[{"page":"811-842","status":"public","day":"01","type":"journal_article","publisher":"Springer Nature","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"quality_controlled":"1","month":"04","corr_author":"1","_id":"10773","article_processing_charge":"Yes (via OA deal)","article_type":"original","author":[{"first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera Di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","last_name":"Cultrera Di Montesano"},{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza"}],"oa":1,"language":[{"iso":"eng"}],"title":"Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics","isi":1,"scopus_import":"1","oa_version":"Published Version","acknowledgement":"Open access funding provided by the Institute of Science and Technology (IST Austria).","has_accepted_license":"1","citation":{"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>","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 67 (2022) 811–842.","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>","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.","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>.","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>."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_created":"2022-02-20T23:01:34Z","year":"2022","date_published":"2022-04-01T00:00:00Z","ddc":["510"],"date_updated":"2024-10-09T21:01:38Z","volume":67,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"department":[{"_id":"HeEd"}],"external_id":{"isi":["000752175300002"]},"publication_status":"published","doi":"10.1007/s00454-022-00371-2","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."}],"file_date_updated":"2022-08-02T06:07:55Z","file":[{"date_updated":"2022-08-02T06:07:55Z","file_size":2518111,"file_name":"2022_DiscreteCompGeometry_Biswas.pdf","access_level":"open_access","creator":"dernst","success":1,"content_type":"application/pdf","date_created":"2022-08-02T06:07:55Z","relation":"main_file","checksum":"9383d3b70561bacee905e335dc922680","file_id":"11718"}],"publication":"Discrete and Computational Geometry","intvolume":"        67"},{"publication_status":"published","external_id":{"arxiv":["1912.12685"]},"department":[{"_id":"HeEd"}],"ec_funded":1,"abstract":[{"lang":"eng","text":"Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law."}],"doi":"10.1007/s40879-020-00405-0","file_date_updated":"2020-07-14T12:48:03Z","file":[{"content_type":"application/pdf","date_created":"2020-05-04T10:33:42Z","relation":"main_file","checksum":"f53e71fd03744075adcd0b8fc1b8423d","file_id":"7796","file_name":"2020_EuropMathematics_Akopyan.pdf","file_size":263926,"date_updated":"2020-07-14T12:48:03Z","access_level":"open_access","creator":"dernst"}],"publication":"European Journal of Mathematics","intvolume":"         8","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","year":"2022","date_created":"2020-05-03T22:00:48Z","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"ddc":["510"],"date_published":"2022-12-01T00:00:00Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"volume":8,"date_updated":"2025-04-14T07:48:36Z","title":"When different norms lead to same billiard trajectories?","acknowledgement":"AA was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). RK was supported by the Federal professorship program Grant 1.456.2016/1.4 and the Russian Foundation for Basic Research Grants 18-01-00036 and 19-01-00169. Open access funding provided by Institute of Science and Technology (IST Austria). The authors thank Alexey Balitskiy, Milena Radnović, and Serge Tabachnikov for useful discussions.","oa_version":"Published Version","scopus_import":"1","citation":{"chicago":"Akopyan, Arseniy, and Roman Karasev. “When Different Norms Lead to Same Billiard Trajectories?” <i>European Journal of Mathematics</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s40879-020-00405-0\">https://doi.org/10.1007/s40879-020-00405-0</a>.","mla":"Akopyan, Arseniy, and Roman Karasev. “When Different Norms Lead to Same Billiard Trajectories?” <i>European Journal of Mathematics</i>, vol. 8, no. 4, Springer Nature, 2022, pp. 1309–12, doi:<a href=\"https://doi.org/10.1007/s40879-020-00405-0\">10.1007/s40879-020-00405-0</a>.","apa":"Akopyan, A., &#38; Karasev, R. (2022). When different norms lead to same billiard trajectories? <i>European Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40879-020-00405-0\">https://doi.org/10.1007/s40879-020-00405-0</a>","ista":"Akopyan A, Karasev R. 2022. When different norms lead to same billiard trajectories? European Journal of Mathematics. 8(4), 1309–1312.","ieee":"A. Akopyan and R. Karasev, “When different norms lead to same billiard trajectories?,” <i>European Journal of Mathematics</i>, vol. 8, no. 4. Springer Nature, pp. 1309–1312, 2022.","ama":"Akopyan A, Karasev R. When different norms lead to same billiard trajectories? <i>European Journal of Mathematics</i>. 2022;8(4):1309-1312. doi:<a href=\"https://doi.org/10.1007/s40879-020-00405-0\">10.1007/s40879-020-00405-0</a>","short":"A. Akopyan, R. Karasev, European Journal of Mathematics 8 (2022) 1309–1312."