[{"title":"Counting blanks in polygonal arrangements","external_id":{"isi":["000450810500036"],"arxiv":["1604.00960"]},"intvolume":"        32","date_created":"2018-12-11T11:44:24Z","year":"2018","publisher":"Society for Industrial and Applied Mathematics ","article_processing_charge":"No","citation":{"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>","short":"A. Akopyan, E. Segal Halevi, SIAM Journal on Discrete Mathematics 32 (2018) 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>.","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>"},"day":"06","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"}],"oa":1,"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","first_name":"Arseniy","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X"},{"last_name":"Segal Halevi","first_name":"Erel","full_name":"Segal Halevi, Erel"}],"arxiv":1,"date_published":"2018-09-06T00:00:00Z","issue":"3","doi":"10.1137/16M110407X","main_file_link":[{"url":"https://arxiv.org/abs/1604.00960","open_access":"1"}],"date_updated":"2025-04-15T06:50:24Z","status":"public","publist_id":"7996","month":"09","volume":32,"publication_status":"published","isi":1,"oa_version":"Preprint","_id":"58","page":"2242 - 2257","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","quality_controlled":"1","project":[{"call_identifier":"FP7","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme"}],"publication":"SIAM Journal on Discrete Mathematics","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"type":"journal_article","scopus_import":"1","ec_funded":1},{"doi":"10.1017/fms.2018.7","date_published":"2018-05-31T00:00:00Z","author":[{"last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","first_name":"Arseniy"},{"orcid":"0000-0002-7840-5062","first_name":"Sergey","full_name":"Avvakumov, Sergey","last_name":"Avvakumov","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87"}],"arxiv":1,"oa":1,"abstract":[{"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.","lang":"eng"}],"day":"31","citation":{"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>","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>.","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.","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>.","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>"},"publisher":"Cambridge University Press","file":[{"date_updated":"2020-07-14T12:47:28Z","file_size":249246,"file_name":"2018_ForumMahtematics_Akopyan.pdf","creator":"dernst","content_type":"application/pdf","date_created":"2019-04-30T06:14:58Z","file_id":"6356","relation":"main_file","checksum":"5a71b24ba712a3eb2e46165a38fbc30a","access_level":"open_access"}],"article_processing_charge":"No","file_date_updated":"2020-07-14T12:47:28Z","ddc":["510"],"year":"2018","date_created":"2019-04-30T06:09:57Z","intvolume":"         6","external_id":{"isi":["000433915500001"],"arxiv":["1712.10205"]},"publication_identifier":{"issn":["2050-5094"]},"title":"Any cyclic quadrilateral can be inscribed in any closed convex smooth curve","has_accepted_license":"1","ec_funded":1,"corr_author":"1","type":"journal_article","department":[{"_id":"UlWa"},{"_id":"HeEd"},{"_id":"JaMa"}],"language":[{"iso":"eng"}],"publication":"Forum of Mathematics, Sigma","project":[{"grant_number":"716117","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics"}],"article_number":"e7","quality_controlled":"1","related_material":{"record":[{"id":"8156","status":"public","relation":"dissertation_contains"}]},"oa_version":"Published Version","_id":"6355","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","isi":1,"month":"05","publication_status":"published","volume":6,"date_updated":"2026-04-08T07:25:54Z","status":"public","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"}},{"date_published":"2018-06-11T00:00:00Z","author":[{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","last_name":"Iglesias Ham","full_name":"Iglesias Ham, Mabel","first_name":"Mabel"}],"oa":1,"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"}],"day":"11","doi":"10.15479/AT:ISTA:th_1026","year":"2018","date_created":"2018-12-11T11:45:10Z","publication_identifier":{"issn":["2663-337X"]},"title":"Multiple covers with balls","has_accepted_license":"1","citation":{"ieee":"M. Iglesias Ham, “Multiple covers with balls,” Institute of Science and Technology Austria, 2018.","ista":"Iglesias Ham M. 2018. Multiple covers with balls. Institute of Science and Technology Austria.","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>","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>.","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>","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>."