[{"title":"On the optimality of the FCC lattice for soft sphere packing","external_id":{"isi":["000428958900038"]},"issue":"1","type":"journal_article","department":[{"_id":"HeEd"}],"quality_controlled":"1","article_type":"original","project":[{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"date_updated":"2026-07-06T14:00:50Z","publication_identifier":{"issn":["0895-4801"]},"oa_version":"Submitted Version","isi":1,"publication":"SIAM J Discrete Math","month":"03","volume":32,"language":[{"iso":"eng"}],"article_processing_charge":"No","day":"29","acknowledgement":"This work was partially supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics,” through grant I02979-N35 of the Austrian Science Fund (FWF).","_id":"312","scopus_import":"1","page":"750 - 782","year":"2018","abstract":[{"text":"Motivated by biological questions, we study configurations of equal spheres that neither pack nor cover. Placing their centers on a lattice, we define the soft density of the configuration by penalizing multiple overlaps. Considering the 1-parameter family of diagonally distorted 3-dimensional integer lattices, we show that the soft density is maximized at the FCC lattice.","lang":"eng"}],"doi":"10.1137/16M1097201","date_created":"2018-12-11T11:45:46Z","publisher":"Society for Industrial and Applied Mathematics","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"first_name":"Mabel","id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","last_name":"Iglesias Ham","full_name":"Iglesias Ham, Mabel"}],"das_tickbox":"1","intvolume":"        32","oa":1,"status":"public","citation":{"chicago":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “On the Optimality of the FCC Lattice for Soft Sphere Packing.” <i>SIAM J Discrete Math</i>. Society for Industrial and Applied Mathematics, 2018. <a href=\"https://doi.org/10.1137/16M1097201\">https://doi.org/10.1137/16M1097201</a>.","ieee":"H. Edelsbrunner and M. Iglesias Ham, “On the optimality of the FCC lattice for soft sphere packing,” <i>SIAM J Discrete Math</i>, vol. 32, no. 1. Society for Industrial and Applied Mathematics, pp. 750–782, 2018.","ista":"Edelsbrunner H, Iglesias Ham M. 2018. On the optimality of the FCC lattice for soft sphere packing. SIAM J Discrete Math. 32(1), 750–782.","mla":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “On the Optimality of the FCC Lattice for Soft Sphere Packing.” <i>SIAM J Discrete Math</i>, vol. 32, no. 1, Society for Industrial and Applied Mathematics, 2018, pp. 750–82, doi:<a href=\"https://doi.org/10.1137/16M1097201\">10.1137/16M1097201</a>.","ama":"Edelsbrunner H, Iglesias Ham M. On the optimality of the FCC lattice for soft sphere packing. <i>SIAM J Discrete Math</i>. 2018;32(1):750-782. doi:<a href=\"https://doi.org/10.1137/16M1097201\">10.1137/16M1097201</a>","apa":"Edelsbrunner, H., &#38; Iglesias Ham, M. (2018). On the optimality of the FCC lattice for soft sphere packing. <i>SIAM J Discrete Math</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/16M1097201\">https://doi.org/10.1137/16M1097201</a>","short":"H. Edelsbrunner, M. Iglesias Ham, SIAM J Discrete Math 32 (2018) 750–782."},"publication_status":"published","main_file_link":[{"url":"http://pdfs.semanticscholar.org/d2d5/6da00fbc674e6a8b1bb9d857167e54200dc6.pdf","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publist_id":"7553","date_published":"2018-03-29T00:00:00Z"},{"external_id":{"arxiv":["1604.00960"],"isi":["000450810500036"]},"title":"Counting blanks in polygonal arrangements","arxiv":1,"ec_funded":1,"type":"journal_article","issue":"3","quality_controlled":"1","department":[{"_id":"HeEd"}],"project":[{"grant_number":"291734","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme"}],"date_updated":"2026-07-07T10:46:26Z","oa_version":"Preprint","volume":32,"publication":"SIAM Journal on Discrete Mathematics","isi":1,"month":"09","language":[{"iso":"eng"}],"article_processing_charge":"No","day":"06","_id":"58","scopus_import":"1","year":"2018","abstract":[{"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.","lang":"eng"}],"page":"2242 - 2257","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan"},{"full_name":"Segal Halevi, Erel","last_name":"Segal Halevi","first_name":"Erel"}],"intvolume":"        32","date_created":"2018-12-11T11:44:24Z","doi":"10.1137/16M110407X","publisher":"Society for Industrial and Applied Mathematics","main_file_link":[{"url":"https://arxiv.org/abs/1604.00960","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"status":"public","citation":{"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>","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>","short":"A. Akopyan, E. Segal Halevi, SIAM Journal on Discrete Mathematics 32 (2018) 2242–2257.","ista":"Akopyan A, Segal Halevi E. 2018. Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. 32(3), 2242–2257.","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>.","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>.","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."},"publication_status":"published","publist_id":"7996","date_published":"2018-09-06T00:00:00Z"},{"supervisor":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"}],"oa_version":"Published Version","publication_identifier":{"issn":["2663-337X"]},"language":[{"iso":"eng"}],"file_date_updated":"2020-07-14T12:45:24Z","month":"06","has_accepted_license":"1","degree_awarded":"PhD","type":"dissertation","ddc":["514","516"],"title":"Multiple covers with balls","alternative_title":["ISTA Thesis"],"date_updated":"2026-07-29T13:24:38Z","department":[{"_id":"HeEd"},{"_id":"GradSch"}],"doi":"10.15479/AT:ISTA:th_1026","date_created":"2018-12-11T11:45:10Z","publisher":"Institute of Science and Technology Austria","author":[{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","first_name":"Mabel","full_name":"Iglesias Ham, Mabel","last_name":"Iglesias Ham"}],"OA_place":"publisher","doi_confirm":"1","date_published":"2018-06-11T00:00:00Z","publist_id":"7712","corr_author":"1","oa":1,"status":"public","citation":{"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>","short":"M. Iglesias Ham, Multiple Covers with Balls, Institute of Science and Technology Austria, 2018.","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>.","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>.","ista":"Iglesias Ham M. 2018. Multiple covers with balls. Institute of Science and Technology Austria."