” <i>Comptes Rendus Mathematique</i>. Elsevier, 2018. <a href=\"https://doi.org/10.1016/j.crma.2018.03.005\">https://doi.org/10.1016/j.crma.2018.03.005</a>.","ieee":"A. Akopyan, “On the number of non-hexagons in a planar tiling,” <i>Comptes Rendus Mathematique</i>, vol. 356, no. 4. Elsevier, pp. 412–414, 2018.","ista":"Akopyan A. 2018. On the number of non-hexagons in a planar tiling. Comptes Rendus Mathematique. 356(4), 412–414.","mla":"Akopyan, Arseniy. “On the Number of Non-Hexagons in a Planar Tiling.” <i>Comptes Rendus Mathematique</i>, vol. 356, no. 4, Elsevier, 2018, pp. 412–14, doi:<a href=\"https://doi.org/10.1016/j.crma.2018.03.005\">10.1016/j.crma.2018.03.005</a>."},"intvolume":"       356","status":"public","quality_controlled":"1","department":[{"_id":"HeEd"}],"publication":"Comptes Rendus Mathematique","isi":1,"publisher":"Elsevier","date_created":"2018-12-11T11:46:19Z","month":"04","page":"412-414"},{"page":"3741 - 3762","month":"05","date_created":"2018-12-11T11:49:59Z","publisher":"American Mathematical Society","isi":1,"quality_controlled":"1","department":[{"_id":"HeEd"}],"publication":"Transactions of the American Mathematical Society","status":"public","intvolume":"       369","ec_funded":1,"citation":{"mla":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Complexes.” <i>Transactions of the American Mathematical Society</i>, vol. 369, no. 5, American Mathematical Society, 2017, pp. 3741–62, doi:<a href=\"https://doi.org/10.1090/tran/6991\">10.1090/tran/6991</a>.","ista":"Bauer U, Edelsbrunner H. 2017. The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. 369(5), 3741–3762.","ieee":"U. Bauer and H. Edelsbrunner, “The Morse theory of Čech and delaunay complexes,” <i>Transactions of the American Mathematical Society</i>, vol. 369, no. 5. American Mathematical Society, pp. 3741–3762, 2017.","chicago":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Complexes.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2017. <a href=\"https://doi.org/10.1090/tran/6991\">https://doi.org/10.1090/tran/6991</a>.","apa":"Bauer, U., &#38; Edelsbrunner, H. (2017). The Morse theory of Čech and delaunay complexes. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/6991\">https://doi.org/10.1090/tran/6991</a>","ama":"Bauer U, Edelsbrunner H. The Morse theory of Čech and delaunay complexes. <i>Transactions of the American Mathematical Society</i>. 2017;369(5):3741-3762. doi:<a href=\"https://doi.org/10.1090/tran/6991\">10.1090/tran/6991</a>","short":"U. Bauer, H. Edelsbrunner, Transactions of the American Mathematical Society 369 (2017) 3741–3762."},"title":"The Morse theory of Čech and delaunay complexes","day":"01","author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9683-0724","last_name":"Bauer","first_name":"Ulrich","full_name":"Bauer, Ulrich"},{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"}],"type":"journal_article","acknowledgement":"This research has been supported by the EU project Toposys(FP7-ICT-318493-STREP), by ESF under the ACAT Research Network Programme, by the Russian Government under mega project 11.G34.31.0053, and by the DFG Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics”.","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493","name":"Topological Complex Systems"}],"language":[{"iso":"eng"}],"doi":"10.1090/tran/6991","arxiv":1,"issue":"5","article_processing_charge":"No","_id":"1072","date_published":"2017-05-01T00:00:00Z","abstract":[{"lang":"eng","text":"Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field."}],"publication_status":"published","main_file_link":[{"url":"https://arxiv.org/abs/1312.1231","open_access":"1"}],"oa":1,"volume":369,"publist_id":"6311","article_type":"original","year":"2017","oa_version":"Preprint","date_updated":"2023-09-20T12:05:56Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","external_id":{"isi":["000398030400024"],"arxiv":["1312.1231"]},"scopus_import":"1"},{"publication_identifier":{"issn":["02099683"]},"date_updated":"2023-09-20T11:23:53Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","scopus_import":"1","external_id":{"isi":["000418056000005"]},"oa_version":"Submitted Version","year":"2017","publist_id":"6182","volume":37,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1411.6337"}],"oa":1,"publication_status":"published","abstract":[{"lang":"eng","text":"We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions."}],"_id":"1173","date_published":"2017-10-01T00:00:00Z","issue":"5","article_processing_charge":"No","doi":"10.1007/s00493-016-3308-y","language":[{"iso":"eng"}],"acknowledgement":"This research is partially supported by the Russian Government under the Mega Project 11.G34.31.0053, by the Toposys project FP7-ICT-318493-STREP, by ESF under the ACAT Research Network Programme, by RFBR grant 11-01-00735, and by NSF grants DMS-1101688, DMS-1400876.","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493","name":"Topological Complex Systems"}],"author":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"full_name":"Glazyrin, Alexey","first_name":"Alexey","last_name":"Glazyrin"},{"last_name":"Musin","first_name":"Oleg","full_name":"Musin, Oleg"},{"orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","full_name":"Nikitenko, Anton","last_name":"Nikitenko"}],"type":"journal_article","day":"01","title":"The Voronoi functional is maximized by the Delaunay triangulation in the plane","ec_funded":1,"citation":{"chicago":"Edelsbrunner, Herbert, Alexey Glazyrin, Oleg Musin, and Anton Nikitenko. “The Voronoi Functional Is Maximized by the Delaunay Triangulation in the Plane.” <i>Combinatorica</i>. Springer, 2017. <a href=\"https://doi.org/10.1007/s00493-016-3308-y\">https://doi.org/10.1007/s00493-016-3308-y</a>.","ieee":"H. Edelsbrunner, A. Glazyrin, O. Musin, and A. Nikitenko, “The Voronoi functional is maximized by the Delaunay triangulation in the plane,” <i>Combinatorica</i>, vol. 37, no. 5. Springer, pp. 887–910, 2017.","ista":"Edelsbrunner H, Glazyrin A, Musin O, Nikitenko A. 2017. The Voronoi functional is maximized by the Delaunay triangulation in the plane. Combinatorica. 37(5), 887–910.","mla":"Edelsbrunner, Herbert, et al. “The Voronoi Functional Is Maximized by the Delaunay Triangulation in the Plane.” <i>Combinatorica</i>, vol. 37, no. 5, Springer, 2017, pp. 887–910, doi:<a href=\"https://doi.org/10.1007/s00493-016-3308-y\">10.1007/s00493-016-3308-y</a>.","short":"H. Edelsbrunner, A. Glazyrin, O. Musin, A. Nikitenko, Combinatorica 37 (2017) 887–910.","ama":"Edelsbrunner H, Glazyrin A, Musin O, Nikitenko A. The Voronoi functional is maximized by the Delaunay triangulation in the plane. <i>Combinatorica</i>. 2017;37(5):887-910. doi:<a href=\"https://doi.org/10.1007/s00493-016-3308-y\">10.1007/s00493-016-3308-y</a>","apa":"Edelsbrunner, H., Glazyrin, A., Musin, O., &#38; Nikitenko, A. (2017). The Voronoi functional is maximized by the Delaunay triangulation in the plane. <i>Combinatorica</i>. Springer. <a href=\"https://doi.org/10.1007/s00493-016-3308-y\">https://doi.org/10.1007/s00493-016-3308-y</a>"},"status":"public","intvolume":"        37","department":[{"_id":"HeEd"}],"quality_controlled":"1","publication":"Combinatorica","isi":1,"publisher":"Springer","date_created":"2018-12-11T11:50:32Z","month":"10","page":"887 - 910"},{"publication_status":"published","oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1705.02045"}],"volume":10424,"article_processing_charge":"No","_id":"833","date_published":"2017-07-28T00:00:00Z","abstract":[{"text":"We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams.","lang":"eng"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2023-09-26T16:10:03Z","external_id":{"isi":["000432085900032"]},"scopus_import":"1","publication_identifier":{"issn":["03029743"]},"editor":[{"full_name":"Felsberg, Michael","first_name":"Michael","last_name":"Felsberg"},{"full_name":"Heyden, Anders","first_name":"Anders","last_name":"Heyden"},{"full_name":"Krüger, Norbert","first_name":"Norbert","last_name":"Krüger"}],"publist_id":"6815","oa_version":"Submitted Version","year":"2017","publisher":"Springer","isi":1,"department":[{"_id":"HeEd"}],"quality_controlled":"1","intvolume":"     10424","status":"public","page":"397 - 409","month":"07","date_created":"2018-12-11T11:48:45Z","language":[{"iso":"eng"}],"doi":"10.1007/978-3-319-64689-3_32","citation":{"mla":"Heiss, Teresa, and Hubert Wagner. <i>Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images</i>. Edited by Michael Felsberg et al., vol. 10424, Springer, 2017, pp. 397–409, doi:<a href=\"https://doi.org/10.1007/978-3-319-64689-3_32\">10.1007/978-3-319-64689-3_32</a>.","ista":"Heiss T, Wagner H. 2017. Streaming algorithm for Euler characteristic curves of multidimensional images. CAIP: Computer Analysis of Images and Patterns, LNCS, vol. 10424, 397–409.","ieee":"T. Heiss and