},"file":[{"date_created":"2019-02-05T07:43:31Z","file_size":11827713,"date_updated":"2020-07-14T12:45:24Z","content_type":"application/zip","file_name":"IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip","creator":"kschuh","access_level":"closed","checksum":"dd699303623e96d1478a6ae07210dd05","relation":"source_file","file_id":"5918"},{"relation":"main_file","access_level":"open_access","checksum":"ba163849a190d2b41d66fef0e4983294","file_id":"5919","date_created":"2019-02-05T07:43:45Z","file_name":"IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf","creator":"kschuh","content_type":"application/pdf","date_updated":"2020-07-14T12:45:24Z","file_size":4783846}],"article_processing_charge":"No","publisher":"Institute of Science and Technology Austria","file_date_updated":"2020-07-14T12:45:24Z","ddc":["514","516"],"pubrep_id":"1026","language":[{"iso":"eng"}],"alternative_title":["ISTA Thesis"],"corr_author":"1","supervisor":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"}],"OA_place":"publisher","type":"dissertation","department":[{"_id":"HeEd"}],"degree_awarded":"PhD","month":"06","publication_status":"published","status":"public","date_updated":"2026-04-08T14:04:03Z","publist_id":"7712","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","_id":"201","page":"171","oa_version":"Published Version"},{"type":"journal_article","department":[{"_id":"HeEd"}],"scopus_import":"1","publication":"Mathematical Intelligencer","language":[{"iso":"eng"}],"_id":"106","page":"26 - 31","oa_version":"Preprint","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","quality_controlled":"1","date_updated":"2023-09-13T08:49:16Z","publist_id":"7948","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1702.05172"}],"isi":1,"volume":40,"publication_status":"published","month":"09","issue":"3","doi":"10.1007/s00283-018-9795-5","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."}],"day":"01","date_published":"2018-09-01T00:00:00Z","author":[{"orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Petrunin","first_name":"Anton","full_name":"Petrunin, Anton"}],"arxiv":1,"oa":1,"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>","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>.","ista":"Akopyan A, Petrunin A. 2018. Long geodesics on convex surfaces. Mathematical Intelligencer. 40(3), 26–31.","ieee":"A. Akopyan and A. Petrunin, “Long geodesics on convex surfaces,” <i>Mathematical Intelligencer</i>, vol. 40, no. 3. Springer, pp. 26–31, 2018.","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>","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>.","short":"A. Akopyan, A. Petrunin, Mathematical Intelligencer 40 (2018) 26–31."},"publisher":"Springer","article_processing_charge":"No","title":"Long geodesics on convex surfaces","date_created":"2018-12-11T11:44:40Z","year":"2018","intvolume":"        40","external_id":{"isi":["000444141200005"],"arxiv":["1702.05172"]}},{"citation":{"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>","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.","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.","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>.","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."},"file":[{"file_id":"5844","access_level":"open_access","relation":"main_file","content_type":"application/pdf","file_name":"2018_DiscreteComp_Akopyan.pdf","success":1,"creator":"dernst","file_size":482518,"date_updated":"2019-01-18T09:27:36Z","date_created":"2019-01-18T09:27:36Z"}],"article_processing_charge":"Yes (via OA deal)","publisher":"Springer","file_date_updated":"2019-01-18T09:27:36Z","ddc":["516","000"],"intvolume":"        59","date_created":"2018-12-11T11:49:57Z","year":"2018","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"external_id":{"isi":["000432205500011"]},"title":"On the circle covering theorem by A.W. Goodman and R.E. Goodman","has_accepted_license":"1","doi":"10.1007/s00454-017-9883-x","article_type":"original","issue":"4","date_published":"2018-06-01T00:00:00Z","oa":1,"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","full_name":"Akopyan, Arseniy","first_name":"Arseniy","orcid":"0000-0002-2548-617X"},{"full_name":"Balitskiy, Alexey","first_name":"Alexey","last_name":"Balitskiy"},{"first_name":"Mikhail","full_name":"Grigorev, Mikhail","last_name":"Grigorev"}],"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."}],"day":"01","quality_controlled":"1","_id":"1064","oa_version":"Published Version","page":"1001-1009","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"06","volume":59,"publication_status":"published","isi":1,"publist_id":"6324","status":"public","date_updated":"2026-05-20T10:19:33Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"ec_funded":1,"corr_author":"1","scopus_import":"1","type":"journal_article","department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"publication":"Discrete & Computational Geometry","project":[{"call_identifier":"FP7","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme"}]},{"article_type":"original","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).","issue":"1","doi":"10.1137/16M1097201","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"}],"day":"29","date_published":"2018-03-29T00:00:00Z","author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","last_name":"Iglesias Ham","full_name":"Iglesias Ham, Mabel","first_name":"Mabel"}],"oa":1,"citation":{"short":"H. Edelsbrunner, M. Iglesias Ham, SIAM J Discrete Math 32 (2018) 750–782.","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>","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>.","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.","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.","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>","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>."