},"file":[{"file_size":11827713,"file_name":"IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip","date_updated":"2020-07-14T12:45:24Z","file_id":"5918","content_type":"application/zip","relation":"source_file","checksum":"dd699303623e96d1478a6ae07210dd05","access_level":"closed","date_created":"2019-02-05T07:43:31Z","creator":"kschuh"},{"file_id":"5919","date_updated":"2020-07-14T12:45:24Z","file_name":"IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf","file_size":4783846,"date_created":"2019-02-05T07:43:45Z","creator":"kschuh","content_type":"application/pdf","access_level":"open_access","checksum":"ba163849a190d2b41d66fef0e4983294","relation":"main_file"}],"publication_status":"published","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","article_processing_charge":"No","day":"11","page":"171","year":"2018","pubrep_id":"1026","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"}],"_id":"201"},{"quality_controlled":"1","department":[{"_id":"HeEd"}],"project":[{"call_identifier":"FP7","grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems"}],"date_updated":"2025-06-04T08:44:44Z","title":"The Voronoi functional is maximized by the Delaunay triangulation in the plane","arxiv":1,"external_id":{"arxiv":["1411.6337"],"isi":["000418056000005"]},"ec_funded":1,"type":"journal_article","issue":"5","volume":37,"month":"10","isi":1,"publication":"Combinatorica","language":[{"iso":"eng"}],"oa_version":"Submitted Version","publication_identifier":{"issn":["0209-9683"]},"_id":"1173","scopus_import":"1","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.","year":"2017","abstract":[{"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.","lang":"eng"}],"page":"887 - 910","day":"01","article_processing_charge":"No","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1411.6337"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"status":"public","publication_status":"published","citation":{"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>","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>.","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.","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>."},"publist_id":"6182","date_published":"2017-10-01T00:00:00Z","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Alexey","last_name":"Glazyrin","full_name":"Glazyrin, Alexey"},{"last_name":"Musin","full_name":"Musin, Oleg","first_name":"Oleg"},{"full_name":"Nikitenko, Anton","last_name":"Nikitenko","orcid":"0000-0002-0659-3201","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"}],"intvolume":"        37","doi":"10.1007/s00493-016-3308-y","date_created":"2018-12-11T11:50:32Z","publisher":"Springer"},{"date_updated":"2025-06-04T08:45:48Z","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","grant_number":"291734"}],"quality_controlled":"1","department":[{"_id":"HeEd"}],"type":"journal_article","arxiv":1,"external_id":{"arxiv":["1508.07594"],"isi":["000409292900015"]},"title":"Algebraic vertices of non-convex polyhedra","ec_funded":1,"language":[{"iso":"eng"}],"volume":308,"isi":1,"month":"02","publication":"Advances in Mathematics","publication_identifier":{"issn":["0001-8708"]},"oa_version":"Submitted Version","abstract":[{"lang":"eng","text":"In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform."}],"year":"2017","page":"627 - 644","scopus_import":"1","_id":"1180","article_processing_charge":"No","day":"21","publist_id":"6173","date_published":"2017-02-21T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"url":"https://arxiv.org/abs/1508.07594","open_access":"1"}],"publication_status":"published","citation":{"mla":"Akopyan, Arseniy, et al. “Algebraic Vertices of Non-Convex Polyhedra.” <i>Advances in Mathematics</i>, vol. 308, Academic Press, 2017, pp. 627–44, doi:<a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">10.1016/j.aim.2016.12.026</a>.","ista":"Akopyan A, Bárány I, Robins S. 2017. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 308, 627–644.","ieee":"A. Akopyan, I. Bárány, and S. Robins, “Algebraic vertices of non-convex polyhedra,” <i>Advances in Mathematics</i>, vol. 308. Academic Press, pp. 627–644, 2017.","chicago":"Akopyan, Arseniy, Imre Bárány, and Sinai Robins. “Algebraic Vertices of Non-Convex Polyhedra.” <i>Advances in Mathematics</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">https://doi.org/10.1016/j.aim.2016.12.026</a>.","short":"A. Akopyan, I. Bárány, S. Robins, Advances in Mathematics 308 (2017) 627–644.","apa":"Akopyan, A., Bárány, I., &#38; Robins, S. (2017). Algebraic vertices of non-convex polyhedra. <i>Advances in Mathematics</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">https://doi.org/10.1016/j.aim.2016.12.026</a>","ama":"Akopyan A, Bárány I, Robins S. Algebraic vertices of non-convex polyhedra. <i>Advances in Mathematics</i>. 2017;308:627-644. doi:<a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">10.1016/j.aim.2016.12.026</a>"},"status":"public","oa":1,"intvolume":"       308","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","last_name":"Akopyan","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy"},{"first_name":"Imre","last_name":"Bárány","full_name":"Bárány, Imre"},{"last_name":"Robins","full_name":"Robins, Sinai","first_name":"Sinai"}],"publisher":"Academic Press","date_created":"2018-12-11T11:50:34Z","doi":"10.1016/j.aim.2016.12.026"},{"oa_version":"Published Version","publication_identifier":{"issn":[" 0747-7171"]},"language":[{"iso":"eng"}],"related_material":{"record":[{"id":"10894","relation":"earlier_version","status":"public"}]},"isi":1,"publication":"Journal of Symbolic Computation","month":"01","volume":78,"type":"journal_article","ec_funded":1,"ddc":["500"],"external_id":{"isi":["000384396000005"]},"title":"Phat - Persistent homology algorithms toolbox","article_type":"original","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7"}],"date_updated":"2026-06-18T17:35:16Z","quality_controlled":"1","department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:51:59Z","doi":"10.1016/j.jsc.2016.03.008","publisher":"Academic Press","author":[{"full_name":"Bauer, Ulrich","last_name":"Bauer","first_name":"Ulrich"},{"last_name":"Kerber","full_name":"Kerber, Michael","first_name":"Michael"},{"full_name":"Reininghaus, Jan","last_name":"Reininghaus","first_name":"Jan"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","orcid":"0009-0009-9111-8429","last_name":"Wagner","full_name":"Wagner, Hubert"}],"intvolume":"        78","publist_id":"5765","date_published":"2017-01-01T00:00:00Z","corr_author":"1","oa":1,"status":"public","citation":{"ama":"Bauer U, Kerber M, Reininghaus J, Wagner H. Phat - Persistent homology algorithms toolbox. <i>Journal of Symbolic Computation</i>. 