},"arxiv":1,"has_accepted_license":"1","type":"journal_article","day":"01","page":"1309 - 1312","status":"public","month":"12","quality_controlled":"1","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"publisher":"Springer Nature","article_type":"original","article_processing_charge":"Yes (via OA deal)","_id":"7791","corr_author":"1","issue":"4","oa":1,"language":[{"iso":"eng"}],"author":[{"first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan"},{"last_name":"Karasev","full_name":"Karasev, Roman","first_name":"Roman"}]},{"intvolume":"         8","publication":"European Journal of Mathematics","doi":"10.1007/s40879-020-00426-9","ec_funded":1,"abstract":[{"lang":"eng","text":"We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods."}],"department":[{"_id":"HeEd"}],"external_id":{"arxiv":["2001.02934"]},"publication_status":"published","date_updated":"2025-04-14T07:48:34Z","volume":8,"date_published":"2022-12-01T00:00:00Z","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended"}],"date_created":"2020-09-20T22:01:38Z","main_file_link":[{"url":"https://arxiv.org/abs/2001.02934","open_access":"1"}],"year":"2022","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","arxiv":1,"citation":{"short":"A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics 8 (2022) 1313–1327.","ama":"Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. <i>European Journal of Mathematics</i>. 2022;8(4):1313-1327. doi:<a href=\"https://doi.org/10.1007/s40879-020-00426-9\">10.1007/s40879-020-00426-9</a>","ieee":"A. Akopyan, R. Schwartz, and S. Tabachnikov, “Billiards in ellipses revisited,” <i>European Journal of Mathematics</i>, vol. 8, no. 4. Springer Nature, pp. 1313–1327, 2022.","apa":"Akopyan, A., Schwartz, R., &#38; Tabachnikov, S. (2022). Billiards in ellipses revisited. <i>European Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40879-020-00426-9\">https://doi.org/10.1007/s40879-020-00426-9</a>","ista":"Akopyan A, Schwartz R, Tabachnikov S. 2022. Billiards in ellipses revisited. European Journal of Mathematics. 8(4), 1313–1327.","mla":"Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” <i>European Journal of Mathematics</i>, vol. 8, no. 4, Springer Nature, 2022, pp. 1313–27, doi:<a href=\"https://doi.org/10.1007/s40879-020-00426-9\">10.1007/s40879-020-00426-9</a>.","chicago":"Akopyan, Arseniy, Richard Schwartz, and Serge Tabachnikov. “Billiards in Ellipses Revisited.” <i>European Journal of Mathematics</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s40879-020-00426-9\">https://doi.org/10.1007/s40879-020-00426-9</a>."},"acknowledgement":" This paper would not be written if not for Dan Reznik’s curiosity and persistence; we are very grateful to him. We also thank R. Garcia and J. Koiller for interesting discussions. It is a pleasure to thank the Mathematical Institute of the University of Heidelberg for its stimulating atmosphere. ST thanks M. Bialy for interesting discussions and the Tel Aviv\r\nUniversity for its invariable hospitality. AA was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 78818 Alpha). RS is supported by NSF Grant DMS-1807320. ST was supported by NSF grant DMS-1510055 and SFB/TRR 191.","scopus_import":"1","oa_version":"Preprint","title":"Billiards in ellipses revisited","author":[{"last_name":"Akopyan","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy"},{"first_name":"Richard","last_name":"Schwartz","full_name":"Schwartz, Richard"},{"full_name":"Tabachnikov, Serge","last_name":"Tabachnikov","first_name":"Serge"}],"oa":1,"language":[{"iso":"eng"}],"issue":"4","_id":"8538","article_type":"original","article_processing_charge":"No","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"publisher":"Springer Nature","quality_controlled":"1","month":"12","page":"1313-1327","status":"public","day":"01","type":"journal_article"},{"file_date_updated":"2021-07-14T06:44:36Z","file":[{"date_created":"2021-07-14T06:44:36Z","content_type":"application/pdf","file_id":"9650","checksum":"f1d372ec3c08ec22e84f8e93e1126b8c","relation":"main_file","access_level":"open_access","date_updated":"2021-07-14T06:44:36Z","file_name":"Boissonnat-Wintraecken2021_Article_TheTopologicalCorrectnessOfPLA.pdf","file_size":1455699,"creator":"mwintrae"}],"publication":"Foundations of Computational Mathematics ","intvolume":"        22","publication_status":"published","department":[{"_id":"HeEd"}],"external_id":{"isi":["000673039600001"]},"ec_funded":1,"abstract":[{"lang":"eng","text":"Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently\r\nfine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary."}],"doi":"10.1007/s10208-021-09520-0","date_published":"2022-01-01T00:00:00Z","project":[{"name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","call_identifier":"H2020"}],"ddc":["516"],"date_updated":"2025-04-22T13:45:18Z","volume":22,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2022","date_created":"2021-07-14T06:44:53Z","citation":{"chicago":"Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL Approximations of Isomanifolds.” <i>Foundations of Computational Mathematics </i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s10208-021-09520-0\">https://doi.org/10.1007/s10208-021-09520-0</a>.","mla":"Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL Approximations of Isomanifolds.” <i>Foundations of Computational Mathematics </i>, vol. 22, Springer Nature, 2022, pp. 967–1012, doi:<a href=\"https://doi.org/10.1007/s10208-021-09520-0\">10.1007/s10208-021-09520-0</a>.","ista":"Boissonnat J-D, Wintraecken M. 2022. The topological correctness of PL approximations of isomanifolds. Foundations of Computational Mathematics . 22, 967–1012.","apa":"Boissonnat, J.