H. Wagner, “Streaming algorithm for Euler characteristic curves of multidimensional images,” presented at the CAIP: Computer Analysis of Images and Patterns, Ystad, Sweden, 2017, vol. 10424, pp. 397–409.","chicago":"Heiss, Teresa, and Hubert Wagner. “Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images.” edited by Michael Felsberg, Anders Heyden, and Norbert Krüger, 10424:397–409. Springer, 2017. <a href=\"https://doi.org/10.1007/978-3-319-64689-3_32\">https://doi.org/10.1007/978-3-319-64689-3_32</a>.","apa":"Heiss, T., &#38; Wagner, H. (2017). Streaming algorithm for Euler characteristic curves of multidimensional images. In M. Felsberg, A. Heyden, &#38; N. Krüger (Eds.) (Vol. 10424, pp. 397–409). Presented at the CAIP: Computer Analysis of Images and Patterns, Ystad, Sweden: Springer. <a href=\"https://doi.org/10.1007/978-3-319-64689-3_32\">https://doi.org/10.1007/978-3-319-64689-3_32</a>","ama":"Heiss T, Wagner H. Streaming algorithm for Euler characteristic curves of multidimensional images. In: Felsberg M, Heyden A, Krüger N, eds. Vol 10424. Springer; 2017:397-409. doi:<a href=\"https://doi.org/10.1007/978-3-319-64689-3_32\">10.1007/978-3-319-64689-3_32</a>","short":"T. Heiss, H. Wagner, in:, M. Felsberg, A. Heyden, N. Krüger (Eds.), Springer, 2017, pp. 397–409."},"conference":{"name":"CAIP: Computer Analysis of Images and Patterns","start_date":"2017-08-22","end_date":"2017-08-24","location":"Ystad, Sweden"},"title":"Streaming algorithm for Euler characteristic curves of multidimensional images","day":"28","alternative_title":["LNCS"],"author":[{"last_name":"Heiss","first_name":"Teresa","full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1780-2689"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","full_name":"Wagner, Hubert","last_name":"Wagner"}],"type":"conference"},{"date_updated":"2023-09-26T15:50:52Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","external_id":{"isi":["000434088200008"]},"scopus_import":"1","publication_identifier":{"isbn":["978-331956930-7"]},"publist_id":"6812","oa_version":"None","year":"2017","publication_status":"published","volume":198,"article_processing_charge":"No","_id":"836","abstract":[{"text":"Recent research has examined how to study the topological features of a continuous self-map by means of the persistence of the eigenspaces, for given eigenvalues, of the endomorphism induced in homology over a field. This raised the question of how to select dynamically significant eigenvalues. The present paper aims to answer this question, giving an algorithm that computes the persistence of eigenspaces for every eigenvalue simultaneously, also expressing said eigenspaces as direct sums of “finite” and “singular” subspaces.","lang":"eng"}],"date_published":"2017-07-27T00:00:00Z","project":[{"grant_number":"318493","name":"Topological Complex Systems","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"language":[{"iso":"eng"}],"doi":"10.1007/978-3-319-56932-1_8","ec_funded":1,"citation":{"mla":"Ethier, Marc, et al. “Finding Eigenvalues of Self-Maps with the Kronecker Canonical Form.” <i>Special Sessions in Applications of Computer Algebra</i>, vol. 198, Springer, 2017, pp. 119–36, doi:<a href=\"https://doi.org/10.1007/978-3-319-56932-1_8\">10.1007/978-3-319-56932-1_8</a>.","chicago":"Ethier, Marc, Grzegorz Jablonski, and Marian Mrozek. “Finding Eigenvalues of Self-Maps with the Kronecker Canonical Form.” In <i>Special Sessions in Applications of Computer Algebra</i>, 198:119–36. Springer, 2017. <a href=\"https://doi.org/10.1007/978-3-319-56932-1_8\">https://doi.org/10.1007/978-3-319-56932-1_8</a>.","ista":"Ethier M, Jablonski G, Mrozek M. 2017. Finding eigenvalues of self-maps with the Kronecker canonical form. Special Sessions in Applications of Computer Algebra. ACA: Applications of Computer Algebra, PROMS, vol. 198, 119–136.","ieee":"M. Ethier, G. Jablonski, and M. Mrozek, “Finding eigenvalues of self-maps with the Kronecker canonical form,” in <i>Special Sessions in Applications of Computer Algebra</i>, Kalamata, Greece, 2017, vol. 198, pp. 119–136.","ama":"Ethier M, Jablonski G, Mrozek M. Finding eigenvalues of self-maps with the Kronecker canonical form. In: <i>Special Sessions in Applications of Computer Algebra</i>. Vol 198. Springer; 2017:119-136. doi:<a href=\"https://doi.org/10.1007/978-3-319-56932-1_8\">10.1007/978-3-319-56932-1_8</a>","apa":"Ethier, M., Jablonski, G., &#38; Mrozek, M. (2017). Finding eigenvalues of self-maps with the Kronecker canonical form. In <i>Special Sessions in Applications of Computer Algebra</i> (Vol. 198, pp. 119–136). Kalamata, Greece: Springer. <a href=\"https://doi.org/10.1007/978-3-319-56932-1_8\">https://doi.org/10.1007/978-3-319-56932-1_8</a>","short":"M. Ethier, G. Jablonski, M. Mrozek, in:, Special Sessions in Applications of Computer Algebra, Springer, 2017, pp. 119–136."