2017;78:76-90. doi:<a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">10.1016/j.jsc.2016.03.008</a>","apa":"Bauer, U., Kerber, M., Reininghaus, J., &#38; Wagner, H. (2017). Phat - Persistent homology algorithms toolbox. <i>Journal of Symbolic Computation</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">https://doi.org/10.1016/j.jsc.2016.03.008</a>","short":"U. Bauer, M. Kerber, J. Reininghaus, H. Wagner, Journal of Symbolic Computation 78 (2017) 76–90.","chicago":"Bauer, Ulrich, Michael Kerber, Jan Reininghaus, and Hubert Wagner. “Phat - Persistent Homology Algorithms Toolbox.” <i>Journal of Symbolic Computation</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">https://doi.org/10.1016/j.jsc.2016.03.008</a>.","ieee":"U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner, “Phat - Persistent homology algorithms toolbox,” <i>Journal of Symbolic Computation</i>, vol. 78. Academic Press, pp. 76–90, 2017.","mla":"Bauer, Ulrich, et al. “Phat - Persistent Homology Algorithms Toolbox.” <i>Journal of Symbolic Computation</i>, vol. 78, Academic Press, 2017, pp. 76–90, doi:<a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">10.1016/j.jsc.2016.03.008</a>.","ista":"Bauer U, Kerber M, Reininghaus J, Wagner H. 2017. Phat - Persistent homology algorithms toolbox. Journal of Symbolic Computation. 78, 76–90."},"publication_status":"published","main_file_link":[{"url":"https://doi.org/10.1016/j.jsc.2016.03.008","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","day":"01","page":"76 - 90","year":"2017","abstract":[{"lang":"eng","text":"Phat is an open-source C. ++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. We provide numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. We compare the different combinations through extensive experimental evaluation and identify optimization techniques that work well in practical situations. We also compare our software with various other publicly available libraries for persistent homology."}],"acknowledgement":"Michael Kerber acknowledges support by the Max Planck Center for Visual Computing and Communications (FKZ-01IMC01 and FKZ-01IM10001). Ulrich Bauer, Jan Reininghaus, and Hubert Wagner acknowledge support by the EU Project TOPOSYS (FP7-ICT-318493-STREP).","_id":"1433","OA_type":"free access","scopus_import":"1"},{"acknowledgement":"Supported by NSERC and the Ross and Muriel Cheriton Fellowship. Research supported by Austrian Science Fund (FWF): P25816-N15.","scopus_import":1,"_id":"481","page":"211 - 229","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)"},"abstract":[{"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.","lang":"eng"}],"pubrep_id":"949","year":"2017","day":"13","publication_status":"published","citation":{"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>","short":"T. Biedl, S. Huber, P. Palfrader, International Journal of Computational Geometry and Applications 26 (2017) 211–229.","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>","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>.","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.","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>."},"file":[{"file_id":"4758","date_updated":"2020-07-14T12:46:35Z","file_name":"IST-2018-949-v1+1_2016_huber_PLanar_matchings.pdf","file_size":769296,"creator":"system","date_created":"2018-12-12T10:09:34Z","access_level":"open_access","relation":"main_file","checksum":"f79e8558bfe4b368dfefeb8eec2e3a5e","content_type":"application/pdf"}],"oa":1,"status":"public","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","corr_author":"1","publist_id":"7338","date_published":"2017-04-13T00:00:00Z","publisher":"World Scientific Publishing","date_created":"2018-12-11T11:46:43Z","doi":"10.1142/S0218195916600050","intvolume":"        26","author":[{"last_name":"Biedl","full_name":"Biedl, Therese","first_name":"Therese"},{"orcid":"0000-0002-8871-5814","last_name":"Huber","full_name":"Huber, Stefan","id":"4700A070-F248-11E8-B48F-1D18A9856A87","first_name":"Stefan"},{"first_name":"Peter","last_name":"Palfrader","full_name":"Palfrader, Peter"}],"department":[{"_id":"HeEd"}],"quality_controlled":"1","date_updated":"2025-09-29T13:22:54Z","ddc":["004","514","516"],"title":"Planar matchings for weighted straight skeletons","issue":"3-4","type":"journal_article","month":"04","publication":"International Journal of Computational Geometry and Applications","has_accepted_license":"1","volume":26,"language":[{"iso":"eng"}],"file_date_updated":"2020-07-14T12:46:35Z","related_material":{"record":[{"status":"public","id":"10892","relation":"earlier_version"}]},"oa_version":"Published Version"},{"quality_controlled":"1","department":[{"_id":"HeEd"}],"date_updated":"2025-09-18T09:47:04Z","arxiv":1,"title":"Higson compactification and dimension raising","external_id":{"arxiv":["1608.03954"],"isi":["000390501400005"]},"type":"journal_article","volume":215,"isi":1,"month":"01","publication":"Topology and its Applications","language":[{"iso":"eng"}],"oa_version":"Submitted Version","publication_identifier":{"issn":["0166-8641"]},"scopus_import":"1","_id":"521","abstract":[{"lang":"eng","text":"Let X and Y be proper metric spaces. We show that a coarsely n-to-1 map f:X→Y induces an n-to-1 map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if f:X→Y is a coarsely n-to-1 map between proper metric spaces X and Y then asdim(Y)≤asdim(X)+n−1. Furthermore we introduce coarsely open coarsely n-to-1 maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension."}],"year":"2017","page":"45 - 57","article_processing_charge":"No","day":"01","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.03954"}],"citation":{"ama":"Austin K, Virk Z. Higson compactification and dimension raising. <i>Topology and its Applications</i>. 2017;215:45-57. doi:<a href=\"https://doi.org/10.1016/j.topol.2016.10.005\">10.1016/j.topol.2016.10.005</a>","apa":"Austin, K., &#38; Virk, Z. (2017). Higson compactification and dimension raising. <i>Topology and Its Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.topol.2016.10.005\">https://doi.org/10.1016/j.topol.2016.10.005</a>","short":"K. Austin, Z. Virk, Topology and Its Applications 215 (2017) 45–57.","mla":"Austin, Kyle, and Ziga Virk. “Higson Compactification and Dimension Raising.” <i>Topology and Its Applications</i>, vol. 215, Elsevier, 2017, pp. 45–57, doi:<a href=\"https://doi.org/10.1016/j.topol.2016.10.005\">10.1016/j.topol.2016.10.005</a>.","ista":"Austin K, Virk Z. 2017. Higson compactification and dimension raising. Topology and its Applications. 215, 45–57.","ieee":"K. Austin and Z. Virk, “Higson compactification and dimension raising,” <i>Topology and its Applications</i>, vol. 215. Elsevier, pp. 45–57, 2017.","chicago":"Austin, Kyle, and Ziga Virk. “Higson Compactification and Dimension Raising.” <i>Topology and Its Applications</i>. Elsevier, 2017. <a href=\"https://doi.org/10.1016/j.topol.2016.10.005\">https://doi.org/10.1016/j.topol.2016.10.005</a>."},"publication_status":"published","status":"public","oa":1,"publist_id":"7299","corr_author":"1","date_published":"2017-01-01T00:00:00Z","intvolume":"       215","author":[{"full_name":"Austin, Kyle","last_name":"Austin","first_name":"Kyle"},{"last_name":"Virk","full_name":"Virk, Ziga","first_name":"Ziga","id":"2E36B656-F248-11E8-B48F-1D18A9856A87"}],"publisher":"Elsevier","doi":"10.1016/j.topol.2016.10.005","date_created":"2018-12-11T11:46:56Z"},{"publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["978-3-319-59107-0","978-3-319-59108-7"]},"oa_version":"None","language":[{"iso":"eng"}],"volume":10256,"place":"Cham","month":"05","publication":"Combinatorial image analysis","type":"book_chapter","conference":{"name":"IWCIA: International Workshop on Combinatorial Image Analysis","end_date":"2017-06-21","location":"Plovdiv, Bulgaria","start_date":"2017-06-19"},"title":"Construction of persistent Voronoi diagram on 3D digital plane","date_updated":"2022-01-28T07:48:24Z","alternative_title":["LNCS"],"quality_controlled":"1","department":[{"_id":"HeEd"}],"intvolume":"     10256","author":[{"last_name":"Biswas","orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Bhowmick","full_name":"Bhowmick, Partha","first_name":"Partha"}],"publisher":"Springer Nature","date_created":"2019-01-08T20:42:56Z","doi":"10.1007/978-3-319-59108-7_8","date_published":"2017-05-17T00:00:00Z","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","citation":{"mla":"Biswas, Ranita, and Partha Bhowmick. “Construction of Persistent Voronoi Diagram on 3D Digital Plane.” <i>Combinatorial Image Analysis</i>, vol. 10256, Springer Nature, 2017, pp. 93–104, doi:<a href=\"https://doi.org/10.1007/978-3-319-59108-7_8\">10.1007/978-3-319-59108-7_8</a>.","ista":"Biswas R, Bhowmick P. 2017.Construction of persistent Voronoi diagram on 3D digital plane. In: Combinatorial image analysis. LNCS, vol. 10256, 93–104.","ieee":"R. Biswas and P. Bhowmick, “Construction of persistent Voronoi diagram on 3D digital plane,” in <i>Combinatorial image analysis</i>, vol. 10256, Cham: Springer Nature, 2017, pp. 93–104.","chicago":"Biswas, Ranita, and Partha Bhowmick. “Construction of Persistent Voronoi Diagram on 3D Digital Plane.” In <i>Combinatorial Image Analysis</i>, 10256:93–104. Cham: Springer Nature, 2017. <a href=\"https://doi.org/10.1007/978-3-319-59108-7_8\">https://doi.org/10.1007/978-3-319-59108-7_8</a>.","short":"R. Biswas, P. Bhowmick, in:, Combinatorial Image Analysis, Springer Nature, Cham, 2017, pp. 93–104.","ama":"Biswas R, Bhowmick P. Construction of persistent Voronoi diagram on 3D digital plane. In: <i>Combinatorial Image Analysis</i>. Vol 10256. Cham: Springer Nature; 2017:93-104. doi:<a href=\"https://doi.org/10.1007/978-3-319-59108-7_8\">10.1007/978-3-319-59108-7_8</a>","apa":"Biswas, R., &#38; Bhowmick, P. (2017). Construction of persistent Voronoi diagram on 3D digital plane. In <i>Combinatorial image analysis</i> (Vol. 10256, pp. 93–104). Cham: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-319-59108-7_8\">https://doi.org/10.1007/978-3-319-59108-7_8</a>"},"publication_status":"published","status":"public","article_processing_charge":"No","day":"17","abstract":[{"text":"Different distance metrics produce Voronoi diagrams with different properties. It is a well-known that on the (real) 2D plane or even on any 3D plane, a Voronoi diagram (VD) based on the Euclidean distance metric produces convex Voronoi regions. In this paper, we first show that this metric produces a persistent VD on the 2D digital plane, as it comprises digitally convex Voronoi regions and hence correctly approximates the corresponding VD on the 2D real plane. Next, we show that on a 3D digital plane D, the Euclidean metric spanning over its voxel set does not guarantee a digital VD which is persistent with the real-space VD. As a solution, we introduce a novel concept of functional-plane-convexity, which is ensured by the Euclidean metric spanning over the pedal set of D. Necessary proofs and some visual result have been provided to adjudge the merit and usefulness of the proposed concept.","lang":"eng"}],"year":"2017","extern":"1","page":"93-104","_id":"5803"},{"article_processing_charge":"No","day":"01","_id":"1022","scopus_import":"1","acknowledgement":"Part of this work has been supported by the 7th Framework Programme for Research of the European Commission, under FETOpen grant number 255827 (CGL Computational Geometry Learning) and ERC advanced grant, URSAT (Understanding Random Systems via Algebraic Topology) number 