-D., &#38; Wintraecken, M. (2022). The topological correctness of PL approximations of isomanifolds. <i>Foundations of Computational Mathematics </i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10208-021-09520-0\">https://doi.org/10.1007/s10208-021-09520-0</a>","short":"J.-D. Boissonnat, M. Wintraecken, Foundations of Computational Mathematics  22 (2022) 967–1012.","ama":"Boissonnat J-D, Wintraecken M. The topological correctness of PL approximations of isomanifolds. <i>Foundations of Computational Mathematics </i>. 2022;22:967-1012. doi:<a href=\"https://doi.org/10.1007/s10208-021-09520-0\">10.1007/s10208-021-09520-0</a>","ieee":"J.-D. Boissonnat and M. Wintraecken, “The topological correctness of PL approximations of isomanifolds,” <i>Foundations of Computational Mathematics </i>, vol. 22. Springer Nature, pp. 967–1012, 2022."},"has_accepted_license":"1","title":"The topological correctness of PL approximations of isomanifolds","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"7952"}]},"scopus_import":"1","acknowledgement":"First and foremost, we acknowledge Siargey Kachanovich for discussions. We thank Herbert Edelsbrunner and all members of his group, all former and current members of the Datashape team (formerly known as Geometrica), and André Lieutier for encouragement. We further thank the reviewers of Foundations of Computational Mathematics and the reviewers and program committee of the Symposium on Computational Geometry for their feedback, which improved the exposition.\r\nThis work was funded by the European Research Council under the European Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). This work was also supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002. Mathijs Wintraecken also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 754411.","oa_version":"Published Version","isi":1,"article_type":"original","article_processing_charge":"Yes (via OA deal)","corr_author":"1","_id":"9649","language":[{"iso":"eng"}],"oa":1,"author":[{"first_name":"Jean-Daniel","full_name":"Boissonnat, Jean-Daniel","last_name":"Boissonnat"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220","last_name":"Wintraecken"}],"day":"01","type":"journal_article","page":"967-1012","status":"public","quality_controlled":"1","month":"01","publication_identifier":{"eissn":["1615-3383"]},"publisher":"Springer Nature"},{"status":"public","page":"2621–2635","day":"01","type":"journal_article","publisher":"Springer Nature","publication_identifier":{"eissn":["1868-5145"],"issn":["1868-5137"]},"month":"05","quality_controlled":"1","_id":"10208","keyword":["general computer science"],"article_processing_charge":"No","article_type":"original","author":[{"last_name":"Goudarzi","full_name":"Goudarzi, Samira","first_name":"Samira"},{"first_name":"Mohammad","full_name":"Sharif, Mohammad","last_name":"Sharif"},{"first_name":"Farid","id":"2A2BCDC4-CF62-11E9-BE5E-3B1EE6697425","last_name":"Karimipour","orcid":"0000-0001-6746-4174","full_name":"Karimipour, Farid"}],"language":[{"iso":"eng"}],"oa":1,"title":"A context-aware dimension reduction framework for trajectory and health signal analyses","isi":1,"oa_version":"Submitted Version","scopus_import":"1","acknowledgement":"The third author acknowledges the funding received from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","has_accepted_license":"1","citation":{"short":"S. Goudarzi, M. Sharif, F. Karimipour, Journal of Ambient Intelligence and Humanized Computing 13 (2022) 2621–2635.","ieee":"S. Goudarzi, M. Sharif, and F. Karimipour, “A context-aware dimension reduction framework for trajectory and health signal analyses,” <i>Journal of Ambient Intelligence and Humanized Computing</i>, vol. 13. Springer Nature, pp. 2621–2635, 2022.","ama":"Goudarzi S, Sharif M, Karimipour F. A context-aware dimension reduction framework for trajectory and health signal analyses. <i>Journal of Ambient Intelligence and Humanized Computing</i>. 2022;13:2621–2635. doi:<a href=\"https://doi.org/10.1007/s12652-021-03569-z\">10.1007/s12652-021-03569-z</a>","ista":"Goudarzi S, Sharif M, Karimipour F. 2022. A context-aware dimension reduction framework for trajectory and health signal analyses. Journal of Ambient Intelligence and Humanized Computing. 13, 2621–2635.","apa":"Goudarzi, S., Sharif, M., &#38; Karimipour, F. (2022). A context-aware dimension reduction framework for trajectory and health signal analyses. <i>Journal of Ambient Intelligence and Humanized Computing</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s12652-021-03569-z\">https://doi.org/10.1007/s12652-021-03569-z</a>","chicago":"Goudarzi, Samira, Mohammad Sharif, and Farid Karimipour. “A Context-Aware Dimension Reduction Framework for Trajectory and Health Signal Analyses.” <i>Journal of Ambient Intelligence and Humanized Computing</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s12652-021-03569-z\">https://doi.org/10.1007/s12652-021-03569-z</a>.","mla":"Goudarzi, Samira, et al. “A Context-Aware Dimension Reduction Framework for Trajectory and Health Signal Analyses.” <i>Journal of Ambient Intelligence and Humanized Computing</i>, vol. 13, Springer Nature, 2022, pp. 2621–2635, doi:<a href=\"https://doi.org/10.1007/s12652-021-03569-z\">10.1007/s12652-021-03569-z</a>."