},"article_processing_charge":"No","publisher":"Society for Industrial and Applied Mathematics ","title":"On the optimality of the FCC lattice for soft sphere packing","date_created":"2018-12-11T11:45:46Z","year":"2018","intvolume":"        32","external_id":{"isi":["000428958900038"]},"publication_identifier":{"issn":["0895-4801"]},"type":"journal_article","department":[{"_id":"HeEd"}],"scopus_import":"1","publication":"SIAM J Discrete Math","project":[{"grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes"}],"language":[{"iso":"eng"}],"oa_version":"Submitted Version","_id":"312","page":"750 - 782","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","quality_controlled":"1","status":"public","date_updated":"2026-04-16T09:53:02Z","publist_id":"7553","main_file_link":[{"url":"http://pdfs.semanticscholar.org/d2d5/6da00fbc674e6a8b1bb9d857167e54200dc6.pdf","open_access":"1"}],"isi":1,"publication_status":"published","volume":32,"month":"03"},{"type":"journal_article","department":[{"_id":"HeEd"}],"scopus_import":"1","corr_author":"1","publication":"Comptes Rendus Mathematique","language":[{"iso":"eng"}],"page":"412-414","_id":"409","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","status":"public","date_updated":"2025-07-10T11:52:35Z","publist_id":"7420","main_file_link":[{"url":"https://arxiv.org/abs/1805.01652","open_access":"1"}],"publication_status":"published","month":"04","volume":356,"isi":1,"article_type":"original","issue":"4","doi":"10.1016/j.crma.2018.03.005","abstract":[{"lang":"eng","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."}],"day":"01","date_published":"2018-04-01T00:00:00Z","oa":1,"arxiv":1,"author":[{"orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"}],"citation":{"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.","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>.","short":"A. Akopyan, Comptes Rendus Mathematique 356 (2018) 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>.","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>"},"publisher":"Elsevier","article_processing_charge":"No","title":"On the number of non-hexagons in a planar tiling","intvolume":"       356","date_created":"2018-12-11T11:46:19Z","year":"2018","publication_identifier":{"issn":["1631-073X"]},"external_id":{"isi":["000430402700009"],"arxiv":["1805.01652"]}},{"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1705.02870"}],"publist_id":"7967","status":"public","date_updated":"2026-04-08T14:19:30Z","isi":1,"publication_status":"published","month":"10","volume":28,"page":"3215 - 3238","_id":"87","oa_version":"Preprint","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"6287"}]},"quality_controlled":"1","project":[{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF"}],"publication":"Annals of Applied Probability","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"type":"journal_article","scopus_import":"1","title":"Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics","external_id":{"isi":["000442893500018"],"arxiv":["1705.02870"]},"date_created":"2018-12-11T11:44:33Z","year":"2018","intvolume":"        28","article_processing_charge":"No","publisher":"Institute of Mathematical Statistics","citation":{"apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2018). Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/18-AAP1389\">https://doi.org/10.1214/18-AAP1389</a>","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2018. <a href=\"https://doi.org/10.1214/18-AAP1389\">https://doi.org/10.1214/18-AAP1389</a>.","ista":"Edelsbrunner H, Nikitenko A. 2018. Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. Annals of Applied Probability. 28(5), 3215–3238.","ieee":"H. Edelsbrunner and A. Nikitenko, “Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics,” <i>Annals of Applied Probability</i>, vol. 28, no. 5. Institute of Mathematical Statistics, pp. 3215–3238, 2018.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics.” <i>Annals of Applied Probability</i>, vol. 28, no. 5, Institute of Mathematical Statistics, 2018, pp. 3215–38, doi:<a href=\"https://doi.org/10.1214/18-AAP1389\">10.1214/18-AAP1389</a>.","ama":"Edelsbrunner H, Nikitenko A. Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. <i>Annals of Applied Probability</i>. 2018;28(5):3215-3238. doi:<a href=\"https://doi.org/10.1214/18-AAP1389\">10.1214/18-AAP1389</a>","short":"H. Edelsbrunner, A. Nikitenko, Annals of Applied Probability 28 (2018) 3215–3238."},"day":"01","abstract":[{"text":"Using the geodesic distance on the n-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson–Delaunay mosaic in n-dimensional Euclidean space. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in Rn+1, so we also get the expected number of faces of a random inscribed polytope. As proved in Antonelli et al. [Adv. in Appl. Probab. 