},"conference":{"name":"ACA: Applications of Computer Algebra","end_date":"2015-07-23","start_date":"2015-07-20","location":"Kalamata, Greece"},"title":"Finding eigenvalues of self-maps with the Kronecker canonical form","day":"27","alternative_title":["PROMS"],"type":"conference","author":[{"last_name":"Ethier","first_name":"Marc","full_name":"Ethier, Marc"},{"last_name":"Jablonski","first_name":"Grzegorz","full_name":"Jablonski, Grzegorz","id":"4483EF78-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-3536-9866"},{"first_name":"Marian","full_name":"Mrozek, Marian","last_name":"Mrozek"}],"publisher":"Springer","isi":1,"quality_controlled":"1","department":[{"_id":"HeEd"}],"publication":"Special Sessions in Applications of Computer Algebra","status":"public","intvolume":"       198","page":"119 - 136","month":"07","date_created":"2018-12-11T11:48:46Z"},{"publisher":"Taylor & Francis","publication_status":"published","publication":"Handbook of Discrete and Computational Geometry, Third Edition","quality_controlled":"1","department":[{"_id":"HeEd"}],"status":"public","page":"1709 - 1735","article_processing_charge":"No","month":"11","series_title":"Handbook of Discrete and Computational Geometry","_id":"84","date_published":"2017-11-09T00:00:00Z","date_created":"2018-12-11T11:44:32Z","abstract":[{"text":"The advent of high-throughput technologies and the concurrent advances in information sciences have led to a data revolution in biology. This revolution is most significant in molecular biology, with an increase in the number and scale of the “omics” projects over the last decade. Genomics projects, for example, have produced impressive advances in our knowledge of the information concealed into genomes, from the many genes that encode for the proteins that are responsible for most if not all cellular functions, to the noncoding regions that are now known to provide regulatory functions. Proteomics initiatives help to decipher the role of post-translation modifications on the protein structures and provide maps of protein-protein interactions, while functional genomics is the field that attempts to make use of the data produced by these projects to understand protein functions. The biggest challenge today is to assimilate the wealth of information provided by these initiatives into a conceptual framework that will help us decipher life. For example, the current views of the relationship between protein structure and function remain fragmented. We know of their sequences, more and more about their structures, we have information on their biological activities, but we have difficulties connecting this dotted line into an informed whole. We lack the experimental and computational tools for directly studying protein structure, function, and dynamics at the molecular and supra-molecular levels. In this chapter, we review some of the current developments in building the computational tools that are needed, focusing on the role that geometry and topology play in these efforts. One of our goals is to raise the general awareness about the importance of geometric methods in elucidating the mysterious foundations of our very existence. Another goal is the broadening of what we consider a geometric algorithm. There is plenty of valuable no-man’s-land between combinatorial and numerical algorithms, and it seems opportune to explore this land with a computational-geometric frame of mind.","lang":"eng"}],"scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"date_updated":"2023-10-16T11:15:22Z","publication_identifier":{"eisbn":["9781498711425"]},"doi":"10.1201/9781315119601","citation":{"apa":"Edelsbrunner, H., &#38; Koehl, P. (2017). Computational topology for structural molecular biology. In C. Toth, J. O’Rourke, &#38; J. Goodman (Eds.), <i>Handbook of Discrete and Computational Geometry, Third Edition</i> (pp. 1709–1735). Taylor &#38; Francis. <a href=\"https://doi.org/10.1201/9781315119601\">https://doi.org/10.1201/9781315119601</a>","ama":"Edelsbrunner