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_created":"2021-11-02T09:28:55Z","year":"2022","date_published":"2022-05-01T00:00:00Z","project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science"}],"ddc":["000"],"date_updated":"2025-04-15T07:16:55Z","volume":13,"department":[{"_id":"HeEd"}],"external_id":{"isi":["000712198000001"]},"publication_status":"published","doi":"10.1007/s12652-021-03569-z","abstract":[{"lang":"eng","text":"It is practical to collect a huge amount of movement data and environmental context information along with the health signals of individuals because there is the emergence of new generations of positioning and tracking technologies and rapid advancements of health sensors. The study of the relations between these datasets and their sequence similarity analysis is of interest to many applications such as health monitoring and recommender systems. However, entering all movement parameters and health signals can lead to the complexity of the problem and an increase in its computational load. In this situation, dimension reduction techniques can be used to avoid consideration of simultaneous dependent parameters in the process of similarity measurement of the trajectories. The present study provides a framework, named CaDRAW, to use spatial–temporal data and movement parameters along with independent context information in the process of measuring the similarity of trajectories. In this regard, the omission of dependent movement characteristic signals is conducted by using an unsupervised feature selection dimension reduction technique. To evaluate the effectiveness of the proposed framework, it was applied to a real contextualized movement and related health signal datasets of individuals. The results indicated the capability of the proposed framework in measuring the similarity and in decreasing the characteristic signals in such a way that the similarity results -before and after reduction of dependent characteristic signals- have small differences. The mean differences between the obtained results before and after reducing the dimension were 0.029 and 0.023 for the round path, respectively."}],"file_date_updated":"2022-12-20T23:30:08Z","file":[{"embargo":"2022-11-12","file_size":1634958,"file_name":"A Context‑aware Dimension Reduction Framework - Journal of Ambient Intelligence 2021 (Preprint version).pdf","date_updated":"2022-12-20T23:30:08Z","access_level":"open_access","creator":"fkarimip","content_type":"application/pdf","date_created":"2021-11-12T19:38:05Z","relation":"main_file","file_id":"10279","checksum":"0a8961416a9bb2be5a1cebda65468bcf"}],"intvolume":"        13","publication":"Journal of Ambient Intelligence and Humanized Computing"},{"publication_status":"published","department":[{"_id":"HeEd"}],"external_id":{"arxiv":["1912.02342"]},"abstract":[{"text":"In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.","lang":"eng"}],"doi":"10.1007/s00493-021-4530-9","intvolume":"        41","publication":"Combinatorica","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2021","date_created":"2024-04-03T07:59:57Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1912.02342","open_access":"1"}],"date_published":"2021-11-20T00:00:00Z","date_updated":"2024-04-09T10:40:08Z","volume":41,"title":"Bounded VC-dimension implies the Schur-Erdős conjecture","oa_version":"Preprint","arxiv":1,"citation":{"chicago":"Fox, Jacob, János Pach, and Andrew Suk. “Bounded VC-Dimension Implies the Schur-Erdős Conjecture.” <i>Combinatorica</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00493-021-4530-9\">https://doi.org/10.1007/s00493-021-4530-9</a>.","mla":"Fox, Jacob, et al. “Bounded VC-Dimension Implies the Schur-Erdős Conjecture.” <i>Combinatorica</i>, vol. 41, no. 6, Springer Nature, 2021, pp. 803–13, doi:<a href=\"https://doi.org/10.1007/s00493-021-4530-9\">10.1007/s00493-021-4530-9</a>.","ista":"Fox J, Pach J, Suk A. 2021. Bounded VC-dimension implies the Schur-Erdős conjecture. Combinatorica. 41(6), 803–813.","apa":"Fox, J., Pach, J., &#38; Suk, A. (2021). Bounded VC-dimension implies the Schur-Erdős conjecture. <i>Combinatorica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00493-021-4530-9\">https://doi.org/10.1007/s00493-021-4530-9</a>","short":"J. Fox, J. Pach, A. Suk, Combinatorica 41 (2021) 803–813.","ama":"Fox J, Pach J, Suk A. Bounded VC-dimension implies the Schur-Erdős conjecture. <i>Combinatorica</i>. 2021;41(6):803-813. doi:<a href=\"https://doi.org/10.1007/s00493-021-4530-9\">10.1007/s00493-021-4530-9</a>","ieee":"J. Fox, J. Pach, and A. Suk, “Bounded VC-dimension implies the Schur-Erdős conjecture,” <i>Combinatorica</i>, vol. 41, no. 6. Springer Nature, pp. 803–813, 2021."},"day":"20","type":"journal_article","status":"public","page":"803-813","month":"11","quality_controlled":"1","publisher":"Springer Nature","publication_identifier":{"eissn":["1439-6912"],"issn":["0209-9683"]},"article_type":"original","article_processing_charge":"No","_id":"15275","issue":"6","keyword":["Computational Mathematics","Discrete Mathematics and Combinatorics"],"language":[{"iso":"eng"}],"oa":1,"author":[{"full_name":"Fox, Jacob","last_name":"Fox","first_name":"Jacob"},{"last_name":"Pach","full_name":"Pach, János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János"},{"last_name":"Suk","full_name":"Suk, Andrew","first_name":"Andrew"}]},{"author":[{"full_name":"Adams, Henry","last_name":"Adams","first_name":"Henry"},{"first_name":"Hana","orcid":"0000-0001-7841-0091","full_name":"Kourimska, Hana","last_name":"Kourimska","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E"},{"first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss","orcid":"0000-0002-1780-2689","full_name":"Heiss, Teresa"},{"first_name":"Sarah","last_name":"Percival","full_name":"Percival, Sarah"},{"full_name":"Ziegelmeier, Lori","last_name":"Ziegelmeier","first_name":"Lori"}],"language":[{"iso":"eng"}],"oa":1,"issue":"9","_id":"10071","article_type":"letter_note","article_processing_charge":"No","publication_identifier":{"issn":["0002-9920"],"eissn":["1088-9477"]},"publisher":"American Mathematical Society","quality_controlled":"1","month":"10","page":"1511-1514","status":"public","day":"01","type":"journal_article","citation":{"ama":"Adams H, Kourimska H, Heiss T, Percival S, Ziegelmeier L. How to tutorial-a-thon. <i>Notices of the American Mathematical Society</i>. 