9–12 (1977–1980)], an orthant section of the n-sphere is isometric to the standard n-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the n-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics.","lang":"eng"}],"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Anton","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","last_name":"Nikitenko"}],"arxiv":1,"oa":1,"date_published":"2018-10-01T00:00:00Z","issue":"5","article_type":"original","doi":"10.1214/18-AAP1389"},{"title":"3-Webs generated by confocal conics and circles","has_accepted_license":"1","year":"2018","date_created":"2018-12-11T11:47:57Z","intvolume":"       194","external_id":{"isi":["000431418800004"]},"file_date_updated":"2020-07-14T12:47:44Z","ddc":["510"],"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>.","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.","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>","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."},"publisher":"Springer","file":[{"creator":"kschuh","file_name":"2018_Springer_Akopyan.pdf","content_type":"application/pdf","date_updated":"2020-07-14T12:47:44Z","file_size":1140860,"date_created":"2020-01-03T11:35:08Z","file_id":"7222","relation":"main_file","checksum":"1febcfc1266486053a069e3425ea3713","access_level":"open_access"}],"article_processing_charge":"Yes (via OA deal)","abstract":[{"text":"We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.","lang":"eng"}],"day":"01","date_published":"2018-06-01T00:00:00Z","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","first_name":"Arseniy","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X"}],"oa":1,"article_type":"original","issue":"1","doi":"10.1007/s10711-017-0265-6","publist_id":"7014","date_updated":"2025-04-15T06:50:29Z","status":"public","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"isi":1,"month":"06","publication_status":"published","volume":194,"_id":"692","page":"55 - 64","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa_version":"Published Version","quality_controlled":"1","publication":"Geometriae Dedicata","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme","grant_number":"291734","call_identifier":"FP7"}],"language":[{"iso":"eng"}],"type":"journal_article","department":[{"_id":"HeEd"}],"ec_funded":1,"scopus_import":"1","corr_author":"1"},{"department":[{"_id":"HeEd"},{"_id":"JaMa"}],"type":"preprint","corr_author":"1","doi":"10.48550/arXiv.1804.03057","ec_funded":1,"day":"13","project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020","grant_number":"716117"}],"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."}],"oa":1,"arxiv":1,"language":[{"iso":"eng"}],"author":[{"last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","full_name":"Akopyan, Arseniy"},{"first_name":"Sergey","full_name":"Avvakumov, Sergey","orcid":"0000-0002-7840-5062","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","last_name":"Avvakumov"},{"full_name":"Karasev, Roman","first_name":"Roman","last_name":"Karasev"}],"date_published":"2018-09-13T00:00:00Z","oa_version":"Preprint","_id":"75","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","publisher":"arXiv","related_material":{"record":[{"id":"8156","status":"public","relation":"dissertation_contains"}]},"citation":{"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>.","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>","ista":"Akopyan A, Avvakumov S, Karasev R. 2018. Convex fair partitions into arbitrary number of pieces. 1804.03057.","ieee":"A. Akopyan, S. Avvakumov, and R. Karasev, “Convex fair partitions into arbitrary number of pieces.” arXiv, 2018.","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>","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>."},"article_number":"1804.03057","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1804.03057"}],"title":"Convex fair partitions into arbitrary number of pieces","date_updated":"2026-04-08T07:25:54Z","status":"public","publication_status":"published","month":"09","external_id":{"arxiv":["1804.03057"]},"date_created":"2018-12-11T11:44:30Z","year":"2018"},{"publication_status":"published","volume":37,"month":"10","isi":1,"main_file_link":[{"url":"https://arxiv.org/abs/1411.6337","open_access":"1"}],"publist_id":"6182","date_updated":"2025-06-04T08:44:44Z","status":"public","quality_controlled":"1","oa_version":"Submitted Version","_id":"1173","page":"887 - 910","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"project":[{"grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems"}],"publication":"Combinatorica","scopus_import":"1","ec_funded":1,"department":[{"_id":"HeEd"}],"type":"journal_article","publication_identifier":{"issn":["0209-9683"]},"external_id":{"isi":["000418056000005"],"arxiv":["1411.6337"]},"intvolume":"        37","year":"2017","date_created":"2018-12-11T11:50:32Z","title":"The Voronoi functional is maximized by the Delaunay triangulation in the plane","publisher":"Springer","article_processing_charge":"No","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>.","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.","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.","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>","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."