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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."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"718","intvolume":" 49","status":"public","title":"Expected sizes of poisson Delaunay mosaics and their discrete Morse functions","oa_version":"Preprint","scopus_import":1,"day":"01","citation":{"short":"H. Edelsbrunner, A. Nikitenko, M. Reitzner, Advances in Applied Probability 49 (2017) 745–767.","mla":"Edelsbrunner, Herbert, et al. “Expected Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” Advances in Applied Probability, vol. 49, no. 3, Cambridge University Press, 2017, pp. 745–67, doi:10.1017/apr.2017.20.","chicago":"Edelsbrunner, Herbert, Anton Nikitenko, and Matthias Reitzner. “Expected Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” Advances in Applied Probability. Cambridge University Press, 2017. https://doi.org/10.1017/apr.2017.20.","ama":"Edelsbrunner H, Nikitenko A, Reitzner M. Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. Advances in Applied Probability. 2017;49(3):745-767. doi:10.1017/apr.2017.20","apa":"Edelsbrunner, H., Nikitenko, A., & Reitzner, M. (2017). Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. Advances in Applied Probability. Cambridge University Press. https://doi.org/10.1017/apr.2017.20","ieee":"H. Edelsbrunner, A. Nikitenko, and M. Reitzner, “Expected sizes of poisson Delaunay mosaics and their discrete Morse functions,” Advances in Applied Probability, vol. 49, no. 3. Cambridge University Press, pp. 745–767, 2017.","ista":"Edelsbrunner H, Nikitenko A, Reitzner M. 2017. Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. Advances in Applied Probability. 49(3), 745–767."},"publication":"Advances in Applied Probability","page":"745 - 767","date_published":"2017-09-01T00:00:00Z"},{"publication_identifier":{"issn":["2663-337X"]},"month":"10","language":[{"iso":"eng"}],"supervisor":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert"}],"degree_awarded":"PhD","doi":"10.15479/AT:ISTA:th_873","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"file_date_updated":"2020-07-14T12:47:26Z","date_updated":"2023-09-15T12:10:34Z","date_created":"2019-04-09T15:04:32Z","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"718"},{"relation":"part_of_dissertation","status":"public","id":"5678"},{"relation":"part_of_dissertation","status":"public","id":"87"}]},"author":[{"last_name":"Nikitenko","first_name":"Anton","orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton"}],"department":[{"_id":"HeEd"}],"publisher":"Institute of Science and Technology Austria","publication_status":"published","year":"2017","has_accepted_license":"1","article_processing_charge":"No","day":"27","date_published":"2017-10-27T00:00:00Z","page":"86","citation":{"mla":"Nikitenko, Anton. Discrete Morse Theory for Random Complexes . Institute of Science and Technology Austria, 2017, doi:10.15479/AT:ISTA:th_873.","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. https://doi.org/10.15479/AT:ISTA:th_873.","ama":"Nikitenko A. Discrete Morse theory for random complexes . 2017. doi:10.15479/AT:ISTA:th_873","ista":"Nikitenko A. 2017. Discrete Morse theory for random complexes . Institute of Science and Technology Austria.","apa":"Nikitenko, A. (2017). Discrete Morse theory for random complexes . Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th_873","ieee":"A. Nikitenko, “Discrete Morse theory for random complexes ,” Institute of Science and Technology Austria, 2017."},"abstract":[{"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.","lang":"eng"}],"alternative_title":["ISTA Thesis"],"type":"dissertation","oa_version":"Published Version","file":[{"access_level":"open_access","file_name":"2017_Thesis_Nikitenko.pdf","content_type":"application/pdf","file_size":2324870,"creator":"dernst","relation":"main_file","file_id":"6289","checksum":"ece7e598a2f060b263c2febf7f3fe7f9","date_updated":"2020-07-14T12:47:26Z","date_created":"2019-04-09T14:54:51Z"},{"relation":"source_file","file_id":"6290","checksum":"99b7ad76e317efd447af60f91e29b49b","date_updated":"2020-07-14T12:47:26Z","date_created":"2019-04-09T14:54:51Z","access_level":"closed","file_name":"2017_Thesis_Nikitenko_source.zip","file_size":2863219,"content_type":"application/zip","creator":"dernst"}],"pubrep_id":"873","ddc":["514","516","519"],"title":"Discrete