2021;68(9):1511-1514. doi:<a href=\"https://doi.org/10.1090/noti2349\">10.1090/noti2349</a>","ieee":"H. Adams, H. Kourimska, T. Heiss, S. Percival, and L. Ziegelmeier, “How to tutorial-a-thon,” <i>Notices of the American Mathematical Society</i>, vol. 68, no. 9. American Mathematical Society, pp. 1511–1514, 2021.","short":"H. Adams, H. Kourimska, T. Heiss, S. Percival, L. Ziegelmeier, Notices of the American Mathematical Society 68 (2021) 1511–1514.","apa":"Adams, H., Kourimska, H., Heiss, T., Percival, S., &#38; Ziegelmeier, L. (2021). How to tutorial-a-thon. <i>Notices of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/noti2349\">https://doi.org/10.1090/noti2349</a>","ista":"Adams H, Kourimska H, Heiss T, Percival S, Ziegelmeier L. 2021. How to tutorial-a-thon. Notices of the American Mathematical Society. 68(9), 1511–1514.","mla":"Adams, Henry, et al. “How to Tutorial-a-Thon.” <i>Notices of the American Mathematical Society</i>, vol. 68, no. 9, American Mathematical Society, 2021, pp. 1511–14, doi:<a href=\"https://doi.org/10.1090/noti2349\">10.1090/noti2349</a>.","chicago":"Adams, Henry, Hana Kourimska, Teresa Heiss, Sarah Percival, and Lori Ziegelmeier. “How to Tutorial-a-Thon.” <i>Notices of the American Mathematical Society</i>. American Mathematical Society, 2021. <a href=\"https://doi.org/10.1090/noti2349\">https://doi.org/10.1090/noti2349</a>."},"oa_version":"Published Version","scopus_import":"1","title":"How to tutorial-a-thon","date_updated":"2026-06-18T08:36:13Z","volume":68,"date_published":"2021-10-01T00:00:00Z","ddc":["500"],"date_created":"2021-10-03T22:01:22Z","main_file_link":[{"open_access":"1","url":"http://www.ams.org/notices/"}],"year":"2021","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication":"Notices of the American Mathematical Society","intvolume":"        68","doi":"10.1090/noti2349","alternative_title":["Early Career"],"department":[{"_id":"HeEd"}],"publication_status":"published"},{"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","date_created":"2021-11-28T23:01:30Z","main_file_link":[{"open_access":"1","url":"https://aclanthology.org/2021.acl-tutorials.6/"}],"year":"2021","date_published":"2021-08-01T00:00:00Z","ddc":["000"],"conference":{"end_date":"2021-08-06","name":"ACL: Association for Computational Linguistics ; IJCNLP: International Joint Conference on Natural Language Processing","start_date":"2021-08-01","location":"Bangkok, Thailand"},"date_updated":"2022-01-26T14:26:36Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"department":[{"_id":"HeEd"}],"publication_status":"published","doi":"10.18653/v1/2021.acl-tutorials.6","abstract":[{"lang":"eng","text":"How information is created, shared and consumed has changed rapidly in recent decades, in part thanks to new social platforms and technologies on the web. With ever-larger amounts of unstructured and limited labels, organizing and reconciling information from different sources and modalities is a central challenge in machine learning. This cutting-edge tutorial aims to introduce the multimodal entailment task, which can be useful for detecting semantic alignments when a single modality alone does not suffice for a whole content understanding. Starting with a brief overview of natural language processing, computer vision, structured data and neural graph learning, we lay the foundations for the multimodal sections to follow. We then discuss recent multimodal learning literature covering visual, audio and language streams, and explore case studies focusing on tasks which require fine-grained understanding of visual and linguistic semantics question answering, veracity and hatred classification. Finally, we introduce a new dataset for recognizing multimodal entailment, exploring it in a hands-on collaborative section. Overall, this tutorial gives an overview of multimodal learning, introduces a multimodal entailment dataset, and encourages future research in the topic."