},"oa":1,"arxiv":1,"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"last_name":"Glazyrin","full_name":"Glazyrin, Alexey","first_name":"Alexey"},{"last_name":"Musin","full_name":"Musin, Oleg","first_name":"Oleg"},{"orcid":"0000-0002-0659-3201","first_name":"Anton","full_name":"Nikitenko, Anton","last_name":"Nikitenko","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"}],"date_published":"2017-10-01T00:00:00Z","day":"01","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."}],"doi":"10.1007/s00493-016-3308-y","issue":"5","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."},{"citation":{"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>","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>.","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.","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>","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>.","short":"A. Akopyan, I. Bárány, S. Robins, Advances in Mathematics 308 (2017) 627–644."},"publisher":"Academic Press","article_processing_charge":"No","year":"2017","date_created":"2018-12-11T11:50:34Z","intvolume":"       308","external_id":{"arxiv":["1508.07594"],"isi":["000409292900015"]},"publication_identifier":{"issn":["0001-8708"]},"title":"Algebraic vertices of non-convex polyhedra","doi":"10.1016/j.aim.2016.12.026","date_published":"2017-02-21T00:00:00Z","arxiv":1,"author":[{"orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Bárány","full_name":"Bárány, Imre","first_name":"Imre"},{"last_name":"Robins","full_name":"Robins, Sinai","first_name":"Sinai"}],"oa":1,"abstract":[{"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.","lang":"eng"}],"day":"21","quality_controlled":"1","page":"627 - 644","_id":"1180","oa_version":"Submitted Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","isi":1,"volume":308,"month":"02","publication_status":"published","status":"public","date_updated":"2025-06-04T08:45:48Z","publist_id":"6173","main_file_link":[{"url":"https://arxiv.org/abs/1508.07594","open_access":"1"}],"ec_funded":1,"scopus_import":"1","type":"journal_article","department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"publication":"Advances in Mathematics","project":[{"name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734"}]},{"date_published":"2017-01-01T00:00:00Z","oa":1,"author":[{"last_name":"Bauer","first_name":"Ulrich","full_name":"Bauer, Ulrich"},{"full_name":"Kerber, Michael","first_name":"Michael","last_name":"Kerber"},{"last_name":"Reininghaus","first_name":"Jan","full_name":"Reininghaus, Jan"},{"last_name":"Wagner","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","orcid":"0009-0009-9111-8429","full_name":"Wagner, Hubert","first_name":"Hubert"}],"abstract":[{"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.","lang":"eng"}],"day":"01","doi":"10.1016/j.jsc.2016.03.008","article_type":"original","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).","intvolume":"        78","date_created":"2018-12-11T11:51:59Z","year":"2017","publication_identifier":{"issn":[" 0747-7171"]},"external_id":{"isi":["000384396000005"]},"title":"Phat - Persistent homology algorithms toolbox","citation":{"ista":"Bauer U, Kerber M, Reininghaus J, Wagner H. 2017. Phat - Persistent homology algorithms toolbox. Journal of Symbolic Computation. 78, 76–90.","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.","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>.","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.","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>","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>."},"publisher":"Academic Press","article_processing_charge":"No","ddc":["500"],"language":[{"iso":"eng"}],"publication":"Journal of Symbolic Computation","project":[{"grant_number":"318493","call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"ec_funded":1,"scopus_import":"1","corr_author":"1","type":"journal_article","department":[{"_id":"HeEd"}],"OA_type":"free access","month":"01","publication_status":"published","volume":78,"isi":1,"publist_id":"5765","status":"public","date_updated":"2026-06-18T17:35:16Z","main_file_link":[{"url":"https://doi.org/10.1016/j.jsc.2016.03.008","open_access":"1"}],"quality_controlled":"1","related_material":{"record":[{"status":"public","id":"10894","relation":"earlier_version"}]},"oa_version":"Published Version","_id":"1433","page":"76 - 90","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"quality_controlled":"1","related_material":{"record":[{"relation":"earlier_version","id":"10892","status":"public"}]},"_id":"481","page":"211 - 229","oa_version":"Published Version","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","month":"04","publication_status":"published","volume":26,"publist_id":"7338","date_updated":"2025-09-29T13:22:54Z","status":"public","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"scopus_import":1,"corr_author":"1","type":"journal_article","department":[{"_id":"HeEd"}],"pubrep_id":"949","language":[{"iso":"eng"}],"publication":"International Journal of Computational Geometry and Applications","citation":{"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.","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.","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>","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>","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>."