Morse theory for random complexes ","status":"public","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"6287"},{"year":"2017","department":[{"_id":"HeEd"}],"publisher":"Academic Press","publication_status":"published","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"10894"}]},"author":[{"last_name":"Bauer","first_name":"Ulrich","full_name":"Bauer, Ulrich"},{"full_name":"Kerber, Michael","last_name":"Kerber","first_name":"Michael"},{"first_name":"Jan","last_name":"Reininghaus","full_name":"Reininghaus, Jan"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","last_name":"Wagner","full_name":"Wagner, Hubert"}],"volume":78,"date_updated":"2023-09-20T09:42:40Z","date_created":"2018-12-11T11:51:59Z","publist_id":"5765","ec_funded":1,"oa":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1016/j.jsc.2016.03.008"}],"external_id":{"isi":["000384396000005"]},"project":[{"call_identifier":"FP7","name":"Topological Complex Systems","grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"isi":1,"quality_controlled":"1","doi":"10.1016/j.jsc.2016.03.008","language":[{"iso":"eng"}],"publication_identifier":{"issn":[" 07477171"]},"month":"01","_id":"1433","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","intvolume":" 78","title":"Phat - Persistent homology algorithms toolbox","status":"public","oa_version":"Published Version","type":"journal_article","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"}],"citation":{"mla":"Bauer, Ulrich, et al. “Phat - Persistent Homology Algorithms Toolbox.” Journal of Symbolic Computation, vol. 78, Academic Press, 2017, pp. 76–90, doi:10.1016/j.jsc.2016.03.008.","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.” Journal of Symbolic Computation. Academic Press, 2017. https://doi.org/10.1016/j.jsc.2016.03.008.","ama":"Bauer U, Kerber M, Reininghaus J, Wagner H. Phat - Persistent homology algorithms toolbox. Journal of Symbolic Computation. 2017;78:76-90. doi:10.1016/j.jsc.2016.03.008","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,” Journal of Symbolic Computation, vol. 78. Academic Press, pp. 76–90, 2017.","apa":"Bauer, U., Kerber, M., Reininghaus, J., & Wagner, H. (2017). Phat - Persistent homology algorithms toolbox. Journal of Symbolic Computation. 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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"}],"citation":{"ama":"Akopyan A, Bárány I, Robins S. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 2017;308:627-644. doi:10.1016/j.aim.2016.12.026","ista":"Akopyan A, Bárány I, Robins S. 2017. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 308, 627–644.","apa":"Akopyan, A., Bárány, I., & Robins, S. (2017). Algebraic vertices of non-convex polyhedra. Advances in Mathematics. Academic Press. https://doi.org/10.1016/j.aim.2016.12.026","ieee":"A. Akopyan, I. Bárány, and S. Robins, “Algebraic vertices of non-convex polyhedra,” Advances in Mathematics, vol. 308. Academic Press, pp. 627–644, 2017.","mla":"Akopyan, Arseniy, et al. “Algebraic Vertices of Non-Convex Polyhedra.” Advances in Mathematics, vol. 308, Academic Press, 2017, pp. 627–44, doi:10.1016/j.aim.2016.12.026.","short":"A. Akopyan, I. Bárány, S. Robins, Advances in Mathematics 308 (2017) 627–644.","chicago":"Akopyan, Arseniy, Imre Bárány, and Sinai Robins. “Algebraic Vertices of Non-Convex Polyhedra.” Advances in Mathematics. Academic Press, 2017. https://doi.org/10.1016/j.aim.2016.12.026."