H, Koehl P. Computational topology for structural molecular biology. In: Toth C, O’Rourke J, Goodman J, eds. <i>Handbook of Discrete and Computational Geometry, Third Edition</i>. Handbook of Discrete and Computational Geometry. Taylor &#38; Francis; 2017:1709-1735. doi:<a href=\"https://doi.org/10.1201/9781315119601\">10.1201/9781315119601</a>","short":"H. Edelsbrunner, P. Koehl, in:, C. Toth, J. O’Rourke, J. Goodman (Eds.), Handbook of Discrete and Computational Geometry, Third Edition, Taylor &#38; Francis, 2017, pp. 1709–1735.","mla":"Edelsbrunner, Herbert, and Patrice Koehl. “Computational Topology for Structural Molecular Biology.” <i>Handbook of Discrete and Computational Geometry, Third Edition</i>, edited by Csaba Toth et al., Taylor &#38; Francis, 2017, pp. 1709–35, doi:<a href=\"https://doi.org/10.1201/9781315119601\">10.1201/9781315119601</a>.","ista":"Edelsbrunner H, Koehl P. 2017.Computational topology for structural molecular biology. In: Handbook of Discrete and Computational Geometry, Third Edition. , 1709–1735.","ieee":"H. Edelsbrunner and P. Koehl, “Computational topology for structural molecular biology,” in <i>Handbook of Discrete and Computational Geometry, Third Edition</i>, C. Toth, J. O’Rourke, and J. Goodman, Eds. Taylor &#38; Francis, 2017, pp. 1709–1735.","chicago":"Edelsbrunner, Herbert, and Patrice Koehl. “Computational Topology for Structural Molecular Biology.” In <i>Handbook of Discrete and Computational Geometry, Third Edition</i>, edited by Csaba Toth, Joseph O’Rourke, and Jacob Goodman, 1709–35. Handbook of Discrete and Computational Geometry. Taylor &#38; Francis, 2017. <a href=\"https://doi.org/10.1201/9781315119601\">https://doi.org/10.1201/9781315119601</a>."},"title":"Computational topology for structural molecular biology","editor":[{"last_name":"Toth","full_name":"Toth, Csaba","first_name":"Csaba"},{"full_name":"O'Rourke, Joseph","first_name":"Joseph","last_name":"O'Rourke"},{"full_name":"Goodman, Jacob","first_name":"Jacob","last_name":"Goodman"}],"day":"09","publist_id":"7970","type":"book_chapter","author":[{"last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Patrice","full_name":"Koehl, Patrice","last_name":"Koehl"}],"oa_version":"None","year":"2017"},{"scopus_import":1,"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_updated":"2021-01-12T08:09:26Z","publication_identifier":{"issn":["18688969"]},"publist_id":"7021","oa_version":"Published Version","year":"2017","has_accepted_license":"1","file_date_updated":"2020-07-14T12:47:42Z","pubrep_id":"895","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"volume":77,"publication_status":"published","oa":1,"file":[{"file_name":"IST-2017-895-v1+1_LIPIcs-SoCG-2017-39.pdf","file_size":990546,"creator":"system","date_updated":"2020-07-14T12:47:42Z","access_level":"open_access","checksum":"067ab0cb3f962bae6c3af6bf0094e0f3","content_type":"application/pdf","file_id":"4856","relation":"main_file","date_created":"2018-12-12T10:11:03Z"}],"_id":"688","abstract":[{"lang":"eng","text":"We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback - Leibler divergence, which is commonly used for comparing text and images, and the Itakura - Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized čech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized čech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory. "}],"date_published":"2017-06-01T00:00:00Z","language":[{"iso":"eng"}],"doi":"10.4230/LIPIcs.SoCG.2017.39","ddc":["514","516"],"day":"01","author":[{"last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"last_name":"Wagner","full_name":"Wagner, Hubert","first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"type":"conference","alternative_title":["LIPIcs"],"citation":{"ama":"Edelsbrunner H, Wagner H. Topological data analysis with Bregman divergences. In: Vol 77. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2017:391-3916. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2017.39\">10.4230/LIPIcs.SoCG.2017.39</a>","apa":"Edelsbrunner, H., &#38; Wagner, H. (2017). Topological data analysis with Bregman divergences (Vol. 77, pp. 391–3916). Presented at the Symposium on Computational Geometry, SoCG, Brisbane, Australia: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2017.39\">https://doi.org/10.4230/LIPIcs.SoCG.2017.39</a>","short":"H. Edelsbrunner, H. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017, pp. 391–3916.","mla":"Edelsbrunner, Herbert, and Hubert Wagner. <i>Topological Data Analysis with Bregman Divergences</i>. Vol. 77, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017, pp. 391–3916, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2017.39\">10.4230/LIPIcs.SoCG.2017.39</a>.","chicago":"Edelsbrunner, Herbert, and Hubert Wagner. “Topological Data Analysis with Bregman Divergences,” 77:391–3916. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2017.39\">https://doi.org/10.4230/LIPIcs.SoCG.2017.39</a>.","ieee":"H. Edelsbrunner and H. Wagner, “Topological data analysis with Bregman divergences,” presented at the Symposium on Computational Geometry, SoCG, Brisbane, Australia, 2017, vol. 77, pp. 391–3916.","ista":"Edelsbrunner H, Wagner H. 2017. Topological data analysis with Bregman divergences. Symposium on Computational Geometry, SoCG, LIPIcs, vol. 77, 391–3916."},"title":"Topological data analysis with Bregman divergences","conference":{"location":"Brisbane, Australia","name":"Symposium on Computational Geometry, SoCG","start_date":"2017-07-04","end_date":"2017-07-07"},"quality_controlled":"1","department":[{"_id":"HeEd"},{"_id":"UlWa"}],"intvolume":"        77","status":"public","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","month":"06","date_created":"2018-12-11T11:47:56Z","page":"391-3916"},{"_id":"707","abstract":[{"text":"We answer a question of M. Gromov on the waist of the unit ball.","lang":"eng"}],"date_published":"2017-08-01T00:00:00Z","issue":"4","volume":49,"publication_status":"published","oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.06279"}],"publist_id":"6982","year":"2017","oa_version":"Preprint","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_updated":"2021-01-12T08:11:41Z","scopus_import":1,"publication_identifier":{"issn":["00246093"]},"month":"08","date_created":"2018-12-11T11:48:02Z","page":"690 - 693","quality_controlled":"1","department":[{"_id":"HeEd"}],"publication":"Bulletin of the London Mathematical Society","intvolume":"        49","status":"public","publisher":"Wiley-Blackwell","day":"01","type":"journal_article","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","full_name":"Akopyan, Arseniy","last_name":"Akopyan"},{"full_name":"Karasev, Roman","first_name":"Roman","last_name":"Karasev"}],"ec_funded":1,"citation":{"ista":"Akopyan A, Karasev R. 2017. A tight estimate for the waist of the ball . Bulletin of the London Mathematical Society. 49(4), 690–693.","ieee":"A. Akopyan and R. Karasev, “A tight estimate for the waist of the ball ,” <i>Bulletin of the London Mathematical Society</i>, vol. 49, no. 4. Wiley-Blackwell, pp. 690–693, 2017.","chicago":"Akopyan, Arseniy, and Roman Karasev. “A Tight Estimate for the Waist of the Ball .” <i>Bulletin of the London Mathematical Society</i>. Wiley-Blackwell, 2017. <a href=\"https://doi.org/10.1112/blms.12062\">https://doi.org/10.1112/blms.12062</a>.","mla":"Akopyan, Arseniy, and Roman Karasev. “A Tight Estimate for the Waist of the Ball .” <i>Bulletin of the London Mathematical Society</i>, vol. 49, no. 4, Wiley-Blackwell, 2017, pp. 690–93, doi:<a href=\"https://doi.org/10.1112/blms.12062\">10.1112/blms.12062</a>.","short":"A. Akopyan, R. Karasev, Bulletin of the London Mathematical Society 49 (2017) 690–693.","apa":"Akopyan, A., &#38; Karasev, R. (2017). A tight estimate for the waist of the ball . <i>Bulletin of the London Mathematical Society</i>. Wiley-Blackwell. <a href=\"https://doi.org/10.1112/blms.12062\">https://doi.org/10.1112/blms.12062</a>","ama":"Akopyan A, Karasev R. A tight estimate for the waist of the ball . <i>Bulletin of the London Mathematical Society</i>. 2017;49(4):690-693. doi:<a href=\"https://doi.org/10.1112/blms.12062\">10.1112/blms.12062</a>"},"title":"A tight estimate for the waist of the ball ","language":[{"iso":"eng"}],"doi":"10.1112/blms.12062","project":[{"grant_number":"291734","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425"}]},{"status":"public","intvolume":"        49","department":[{"_id":"HeEd"}],"quality_controlled":"1","publication":"Advances in Applied Probability","publisher":"Cambridge University Press","date_created":"2018-12-11T11:48:07Z","month":"09","page":"745 - 767","doi":"10.1017/apr.2017.20","language":[{"iso":"eng"}],"project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Topological Complex