}],"file_date_updated":"2021-11-29T08:41:00Z","publication":"59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts","file":[{"relation":"main_file","file_id":"10368","checksum":"b14052a025a6ecf675bdfe51db98c0d7","success":1,"content_type":"application/pdf","date_created":"2021-11-29T08:41:00Z","creator":"cchlebak","date_updated":"2021-11-29T08:41:00Z","file_name":"2021_ACL_Ilharco.pdf","file_size":1227703,"access_level":"open_access"}],"page":"29-30","status":"public","day":"01","type":"conference","publisher":"Association for Computational Linguistics","publication_identifier":{"isbn":["9-781-9540-8557-2"]},"month":"08","quality_controlled":"1","_id":"10367","article_processing_charge":"No","author":[{"full_name":"Ilharco, Cesar","last_name":"Ilharco","first_name":"Cesar"},{"first_name":"Afsaneh","full_name":"Shirazi, Afsaneh","last_name":"Shirazi"},{"full_name":"Gopalan, Arjun","last_name":"Gopalan","first_name":"Arjun"},{"first_name":"Arsha","last_name":"Nagrani","full_name":"Nagrani, Arsha"},{"full_name":"Bratanič, Blaž","last_name":"Bratanič","first_name":"Blaž"},{"first_name":"Chris","full_name":"Bregler, Chris","last_name":"Bregler"},{"full_name":"Liu, Christina","last_name":"Liu","first_name":"Christina"},{"full_name":"Ferreira, Felipe","last_name":"Ferreira","first_name":"Felipe"},{"full_name":"Barcik, Gabriek","last_name":"Barcik","first_name":"Gabriek"},{"last_name":"Ilharco","full_name":"Ilharco, Gabriel","first_name":"Gabriel"},{"first_name":"Georg F","full_name":"Osang, Georg F","last_name":"Osang","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Bulian, Jannis","last_name":"Bulian","first_name":"Jannis"},{"last_name":"Frank","full_name":"Frank, Jared","first_name":"Jared"},{"first_name":"Lucas","last_name":"Smaira","full_name":"Smaira, Lucas"},{"full_name":"Cao, Qin","last_name":"Cao","first_name":"Qin"},{"first_name":"Ricardo","last_name":"Marino","full_name":"Marino, Ricardo"},{"first_name":"Roma","last_name":"Patel","full_name":"Patel, Roma"},{"full_name":"Leung, Thomas","last_name":"Leung","first_name":"Thomas"},{"full_name":"Imbrasaite, Vaiva","last_name":"Imbrasaite","first_name":"Vaiva"}],"language":[{"iso":"eng"}],"oa":1,"title":"Recognizing multimodal entailment","acknowledgement":"We would like to thank Abby Schantz, Abe Ittycheriah, Aliaksei Severyn, Allan Heydon, Aly\r\nGrealish, Andrey Vlasov, Arkaitz Zubiaga, Ashwin Kakarla, Chen Sun, Clayton Williams, Cong\r\nYu, Cordelia Schmid, Da-Cheng Juan, Dan Finnie, Dani Valevski, Daniel Rocha, David Price, David Sklar, Devi Krishna, Elena Kochkina, Enrique Alfonseca, Franc¸oise Beaufays, Isabelle Augenstein, Jialu Liu, John Cantwell, John Palowitch, Jordan Boyd-Graber, Lei Shi, Luis Valente, Maria Voitovich, Mehmet Aktuna, Mogan Brown, Mor Naaman, Natalia P, Nidhi Hebbar, Pete Aykroyd, Rahul Sukthankar, Richa Dixit, Steve Pucci, Tania Bedrax-Weiss, Tobias Kaufmann, Tom Boulos, Tu Tsao, Vladimir Chtchetkine, Yair Kurzion, Yifan Xu and Zach Hynes.","scopus_import":"1","oa_version":"Published Version","has_accepted_license":"1","citation":{"mla":"Ilharco, Cesar, et al. “Recognizing Multimodal Entailment.” <i>59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts</i>, Association for Computational Linguistics, 2021, pp. 29–30, doi:<a href=\"https://doi.org/10.18653/v1/2021.acl-tutorials.6\">10.18653/v1/2021.acl-tutorials.6</a>.","chicago":"Ilharco, Cesar, Afsaneh Shirazi, Arjun Gopalan, Arsha Nagrani, Blaž Bratanič, Chris Bregler, Christina Liu, et al. “Recognizing Multimodal Entailment.” In <i>59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts</i>, 29–30. Association for Computational Linguistics, 2021. <a href=\"https://doi.org/10.18653/v1/2021.acl-tutorials.6\">https://doi.org/10.18653/v1/2021.acl-tutorials.6</a>.","ista":"Ilharco C, Shirazi A, Gopalan A, Nagrani A, Bratanič B, Bregler C, Liu C, Ferreira F, Barcik G, Ilharco G, Osang GF, Bulian J, Frank J, Smaira L, Cao Q, Marino R, Patel R, Leung T, Imbrasaite V. 2021. Recognizing multimodal entailment. 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts. ACL: Association for Computational Linguistics ; IJCNLP: International Joint Conference on Natural Language Processing, 29–30.","apa":"Ilharco, C., Shirazi, A., Gopalan, A., Nagrani, A., Bratanič, B., Bregler, C., … Imbrasaite, V. (2021). Recognizing multimodal entailment. In <i>59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts</i> (pp. 29–30). Bangkok, Thailand: Association for Computational Linguistics. <a href=\"https://doi.org/10.18653/v1/2021.acl-tutorials.6\">https://doi.org/10.18653/v1/2021.acl-tutorials.6</a>","ieee":"C. Ilharco <i>et al.</i>, “Recognizing multimodal entailment,” in <i>59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts</i>, Bangkok, Thailand, 2021, pp. 29–30.","short":"C. Ilharco, A. Shirazi, A. Gopalan, A. Nagrani, B. Bratanič, C. Bregler, C. Liu, F. Ferreira, G. Barcik, G. Ilharco, G.F. Osang, J. Bulian, J. Frank, L. Smaira, Q. Cao, R. Marino, R. Patel, T. Leung, V. Imbrasaite, in:, 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts, Association for Computational Linguistics, 2021, pp. 29–30.","ama":"Ilharco C, Shirazi A, Gopalan A, et al. Recognizing multimodal entailment. In: <i>59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts</i>. Association for Computational Linguistics; 2021:29-30. doi:<a href=\"https://doi.org/10.18653/v1/2021.acl-tutorials.6\">10.18653/v1/2021.acl-tutorials.6</a>"}},{"oa":1,"language":[{"iso":"eng"}],"author":[{"first_name":"Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","last_name":"Brown","full_name":"Brown, Adam"},{"first_name":"Bei","full_name":"Wang, Bei","last_name":"Wang"}],"article_type":"original","article_processing_charge":"Yes (via OA deal)","corr_author":"1","_id":"7905","quality_controlled":"1","month":"06","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"publisher":"Springer Nature","day":"01","type":"journal_article","page":"1166-1198","status":"public","arxiv":1,"citation":{"ista":"Brown A, Wang B. 2021. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. 65, 1166–1198.","apa":"Brown, A., &#38; Wang, B. (2021). Sheaf-theoretic stratification learning from geometric and topological perspectives. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00206-y\">https://doi.org/10.1007/s00454-020-00206-y</a>","ama":"Brown A, Wang B. Sheaf-theoretic stratification learning from geometric and topological perspectives. <i>Discrete and Computational Geometry</i>. 2021;65:1166-1198. doi:<a href=\"https://doi.org/10.1007/s00454-020-00206-y\">10.1007/s00454-020-00206-y</a>","short":"A. Brown, B. Wang, Discrete and Computational Geometry 65 (2021) 1166–1198.","ieee":"A. Brown and B. Wang, “Sheaf-theoretic stratification learning from geometric and topological perspectives,” <i>Discrete and Computational Geometry</i>, vol. 65. Springer Nature, pp. 1166–1198, 2021.","chicago":"Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00206-y\">https://doi.org/10.1007/s00454-020-00206-y</a>.","mla":"Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” <i>Discrete and Computational Geometry</i>, vol. 65, Springer Nature, 2021, pp. 1166–98, doi:<a href=\"https://doi.org/10.1007/s00454-020-00206-y\">10.1007/s00454-020-00206-y</a>."},"has_accepted_license":"1","scopus_import":"1","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). This work was partially supported by NSF IIS-1513616 and NSF ABI-1661375. The authors would like to thank the anonymous referees for their insightful comments.","oa_version":"Published Version","isi":1,"title":"Sheaf-theoretic stratification learning from geometric and topological perspectives","date_updated":"2025-04-15T06:53:15Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"volume":65,"date_published":"2021-06-01T00:00:00Z","ddc":["510"],"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"year":"2021","date_created":"2020-05-30T10:26:04Z","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","file":[{"relation":"main_file","checksum":"487a84ea5841b75f04f66d7ebd71b67e","file_id":"8803","content_type":"application/pdf","success":1,"date_created":"2020-11-25T09:06:41Z","creator":"dernst","file_size":1013730,"file_name":"2020_DiscreteCompGeometry_Brown.pdf","date_updated":"2020-11-25T09:06:41Z","access_level":"open_access"}],"publication":"Discrete and Computational Geometry","intvolume":"        65","file_date_updated":"2020-11-25T09:06:41Z","abstract":[{"text":"We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.","lang":"eng"}],"doi":"10.1007/s00454-020-00206-y","publication_status":"published","department":[{"_id":"HeEd"}],"external_id":{"isi":["000536324700001"],"arxiv":["1712.07734"]}},{"year":"2021","main_file_link":[{"url":"https://doi.org/10.1007/s00454-020-00233-9","open_access":"1"}],"date_created":"2020-08-11T07:11:51Z","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"volume":66,"date_updated":"2025-04-14T07:44:05Z","ddc":["510"],"project":[{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"date_published":"2021-09-01T00:00:00Z","ec_funded":1,"abstract":[{"lang":"eng","text":"We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic."}],"doi":"10.1007/s00454-020-00233-9","publication_status":"published","external_id":{"isi":["000558119300001"]},"department":[{"_id":"HeEd"}],"publication":"Discrete and Computational Geometry","intvolume":"        66","quality_controlled":"1","month":"09","publisher":"Springer Nature","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"type":"journal_article","day":"01","page":"666-686","status":"public","language":[{"iso":"eng"}],"oa":1,"author":[{"last_name":"Boissonnat","full_name":"Boissonnat, Jean-Daniel","first_name":"Jean-Daniel"},{"first_name":"Ramsay","full_name":"Dyer, Ramsay","last_name":"Dyer"},{"first_name":"Arijit","full_name":"Ghosh, Arijit","last_name":"Ghosh"},{"last_name":"Lieutier","full_name":"Lieutier, Andre","first_name":"Andre"},{"first_name":"Mathijs","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"Yes (via OA deal)","article_type":"original","_id":"8248","corr_author":"1","acknowledgement":"Open access funding provided by the Institute of Science and Technology (IST Austria). Arijit Ghosh is supported by the Ramanujan Fellowship (No. SB/S2/RJN-064/2015), India.\r\nThis work has been funded by the European Research Council under the European Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). The third author is supported by Ramanujan Fellowship (No. SB/S2/RJN-064/2015), India. The fifth author also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411.","scopus_import":"1","oa_version":"Published Version","isi":1,"title":"Local conditions for triangulating submanifolds of Euclidean space","citation":{"ama":"Boissonnat J-D, Dyer R, Ghosh A, Lieutier A, Wintraecken M. Local conditions for triangulating submanifolds of Euclidean space. <i>Discrete and Computational Geometry</i>. 2021;66:666-686. doi:<a href=\"https://doi.org/10.1007/s00454-020-00233-9\">10.1007/s00454-020-00233-9</a>","short":"J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, M. Wintraecken, Discrete and Computational Geometry 66 (2021) 666–686.","ieee":"J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, and M. Wintraecken, “Local conditions for triangulating submanifolds of Euclidean space,” <i>Discrete and Computational Geometry</i>, vol. 66. Springer Nature, pp. 666–686, 2021.","ista":"Boissonnat J-D, Dyer R, Ghosh A, Lieutier A, Wintraecken M. 2021. Local conditions for triangulating submanifolds of Euclidean space. Discrete and Computational Geometry. 66, 666–686.","apa":"Boissonnat, J.