},"publication":"Advances in Mathematics","page":"627 - 644","date_published":"2017-02-21T00:00:00Z","scopus_import":"1","article_processing_charge":"No","day":"21"},{"project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493","name":"Topological Complex Systems","call_identifier":"FP7"}],"isi":1,"quality_controlled":"1","oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1411.6337","open_access":"1"}],"external_id":{"isi":["000418056000005"]},"language":[{"iso":"eng"}],"doi":"10.1007/s00493-016-3308-y","publication_identifier":{"issn":["02099683"]},"month":"10","department":[{"_id":"HeEd"}],"publisher":"Springer","publication_status":"published","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","volume":37,"date_updated":"2023-09-20T11:23:53Z","date_created":"2018-12-11T11:50:32Z","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"first_name":"Alexey","last_name":"Glazyrin","full_name":"Glazyrin, Alexey"},{"first_name":"Oleg","last_name":"Musin","full_name":"Musin, Oleg"},{"first_name":"Anton","last_name":"Nikitenko","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0659-3201","full_name":"Nikitenko, Anton"}],"ec_funded":1,"publist_id":"6182","page":"887 - 910","citation":{"apa":"Edelsbrunner, H., Glazyrin, A., Musin, O., & Nikitenko, A. (2017). The Voronoi functional is maximized by the Delaunay triangulation in the plane. Combinatorica. Springer. https://doi.org/10.1007/s00493-016-3308-y","ieee":"H. Edelsbrunner, A. Glazyrin, O. Musin, and A. Nikitenko, “The Voronoi functional is maximized by the Delaunay triangulation in the plane,” Combinatorica, vol. 37, no. 5. Springer, pp. 887–910, 2017.","ista":"Edelsbrunner H, Glazyrin A, Musin O, Nikitenko A. 2017. The Voronoi functional is maximized by the Delaunay triangulation in the plane. Combinatorica. 37(5), 887–910.","ama":"Edelsbrunner H, Glazyrin A, Musin O, Nikitenko A. The Voronoi functional is maximized by the Delaunay triangulation in the plane. Combinatorica. 2017;37(5):887-910. doi:10.1007/s00493-016-3308-y","chicago":"Edelsbrunner, Herbert, Alexey Glazyrin, Oleg Musin, and Anton Nikitenko. “The Voronoi Functional Is Maximized by the Delaunay Triangulation in the Plane.” Combinatorica. Springer, 2017. https://doi.org/10.1007/s00493-016-3308-y.","short":"H. Edelsbrunner, A. Glazyrin, O. Musin, A. Nikitenko, Combinatorica 37 (2017) 887–910.","mla":"Edelsbrunner, Herbert, et al. “The Voronoi Functional Is Maximized by the Delaunay Triangulation in the Plane.” Combinatorica, vol. 37, no. 5, Springer, 2017, pp. 887–910, doi:10.1007/s00493-016-3308-y."},"publication":"Combinatorica","date_published":"2017-10-01T00:00:00Z","scopus_import":"1","article_processing_charge":"No","day":"01","intvolume":" 37","title":"The Voronoi functional is maximized by the Delaunay triangulation in the plane","status":"public","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"1173","oa_version":"Submitted Version","type":"journal_article","issue":"5","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"}]},{"article_processing_charge":"No","day":"01","scopus_import":"1","date_published":"2017-05-01T00:00:00Z","page":"3741 - 3762","article_type":"original","citation":{"ieee":"U. Bauer and H. Edelsbrunner, “The Morse theory of Čech and delaunay complexes,” Transactions of the American Mathematical Society, vol. 369, no. 5. American Mathematical Society, pp. 3741–3762, 2017.","apa":"Bauer, U., & Edelsbrunner, H. (2017). The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/tran/6991","ista":"Bauer U, Edelsbrunner H. 2017. The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. 369(5), 3741–3762.","ama":"Bauer U, Edelsbrunner H. The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. 2017;369(5):3741-3762. doi:10.1090/tran/6991","chicago":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Complexes.” Transactions of the American Mathematical Society. American Mathematical Society, 2017. https://doi.org/10.1090/tran/6991.","short":"U. Bauer, H. Edelsbrunner, Transactions of the American Mathematical Society 369 (2017) 3741–3762.","mla":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Complexes.” Transactions of the American Mathematical Society, vol. 369, no. 5, American Mathematical Society, 2017, pp. 3741–62, doi:10.1090/tran/6991."