Systems","grant_number":"318493"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"related_material":{"record":[{"id":"6287","relation":"dissertation_contains","status":"public"}]},"type":"journal_article","author":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"last_name":"Nikitenko","full_name":"Nikitenko, Anton","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0659-3201"},{"last_name":"Reitzner","first_name":"Matthias","full_name":"Reitzner, Matthias"}],"day":"01","title":"Expected sizes of poisson Delaunay mosaics and their discrete Morse functions","ec_funded":1,"citation":{"mla":"Edelsbrunner, Herbert, et al. “Expected Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” <i>Advances in Applied Probability</i>, vol. 49, no. 3, Cambridge University Press, 2017, pp. 745–67, doi:<a href=\"https://doi.org/10.1017/apr.2017.20\">10.1017/apr.2017.20</a>.","chicago":"Edelsbrunner, Herbert, Anton Nikitenko, and Matthias Reitzner. “Expected Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” <i>Advances in Applied Probability</i>. Cambridge University Press, 2017. <a href=\"https://doi.org/10.1017/apr.2017.20\">https://doi.org/10.1017/apr.2017.20</a>.","ieee":"H. Edelsbrunner, A. Nikitenko, and M. Reitzner, “Expected sizes of poisson Delaunay mosaics and their discrete Morse functions,” <i>Advances in Applied Probability</i>, vol. 49, no. 3. Cambridge University Press, pp. 745–767, 2017.","ista":"Edelsbrunner H, Nikitenko A, Reitzner M. 2017. Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. Advances in Applied Probability. 49(3), 745–767.","ama":"Edelsbrunner H, Nikitenko A, Reitzner M. Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. <i>Advances in Applied Probability</i>. 2017;49(3):745-767. doi:<a href=\"https://doi.org/10.1017/apr.2017.20\">10.1017/apr.2017.20</a>","apa":"Edelsbrunner, H., Nikitenko, A., &#38; Reitzner, M. (2017). Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. <i>Advances in Applied Probability</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/apr.2017.20\">https://doi.org/10.1017/apr.2017.20</a>","short":"H. Edelsbrunner, A. Nikitenko, M. Reitzner, Advances in Applied Probability 49 (2017) 745–767."},"volume":49,"main_file_link":[{"url":"https://arxiv.org/abs/1607.05915","open_access":"1"}],"oa":1,"publication_status":"published","_id":"718","date_published":"2017-09-01T00:00:00Z","abstract":[{"text":"Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝ n , we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.","lang":"eng"}],"issue":"3","arxiv":1,"publication_identifier":{"issn":["00018678"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2023-09-07T12:07:12Z","scopus_import":1,"external_id":{"arxiv":["1607.05915"]},"year":"2017","oa_version":"Preprint","publist_id":"6962"},{"quality_controlled":"1","department":[{"_id":"HeEd"}],"publication":"Topology and its Applications","status":"public","intvolume":"       231","publisher":"Elsevier","isi":1,"month":"11","date_created":"2018-12-11T11:48:14Z","page":"186 - 196","language":[{"iso":"eng"}],"doi":"10.1016/j.topol.2017.09.015","day":"01","author":[{"first_name":"Ziga","full_name":"Virk, Ziga","last_name":"Virk","id":"2E36B656-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Zastrow","full_name":"Zastrow, Andreas","first_name":"Andreas"}],"type":"journal_article","citation":{"mla":"Virk, Ziga, and Andreas Zastrow. “A New Topology on the Universal Path Space.” <i>Topology and Its Applications</i>, vol. 231, Elsevier, 2017, pp. 186–96, doi:<a href=\"https://doi.org/10.1016/j.topol.2017.09.015\">10.1016/j.topol.2017.09.015</a>.","ista":"Virk Z, Zastrow A. 2017. A new topology on the universal path space. Topology and its Applications. 231, 186–196.","ieee":"Z. Virk and A. Zastrow, “A new topology on the universal path space,” <i>Topology and its Applications</i>, vol. 231. Elsevier, pp. 186–196, 2017.","chicago":"Virk, Ziga, and Andreas Zastrow. “A New Topology on the Universal Path Space.” <i>Topology and Its Applications</i>. Elsevier, 2017. <a href=\"https://doi.org/10.1016/j.topol.2017.09.015\">https://doi.org/10.1016/j.topol.2017.09.015</a>.","apa":"Virk, Z., &#38; Zastrow, A. (2017). A new topology on the universal path space. <i>Topology and Its Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.topol.2017.09.015\">https://doi.org/10.1016/j.topol.2017.09.015</a>","ama":"Virk Z, Zastrow A. A new topology on the universal path space. <i>Topology and its Applications</i>. 