[{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"Work by all authors but A. Garber is supported by the European Research Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation.","oa_version":"Published Version","quality_controlled":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"_id":"14345","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"date_updated":"2023-12-13T12:25:06Z","oa":1,"article_processing_charge":"Yes (via OA deal)","arxiv":1,"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Alexey","full_name":"Garber, Alexey","last_name":"Garber"},{"full_name":"Ghafari, Mohadese","last_name":"Ghafari","first_name":"Mohadese"},{"id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1780-2689","last_name":"Heiss","full_name":"Heiss, Teresa","first_name":"Teresa"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza"}],"abstract":[{"lang":"eng","text":"For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2  is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970))."}],"publication_status":"epub_ahead","citation":{"ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>","mla":"Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2023. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry (2023).","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2023). On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023.","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>."},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00454-023-00566-1"}],"isi":1,"title":"On angles in higher order Brillouin tessellations and related tilings in the plane","external_id":{"isi":["001060727600004"],"arxiv":["2204.01076"]},"ec_funded":1,"doi":"10.1007/s00454-023-00566-1","year":"2023","publication":"Discrete and Computational Geometry","status":"public","type":"journal_article","day":"07","date_created":"2023-09-17T22:01:10Z","department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","date_published":"2023-09-07T00:00:00Z","article_type":"original","month":"09"},{"status":"public","day":"17","type":"journal_article","publication":"Journal of Applied and Computational Topology","file_date_updated":"2023-07-03T09:41:05Z","publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"06","article_type":"original","date_published":"2023-06-17T00:00:00Z","file":[{"success":1,"creator":"alisjak","file_id":"13185","content_type":"application/pdf","relation":"main_file","date_created":"2023-07-03T09:41:05Z","checksum":"697249d5d1c61dea4410b9f021b70fce","file_name":"2023_Journal of Applied and Computational Topology_Biswas.pdf","file_size":487355,"date_updated":"2023-07-03T09:41:05Z","access_level":"open_access"}],"date_created":"2023-07-02T22:00:44Z","has_accepted_license":"1","department":[{"_id":"HeEd"}],"abstract":[{"lang":"eng","text":"We characterize critical points of 1-dimensional maps paired in persistent homology\r\ngeometrically and this way get elementary proofs of theorems about the symmetry\r\nof persistence diagrams and the variation of such maps. In particular, we identify\r\nbranching points and endpoints of networks as the sole source of asymmetry and\r\nrelate the cycle basis in persistent homology with a version of the stable marriage\r\nproblem. Our analysis provides the foundations of fast algorithms for maintaining a\r\ncollection of sorted lists together with its persistence diagram."}],"author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","last_name":"Biswas","orcid":"0000-0002-5372-7890","first_name":"Ranita"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","orcid":"0000-0001-6249-0832","full_name":"Cultrera Di Montesano, Sebastiano","last_name":"Cultrera Di Montesano"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"publication_status":"epub_ahead","citation":{"ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>","mla":"Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology (2023).","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2023. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2023.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2023). Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>."},"_id":"13182","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"acknowledgement":"Open access funding provided by Austrian Science Fund (FWF). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35. The authors of this paper thank anonymous reviewers for their constructive criticism and Monika Henzinger for detailed comments on an earlier version of this paper.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF","grant_number":"Z00342"}],"quality_controlled":"1","date_updated":"2023-10-18T08:13:10Z","oa":1,"article_processing_charge":"Yes (via OA deal)","title":"Geometric characterization of the persistence of 1D maps","year":"2023","doi":"10.1007/s41468-023-00126-9","ec_funded":1,"ddc":["000"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"}},{"quality_controlled":"1","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35"}],"oa_version":"Published Version","acknowledgement":"Open access funding provided by Austrian Science Fund (FWF). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.","user_id":"2EBD1598-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["0178-4617"],"eissn":["1432-0541"]},"_id":"12086","article_processing_charge":"Yes (via OA deal)","volume":85,"date_updated":"2023-06-27T12:53:43Z","oa":1,"author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Osang, Georg F","last_name":"Osang","first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"text":"We present a simple algorithm for computing higher-order Delaunay mosaics that works in Euclidean spaces of any finite dimensions. The algorithm selects the vertices of the order-k mosaic from incrementally constructed lower-order mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box to construct the order-k mosaic from its vertices. Beyond this black-box, the algorithm uses only combinatorial operations, thus facilitating easy implementation. We extend this algorithm to compute higher-order α-shapes and provide open-source implementations. We present experimental results for properties of higher-order Delaunay mosaics of random point sets.","lang":"eng"}],"citation":{"ista":"Edelsbrunner H, Osang GF. 2023. A simple algorithm for higher-order Delaunay mosaics and alpha shapes. Algorithmica. 85, 277–295.","short":"H. Edelsbrunner, G.F. Osang, Algorithmica 85 (2023) 277–295.","mla":"Edelsbrunner, Herbert, and Georg F. Osang. “A Simple Algorithm for Higher-Order Delaunay Mosaics and Alpha Shapes.” <i>Algorithmica</i>, vol. 85, Springer Nature, 2023, pp. 277–95, doi:<a href=\"https://doi.org/10.1007/s00453-022-01027-6\">10.1007/s00453-022-01027-6</a>.","ama":"Edelsbrunner H, Osang GF. A simple algorithm for higher-order Delaunay mosaics and alpha shapes. <i>Algorithmica</i>. 2023;85:277-295. doi:<a href=\"https://doi.org/10.1007/s00453-022-01027-6\">10.1007/s00453-022-01027-6</a>","chicago":"Edelsbrunner, Herbert, and Georg F Osang. “A Simple Algorithm for Higher-Order Delaunay Mosaics and Alpha Shapes.” <i>Algorithmica</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00453-022-01027-6\">https://doi.org/10.1007/s00453-022-01027-6</a>.","apa":"Edelsbrunner, H., &#38; Osang, G. F. (2023). A simple algorithm for higher-order Delaunay mosaics and alpha shapes. <i>Algorithmica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00453-022-01027-6\">https://doi.org/10.1007/s00453-022-01027-6</a>","ieee":"H. Edelsbrunner and G. F. Osang, “A simple algorithm for higher-order Delaunay mosaics and alpha shapes,” <i>Algorithmica</i>, vol. 85. Springer Nature, pp. 277–295, 2023."},"publication_status":"published","ddc":["510"],"isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"A simple algorithm for higher-order Delaunay mosaics and alpha shapes","external_id":{"isi":["000846967100001"]},"ec_funded":1,"year":"2023","doi":"10.1007/s00453-022-01027-6","file_date_updated":"2023-01-20T10:02:48Z","page":"277-295","publication":"Algorithmica","status":"public","intvolume":"        85","type":"journal_article","day":"01","file":[{"date_updated":"2023-01-20T10:02:48Z","access_level":"open_access","file_name":"2023_Algorithmica_Edelsbrunner.pdf","file_size":911017,"date_created":"2023-01-20T10:02:48Z","checksum":"71685ca5121f4c837f40c3f8eb50c915","content_type":"application/pdf","relation":"main_file","file_id":"12322","creator":"dernst","success":1}],"date_created":"2022-09-11T22:01:57Z","department":[{"_id":"HeEd"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature","date_published":"2023-01-01T00:00:00Z","article_type":"original","month":"01"},{"citation":{"ista":"Koehl P, Akopyan A, Edelsbrunner H. 2023. Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. Journal of Chemical Information and Modeling. 63(3), 973–985.","short":"P. Koehl, A. Akopyan, H. Edelsbrunner, Journal of Chemical Information and Modeling 63 (2023) 973–985.","ama":"Koehl P, Akopyan A, Edelsbrunner H. Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. <i>Journal of Chemical Information and Modeling</i>. 2023;63(3):973-985. doi:<a href=\"https://doi.org/10.1021/acs.jcim.2c01346\">10.1021/acs.jcim.2c01346</a>","mla":"Koehl, Patrice, et al. “Computing the Volume, Surface Area, Mean, and Gaussian Curvatures of Molecules and Their Derivatives.” <i>Journal of Chemical Information and Modeling</i>, vol. 63, no. 3, American Chemical Society, 2023, pp. 973–85, doi:<a href=\"https://doi.org/10.1021/acs.jcim.2c01346\">10.1021/acs.jcim.2c01346</a>.","chicago":"Koehl, Patrice, Arseniy Akopyan, and Herbert Edelsbrunner. “Computing the Volume, Surface Area, Mean, and Gaussian Curvatures of Molecules and Their Derivatives.” <i>Journal of Chemical Information and Modeling</i>. American Chemical Society, 2023. <a href=\"https://doi.org/10.1021/acs.jcim.2c01346\">https://doi.org/10.1021/acs.jcim.2c01346</a>.","apa":"Koehl, P., Akopyan, A., &#38; Edelsbrunner, H. (2023). Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. <i>Journal of Chemical Information and Modeling</i>. American Chemical Society. <a href=\"https://doi.org/10.1021/acs.jcim.2c01346\">https://doi.org/10.1021/acs.jcim.2c01346</a>","ieee":"P. Koehl, A. Akopyan, and H. Edelsbrunner, “Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives,” <i>Journal of Chemical Information and Modeling</i>, vol. 63, no. 3. American Chemical Society, pp. 973–985, 2023."},"publication_status":"published","author":[{"last_name":"Koehl","full_name":"Koehl, Patrice","first_name":"Patrice"},{"last_name":"Akopyan","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert"}],"abstract":[{"text":"Geometry is crucial in our efforts to comprehend the structures and dynamics of biomolecules. For example, volume, surface area, and integrated mean and Gaussian curvature of the union of balls representing a molecule are used to quantify its interactions with the water surrounding it in the morphometric implicit solvent models. The Alpha Shape theory provides an accurate and reliable method for computing these geometric measures. In this paper, we derive homogeneous formulas for the expressions of these measures and their derivatives with respect to the atomic coordinates, and we provide algorithms that implement them into a new software package, AlphaMol. The only variables in these formulas are the interatomic distances, making them insensitive to translations and rotations. AlphaMol includes a sequential algorithm and a parallel algorithm. In the parallel version, we partition the atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented by a buffer zone to account for atoms whose ball representations may partially cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up compared to an independent serial implementation when using 32 processors. For instance, it takes 31 s to compute the geometric measures and derivatives of each atom in a viral capsid with more than 26 million atoms on 32 Intel processors running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant computations, which ultimately limit the impact of using multiple processors. AlphaMol is available as an OpenSource software.","lang":"eng"}],"article_processing_charge":"No","oa":1,"volume":63,"date_updated":"2023-08-16T12:22:07Z","oa_version":"Published Version","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35"}],"quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"P.K. acknowledges support from the University of California Multicampus Research Programs and Initiatives (Grant No. M21PR3267) and from the NSF (Grant No.1760485). H.E. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.\r\nOpen Access is funded by the Austrian Science Fund (FWF).","publication_identifier":{"eissn":["1549-960X"],"issn":["1549-9596"]},"pmid":1,"_id":"12544","ec_funded":1,"year":"2023","doi":"10.1021/acs.jcim.2c01346","title":"Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives","external_id":{"isi":["000920370700001"],"pmid":["36638318"]},"isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["510","540"],"type":"journal_article","day":"13","status":"public","intvolume":"        63","file_date_updated":"2023-08-16T12:21:13Z","page":"973-985","publication":"Journal of Chemical Information and Modeling","issue":"3","article_type":"original","date_published":"2023-02-13T00:00:00Z","month":"02","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"American Chemical Society","department":[{"_id":"HeEd"}],"has_accepted_license":"1","file":[{"date_created":"2023-08-16T12:21:13Z","checksum":"7d20562269edff1e31b9d6019d4983b0","file_name":"2023_JCIM_Koehl.pdf","file_size":8069223,"date_updated":"2023-08-16T12:21:13Z","access_level":"open_access","success":1,"file_id":"14070","creator":"dernst","content_type":"application/pdf","relation":"main_file"}],"date_created":"2023-02-12T23:00:59Z"},{"ddc":["510"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"isi":1,"title":"Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics","external_id":{"isi":["000752175300002"]},"year":"2022","doi":"10.1007/s00454-022-00371-2","_id":"10773","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"acknowledgement":"Open access funding provided by the Institute of Science and Technology (IST Austria).","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa_version":"Published Version","quality_controlled":"1","volume":67,"oa":1,"date_updated":"2023-08-02T14:31:25Z","article_processing_charge":"Yes (via OA deal)","abstract":[{"lang":"eng","text":"The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function."}],"author":[{"first_name":"Ranita","full_name":"Biswas, Ranita","last_name":"Biswas","orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Cultrera Di Montesano, Sebastiano","last_name":"Cultrera Di Montesano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza"}],"publication_status":"published","citation":{"ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2022. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. 67, 811–842.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 67 (2022) 811–842.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. <i>Discrete and Computational Geometry</i>. 2022;67:811-842. doi:<a href=\"https://doi.org/10.1007/s00454-022-00371-2\">10.1007/s00454-022-00371-2</a>","mla":"Biswas, Ranita, et al. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>, vol. 67, Springer Nature, 2022, pp. 811–42, doi:<a href=\"https://doi.org/10.1007/s00454-022-00371-2\">10.1007/s00454-022-00371-2</a>.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00454-022-00371-2\">https://doi.org/10.1007/s00454-022-00371-2</a>.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics,” <i>Discrete and Computational Geometry</i>, vol. 67. Springer Nature, pp. 811–842, 2022.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2022). Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-022-00371-2\">https://doi.org/10.1007/s00454-022-00371-2</a>"},"file":[{"access_level":"open_access","date_updated":"2022-08-02T06:07:55Z","checksum":"9383d3b70561bacee905e335dc922680","date_created":"2022-08-02T06:07:55Z","file_size":2518111,"file_name":"2022_DiscreteCompGeometry_Biswas.pdf","file_id":"11718","creator":"dernst","relation":"main_file","content_type":"application/pdf","success":1}],"date_created":"2022-02-20T23:01:34Z","has_accepted_license":"1","department":[{"_id":"HeEd"}],"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"04","date_published":"2022-04-01T00:00:00Z","article_type":"original","publication":"Discrete and Computational Geometry","page":"811-842","file_date_updated":"2022-08-02T06:07:55Z","intvolume":"        67","status":"public","day":"01","type":"journal_article"},{"month":"07","date_published":"2022-07-27T00:00:00Z","publisher":"Schloss Dagstuhl - Leibniz Zentrum für Informatik","language":[{"iso":"eng"}],"has_accepted_license":"1","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"date_created":"2022-07-27T09:27:34Z","file":[{"creator":"scultrer","file_id":"11659","content_type":"application/pdf","relation":"main_file","date_updated":"2022-07-27T09:25:53Z","access_level":"open_access","date_created":"2022-07-27T09:25:53Z","checksum":"b2f511e8b1cae5f1892b0cdec341acac","file_name":"D-S-E.pdf","file_size":639266}],"day":"27","type":"journal_article","status":"public","publication":"Leibniz International Proceedings on Mathematics","file_date_updated":"2022-07-27T09:25:53Z","year":"2022","ec_funded":1,"title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["510"],"publication_status":"submitted","citation":{"short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Leibniz International Proceedings on Mathematics (n.d.).","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Leibniz International Proceedings on Mathematics</i>.","mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Leibniz International Proceedings on Mathematics</i>, Schloss Dagstuhl - Leibniz Zentrum für Informatik.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz Zentrum für Informatik, n.d.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz Zentrum für Informatik.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz Zentrum für Informatik."},"abstract":[{"lang":"eng","text":"The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements."}],"author":[{"first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0001-6249-0832","full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza"}],"date_updated":"2022-07-28T07:57:48Z","oa":1,"article_processing_charge":"No","_id":"11658","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.","quality_controlled":"1","oa_version":"Submitted Version","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}]},{"date_created":"2022-07-27T09:31:15Z","file":[{"checksum":"95903f9d1649e8e437a967b6f2f64730","date_created":"2022-07-27T09:30:30Z","file_size":564836,"file_name":"window 1.pdf","access_level":"open_access","date_updated":"2022-07-27T09:30:30Z","creator":"scultrer","file_id":"11661","relation":"main_file","content_type":"application/pdf"}],"has_accepted_license":"1","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","language":[{"iso":"eng"}],"month":"07","date_published":"2022-07-25T00:00:00Z","publication":"LIPIcs","file_date_updated":"2022-07-27T09:30:30Z","status":"public","day":"25","type":"journal_article","ddc":["510"],"alternative_title":["LIPIcs"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs","year":"2022","ec_funded":1,"_id":"11660","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. ","project":[{"grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","grant_number":"Z00342"},{"grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes"}],"oa_version":"Submitted Version","quality_controlled":"1","oa":1,"date_updated":"2022-07-28T08:05:34Z","article_processing_charge":"No","abstract":[{"text":"We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. ","lang":"eng"}],"author":[{"first_name":"Ranita","full_name":"Biswas, Ranita","last_name":"Biswas","orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"publication_status":"submitted","citation":{"apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs,” <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, n.d.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. <i>LIPIcs</i>.","mla":"Biswas, Ranita, et al. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” <i>LIPIcs</i>, Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, LIPIcs (n.d.).","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs."}},{"date_published":"2021-03-31T00:00:00Z","article_type":"original","month":"03","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature","department":[{"_id":"HeEd"}],"has_accepted_license":"1","file":[{"content_type":"application/pdf","relation":"main_file","creator":"cchlebak","file_id":"10394","success":1,"date_updated":"2021-12-01T10:56:53Z","access_level":"open_access","file_name":"2021_DisCompGeo_Edelsbrunner_Osang.pdf","file_size":677704,"date_created":"2021-12-01T10:56:53Z","checksum":"59b4e1e827e494209bcb4aae22e1d347"}],"date_created":"2021-04-11T22:01:15Z","type":"journal_article","day":"31","status":"public","intvolume":"        65","page":"1296–1313","file_date_updated":"2021-12-01T10:56:53Z","publication":"Discrete and Computational Geometry","ec_funded":1,"year":"2021","doi":"10.1007/s00454-021-00281-9","title":"The multi-cover persistence of Euclidean balls","external_id":{"isi":["000635460400001"]},"isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["516"],"related_material":{"record":[{"status":"public","relation":"earlier_version","id":"187"}]},"citation":{"ista":"Edelsbrunner H, Osang GF. 2021. The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. 65, 1296–1313.","short":"H. Edelsbrunner, G.F. Osang, Discrete and Computational Geometry 65 (2021) 1296–1313.","mla":"Edelsbrunner, Herbert, and Georg F. Osang. “The Multi-Cover Persistence of Euclidean Balls.” <i>Discrete and Computational Geometry</i>, vol. 65, Springer Nature, 2021, pp. 1296–1313, doi:<a href=\"https://doi.org/10.1007/s00454-021-00281-9\">10.1007/s00454-021-00281-9</a>.","ama":"Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. <i>Discrete and Computational Geometry</i>. 2021;65:1296–1313. doi:<a href=\"https://doi.org/10.1007/s00454-021-00281-9\">10.1007/s00454-021-00281-9</a>","chicago":"Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence of Euclidean Balls.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-021-00281-9\">https://doi.org/10.1007/s00454-021-00281-9</a>.","apa":"Edelsbrunner, H., &#38; Osang, G. F. (2021). The multi-cover persistence of Euclidean balls. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-021-00281-9\">https://doi.org/10.1007/s00454-021-00281-9</a>","ieee":"H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean balls,” <i>Discrete and Computational Geometry</i>, vol. 65. Springer Nature, pp. 1296–1313, 2021."},"publication_status":"published","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Georg F","last_name":"Osang","full_name":"Osang, Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"text":"Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r consists of all points in Rd that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in Rd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.","lang":"eng"}],"article_processing_charge":"Yes (via OA deal)","oa":1,"date_updated":"2023-08-07T14:35:44Z","volume":65,"oa_version":"Published Version","quality_controlled":"1","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha), and by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF)\r\nOpen Access funding provided by the Institute of Science and Technology (IST Austria).","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"_id":"9317"},{"status":"public","intvolume":"       189","type":"conference","day":"02","file_date_updated":"2021-04-22T08:08:14Z","page":"32:1-32:16","publication":"37th International Symposium on Computational Geometry (SoCG 2021)","language":[{"iso":"eng"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","date_published":"2021-06-02T00:00:00Z","month":"06","file":[{"success":1,"relation":"main_file","content_type":"application/pdf","creator":"mwintrae","file_id":"9346","file_size":3117435,"file_name":"df_socg_final_version.pdf","checksum":"1787baef1523d6d93753b90d0c109a6d","date_created":"2021-04-22T08:08:14Z","access_level":"open_access","date_updated":"2021-04-22T08:08:14Z"}],"date_created":"2021-04-22T08:09:58Z","conference":{"start_date":"2021-06-07","end_date":"2021-06-11","location":"Virtual","name":"SoCG: Symposium on Computational Geometry"},"department":[{"_id":"HeEd"}],"has_accepted_license":"1","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss","full_name":"Heiss, Teresa","orcid":"0000-0002-1780-2689","first_name":"Teresa"},{"full_name":" Kurlin , Vitaliy","last_name":" Kurlin ","first_name":"Vitaliy"},{"last_name":"Smith","full_name":"Smith, Philip","first_name":"Philip"},{"orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"text":"Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction.","lang":"eng"}],"citation":{"apa":"Edelsbrunner, H., Heiss, T.,  Kurlin , V., Smith, P., &#38; Wintraecken, M. (2021). The density fingerprint of a periodic point set. In <i>37th International Symposium on Computational Geometry (SoCG 2021)</i> (Vol. 189, p. 32:1-32:16). Virtual: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">https://doi.org/10.4230/LIPIcs.SoCG.2021.32</a>","ieee":"H. Edelsbrunner, T. Heiss, V.  