},"publication":"Transactions of the American Mathematical Society","issue":"5","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"}],"type":"journal_article","oa_version":"Preprint","intvolume":" 369","title":"The Morse theory of Čech and delaunay complexes","status":"public","_id":"1072","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","month":"05","language":[{"iso":"eng"}],"doi":"10.1090/tran/6991","project":[{"grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","call_identifier":"FP7"}],"isi":1,"quality_controlled":"1","oa":1,"external_id":{"arxiv":["1312.1231"],"isi":["000398030400024"]},"main_file_link":[{"url":"https://arxiv.org/abs/1312.1231","open_access":"1"}],"ec_funded":1,"publist_id":"6311","volume":369,"date_updated":"2023-09-20T12:05:56Z","date_created":"2018-12-11T11:49:59Z","author":[{"first_name":"Ulrich","last_name":"Bauer","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich"},{"last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"}],"department":[{"_id":"HeEd"}],"publisher":"American Mathematical Society","publication_status":"published","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”.","year":"2017"},{"publist_id":"6323","ec_funded":1,"file_date_updated":"2019-10-15T07:44:51Z","department":[{"_id":"KrCh"},{"_id":"HeEd"}],"publisher":"Elsevier","publication_status":"published","year":"2017","volume":122,"date_updated":"2023-09-20T12:08:18Z","date_created":"2018-12-11T11:49:57Z","author":[{"full_name":"Chatterjee, Krishnendu","last_name":"Chatterjee","first_name":"Krishnendu","orcid":"0000-0002-4561-241X","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-8882-5116","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","first_name":"Georg F","full_name":"Osang, Georg F"}],"publication_identifier":{"issn":["00200190"]},"month":"06","project":[{"_id":"2584A770-B435-11E9-9278-68D0E5697425","grant_number":"P 23499-N23","call_identifier":"FWF","name":"Modern Graph Algorithmic Techniques in Formal Verification"},{"_id":"25863FF4-B435-11E9-9278-68D0E5697425","grant_number":"S11407","name":"Game Theory","call_identifier":"FWF"},{"grant_number":"279307","_id":"2581B60A-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Quantitative Graph Games: Theory and Applications"}],"isi":1,"quality_controlled":"1","external_id":{"isi":["000399506600005"]},"oa":1,"language":[{"iso":"eng"}],"doi":"10.1016/j.ipl.2017.02.003","type":"journal_article","abstract":[{"lang":"eng","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."}],"intvolume":" 122","title":"Pushdown reachability with constant treewidth","ddc":["000"],"status":"public","_id":"1065","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","file":[{"date_updated":"2019-10-15T07:44:51Z","date_created":"2018-12-12T10:13:17Z","relation":"main_file","file_id":"4998","content_type":"application/pdf","file_size":247657,"creator":"system","file_name":"IST-2018-991-v1+2_2018_Chatterjee_Pushdown_PREPRINT.pdf","access_level":"open_access"}],"oa_version":"Submitted Version","pubrep_id":"991","scopus_import":"1","has_accepted_license":"1","article_processing_charge":"No","day":"01","page":"25 - 29","citation":{"apa":"Chatterjee, K., & Osang, G. F. (2017). Pushdown reachability with constant treewidth. Information Processing Letters. Elsevier. https://doi.org/10.1016/j.ipl.2017.02.003","ieee":"K. Chatterjee and G. F. Osang, “Pushdown reachability with constant treewidth,” Information Processing Letters, vol. 122. Elsevier, pp. 25–29, 2017.","ista":"Chatterjee K, Osang GF. 2017. Pushdown reachability with constant treewidth. Information Processing Letters. 122, 25–29.","ama":"Chatterjee K, Osang GF. Pushdown reachability with constant treewidth. Information Processing Letters. 2017;122:25-29. doi:10.1016/j.ipl.2017.02.003","chicago":"Chatterjee, Krishnendu, and Georg F Osang. “Pushdown Reachability with Constant Treewidth.” Information Processing Letters. Elsevier, 2017. https://doi.org/10.1016/j.ipl.2017.02.003.","short":"K. Chatterjee, G.F. Osang, Information Processing Letters 122 (2017) 25–29.","mla":"Chatterjee, Krishnendu, and Georg F. Osang. “Pushdown Reachability with Constant Treewidth.” Information Processing Letters, vol. 122, Elsevier, 2017, pp. 25–29, doi:10.1016/j.ipl.2017.02.003."