2017;231:186-196. doi:<a href=\"https://doi.org/10.1016/j.topol.2017.09.015\">10.1016/j.topol.2017.09.015</a>","short":"Z. Virk, A. Zastrow, Topology and Its Applications 231 (2017) 186–196."},"title":"A new topology on the universal path space","volume":231,"publication_status":"published","_id":"737","abstract":[{"text":"We generalize Brazas’ topology on the fundamental group to the whole universal path space X˜ i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action.","lang":"eng"}],"date_published":"2017-11-01T00:00:00Z","article_processing_charge":"No","date_updated":"2023-09-27T12:53:01Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","scopus_import":"1","external_id":{"isi":["000413889100012"]},"publication_identifier":{"issn":["01668641"]},"publist_id":"6930","oa_version":"None","year":"2017"},{"oa":1,"publication_status":"published","volume":26,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"pubrep_id":"949","file_date_updated":"2020-07-14T12:46:35Z","issue":"3-4","abstract":[{"lang":"eng","text":"We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings."}],"_id":"481","date_published":"2017-04-13T00:00:00Z","file":[{"file_id":"4758","relation":"main_file","content_type":"application/pdf","date_created":"2018-12-12T10:09:34Z","creator":"system","file_size":769296,"file_name":"IST-2018-949-v1+1_2016_huber_PLanar_matchings.pdf","checksum":"f79e8558bfe4b368dfefeb8eec2e3a5e","access_level":"open_access","date_updated":"2020-07-14T12:46:35Z"}],"date_updated":"2023-02-21T16:06:22Z","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","scopus_import":1,"year":"2017","oa_version":"Published Version","has_accepted_license":"1","publist_id":"7338","publisher":"World Scientific Publishing","intvolume":"        26","status":"public","department":[{"_id":"HeEd"}],"quality_controlled":"1","publication":"International Journal of Computational Geometry and Applications","page":"211 - 229","date_created":"2018-12-11T11:46:43Z","month":"04","acknowledgement":"Supported by NSERC and the Ross and Muriel Cheriton Fellowship. Research supported by Austrian Science Fund (FWF): P25816-N15.","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"10892"}]},"doi":"10.1142/S0218195916600050","ddc":["004","514","516"],"language":[{"iso":"eng"}],"title":"Planar matchings for weighted straight skeletons","citation":{"mla":"Biedl, Therese, et al. “Planar Matchings for Weighted Straight Skeletons.” <i>International Journal of Computational Geometry and Applications</i>, vol. 26, no. 3–4, World Scientific Publishing, 2017, pp. 211–29, doi:<a href=\"https://doi.org/10.1142/S0218195916600050\">10.1142/S0218195916600050</a>.","ista":"Biedl T, Huber S, Palfrader P. 2017. Planar matchings for weighted straight skeletons. International Journal of Computational Geometry and Applications. 26(3–4), 211–229.","ieee":"T. Biedl, S. Huber, and P. Palfrader, “Planar matchings for weighted straight skeletons,” <i>International Journal of Computational Geometry and Applications</i>, vol. 26, no. 3–4. World Scientific Publishing, pp. 211–229, 2017.","chicago":"Biedl, Therese, Stefan Huber, and Peter Palfrader. “Planar Matchings for Weighted Straight Skeletons.” <i>International Journal of Computational Geometry and Applications</i>. World Scientific Publishing, 2017. <a href=\"https://doi.org/10.1142/S0218195916600050\">https://doi.org/10.1142/S0218195916600050</a>.","apa":"Biedl, T., Huber, S., &#38; Palfrader, P. (2017). Planar matchings for weighted straight skeletons. <i>International Journal of Computational Geometry and Applications</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/S0218195916600050\">https://doi.org/10.1142/S0218195916600050</a>","ama":"Biedl T, Huber S, Palfrader P. Planar matchings for weighted straight skeletons. <i>International Journal of Computational Geometry and Applications</i>. 2017;26(3-4):211-229. doi:<a href=\"https://doi.org/10.1142/S0218195916600050\">10.1142/S0218195916600050</a>","short":"T. Biedl, S. Huber, P. Palfrader, International Journal of Computational Geometry and Applications 26 (2017) 211–229."},"type":"journal_article","author":[{"last_name":"Biedl","first_name":"Therese","full_name":"Biedl, Therese"},{"id":"4700A070-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8871-5814","full_name":"Huber, Stefan","first_name":"Stefan","last_name":"Huber"},{"full_name":"Palfrader, Peter","first_name":"Peter","last_name":"Palfrader"}],"day":"13"}]