Kurlin , P. Smith, and M. Wintraecken, “The density fingerprint of a periodic point set,” in <i>37th International Symposium on Computational Geometry (SoCG 2021)</i>, Virtual, 2021, vol. 189, p. 32:1-32:16.","chicago":"Edelsbrunner, Herbert, Teresa Heiss, Vitaliy  Kurlin , Philip Smith, and Mathijs Wintraecken. “The Density Fingerprint of a Periodic Point Set.” In <i>37th International Symposium on Computational Geometry (SoCG 2021)</i>, 189:32:1-32:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">https://doi.org/10.4230/LIPIcs.SoCG.2021.32</a>.","mla":"Edelsbrunner, Herbert, et al. “The Density Fingerprint of a Periodic Point Set.” <i>37th International Symposium on Computational Geometry (SoCG 2021)</i>, vol. 189, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 32:1-32:16, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">10.4230/LIPIcs.SoCG.2021.32</a>.","ama":"Edelsbrunner H, Heiss T,  Kurlin  V, Smith P, Wintraecken M. The density fingerprint of a periodic point set. In: <i>37th International Symposium on Computational Geometry (SoCG 2021)</i>. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021:32:1-32:16. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">10.4230/LIPIcs.SoCG.2021.32</a>","short":"H. Edelsbrunner, T. Heiss, V.  Kurlin , P. Smith, M. Wintraecken, in:, 37th International Symposium on Computational Geometry (SoCG 2021), Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 32:1-32:16.","ista":"Edelsbrunner H, Heiss T,  Kurlin  V, Smith P, Wintraecken M. 2021. The density fingerprint of a periodic point set. 37th International Symposium on Computational Geometry (SoCG 2021). SoCG: Symposium on Computational Geometry, LIPIcs, vol. 189, 32:1-32:16."},"publication_status":"published","quality_controlled":"1","oa_version":"Published Version","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887"},{"name":"The Wittgenstein Prize","_id":"25C5A090-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00312"},{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020"}],"acknowledgement":"The authors thank Janos Pach for insightful discussions on the topic of thispaper, Morteza Saghafian for finding the one-dimensional counterexample mentioned in Section 5,and Larry Andrews for generously sharing his crystallographic perspective.","user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","publication_identifier":{"issn":["1868-8969"]},"_id":"9345","article_processing_charge":"No","oa":1,"date_updated":"2023-02-23T13:55:40Z","volume":189,"title":"The density fingerprint of a periodic point set","ec_funded":1,"year":"2021","doi":"10.4230/LIPIcs.SoCG.2021.32","ddc":["004","516"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"alternative_title":["LIPIcs"]},{"file_date_updated":"2021-06-11T13:16:26Z","issue":"1","publication":"Journal of Geometry","status":"public","intvolume":"       112","type":"journal_article","day":"01","date_created":"2021-06-06T22:01:29Z","file":[{"access_level":"open_access","date_updated":"2021-06-11T13:16:26Z","checksum":"e52a832f1def52a2b23d21bcc09e646f","date_created":"2021-06-11T13:16:26Z","file_size":694706,"file_name":"2021_Geometry_Edelsbrunner.pdf","file_id":"9544","creator":"kschuh","relation":"main_file","content_type":"application/pdf","success":1}],"department":[{"_id":"HeEd"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","date_published":"2021-04-01T00:00:00Z","article_type":"original","month":"04","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","oa_version":"Published Version","_id":"9465","publication_identifier":{"issn":["00472468"],"eissn":["14208997"]},"oa":1,"volume":112,"date_updated":"2022-05-12T11:41:45Z","article_processing_charge":"Yes (via OA deal)","author":[{"first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Nikitenko, Anton","last_name":"Nikitenko","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"},{"id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","full_name":"Osang, Georg F","first_name":"Georg F"}],"abstract":[{"lang":"eng","text":"Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space."}],"publication_status":"published","citation":{"chicago":"Edelsbrunner, Herbert, Anton Nikitenko, and Georg F Osang. “A Step in the Delaunay Mosaic of Order K.” <i>Journal of Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00022-021-00577-4\">https://doi.org/10.1007/s00022-021-00577-4</a>.","apa":"Edelsbrunner, H., Nikitenko, A., &#38; Osang, G. F. (2021). A step in the Delaunay mosaic of order k. <i>Journal of Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00022-021-00577-4\">https://doi.org/10.1007/s00022-021-00577-4</a>","ieee":"H. Edelsbrunner, A. Nikitenko, and G. F. Osang, “A step in the Delaunay mosaic of order k,” <i>Journal of Geometry</i>, vol. 112, no. 1. Springer Nature, 2021.","short":"H. Edelsbrunner, A. Nikitenko, G.F. Osang, Journal of Geometry 112 (2021).","ista":"Edelsbrunner H, Nikitenko A, Osang GF. 2021. A step in the Delaunay mosaic of order k. Journal of Geometry. 112(1), 15.","ama":"Edelsbrunner H, Nikitenko A, Osang GF. A step in the Delaunay mosaic of order k. <i>Journal of Geometry</i>. 2021;112(1). doi:<a href=\"https://doi.org/10.1007/s00022-021-00577-4\">10.1007/s00022-021-00577-4</a>","mla":"Edelsbrunner, Herbert, et al. “A Step in the Delaunay Mosaic of Order K.” <i>Journal of Geometry</i>, vol. 112, no. 1, 15, Springer Nature, 2021, doi:<a href=\"https://doi.org/10.1007/s00022-021-00577-4\">10.1007/s00022-021-00577-4</a>."},"ddc":["510"],"article_number":"15","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"A step in the Delaunay mosaic of order k","year":"2021","doi":"10.1007/s00022-021-00577-4"},{"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":"1","language":[{"iso":"eng"}],"month":"06","date_published":"2021-06-02T00:00:00Z","conference":{"end_date":"2021-06-11","name":"SoCG: International Symposium on Computational Geometry","location":"Online","start_date":"2021-06-07"},"date_created":"2021-06-27T22:01:48Z","file":[{"file_size":727817,"file_name":"2021_LIPIcs_Biswas.pdf","checksum":"22b11a719018b22ecba2471b51f2eb40","date_created":"2021-06-28T13:11:39Z","access_level":"open_access","date_updated":"2021-06-28T13:11:39Z","success":1,"relation":"main_file","content_type":"application/pdf","file_id":"9611","creator":"asandaue"}],"has_accepted_license":"1","department":[{"_id":"HeEd"}],"intvolume":"       189","status":"public","day":"02","type":"conference","publication":"Leibniz International Proceedings in Informatics","file_date_updated":"2021-06-28T13:11:39Z","title":"Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory","year":"2021","doi":"10.4230/LIPIcs.SoCG.2021.16","ec_funded":1,"ddc":["516"],"alternative_title":["LIPIcs"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_number":"16","abstract":[{"lang":"eng","text":"Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation."}],"author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","orcid":"0000-0002-5372-7890","last_name":"Biswas","full_name":"Biswas, Ranita"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832"},{"first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza"}],"publication_status":"published","citation":{"ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory,” in <i>Leibniz International Proceedings in Informatics</i>, Online, 2021, vol. 189.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2021). Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. In <i>Leibniz International Proceedings in Informatics</i> (Vol. 189). Online: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">https://doi.org/10.4230/LIPIcs.SoCG.2021.16</a>","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Counting Cells of Order-k Voronoi Tessellations in ℝ<sup>3</sup> with Morse Theory.” In <i>Leibniz International Proceedings in Informatics</i>, Vol. 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">https://doi.org/10.4230/LIPIcs.SoCG.2021.16</a>.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. In: <i>Leibniz International Proceedings in Informatics</i>. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">10.4230/LIPIcs.SoCG.2021.16</a>","mla":"Biswas, Ranita, et al. “Counting Cells of Order-k Voronoi Tessellations in ℝ<sup>3</sup> with Morse Theory.” <i>Leibniz International Proceedings in Informatics</i>, vol. 189, 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">10.4230/LIPIcs.SoCG.2021.16</a>.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2021. Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. Leibniz International Proceedings in Informatics. SoCG: International Symposium on Computational Geometry, LIPIcs, vol. 189, 16.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021."},"_id":"9604","publication_identifier":{"isbn":["9783959771849"],"issn":["18688969"]},"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"Z00342","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics","grant_number":"I4887"}],"quality_controlled":"1","oa_version":"Published Version","volume":189,"date_updated":"2023-02-23T14:02:28Z","oa":1,"article_processing_charge":"No"},{"date_updated":"2023-10-03T09:24:27Z","volume":17,"oa":1,"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"MS acknowledges the support by Australian Research Council funding through the ARC Training Centre for M3D Innovation (IC180100008). MS thanks M. Hanifpour and N. Francois for their input and valuable discussions. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, grant no. 788183 and from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","quality_controlled":"1","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"Z00342","call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"oa_version":"Submitted Version","pmid":1,"_id":"10204","publication_identifier":{"issn":["1744-683X"],"eissn":["1744-6848"]},"publication_status":"published","citation":{"ama":"Osang GF, Edelsbrunner H, Saadatfar M. Topological signatures and stability of hexagonal close packing and Barlow stackings. <i>Soft Matter</i>. 2021;17(40):9107-9115. doi:<a href=\"https://doi.org/10.1039/d1sm00774b\">10.1039/d1sm00774b</a>","mla":"Osang, Georg F., et al. “Topological Signatures and Stability of Hexagonal Close Packing and Barlow Stackings.” <i>Soft Matter</i>, vol. 17, no. 40, Royal Society of Chemistry , 2021, pp. 9107–15, doi:<a href=\"https://doi.org/10.1039/d1sm00774b\">10.1039/d1sm00774b</a>.","short":"G.F. Osang, H. Edelsbrunner, M. Saadatfar, Soft Matter 17 (2021) 9107–9115.","ista":"Osang GF, Edelsbrunner H, Saadatfar M. 2021. Topological signatures and stability of hexagonal close packing and Barlow stackings. Soft Matter. 17(40), 9107–9115.","apa":"Osang, G. F., Edelsbrunner, H., &#38; Saadatfar, M. (2021). Topological signatures and stability of hexagonal close packing and Barlow stackings. <i>Soft Matter</i>. Royal Society of Chemistry . <a href=\"https://doi.org/10.1039/d1sm00774b\">https://doi.org/10.1039/d1sm00774b</a>","ieee":"G. F. Osang, H. Edelsbrunner, and M. Saadatfar, “Topological signatures and stability of hexagonal close packing and Barlow stackings,” <i>Soft Matter</i>, vol. 17, no. 40. Royal Society of Chemistry , pp. 9107–9115, 2021.","chicago":"Osang, Georg F, Herbert Edelsbrunner, and Mohammad Saadatfar. “Topological Signatures and Stability of Hexagonal Close Packing and Barlow Stackings.” <i>Soft Matter</i>. Royal Society of Chemistry , 2021. <a href=\"https://doi.org/10.1039/d1sm00774b\">https://doi.org/10.1039/d1sm00774b</a>."},"author":[{"first_name":"Georg F","full_name":"Osang, Georg F","last_name":"Osang","orcid":"0000-0002-8882-5116","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Saadatfar, Mohammad","last_name":"Saadatfar","first_name":"Mohammad"}],"abstract":[{"lang":"eng","text":"Two common representations of close packings of identical spheres consisting of hexagonal layers, called Barlow stackings, appear abundantly in minerals and metals. These motifs, however, occupy an identical portion of space and bear identical first-order topological signatures as measured by persistent homology. Here we present a novel method based on k-fold covers that unambiguously distinguishes between these patterns. Moreover, our approach provides topological evidence that the FCC motif is the more stable of the two in the context of evolving experimental sphere packings during the transition from disordered to an ordered state. We conclude that our approach can be generalised to distinguish between various Barlow stackings manifested in minerals and metals."}],"isi":1,"ddc":["540"],"ec_funded":1,"year":"2021","doi":"10.1039/d1sm00774b","title":"Topological signatures and stability of hexagonal close packing and Barlow stackings","external_id":{"isi":["000700090000001"],"pmid":["34569592"]},"page":"9107-9115","file_date_updated":"2023-10-03T09:21:42Z","issue":"40","publication":"Soft Matter","type":"journal_article","day":"20","status":"public","intvolume":"        17","department":[{"_id":"HeEd"}],"has_accepted_license":"1","date_created":"2021-10-31T23:01:30Z","file":[{"relation":"main_file","content_type":"application/pdf","creator":"dernst","file_id":"14385","success":1,"access_level":"open_access","date_updated":"2023-10-03T09:21:42Z","file_size":4678788,"file_name":"2021_SoftMatter_acceptedversion_Osang.pdf","checksum":"b4da0c420530295e61b153960f6cb350","date_created":"2023-10-03T09:21:42Z"}],"article_type":"original","date_published":"2021-10-20T00:00:00Z","month":"10","language":[{"iso":"eng"}],"publisher":"Royal Society of Chemistry ","scopus_import":"1"},{"scopus_import":"1","publisher":"Taylor and Francis","language":[{"iso":"eng"}],"month":"10","article_type":"original","date_published":"2021-10-25T00:00:00Z","date_created":"2021-11-07T23:01:25Z","file":[{"relation":"main_file","content_type":"application/pdf","creator":"dernst","file_id":"14053","success":1,"access_level":"open_access","date_updated":"2023-08-14T11:55:10Z","file_size":1966019,"file_name":"2023_ExperimentalMath_Akopyan.pdf","checksum":"3514382e3a1eb87fa6c61ad622874415","date_created":"2023-08-14T11:55:10Z"}],"has_accepted_license":"1","department":[{"_id":"HeEd"}],"status":"public","day":"25","type":"journal_article","publication":"Experimental Mathematics","page":"1-15","file_date_updated":"2023-08-14T11:55:10Z","title":"The beauty of random polytopes inscribed in the 2-sphere","external_id":{"arxiv":["2007.07783"],"isi":["000710893500001"]},"year":"2021","doi":"10.1080/10586458.2021.1980459","ec_funded":1,"ddc":["510"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"isi":1,"abstract":[{"lang":"eng","text":"Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density."}],"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan","first_name":"Arseniy"},{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko","orcid":"0000-0002-0659-3201","first_name":"Anton"}],"citation":{"chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, and Anton Nikitenko. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>. Taylor and Francis, 2021. <a href=\"https://doi.org/10.1080/10586458.2021.1980459\">https://doi.org/10.1080/10586458.2021.1980459</a>.","apa":"Akopyan, A., Edelsbrunner, H., &#38; Nikitenko, A. (2021). The beauty of random polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. Taylor and Francis. <a href=\"https://doi.org/10.1080/10586458.2021.1980459\">https://doi.org/10.1080/10586458.2021.1980459</a>","ieee":"A. Akopyan, H. Edelsbrunner, and A. Nikitenko, “The beauty of random polytopes inscribed in the 2-sphere,” <i>Experimental Mathematics</i>. Taylor and Francis, pp. 1–15, 2021.","short":"A. Akopyan, H. Edelsbrunner, A. Nikitenko, Experimental Mathematics (2021) 1–15.","ista":"Akopyan A, Edelsbrunner H, Nikitenko A. 2021. The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics., 1–15.","mla":"Akopyan, Arseniy, et al. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>, Taylor and Francis, 2021, pp. 1–15, doi:<a href=\"https://doi.org/10.1080/10586458.2021.1980459\">10.1080/10586458.2021.1980459</a>.","ama":"Akopyan A, Edelsbrunner H, Nikitenko A. The beauty of random polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. 2021:1-15. doi:<a href=\"https://doi.org/10.1080/10586458.2021.1980459\">10.1080/10586458.2021.1980459</a>"},"publication_status":"published","publication_identifier":{"eissn":["1944-950X"],"issn":["1058-6458"]},"_id":"10222","oa_version":"Published Version","quality_controlled":"1","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.\r\nWe are grateful to Dmitry Zaporozhets and Christoph Thäle for valuable comments and for directing us to relevant references. We also thank to Anton Mellit for a useful discussion on Bessel functions.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","arxiv":1,"article_processing_charge":"Yes (via OA deal)","oa":1,"date_updated":"2023-08-14T11:57:07Z"},{"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"call_identifier":"H2020","_id":"2533E772-B435-11E9-9278-68D0E5697425","name":"Efficient Simulation of Natural Phenomena at Extremely Large Scales","grant_number":"638176"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"quality_controlled":"1","oa_version":"Submitted Version","_id":"8135","publication_identifier":{"eissn":["21978549"],"isbn":["9783030434076"],"issn":["21932808"]},"date_updated":"2021-01-12T08:17:06Z","volume":15,"oa":1,"article_processing_charge":"No","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko","first_name":"Anton"},{"first_name":"Katharina","full_name":"Ölsböck, Katharina","last_name":"Ölsböck","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87"},{"id":"331776E2-F248-11E8-B48F-1D18A9856A87","first_name":"Peter","last_name":"Synak","full_name":"Synak, Peter"}],"abstract":[{"lang":"eng","text":"Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics."}],"publication_status":"published","citation":{"ieee":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions on Poisson–Delaunay mosaics and related complexes experimentally,” in <i>Topological Data Analysis</i>, 2020, vol. 15, pp. 181–218.","apa":"Edelsbrunner, H., Nikitenko, A., Ölsböck, K., &#38; Synak, P. (2020). Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In <i>Topological Data Analysis</i> (Vol. 15, pp. 181–218). Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">https://doi.org/10.1007/978-3-030-43408-3_8</a>","chicago":"Edelsbrunner, Herbert, Anton Nikitenko, Katharina Ölsböck, and Peter Synak. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” In <i>Topological Data Analysis</i>, 15:181–218. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">https://doi.org/10.1007/978-3-030-43408-3_8</a>.","ama":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In: <i>Topological Data Analysis</i>. Vol 15. Springer Nature; 2020:181-218. doi:<a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">10.1007/978-3-030-43408-3_8</a>","mla":"Edelsbrunner, Herbert, et al. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” <i>Topological Data Analysis</i>, vol. 15, Springer Nature, 2020, pp. 181–218, doi:<a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">10.1007/978-3-030-43408-3_8</a>.","ista":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. 2020. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. Topological Data Analysis. , Abel Symposia, vol. 15, 181–218.","short":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, P. Synak, in:, Topological Data Analysis, Springer Nature, 2020, pp. 181–218."},"ddc":["510"],"alternative_title":["Abel Symposia"],"title":"Radius functions on Poisson–Delaunay mosaics and related complexes experimentally","ec_funded":1,"year":"2020","doi":"10.1007/978-3-030-43408-3_8","file_date_updated":"2020-10-08T08:56:14Z","page":"181-218","publication":"Topological Data Analysis","status":"public","intvolume":"        15","type":"conference","day":"22","date_created":"2020-07-19T22:00:59Z","file":[{"content_type":"application/pdf","relation":"main_file","creator":"dernst","file_id":"8628","success":1,"date_updated":"2020-10-08T08:56:14Z","access_level":"open_access","file_name":"2020-B-01-PoissonExperimentalSurvey.pdf","file_size":2207071,"date_created":"2020-10-08T08:56:14Z","checksum":"7b5e0de10675d787a2ddb2091370b8d8"}],"department":[{"_id":"HeEd"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","date_published":"2020-06-22T00:00:00Z","month":"06"},{"abstract":[{"lang":"eng","text":"Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$."}],"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"orcid":"0000-0002-0659-3201","full_name":"Nikitenko, Anton","last_name":"Nikitenko","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"}],"citation":{"short":"H. Edelsbrunner, A. Nikitenko, Theory of Probability and Its Applications 64 (2020) 595–614.","ista":"Edelsbrunner H, Nikitenko A. 2020. Weighted Poisson–Delaunay mosaics. Theory of Probability and its Applications. 64(4), 595–614.","ama":"Edelsbrunner H, Nikitenko A. Weighted Poisson–Delaunay mosaics. <i>Theory of Probability and its Applications</i>. 2020;64(4):595-614. doi:<a href=\"https://doi.org/10.1137/S0040585X97T989726\">10.1137/S0040585X97T989726</a>","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” <i>Theory of Probability and Its Applications</i>, vol. 64, no. 4, SIAM, 2020, pp. 595–614, doi:<a href=\"https://doi.org/10.1137/S0040585X97T989726\">10.1137/S0040585X97T989726</a>.","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” <i>Theory of Probability and Its Applications</i>. SIAM, 2020. <a href=\"https://doi.org/10.1137/S0040585X97T989726\">https://doi.org/10.1137/S0040585X97T989726</a>.","ieee":"H. Edelsbrunner and A. Nikitenko, “Weighted Poisson–Delaunay mosaics,” <i>Theory of Probability and its Applications</i>, vol. 64, no. 4. SIAM, pp. 595–614, 2020.","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2020). Weighted Poisson–Delaunay mosaics. <i>Theory of Probability and Its Applications</i>. SIAM. <a href=\"https://doi.org/10.1137/S0040585X97T989726\">https://doi.org/10.1137/S0040585X97T989726</a>"},"publication_status":"published","publication_identifier":{"issn":["0040585X"],"eissn":["10957219"]},"_id":"7554","quality_controlled":"1","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35"}],"oa_version":"Preprint","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","arxiv":1,"article_processing_charge":"No","date_updated":"2023-08-18T06:45:48Z","volume":64,"oa":1,"external_id":{"isi":["000551393100007"],"arxiv":["1705.08735"]},"title":"Weighted Poisson–Delaunay mosaics","year":"2020","doi":"10.1137/S0040585X97T989726","ec_funded":1,"isi":1,"main_file_link":[{"url":"https://arxiv.org/abs/1705.08735","open_access":"1"}],"intvolume":"        64","status":"public","day":"13","type":"journal_article","publication":"Theory of Probability and its Applications","issue":"4","page":"595-614","scopus_import":"1","publisher":"SIAM","language":[{"iso":"eng"}],"month":"02","article_type":"original","date_published":"2020-02-13T00:00:00Z","date_created":"2020-03-01T23:00:39Z","department":[{"_id":"HeEd"}]},{"publication":"Discrete and Computational Geometry","file_date_updated":"2020-11-20T13:22:21Z","page":"759-775","intvolume":"        64","status":"public","day":"20","type":"journal_article","date_created":"2020-04-19T22:00:56Z","file":[{"relation":"main_file","content_type":"application/pdf","file_id":"8786","creator":"dernst","success":1,"access_level":"open_access","date_updated":"2020-11-20T13:22:21Z","file_size":701673,"file_name":"2020_DiscreteCompGeo_Edelsbrunner.pdf","checksum":"f8cc96e497f00c38340b5dafe0cb91d7","date_created":"2020-11-20T13:22:21Z"}],"has_accepted_license":"1","department":[{"_id":"HeEd"}],"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"03","article_type":"original","date_published":"2020-03-20T00:00:00Z","_id":"7666","publication_identifier":{"eissn":["14320444"],"issn":["01795376"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","acknowledgement":"This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF).","quality_controlled":"1","oa_version":"Published Version","project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"},{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"oa":1,"date_updated":"2023-08-21T06:13:48Z","volume":64,"article_processing_charge":"Yes (via OA deal)","abstract":[{"lang":"eng","text":"Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups."}],"author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Katharina","last_name":"Ölsböck","full_name":"Ölsböck, Katharina","orcid":"0000-0002-4672-8297","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87"}],"publication_status":"published","citation":{"chicago":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s00454-020-00188-x\">https://doi.org/10.1007/s00454-020-00188-x</a>.","ieee":"H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,” <i>Discrete and Computational Geometry</i>, vol. 64. Springer Nature, pp. 759–775, 2020.","apa":"Edelsbrunner, H., &#38; Ölsböck, K. (2020). Tri-partitions and bases of an ordered complex. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00188-x\">https://doi.org/10.1007/s00454-020-00188-x</a>","ista":"Edelsbrunner H, Ölsböck K. 2020. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 64, 759–775.","short":"H. Edelsbrunner, K. Ölsböck, Discrete and Computational Geometry 64 (2020) 759–775.","ama":"Edelsbrunner H, Ölsböck K. Tri-partitions and bases of an ordered complex. <i>Discrete and Computational Geometry</i>. 2020;64:759-775. doi:<a href=\"https://doi.org/10.1007/s00454-020-00188-x\">10.1007/s00454-020-00188-x</a>","mla":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” <i>Discrete and Computational Geometry</i>, vol. 64, Springer Nature, 2020, pp. 759–75, doi:<a href=\"https://doi.org/10.1007/s00454-020-00188-x\">10.1007/s00454-020-00188-x</a>."},"ddc":["510"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"isi":1,"title":"Tri-partitions and bases of an ordered complex","external_id":{"isi":["000520918800001"]},"doi":"10.1007/s00454-020-00188-x","year":"2020","ec_funded":1},{"issue":"4","publication":"Journal of Applied and Computational Topology","file_date_updated":"2024-03-04T10:52:42Z","page":"455-480","intvolume":"         4","status":"public","day":"01","type":"journal_article","file":[{"date_updated":"2024-03-04T10:52:42Z","access_level":"open_access","file_name":"2020_JourApplCompTopology_Bauer.pdf","file_size":851190,"date_created":"2024-03-04T10:52:42Z","checksum":"eed1168b6e66cd55272c19bb7fca8a1c","content_type":"application/pdf","relation":"main_file","creator":"dernst","file_id":"15065","success":1}],"date_created":"2024-03-04T10:47:49Z","has_accepted_license":"1","department":[{"_id":"HeEd"}],"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"12","article_type":"original","date_published":"2020-12-01T00:00:00Z","_id":"15064","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"acknowledgement":"This research has been supported by the DFG Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics”, by Polish MNiSzW Grant No. 2621/7.PR/12/2013/2, by the Polish National Science Center under Maestro Grant No. 2014/14/A/ST1/00453 and Grant No. DEC-2013/09/N/ST6/02995. Open Access funding provided by Projekt DEAL.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","quality_controlled":"1","oa":1,"volume":4,"date_updated":"2024-03-04T10:54:04Z","article_processing_charge":"Yes (via OA deal)","abstract":[{"text":"We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspaces of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.","lang":"eng"}],"author":[{"last_name":"Bauer","full_name":"Bauer, U.","first_name":"U."},{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-3536-9866","full_name":"Jablonski, Grzegorz","last_name":"Jablonski","first_name":"Grzegorz","id":"4483EF78-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Mrozek, M.","last_name":"Mrozek","first_name":"M."}],"publication_status":"published","citation":{"ista":"Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. 2020. Čech-Delaunay gradient flow and homology inference for self-maps. Journal of Applied and Computational Topology. 4(4), 455–480.","short":"U. Bauer, H. Edelsbrunner, G. Jablonski, M. Mrozek, Journal of Applied and Computational Topology 4 (2020) 455–480.","mla":"Bauer, U., et al. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.” <i>Journal of Applied and Computational Topology</i>, vol. 4, no. 4, Springer Nature, 2020, pp. 455–80, doi:<a href=\"https://doi.org/10.1007/s41468-020-00058-8\">10.1007/s41468-020-00058-8</a>.","ama":"Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. Čech-Delaunay gradient flow and homology inference for self-maps. <i>Journal of Applied and Computational Topology</i>. 2020;4(4):455-480. doi:<a href=\"https://doi.org/10.1007/s41468-020-00058-8\">10.1007/s41468-020-00058-8</a>","chicago":"Bauer, U., Herbert Edelsbrunner, Grzegorz Jablonski, and M. Mrozek. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s41468-020-00058-8\">https://doi.org/10.1007/s41468-020-00058-8</a>.","apa":"Bauer, U., Edelsbrunner, H., Jablonski, G., &#38; Mrozek, M. (2020). Čech-Delaunay gradient flow and homology inference for self-maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-020-00058-8\">https://doi.org/10.1007/s41468-020-00058-8</a>","ieee":"U. Bauer, H. Edelsbrunner, G. Jablonski, and M. Mrozek, “Čech-Delaunay gradient flow and homology inference for self-maps,” <i>Journal of Applied and Computational Topology</i>, vol. 4, no. 4. Springer Nature, pp. 455–480, 2020."},"ddc":["500"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"Čech-Delaunay gradient flow and homology inference for self-maps","doi":"10.1007/s41468-020-00058-8","year":"2020"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"The authors of this paper thank Roland Roth for suggesting the analysis of theweighted\r\ncurvature derivatives for the purpose of improving molecular dynamics simulations. They also thank Patrice Koehl for the implementation of the formulas and for his encouragement and advise along the road. Finally, they thank two anonymous reviewers for their constructive criticism.\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","oa_version":"Published Version","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"quality_controlled":"1","_id":"9156","publication_identifier":{"issn":["2544-7297"]},"oa":1,"date_updated":"2023-10-17T12:35:10Z","volume":8,"article_processing_charge":"No","arxiv":1,"author":[{"first_name":"Arseniy","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy."}],"publication_status":"published","citation":{"ieee":"A. Akopyan and H. Edelsbrunner, “The weighted Gaussian curvature derivative of a space-filling diagram,” <i>Computational and Mathematical Biophysics</i>, vol. 8, no. 1. De Gruyter, pp. 74–88, 2020.","apa":"Akopyan, A., &#38; Edelsbrunner, H. (2020). The weighted Gaussian curvature derivative of a space-filling diagram. <i>Computational and Mathematical Biophysics</i>. De Gruyter. <a href=\"https://doi.org/10.1515/cmb-2020-0101\">https://doi.org/10.1515/cmb-2020-0101</a>","chicago":"Akopyan, Arseniy, and Herbert Edelsbrunner. “The Weighted Gaussian Curvature Derivative of a Space-Filling Diagram.” <i>Computational and Mathematical Biophysics</i>. De Gruyter, 2020. <a href=\"https://doi.org/10.1515/cmb-2020-0101\">https://doi.org/10.1515/cmb-2020-0101</a>.","mla":"Akopyan, Arseniy, and Herbert Edelsbrunner. “The Weighted Gaussian Curvature Derivative of a Space-Filling Diagram.” <i>Computational and Mathematical Biophysics</i>, vol. 8, no. 1, De Gruyter, 2020, pp. 74–88, doi:<a href=\"https://doi.org/10.1515/cmb-2020-0101\">10.1515/cmb-2020-0101</a>.","ama":"Akopyan A, Edelsbrunner H. The weighted Gaussian curvature derivative of a space-filling diagram. <i>Computational and Mathematical Biophysics</i>. 2020;8(1):74-88. doi:<a href=\"https://doi.org/10.1515/cmb-2020-0101\">10.1515/cmb-2020-0101</a>","short":"A. Akopyan, H. Edelsbrunner, Computational and Mathematical Biophysics 8 (2020) 74–88.","ista":"Akopyan A, Edelsbrunner H. 2020. The weighted Gaussian curvature derivative of a space-filling diagram. Computational and Mathematical Biophysics. 