320422.","year":"2017","abstract":[{"text":"We introduce a multiscale topological description of the Megaparsec web-like cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different web-like morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of web-like configurations and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.","lang":"eng"}],"page":"4281 - 4310","author":[{"full_name":"Pranav, Pratyush","last_name":"Pranav","first_name":"Pratyush"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"first_name":"Rien","last_name":"Van De Weygaert","full_name":"Van De Weygaert, Rien"},{"first_name":"Gert","full_name":"Vegter, Gert","last_name":"Vegter"},{"first_name":"Michael","full_name":"Kerber, Michael","last_name":"Kerber"},{"first_name":"Bernard","full_name":"Jones, Bernard","last_name":"Jones"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs"}],"intvolume":"       465","doi":"10.1093/mnras/stw2862","date_created":"2018-12-11T11:49:44Z","publisher":"Oxford University Press","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.04519"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"status":"public","citation":{"apa":"Pranav, P., Edelsbrunner, H., Van De Weygaert, R., Vegter, G., Kerber, M., Jones, B., &#38; Wintraecken, M. (2017). The topology of the cosmic web in terms of persistent Betti numbers. <i>Monthly Notices of the Royal Astronomical Society</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/mnras/stw2862\">https://doi.org/10.1093/mnras/stw2862</a>","short":"P. Pranav, H. Edelsbrunner, R. Van De Weygaert, G. Vegter, M. Kerber, B. Jones, M. Wintraecken, Monthly Notices of the Royal Astronomical Society 465 (2017) 4281–4310.","ama":"Pranav P, Edelsbrunner H, Van De Weygaert R, et al. The topology of the cosmic web in terms of persistent Betti numbers. <i>Monthly Notices of the Royal Astronomical Society</i>. 2017;465(4):4281-4310. doi:<a href=\"https://doi.org/10.1093/mnras/stw2862\">10.1093/mnras/stw2862</a>","ieee":"P. Pranav <i>et al.</i>, “The topology of the cosmic web in terms of persistent Betti numbers,” <i>Monthly Notices of the Royal Astronomical Society</i>, vol. 465, no. 4. Oxford University Press, pp. 4281–4310, 2017.","chicago":"Pranav, Pratyush, Herbert Edelsbrunner, Rien Van De Weygaert, Gert Vegter, Michael Kerber, Bernard Jones, and Mathijs Wintraecken. “The Topology of the Cosmic Web in Terms of Persistent Betti Numbers.” <i>Monthly Notices of the Royal Astronomical Society</i>. Oxford University Press, 2017. <a href=\"https://doi.org/10.1093/mnras/stw2862\">https://doi.org/10.1093/mnras/stw2862</a>.","ista":"Pranav P, Edelsbrunner H, Van De Weygaert R, Vegter G, Kerber M, Jones B, Wintraecken M. 2017. The topology of the cosmic web in terms of persistent Betti numbers. Monthly Notices of the Royal Astronomical Society. 465(4), 4281–4310.","mla":"Pranav, Pratyush, et al. “The Topology of the Cosmic Web in Terms of Persistent Betti Numbers.” <i>Monthly Notices of the Royal Astronomical Society</i>, vol. 465, no. 4, Oxford University Press, 2017, pp. 4281–310, doi:<a href=\"https://doi.org/10.1093/mnras/stw2862\">10.1093/mnras/stw2862</a>."},"publication_status":"published","publist_id":"6373","date_published":"2017-01-01T00:00:00Z","title":"The topology of the cosmic web in terms of persistent Betti numbers","arxiv":1,"external_id":{"isi":["000395170200039"],"arxiv":["1608.04519"]},"type":"journal_article","issue":"4","quality_controlled":"1","department":[{"_id":"HeEd"}],"date_updated":"2025-06-04T08:10:31Z","publication_identifier":{"issn":["0035-8711"]},"oa_version":"Submitted Version","volume":465,"month":"01","publication":"Monthly Notices of the Royal Astronomical Society","isi":1,"language":[{"iso":"eng"}]},{"date_updated":"2025-07-10T11:49:53Z","project":[{"name":"Modern Graph Algorithmic Techniques in Formal Verification","_id":"2584A770-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"P 23499-N23"},{"call_identifier":"FWF","grant_number":"S11407","_id":"25863FF4-B435-11E9-9278-68D0E5697425","name":"Game Theory"},{"_id":"2581B60A-B435-11E9-9278-68D0E5697425","name":"Quantitative Graph Games: Theory and Applications","grant_number":"279307","call_identifier":"FP7"}],"quality_controlled":"1","department":[{"_id":"KrCh"},{"_id":"HeEd"}],"type":"journal_article","title":"Pushdown reachability with constant treewidth","ddc":["000"],"external_id":{"isi":["000399506600005"]},"ec_funded":1,"file_date_updated":"2019-10-15T07:44:51Z","language":[{"iso":"eng"}],"volume":122,"has_accepted_license":"1","month":"06","isi":1,"publication":"Information Processing Letters","oa_version":"Submitted Version","publication_identifier":{"issn":["0020-0190"]},"pubrep_id":"991","abstract":[{"text":"We consider the problem of reachability in pushdown graphs. We study the problem for pushdown graphs with constant treewidth. Even for pushdown graphs with treewidth 1, for the reachability problem we establish the following: (i) the problem is PTIME-complete, and (ii) any subcubic algorithm for the problem would contradict the k-clique conjecture and imply faster combinatorial algorithms for cliques in graphs.","lang":"eng"}],"year":"2017","page":"25 - 29","scopus_import":"1","_id":"1065","article_processing_charge":"No","day":"01","publist_id":"6323","date_published":"2017-06-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","citation":{"ama":"Chatterjee K, Osang GF. Pushdown reachability with constant treewidth. <i>Information Processing Letters</i>. 2017;122:25-29. doi:<a href=\"https://doi.org/10.1016/j.ipl.2017.02.003\">10.1016/j.ipl.2017.02.003</a>","apa":"Chatterjee, K., &#38; Osang, G. F. (2017). Pushdown reachability with constant treewidth. <i>Information Processing Letters</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ipl.2017.02.003\">https://doi.org/10.1016/j.ipl.2017.02.003</a>","short":"K. Chatterjee, G.F. Osang, Information Processing Letters 122 (2017) 25–29.","ieee":"K. Chatterjee and G. F. Osang, “Pushdown reachability with constant treewidth,” <i>Information Processing Letters</i>, vol. 122. Elsevier, pp. 25–29, 2017.","chicago":"Chatterjee, Krishnendu, and Georg F Osang. “Pushdown Reachability with Constant Treewidth.” <i>Information Processing Letters</i>. Elsevier, 2017. <a href=\"https://doi.org/10.1016/j.ipl.2017.02.003\">https://doi.org/10.1016/j.ipl.2017.02.003</a>.","mla":"Chatterjee, Krishnendu, and Georg F. Osang. “Pushdown Reachability with Constant Treewidth.” <i>Information Processing Letters</i>, vol. 122, Elsevier, 2017, pp. 25–29, doi:<a href=\"https://doi.org/10.1016/j.ipl.2017.02.003\">10.1016/j.ipl.2017.02.003</a>.","ista":"Chatterjee K, Osang GF. 2017. Pushdown reachability with constant treewidth. Information Processing Letters. 