},"publisher":"World Scientific Publishing","file":[{"relation":"main_file","access_level":"open_access","checksum":"f79e8558bfe4b368dfefeb8eec2e3a5e","file_id":"4758","date_created":"2018-12-12T10:09:34Z","date_updated":"2020-07-14T12:46:35Z","file_size":769296,"creator":"system","file_name":"IST-2018-949-v1+1_2016_huber_PLanar_matchings.pdf","content_type":"application/pdf"}],"file_date_updated":"2020-07-14T12:46:35Z","ddc":["004","514","516"],"year":"2017","date_created":"2018-12-11T11:46:43Z","intvolume":"        26","title":"Planar matchings for weighted straight skeletons","has_accepted_license":"1","doi":"10.1142/S0218195916600050","acknowledgement":"Supported by NSERC and the Ross and Muriel Cheriton Fellowship. Research supported by Austrian Science Fund (FWF): P25816-N15.","issue":"3-4","date_published":"2017-04-13T00:00:00Z","author":[{"full_name":"Biedl, Therese","first_name":"Therese","last_name":"Biedl"},{"full_name":"Huber, Stefan","first_name":"Stefan","orcid":"0000-0002-8871-5814","id":"4700A070-F248-11E8-B48F-1D18A9856A87","last_name":"Huber"},{"last_name":"Palfrader","first_name":"Peter","full_name":"Palfrader, Peter"}],"oa":1,"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"}],"day":"13"},{"month":"01","publication_status":"published","volume":215,"isi":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.03954"}],"publist_id":"7299","status":"public","date_updated":"2025-09-18T09:47:04Z","quality_controlled":"1","page":"45 - 57","_id":"521","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa_version":"Submitted Version","language":[{"iso":"eng"}],"publication":"Topology and its Applications","corr_author":"1","scopus_import":"1","department":[{"_id":"HeEd"}],"type":"journal_article","publication_identifier":{"issn":["0166-8641"]},"external_id":{"isi":["000390501400005"],"arxiv":["1608.03954"]},"intvolume":"       215","year":"2017","date_created":"2018-12-11T11:46:56Z","title":"Higson compactification and dimension raising","article_processing_charge":"No","publisher":"Elsevier","citation":{"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>","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>.","ieee":"K. Austin and Z. Virk, “Higson compactification and dimension raising,” <i>Topology and its Applications</i>, vol. 215. Elsevier, pp. 45–57, 2017.","ista":"Austin K, Virk Z. 2017. Higson compactification and dimension raising. Topology and its Applications. 215, 45–57.","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>","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>.","short":"K. Austin, Z. Virk, Topology and Its Applications 215 (2017) 45–57."},"oa":1,"author":[{"last_name":"Austin","full_name":"Austin, Kyle","first_name":"Kyle"},{"full_name":"Virk, Ziga","first_name":"Ziga","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","last_name":"Virk"}],"arxiv":1,"date_published":"2017-01-01T00:00:00Z","day":"01","abstract":[{"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.","lang":"eng"}],"doi":"10.1016/j.topol.2016.10.005"},{"type":"journal_article","department":[{"_id":"UlWa"},{"_id":"HeEd"}],"ec_funded":1,"corr_author":"1","scopus_import":"1","publication":"Homology, Homotopy and Applications","project":[{"name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734"},{"grant_number":"701309","call_identifier":"H2020","_id":"2590DB08-B435-11E9-9278-68D0E5697425","name":"Atomic Resolution Structures of Mitochondrial Respiratory Chain Supercomplexes"}],"language":[{"iso":"eng"}],"page":"313 - 342","_id":"568","oa_version":"Submitted Version","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","quality_controlled":"1","date_updated":"2025-09-11T07:41:51Z","publist_id":"7246","status":"public","main_file_link":[{"url":"https://arxiv.org/abs/1507.04310","open_access":"1"}],"isi":1,"volume":19,"publication_status":"published","month":"01","issue":"2","doi":"10.4310/HHA.2017.v19.n2.a16","abstract":[{"lang":"eng","text":"We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z&lt; r(f) := (g-1(0): ||g - f|| &lt; r) of all zero sets of all continuous maps g closer to f than r in the max-norm. All of these sets are outside A := (x: |f(x)| ≥ r) and we claim that Z&lt; r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n - 3. By considering all r &gt; 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C)."}],"day":"01","date_published":"2017-01-01T00:00:00Z","arxiv":1,"author":[{"first_name":"Peter","full_name":"Franek, Peter","orcid":"0000-0001-8878-8397","id":"473294AE-F248-11E8-B48F-1D18A9856A87","last_name":"Franek"},{"last_name":"Krcál","id":"33E21118-F248-11E8-B48F-1D18A9856A87","first_name":"Marek","full_name":"Krcál, Marek"}],"oa":1,"citation":{"apa":"Franek, P., &#38; Krcál, M. (2017). Persistence of zero sets. <i>Homology, Homotopy and Applications</i>. International Press. <a href=\"https://doi.org/10.4310/HHA.2017.v19.n2.a16\">https://doi.org/10.4310/HHA.2017.v19.n2.a16</a>","chicago":"Franek, Peter, and Marek Krcál. “Persistence of Zero Sets.” <i>Homology, Homotopy and Applications</i>. International Press, 2017. <a href=\"https://doi.org/10.4310/HHA.2017.v19.n2.a16\">https://doi.org/10.4310/HHA.2017.v19.n2.a16</a>.","ista":"Franek P, Krcál M. 2017. Persistence of zero sets. Homology, Homotopy and Applications. 