8(1), 74–88."},"ddc":["510"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"The weighted Gaussian curvature derivative of a space-filling diagram","external_id":{"arxiv":["1908.06777"]},"ec_funded":1,"doi":"10.1515/cmb-2020-0101","year":"2020","file_date_updated":"2021-02-19T13:33:19Z","page":"74-88","issue":"1","publication":"Computational and Mathematical Biophysics","status":"public","intvolume":"         8","type":"journal_article","day":"21","file":[{"access_level":"open_access","date_updated":"2021-02-19T13:33:19Z","checksum":"ca43a7440834eab6bbea29c59b56ef3a","date_created":"2021-02-19T13:33:19Z","file_size":707452,"file_name":"2020_CompMathBiophysics_Akopyan.pdf","creator":"dernst","file_id":"9170","relation":"main_file","content_type":"application/pdf","success":1}],"date_created":"2021-02-17T15:12:44Z","department":[{"_id":"HeEd"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"publisher":"De Gruyter","date_published":"2020-07-21T00:00:00Z","article_type":"original","month":"07"},{"ddc":["510"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"title":"The weighted mean curvature derivative of a space-filling diagram","doi":"10.1515/cmb-2020-0100","year":"2020","ec_funded":1,"publication_identifier":{"issn":["2544-7297"]},"_id":"9157","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","oa_version":"Published Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"The authors of this paper thank Roland Roth for suggesting the analysis of the weighted\r\ncurvature derivatives for the purpose of improving molecular dynamics simulations and for his continued encouragement. They also thank Patrice Koehl for the implementation of the formulas and for his encouragement and advise along the road. Finally, they thank two anonymous reviewers for their constructive criticism.\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","article_processing_charge":"No","date_updated":"2023-10-17T12:34:51Z","volume":8,"oa":1,"abstract":[{"lang":"eng","text":"Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy."}],"author":[{"orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"citation":{"ama":"Akopyan A, Edelsbrunner H. The weighted mean curvature derivative of a space-filling diagram. <i>Computational and Mathematical Biophysics</i>. 2020;8(1):51-67. doi:<a href=\"https://doi.org/10.1515/cmb-2020-0100\">10.1515/cmb-2020-0100</a>","mla":"Akopyan, Arseniy, and Herbert Edelsbrunner. “The Weighted Mean Curvature Derivative of a Space-Filling Diagram.” <i>Computational and Mathematical Biophysics</i>, vol. 8, no. 1, De Gruyter, 2020, pp. 51–67, doi:<a href=\"https://doi.org/10.1515/cmb-2020-0100\">10.1515/cmb-2020-0100</a>.","short":"A. Akopyan, H. Edelsbrunner, Computational and Mathematical Biophysics 8 (2020) 51–67.","ista":"Akopyan A, Edelsbrunner H. 2020. The weighted mean curvature derivative of a space-filling diagram. Computational and Mathematical Biophysics. 8(1), 51–67.","ieee":"A. Akopyan and H. Edelsbrunner, “The weighted mean curvature derivative of a space-filling diagram,” <i>Computational and Mathematical Biophysics</i>, vol. 8, no. 1. De Gruyter, pp. 51–67, 2020.","apa":"Akopyan, A., &#38; Edelsbrunner, H. (2020). The weighted mean curvature derivative of a space-filling diagram. <i>Computational and Mathematical Biophysics</i>. De Gruyter. <a href=\"https://doi.org/10.1515/cmb-2020-0100\">https://doi.org/10.1515/cmb-2020-0100</a>","chicago":"Akopyan, Arseniy, and Herbert Edelsbrunner. “The Weighted Mean Curvature Derivative of a Space-Filling Diagram.” <i>Computational and Mathematical Biophysics</i>. De Gruyter, 2020. <a href=\"https://doi.org/10.1515/cmb-2020-0100\">https://doi.org/10.1515/cmb-2020-0100</a>."},"publication_status":"published","date_created":"2021-02-17T15:13:01Z","file":[{"relation":"main_file","content_type":"application/pdf","file_id":"9171","creator":"dernst","success":1,"access_level":"open_access","date_updated":"2021-02-19T13:56:24Z","file_size":562359,"file_name":"2020_CompMathBiophysics_Akopyan2.pdf","checksum":"cea41de9937d07a3b927d71ee8b4e432","date_created":"2021-02-19T13:56:24Z"}],"has_accepted_license":"1","department":[{"_id":"HeEd"}],"publisher":"De Gruyter","language":[{"iso":"eng"}],"month":"06","date_published":"2020-06-20T00:00:00Z","article_type":"original","publication":"Computational and Mathematical Biophysics","issue":"1","file_date_updated":"2021-02-19T13:56:24Z","page":"51-67","intvolume":"         8","status":"public","day":"20","type":"journal_article"},{"status":"public","intvolume":"        11","type":"journal_article","day":"14","file_date_updated":"2021-08-11T11:55:11Z","page":"162-182","issue":"2","publication":"Journal of Computational Geometry","language":[{"iso":"eng"}],"publisher":"Carleton University","scopus_import":"1","article_type":"original","date_published":"2020-12-14T00:00:00Z","month":"12","file":[{"access_level":"open_access","date_updated":"2021-08-11T11:55:11Z","checksum":"f02d0b2b3838e7891a6c417fc34ffdcd","date_created":"2021-08-11T11:55:11Z","file_size":1449234,"file_name":"2020_JournalOfComputationalGeometry_Edelsbrunner.pdf","creator":"asandaue","file_id":"9882","relation":"main_file","content_type":"application/pdf","success":1}],"date_created":"2021-07-04T22:01:26Z","department":[{"_id":"HeEd"}],"license":"https://creativecommons.org/licenses/by/3.0/","has_accepted_license":"1","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"2E36B656-F248-11E8-B48F-1D18A9856A87","full_name":"Virk, Ziga","last_name":"Virk","first_name":"Ziga"},{"first_name":"Hubert","last_name":"Wagner","full_name":"Wagner, Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms.  Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context."}],"publication_status":"published","citation":{"mla":"Edelsbrunner, Herbert, et al. “Topological Data Analysis in Information Space.” <i>Journal of Computational Geometry</i>, vol. 11, no. 2, Carleton University, 2020, pp. 162–82, doi:<a href=\"https://doi.org/10.20382/jocg.v11i2a7\">10.20382/jocg.v11i2a7</a>.","ama":"Edelsbrunner H, Virk Z, Wagner H. Topological data analysis in information space. <i>Journal of Computational Geometry</i>. 2020;11(2):162-182. doi:<a href=\"https://doi.org/10.20382/jocg.v11i2a7\">10.20382/jocg.v11i2a7</a>","ista":"Edelsbrunner H, Virk Z, Wagner H. 2020. Topological data analysis in information space. Journal of Computational Geometry. 11(2), 162–182.","short":"H. Edelsbrunner, Z. Virk, H. Wagner, Journal of Computational Geometry 11 (2020) 162–182.","ieee":"H. Edelsbrunner, Z. Virk, and H. Wagner, “Topological data analysis in information space,” <i>Journal of Computational Geometry</i>, vol. 11, no. 2. Carleton University, pp. 162–182, 2020.","apa":"Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2020). Topological data analysis in information space. <i>Journal of Computational Geometry</i>. Carleton University. <a href=\"https://doi.org/10.20382/jocg.v11i2a7\">https://doi.org/10.20382/jocg.v11i2a7</a>","chicago":"Edelsbrunner, Herbert, Ziga Virk, and Hubert Wagner. “Topological Data Analysis in Information Space.” <i>Journal of Computational Geometry</i>. Carleton University, 2020. <a href=\"https://doi.org/10.20382/jocg.v11i2a7\">https://doi.org/10.20382/jocg.v11i2a7</a>."},"acknowledgement":"This research is partially supported by the Office of Naval Research, through grant no. N62909-18-1-2038, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","user_id":"6785fbc1-c503-11eb-8a32-93094b40e1cf","project":[{"grant_number":"I4887","name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"}],"quality_controlled":"1","oa_version":"Published Version","_id":"9630","publication_identifier":{"eissn":["1920180X"]},"date_updated":"2021-08-11T12:26:34Z","volume":11,"oa":1,"article_processing_charge":"Yes","title":"Topological data analysis in information space","doi":"10.20382/jocg.v11i2a7","year":"2020","ddc":["510","000"],"tmp":{"name":"Creative Commons Attribution 3.0 Unported (CC BY 3.0)","short":"CC BY (3.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/3.0/legalcode"}}]