},"publication":"Information Processing Letters","date_published":"2017-06-01T00:00:00Z"},{"type":"journal_article","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"}],"issue":"4","status":"public","title":"The topology of the cosmic web in terms of persistent Betti numbers","intvolume":" 465","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"1022","oa_version":"Submitted Version","scopus_import":"1","day":"01","article_processing_charge":"No","page":"4281 - 4310","publication":"Monthly Notices of the Royal Astronomical Society","citation":{"ama":"Pranav P, Edelsbrunner H, Van De Weygaert R, et al. The topology of the cosmic web in terms of persistent Betti numbers. Monthly Notices of the Royal Astronomical Society. 2017;465(4):4281-4310. doi:10.1093/mnras/stw2862","ieee":"P. Pranav et al., “The topology of the cosmic web in terms of persistent Betti numbers,” Monthly Notices of the Royal Astronomical Society, vol. 465, no. 4. Oxford University Press, pp. 4281–4310, 2017.","apa":"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. Oxford University Press. https://doi.org/10.1093/mnras/stw2862","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.","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.","mla":"Pranav, Pratyush, et al. “The Topology of the Cosmic Web in Terms of Persistent Betti Numbers.” Monthly Notices of the Royal Astronomical Society, vol. 465, no. 4, Oxford University Press, 2017, pp. 4281–310, doi:10.1093/mnras/stw2862.","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.” Monthly Notices of the Royal Astronomical Society. Oxford University Press, 2017. https://doi.org/10.1093/mnras/stw2862."},"date_published":"2017-01-01T00:00:00Z","publist_id":"6373","publication_status":"published","publisher":"Oxford University Press","department":[{"_id":"HeEd"}],"year":"2017","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_updated":"2023-09-22T09:40:55Z","date_created":"2018-12-11T11:49:44Z","volume":465,"author":[{"full_name":"Pranav, Pratyush","first_name":"Pratyush","last_name":"Pranav"},{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Van De Weygaert, Rien","first_name":"Rien","last_name":"Van De Weygaert"},{"full_name":"Vegter, Gert","last_name":"Vegter","first_name":"Gert"},{"full_name":"Kerber, Michael","last_name":"Kerber","first_name":"Michael"},{"last_name":"Jones","first_name":"Bernard","full_name":"Jones, Bernard"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220","first_name":"Mathijs","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs"}],"month":"01","publication_identifier":{"issn":["00358711"]},"isi":1,"quality_controlled":"1","external_id":{"isi":["000395170200039"]},"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1608.04519","open_access":"1"}],"language":[{"iso":"eng"}],"doi":"10.1093/mnras/stw2862"},{"year":"2017","publisher":"Elsevier","department":[{"_id":"HeEd"}],"publication_status":"published","author":[{"last_name":"Virk","first_name":"Ziga","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","full_name":"Virk, Ziga"},{"last_name":"Zastrow","first_name":"Andreas","full_name":"Zastrow, Andreas"}],"volume":231,"date_created":"2018-12-11T11:48:14Z","date_updated":"2023-09-27T12:53:01Z","publist_id":"6930","external_id":{"isi":["000413889100012"]},"isi":1,"quality_controlled":"1","doi":"10.1016/j.topol.2017.09.015","language":[{"iso":"eng"}],"publication_identifier":{"issn":["01668641"]},"month":"11","_id":"737","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","intvolume":" 231","status":"public","title":"A new topology on the universal path space","oa_version":"None","type":"journal_article","abstract":[{"text":"We generalize Brazas’ topology on the fundamental group to the whole universal path space X˜ i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action.","lang":"eng"}],"citation":{"ista":"Virk Z, Zastrow A. 2017. A new topology on the universal path space. Topology and its Applications. 231, 186–196.","ieee":"Z. Virk and A. Zastrow, “A new topology on the universal path space,” Topology and its Applications, vol. 231. Elsevier, pp. 186–196, 2017.","apa":"Virk, Z., & Zastrow, A. (2017). A new topology on the universal path space. Topology and Its Applications. Elsevier. https://doi.org/10.1016/j.topol.2017.09.015","ama":"Virk Z, Zastrow A. A new topology on the universal path space. Topology and its Applications. 