19(2), 313–342.","ieee":"P. Franek and M. Krcál, “Persistence of zero sets,” <i>Homology, Homotopy and Applications</i>, vol. 19, no. 2. International Press, pp. 313–342, 2017.","mla":"Franek, Peter, and Marek Krcál. “Persistence of Zero Sets.” <i>Homology, Homotopy and Applications</i>, vol. 19, no. 2, International Press, 2017, pp. 313–42, doi:<a href=\"https://doi.org/10.4310/HHA.2017.v19.n2.a16\">10.4310/HHA.2017.v19.n2.a16</a>.","ama":"Franek P, Krcál M. Persistence of zero sets. <i>Homology, Homotopy and Applications</i>. 2017;19(2):313-342. doi:<a href=\"https://doi.org/10.4310/HHA.2017.v19.n2.a16\">10.4310/HHA.2017.v19.n2.a16</a>","short":"P. Franek, M. Krcál, Homology, Homotopy and Applications 19 (2017) 313–342."},"publisher":"International Press","article_processing_charge":"No","title":"Persistence of zero sets","date_created":"2018-12-11T11:47:14Z","year":"2017","intvolume":"        19","external_id":{"arxiv":["1507.04310"],"isi":["000440749400010"]},"publication_identifier":{"issn":["1532-0073"]}},{"publication_status":"published","month":"05","volume":10256,"status":"public","date_updated":"2022-01-28T07:48:24Z","quality_controlled":"1","oa_version":"None","_id":"5803","page":"93-104","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","language":[{"iso":"eng"}],"place":"Cham","publication":"Combinatorial image analysis","alternative_title":["LNCS"],"department":[{"_id":"HeEd"}],"type":"book_chapter","publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["978-3-319-59107-0","978-3-319-59108-7"]},"intvolume":"     10256","year":"2017","date_created":"2019-01-08T20:42:56Z","title":"Construction of persistent Voronoi diagram on 3D digital plane","publisher":"Springer Nature","article_processing_charge":"No","citation":{"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>","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>.","short":"R. Biswas, P. Bhowmick, in:, Combinatorial Image Analysis, Springer Nature, Cham, 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>.","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>","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."},"extern":"1","author":[{"last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita","first_name":"Ranita"},{"last_name":"Bhowmick","first_name":"Partha","full_name":"Bhowmick, Partha"}],"date_published":"2017-05-17T00:00:00Z","day":"17","abstract":[{"lang":"eng","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."}],"conference":{"location":"Plovdiv, Bulgaria","start_date":"2017-06-19","name":"IWCIA: International Workshop on Combinatorial Image Analysis","end_date":"2017-06-21"},"doi":"10.1007/978-3-319-59108-7_8"},{"file_date_updated":"2020-07-14T12:47:26Z","ddc":["514","516","519"],"citation":{"ama":"Nikitenko A. Discrete Morse theory for random complexes . 2017. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">10.15479/AT:ISTA:th_873</a>","mla":"Nikitenko, Anton. <i>Discrete Morse Theory for Random Complexes </i>. Institute of Science and Technology Austria, 2017, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">10.15479/AT:ISTA:th_873</a>.","short":"A. Nikitenko, Discrete Morse Theory for Random Complexes , Institute of Science and Technology Austria, 2017.","chicago":"Nikitenko, Anton. “Discrete Morse Theory for Random Complexes .” Institute of Science and Technology Austria, 2017. <a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">https://doi.org/10.15479/AT:ISTA:th_873</a>.","apa":"Nikitenko, A. (2017). <i>Discrete Morse theory for random complexes </i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">https://doi.org/10.15479/AT:ISTA:th_873</a>","ieee":"A. Nikitenko, “Discrete Morse theory for random complexes ,” Institute of Science and Technology Austria, 2017.","ista":"Nikitenko A. 2017. Discrete Morse theory for random complexes . Institute of Science and Technology Austria."},"file":[{"date_created":"2019-04-09T14:54:51Z","file_size":2324870,"date_updated":"2020-07-14T12:47:26Z","content_type":"application/pdf","creator":"dernst","file_name":"2017_Thesis_Nikitenko.pdf","access_level":"open_access","checksum":"ece7e598a2f060b263c2febf7f3fe7f9","relation":"main_file","file_id":"6289"},{"file_id":"6290","relation":"source_file","checksum":"99b7ad76e317efd447af60f91e29b49b","access_level":"closed","date_updated":"2020-07-14T12:47:26Z","file_size":2863219,"file_name":"2017_Thesis_Nikitenko_source.zip","creator":"dernst","content_type":"application/zip","date_created":"2019-04-09T14:54:51Z"}],"article_processing_charge":"No","publisher":"Institute of Science and Technology Austria","title":"Discrete Morse theory for random complexes ","has_accepted_license":"1","date_created":"2019-04-09T15:04:32Z","year":"2017","publication_identifier":{"issn":["2663-337X"]},"doi":"10.15479/AT:ISTA:th_873","abstract":[{"lang":"eng","text":"The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's."