122, 25–29."},"file":[{"file_id":"4998","date_updated":"2019-10-15T07:44:51Z","file_name":"IST-2018-991-v1+2_2018_Chatterjee_Pushdown_PREPRINT.pdf","file_size":247657,"date_created":"2018-12-12T10:13:17Z","creator":"system","content_type":"application/pdf","relation":"main_file","access_level":"open_access"}],"status":"public","oa":1,"intvolume":"       122","author":[{"id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","first_name":"Krishnendu","full_name":"Chatterjee, Krishnendu","orcid":"0000-0002-4561-241X","last_name":"Chatterjee"},{"id":"464B40D6-F248-11E8-B48F-1D18A9856A87","first_name":"Georg F","full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116","last_name":"Osang"}],"publisher":"Elsevier","date_created":"2018-12-11T11:49:57Z","doi":"10.1016/j.ipl.2017.02.003"},{"publisher":"American Mathematical Society","date_created":"2018-12-11T11:49:59Z","doi":"10.1090/tran/6991","intvolume":"       369","author":[{"first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","last_name":"Bauer","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich"},{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"}],"publist_id":"6311","date_published":"2017-05-01T00:00:00Z","publication_status":"published","citation":{"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>","short":"U. Bauer, H. Edelsbrunner, Transactions of the American Mathematical Society 369 (2017) 3741–3762.","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>","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>.","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."},"oa":1,"status":"public","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1312.1231"}],"article_processing_charge":"No","day":"01","page":"3741 - 3762","abstract":[{"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.","lang":"eng"}],"year":"2017","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”.","scopus_import":"1","_id":"1072","oa_version":"Preprint","language":[{"iso":"eng"}],"month":"05","isi":1,"publication":"Transactions of the American Mathematical Society","volume":369,"issue":"5","type":"journal_article","ec_funded":1,"arxiv":1,"title":"The Morse theory of Čech and delaunay complexes","external_id":{"isi":["000398030400024"],"arxiv":["1312.1231"]},"date_updated":"2025-04-15T08:37:54Z","article_type":"original","project":[{"name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493"}],"department":[{"_id":"HeEd"}],"quality_controlled":"1"},{"intvolume":"     10424","author":[{"last_name":"Heiss","orcid":"0000-0002-1780-2689","full_name":"Heiss, Teresa","first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Hubert","last_name":"Wagner"}],"publisher":"Springer","date_created":"2018-12-11T11:48:45Z","doi":"10.1007/978-3-319-64689-3_32","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"url":"https://arxiv.org/abs/1705.02045","open_access":"1"}],"citation":{"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>","short":"T. Heiss, H. Wagner, in:, M. Felsberg, A. Heyden, N. Krüger (Eds.), Springer, 2017, pp. 397–409.","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>","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>.","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.","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."},"publication_status":"published","status":"public","oa":1,"corr_author":"1","publist_id":"6815","date_published":"2017-07-28T00:00:00Z","day":"28","article_processing_charge":"No","scopus_import":"1","_id":"833","abstract":[{"lang":"eng","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."}],"year":"2017","page":"397 - 409","editor":[{"full_name":"Felsberg, Michael","last_name":"Felsberg","first_name":"Michael"},{"full_name":"Heyden, Anders","last_name":"Heyden","first_name":"Anders"},{"first_name":"Norbert","last_name":"Krüger","full_name":"Krüger, Norbert"}],"oa_version":"Submitted Version","publication_identifier":{"issn":["0302-9743"]},"volume":10424,"month":"07","isi":1,"language":[{"iso":"eng"}],"title":"Streaming algorithm for Euler characteristic curves of multidimensional images","arxiv":1,"external_id":{"arxiv":["1705.02045"],"isi":["000432085900032"]},"conference":{"name":"CAIP: Computer Analysis of Images and Patterns","location":"Ystad, Sweden","end_date":"2017-08-24","start_date":"2017-08-22"},"type":"conference","quality_controlled":"1","department":[{"_id":"HeEd"}],"date_updated":"2025-06-04T09:54:22Z","alternative_title":["LNCS"]},{"oa_version":"None","publication_identifier":{"isbn":["978-331956930-7"]},"language":[{"iso":"eng"}],"volume":198,"isi":1,"month":"07","publication":"Special Sessions in Applications of Computer Algebra","type":"conference","conference":{"name":"ACA: Applications of Computer Algebra","location":"Kalamata, Greece","end_date":"2015-07-23","start_date":"2015-07-20"},"title":"Finding eigenvalues of self-maps with the Kronecker canonical form","external_id":{"isi":["000434088200008"]},"ec_funded":1,"project":[{"name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493"}],"date_updated":"2025-04-15T08:37:55Z","alternative_title":["PROMS"],"quality_controlled":"1","department":[{"_id":"HeEd"}],"author":[{"full_name":"Ethier, Marc","last_name":"Ethier","first_name":"Marc"},{"orcid":"0000-0002-3536-9866","last_name":"Jablonski","full_name":"Jablonski, Grzegorz","first_name":"Grzegorz","id":"4483EF78-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Marian","last_name":"Mrozek","full_name":"Mrozek, Marian"}],"intvolume":"       198","date_created":"2018-12-11T11:48:46Z","doi":"10.1007/978-3-319-56932-1_8","publisher":"Springer","date_published":"2017-07-27T00:00:00Z","publist_id":"6812","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","status":"public","publication_status":"published","citation":{"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.","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.","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>.","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>.","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."