2017;231:186-196. doi:10.1016/j.topol.2017.09.015","chicago":"Virk, Ziga, and Andreas Zastrow. “A New Topology on the Universal Path Space.” Topology and Its Applications. Elsevier, 2017. https://doi.org/10.1016/j.topol.2017.09.015.","mla":"Virk, Ziga, and Andreas Zastrow. “A New Topology on the Universal Path Space.” Topology and Its Applications, vol. 231, Elsevier, 2017, pp. 186–96, doi:10.1016/j.topol.2017.09.015.","short":"Z. Virk, A. Zastrow, Topology and Its Applications 231 (2017) 186–196."},"publication":"Topology and its Applications","page":"186 - 196","date_published":"2017-11-01T00:00:00Z","scopus_import":"1","article_processing_charge":"No","day":"01"},{"scopus_import":"1","article_processing_charge":"No","day":"27","page":"119 - 136","citation":{"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.","apa":"Ethier, M., Jablonski, G., & Mrozek, M. (2017). Finding eigenvalues of self-maps with the Kronecker canonical form. In Special Sessions in Applications of Computer Algebra (Vol. 198, pp. 119–136). Kalamata, Greece: Springer. https://doi.org/10.1007/978-3-319-56932-1_8","ieee":"M. Ethier, G. Jablonski, and M. Mrozek, “Finding eigenvalues of self-maps with the Kronecker canonical form,” in Special Sessions in Applications of Computer Algebra, Kalamata, Greece, 2017, vol. 198, pp. 119–136.","ama":"Ethier M, Jablonski G, Mrozek M. Finding eigenvalues of self-maps with the Kronecker canonical form. In: Special Sessions in Applications of Computer Algebra. Vol 198. Springer; 2017:119-136. doi:10.1007/978-3-319-56932-1_8","chicago":"Ethier, Marc, Grzegorz Jablonski, and Marian Mrozek. “Finding Eigenvalues of Self-Maps with the Kronecker Canonical Form.” In Special Sessions in Applications of Computer Algebra, 198:119–36. Springer, 2017. https://doi.org/10.1007/978-3-319-56932-1_8.","mla":"Ethier, Marc, et al. “Finding Eigenvalues of Self-Maps with the Kronecker Canonical Form.” Special Sessions in Applications of Computer Algebra, vol. 198, Springer, 2017, pp. 119–36, doi:10.1007/978-3-319-56932-1_8.","short":"M. Ethier, G. Jablonski, M. Mrozek, in:, Special Sessions in Applications of Computer Algebra, Springer, 2017, pp. 119–136."},"publication":"Special Sessions in Applications of Computer Algebra","date_published":"2017-07-27T00:00:00Z","alternative_title":["PROMS"],"type":"conference","abstract":[{"text":"Recent research has examined how to study the topological features of a continuous self-map by means of the persistence of the eigenspaces, for given eigenvalues, of the endomorphism induced in homology over a field. This raised the question of how to select dynamically significant eigenvalues. The present paper aims to answer this question, giving an algorithm that computes the persistence of eigenspaces for every eigenvalue simultaneously, also expressing said eigenspaces as direct sums of “finite” and “singular” subspaces.","lang":"eng"}],"intvolume":" 198","status":"public","title":"Finding eigenvalues of self-maps with the Kronecker canonical form","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"836","oa_version":"None","publication_identifier":{"isbn":["978-331956930-7"]},"month":"07","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493","name":"Topological Complex Systems","call_identifier":"FP7"}],"isi":1,"quality_controlled":"1","external_id":{"isi":["000434088200008"]},"language":[{"iso":"eng"}],"doi":"10.1007/978-3-319-56932-1_8","conference":{"name":"ACA: Applications of Computer Algebra","end_date":"2015-07-23","start_date":"2015-07-20","location":"Kalamata, Greece"},"ec_funded":1,"publist_id":"6812","publisher":"Springer","department":[{"_id":"HeEd"}],"publication_status":"published","year":"2017","volume":198,"date_created":"2018-12-11T11:48:46Z","date_updated":"2023-09-26T15:50:52Z","author":[{"last_name":"Ethier","first_name":"Marc","full_name":"Ethier, Marc"},{"full_name":"Jablonski, Grzegorz","first_name":"Grzegorz","last_name":"Jablonski","id":"4483EF78-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-3536-9866"},{"full_name":"Mrozek, Marian","last_name":"Mrozek","first_name":"Marian"}]}]