}],"day":"27","date_published":"2017-10-27T00:00:00Z","oa":1,"author":[{"first_name":"Anton","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","last_name":"Nikitenko"}],"_id":"6287","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","oa_version":"Published Version","page":"86","related_material":{"record":[{"id":"87","status":"public","relation":"part_of_dissertation"},{"status":"public","id":"5678","relation":"part_of_dissertation"},{"status":"public","id":"718","relation":"part_of_dissertation"}]},"status":"public","date_updated":"2026-04-08T14:19:31Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"degree_awarded":"PhD","month":"10","publication_status":"published","OA_place":"publisher","type":"dissertation","department":[{"_id":"HeEd"}],"alternative_title":["ISTA Thesis"],"supervisor":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"corr_author":"1","pubrep_id":"873","language":[{"iso":"eng"}]},{"citation":{"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.","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.","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>","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>.","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>","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>."},"article_processing_charge":"No","publisher":"Oxford University Press","intvolume":"       465","year":"2017","date_created":"2018-12-11T11:49:44Z","publication_identifier":{"issn":["0035-8711"]},"external_id":{"arxiv":["1608.04519"],"isi":["000395170200039"]},"title":"The topology of the cosmic web in terms of persistent Betti numbers","doi":"10.1093/mnras/stw2862","issue":"4","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.","date_published":"2017-01-01T00:00:00Z","oa":1,"author":[{"full_name":"Pranav, Pratyush","first_name":"Pratyush","last_name":"Pranav"},{"orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Van De Weygaert","first_name":"Rien","full_name":"Van De Weygaert, Rien"},{"last_name":"Vegter","first_name":"Gert","full_name":"Vegter, Gert"},{"first_name":"Michael","full_name":"Kerber, Michael","last_name":"Kerber"},{"last_name":"Jones","full_name":"Jones, Bernard","first_name":"Bernard"},{"last_name":"Wintraecken","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","first_name":"Mathijs"}],"arxiv":1,"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"}],"day":"01","quality_controlled":"1","_id":"1022","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"4281 - 4310","oa_version":"Submitted Version","publication_status":"published","volume":465,"month":"01","isi":1,"publist_id":"6373","date_updated":"2025-06-04T08:10:31Z","status":"public","main_file_link":[{"url":"https://arxiv.org/abs/1608.04519","open_access":"1"}],"scopus_import":"1","type":"journal_article","department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"publication":"Monthly Notices of the Royal Astronomical Society"},{"quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"1065","page":"25 - 29","oa_version":"Submitted Version","isi":1,"volume":122,"publication_status":"published","month":"06","status":"public","date_updated":"2025-07-10T11:49:53Z","publist_id":"6323","scopus_import":"1","ec_funded":1,"department":[{"_id":"KrCh"},{"_id":"HeEd"}],"type":"journal_article","language":[{"iso":"eng"}],"pubrep_id":"991","project":[{"name":"Modern Graph Algorithmic Techniques in Formal Verification","_id":"2584A770-B435-11E9-9278-68D0E5697425","grant_number":"P 23499-N23","call_identifier":"FWF"},{"_id":"25863FF4-B435-11E9-9278-68D0E5697425","name":"Game Theory","grant_number":"S11407","call_identifier":"FWF"},{"call_identifier":"FP7","grant_number":"279307","name":"Quantitative Graph Games: Theory and Applications","_id":"2581B60A-B435-11E9-9278-68D0E5697425"}],"publication":"Information Processing Letters","file":[{"relation":"main_file","access_level":"open_access","file_id":"4998","date_created":"2018-12-12T10:13:17Z","date_updated":"2019-10-15T07:44:51Z","file_size":247657,"creator":"system","file_name":"IST-2018-991-v1+2_2018_Chatterjee_Pushdown_PREPRINT.pdf","content_type":"application/pdf"}],"article_processing_charge":"No","publisher":"Elsevier","citation":{"short":"K. Chatterjee, G.F. Osang, Information Processing Letters 122 (2017) 25–29.","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>","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>.","ieee":"K. Chatterjee and G. F. Osang, “Pushdown reachability with constant treewidth,” <i>Information Processing Letters</i>, vol. 122. Elsevier, pp. 25–29, 2017.","ista":"Chatterjee K, Osang GF. 2017. Pushdown reachability with constant treewidth. Information Processing Letters. 122, 25–29.","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>.","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>"},"ddc":["000"],"file_date_updated":"2019-10-15T07:44:51Z","external_id":{"isi":["000399506600005"]},"publication_identifier":{"issn":["0020-0190"]},"year":"2017","date_created":"2018-12-11T11:49:57Z","intvolume":"       122","has_accepted_license":"1","title":"Pushdown reachability with constant treewidth","doi":"10.1016/j.ipl.2017.02.003","author":[{"last_name":"Chatterjee","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4561-241X","full_name":"Chatterjee, Krishnendu","first_name":"Krishnendu"},{"first_name":"Georg F","full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang"}],"oa":1,"date_published":"2017-06-01T00:00:00Z","day":"01","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"}]}]