},"day":"27","article_processing_charge":"No","year":"2017","abstract":[{"lang":"eng","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."}],"page":"119 - 136","_id":"836","scopus_import":"1"},{"language":[{"iso":"eng"}],"date_published":"2017-11-09T00:00:00Z","publist_id":"7970","publication_status":"published","citation":{"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>.","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.","ista":"Edelsbrunner H, Koehl P. 2017.Computational topology for structural molecular biology. In: Handbook of Discrete and Computational Geometry, Third Edition. , 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>.","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.","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>","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>"},"month":"11","publication":"Handbook of Discrete and Computational Geometry, Third Edition","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Taylor & Francis","doi":"10.1201/9781315119601","date_created":"2018-12-11T11:44:32Z","publication_identifier":{"eisbn":["9781498711425"]},"oa_version":"None","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"first_name":"Patrice","last_name":"Koehl","full_name":"Koehl, Patrice"}],"editor":[{"last_name":"Toth","full_name":"Toth, Csaba","first_name":"Csaba"},{"first_name":"Joseph","full_name":"O'Rourke, Joseph","last_name":"O'Rourke"},{"last_name":"Goodman","full_name":"Goodman, Jacob","first_name":"Jacob"}],"series_title":"Handbook of Discrete and Computational Geometry","page":"1709 - 1735","date_updated":"2023-10-16T11:15:22Z","abstract":[{"lang":"eng","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."}],"year":"2017","quality_controlled":"1","department":[{"_id":"HeEd"}],"scopus_import":"1","_id":"84","type":"book_chapter","article_processing_charge":"No","day":"09","title":"Computational topology for structural molecular biology"},{"page":"588 - 596","year":"2017","abstract":[{"lang":"eng","text":"We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor &amp;#xbd; cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape."}],"_id":"909","scopus_import":"1","article_processing_charge":"No","day":"01","date_published":"2017-01-01T00:00:00Z","publist_id":"6534","status":"public","oa":1,"citation":{"short":"A. Akopyan, V. Vysotsky, The American Mathematical Monthly 124 (2017) 588–596.","ama":"Akopyan A, Vysotsky V. On the lengths of curves passing through boundary points of a planar convex shape. <i>The American Mathematical Monthly</i>. 2017;124(7):588-596. doi:<a href=\"https://doi.org/10.4169/amer.math.monthly.124.7.588\">10.4169/amer.math.monthly.124.7.588</a>","apa":"Akopyan, A., &#38; Vysotsky, V. (2017). On the lengths of curves passing through boundary points of a planar convex shape. <i>The American Mathematical Monthly</i>. Mathematical Association of America. <a href=\"https://doi.org/10.4169/amer.math.monthly.124.7.588\">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>","mla":"Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical Monthly</i>, vol. 124, no. 7, Mathematical Association of America, 2017, pp. 588–96, doi:<a href=\"https://doi.org/10.4169/amer.math.monthly.124.7.588\">10.4169/amer.math.monthly.124.7.588</a>.","ista":"Akopyan A, Vysotsky V. 2017. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 124(7), 588–596.","ieee":"A. Akopyan and V. Vysotsky, “On the lengths of curves passing through boundary points of a planar convex shape,” <i>The American Mathematical Monthly</i>, vol. 124, no. 7. Mathematical Association of America, pp. 588–596, 2017.","chicago":"Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical Monthly</i>. Mathematical Association of America, 2017. <a href=\"https://doi.org/10.4169/amer.math.monthly.124.7.588\">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>."},"publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1605.07997"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:49:09Z","doi":"10.4169/amer.math.monthly.124.7.588","publisher":"Mathematical Association of America","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan","full_name":"Akopyan, Arseniy"},{"first_name":"Vladislav","full_name":"Vysotsky, Vladislav","last_name":"Vysotsky"}],"intvolume":"       124","project":[{"call_identifier":"FP7","grant_number":"291734","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"article_type":"original","date_updated":"2025-07-10T12:01:35Z","department":[{"_id":"HeEd"}],"quality_controlled":"1","issue":"7","type":"journal_article","ec_funded":1,"external_id":{"isi":["000413947300002"],"arxiv":["1605.07997"]},"title":"On the lengths of curves passing through boundary points of a planar convex shape","arxiv":1,"language":[{"iso":"eng"}],"month":"01","isi":1,"publication":"The American Mathematical Monthly","volume":124,"publication_identifier":{"issn":["0002-9890"]},"oa_version":"Submitted Version"},{"doi":"10.4230/LIPIcs.SoCG.2017.39","date_created":"2018-12-11T11:47:56Z","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"full_name":"Wagner, Hubert","last_name":"Wagner","first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"intvolume":"        77","oa":1,"status":"public","file":[{"access_level":"open_access","checksum":"067ab0cb3f962bae6c3af6bf0094e0f3","relation":"main_file","content_type":"application/pdf","creator":"system","date_created":"2018-12-12T10:11:03Z","file_id":"4856","date_updated":"2020-07-14T12:47:42Z","file_size":990546,"file_name":"IST-2017-895-v1+1_LIPIcs-SoCG-2017-39.pdf"}],"publication_status":"published","citation":{"short":"H. Edelsbrunner, H. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017, pp. 391–3916.","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>","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>","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.","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>.","ista":"Edelsbrunner H, Wagner H. 2017. Topological data analysis with Bregman divergences. Symposium on Computational Geometry, SoCG, LIPIcs, vol. 77, 391–3916."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publist_id":"7021","corr_author":"1","date_published":"2017-06-01T00:00:00Z","day":"01","article_processing_charge":"No","_id":"688","scopus_import":"1","page":"391-3916","year":"2017","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. "}],"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)"},"pubrep_id":"895","oa_version":"Published Version","publication_identifier":{"issn":["1868-8969"]},"month":"06","has_accepted_license":"1","volume":77,"language":[{"iso":"eng"}],"file_date_updated":"2020-07-14T12:47:42Z","title":"Topological data analysis with Bregman divergences","ddc":["514","516"],"conference":{"end_date":"2017-07-07","location":"Brisbane, Australia","start_date":"2017-07-04","name":"Symposium on Computational Geometry, SoCG"},"type":"conference","department":[{"_id":"HeEd"},{"_id":"UlWa"}],"quality_controlled":"1","alternative_title":["LIPIcs"],"date_updated":"2025-07-10T11:53:56Z"},{"day":"01","article_processing_charge":"No","page":"690 - 693","abstract":[{"lang":"eng","text":"We answer a question of M. Gromov on the waist of the unit ball."}],"year":"2017","scopus_import":"1","_id":"707","publisher":"Wiley","doi":"10.1112/blms.12062","date_created":"2018-12-11T11:48:02Z","intvolume":"        49","author":[{"last_name":"Akopyan","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Karasev, Roman","last_name":"Karasev","first_name":"Roman"}],"date_published":"2017-08-01T00:00:00Z","publist_id":"6982","corr_author":"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.","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, 2017, pp. 690–93, doi:<a href=\"https://doi.org/10.1112/blms.12062\">10.1112/blms.12062</a>.","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, 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, 2017. <a href=\"https://doi.org/10.1112/blms.12062\">https://doi.org/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. <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>"},"publication_status":"published","status":"public","oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.06279"}],"issue":"4","type":"journal_article","ec_funded":1,"title":"A tight estimate for the waist of the ball ","arxiv":1,"external_id":{"arxiv":["1608.06279"],"isi":["000407045900012"]},"date_updated":"2025-09-10T11:04:43Z","project":[{"name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734"}],"department":[{"_id":"HeEd"}],"quality_controlled":"1","oa_version":"Preprint","publication_identifier":{"issn":["0024-6093"]},"language":[{"iso":"eng"}],"isi":1,"publication":"Bulletin of the London Mathematical Society","month":"08","volume":49},{"date_updated":"2026-04-08T14:19:30Z","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35"}],"quality_controlled":"1","department":[{"_id":"HeEd"}],"type":"journal_article","issue":"3","arxiv":1,"title":"Expected sizes of poisson Delaunay mosaics and their discrete Morse functions","external_id":{"isi":["000416417500004"],"arxiv":["1607.05915"]},"ec_funded":1,"related_material":{"record":[{"relation":"dissertation_contains","id":"6287","status":"public"}]},"language":[{"iso":"eng"}],"volume":49,"publication":"Advances in Applied Probability","month":"09","isi":1,"oa_version":"Preprint","publication_identifier":{"issn":["0001-8678"]},"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"}],"year":"2017","page":"745 - 767","scopus_import":"1","_id":"718","article_processing_charge":"No","day":"01","date_published":"2017-09-01T00:00:00Z","publist_id":"6962","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1607.05915"}],"publication_status":"published","citation":{"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.","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>","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.","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>.","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.","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>."},"oa":1,"status":"public","intvolume":"        49","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"full_name":"Nikitenko, Anton","last_name":"Nikitenko","orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton"},{"first_name":"Matthias","full_name":"Reitzner, Matthias","last_name":"Reitzner"}],"publisher":"Cambridge University Press","doi":"10.1017/apr.2017.20","date_created":"2018-12-11T11:48:07Z"},{"volume":231,"isi":1,"publication":"Topology and its Applications","month":"11","language":[{"iso":"eng"}],"publication_identifier":{"issn":["0166-8641"]},"oa_version":"None","department":[{"_id":"HeEd"}],"quality_controlled":"1","date_updated":"2026-04-16T10:04:39Z","title":"A new topology on the universal path space","external_id":{"isi":["000413889100012"]},"type":"journal_article","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","status":"public","citation":{"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>","short":"Z. Virk, A. Zastrow, Topology and Its Applications 231 (2017) 186–196.","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>","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>.","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."},"publication_status":"published","date_published":"2017-11-01T00:00:00Z","corr_author":"1","publist_id":"6930","author":[{"full_name":"Virk, Ziga","last_name":"Virk","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","first_name":"Ziga"},{"first_name":"Andreas","last_name":"Zastrow","full_name":"Zastrow, Andreas"}],"intvolume":"       231","doi":"10.1016/j.topol.2017.09.015","date_created":"2018-12-11T11:48:14Z","publisher":"Elsevier","_id":"737","scopus_import":"1","year":"2017","abstract":[{"lang":"eng","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."}],"page":"186 - 196","day":"01","article_processing_charge":"No"}]