-D., Dyer, R., Ghosh, A., Lieutier, A., &#38; Wintraecken, M. (2021). Local conditions for triangulating submanifolds of Euclidean space. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00233-9\">https://doi.org/10.1007/s00454-020-00233-9</a>","chicago":"Boissonnat, Jean-Daniel, Ramsay Dyer, Arijit Ghosh, Andre Lieutier, and Mathijs Wintraecken. “Local Conditions for Triangulating Submanifolds of Euclidean Space.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00233-9\">https://doi.org/10.1007/s00454-020-00233-9</a>.","mla":"Boissonnat, Jean-Daniel, et al. “Local Conditions for Triangulating Submanifolds of Euclidean Space.” <i>Discrete and Computational Geometry</i>, vol. 66, Springer Nature, 2021, pp. 666–86, doi:<a href=\"https://doi.org/10.1007/s00454-020-00233-9\">10.1007/s00454-020-00233-9</a>."},"has_accepted_license":"1"},{"oa":1,"language":[{"iso":"eng"}],"author":[{"first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X"},{"first_name":"Alexander I.","full_name":"Bobenko, Alexander I.","last_name":"Bobenko"},{"first_name":"Wolfgang K.","full_name":"Schief, Wolfgang K.","last_name":"Schief"},{"first_name":"Jan","last_name":"Techter","full_name":"Techter, Jan"}],"article_type":"original","article_processing_charge":"No","_id":"8338","quality_controlled":"1","month":"10","publisher":"Springer Nature","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"day":"01","type":"journal_article","page":"938-976","status":"public","arxiv":1,"citation":{"short":"A. Akopyan, A.I. Bobenko, W.K. Schief, J. Techter, Discrete and Computational Geometry 66 (2021) 938–976.","ama":"Akopyan A, Bobenko AI, Schief WK, Techter J. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. <i>Discrete and Computational Geometry</i>. 2021;66:938-976. doi:<a href=\"https://doi.org/10.1007/s00454-020-00240-w\">10.1007/s00454-020-00240-w</a>","ieee":"A. Akopyan, A. I. Bobenko, W. K. Schief, and J. Techter, “On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs,” <i>Discrete and Computational Geometry</i>, vol. 66. Springer Nature, pp. 938–976, 2021.","ista":"Akopyan A, Bobenko AI, Schief WK, Techter J. 2021. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. 66, 938–976.","apa":"Akopyan, A., Bobenko, A. I., Schief, W. K., &#38; Techter, J. (2021). On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00240-w\">https://doi.org/10.1007/s00454-020-00240-w</a>","chicago":"Akopyan, Arseniy, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00240-w\">https://doi.org/10.1007/s00454-020-00240-w</a>.","mla":"Akopyan, Arseniy, et al. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” <i>Discrete and Computational Geometry</i>, vol. 66, Springer Nature, 2021, pp. 938–76, doi:<a href=\"https://doi.org/10.1007/s00454-020-00240-w\">10.1007/s00454-020-00240-w</a>."},"oa_version":"Preprint","acknowledgement":"This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. W.K.S. was also supported by the Australian Research Council (DP1401000851). A.V.A. was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha).","scopus_import":"1","isi":1,"title":"On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs","date_updated":"2025-04-14T07:48:36Z","volume":66,"date_published":"2021-10-01T00:00:00Z","project":[{"call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"}],"year":"2021","date_created":"2020-09-06T22:01:13Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1908.00856"}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","publication":"Discrete and Computational Geometry","intvolume":"        66","abstract":[{"lang":"eng","text":"Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory."}],"ec_funded":1,"doi":"10.1007/s00454-020-00240-w","publication_status":"published","department":[{"_id":"HeEd"}],"external_id":{"arxiv":["1908.00856"],"isi":["000564488500002"]}},{"date_published":"2021-01-01T00:00:00Z","project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"date_updated":"2025-04-14T07:43:50Z","volume":149,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_created":"2020-11-19T10:17:40Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.08286"}],"year":"2021","publication":"Proceedings of the American Mathematical Society","intvolume":"       149","department":[{"_id":"HeEd"}],"external_id":{"isi":["000600416300004"],"arxiv":["1910.08286"]},"publication_status":"published","doi":"10.1090/proc/15205","ec_funded":1,"abstract":[{"lang":"eng","text":"Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell."}],"keyword":["Applied Mathematics","General Mathematics"],"_id":"8773","issue":"1","article_processing_charge":"No","article_type":"original","author":[{"last_name":"Brown","full_name":"Brown, Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam"},{"full_name":"Romanov, Anna","last_name":"Romanov","first_name":"Anna"}],"oa":1,"language":[{"iso":"eng"}],"status":"public","page":"37-52","day":"01","type":"journal_article","publisher":"American Mathematical Society","publication_identifier":{"issn":["0002-9939"],"eissn":["1088-6826"]},"quality_controlled":"1","month":"01","arxiv":1,"citation":{"short":"A. Brown, A. Romanov, Proceedings of the American Mathematical Society 149 (2021) 37–52.","ieee":"A. Brown and A. Romanov, “Contravariant forms on Whittaker modules,” <i>Proceedings of the American Mathematical Society</i>, vol. 149, no. 1. American Mathematical Society, pp. 37–52, 2021.","ama":"Brown A, Romanov A. Contravariant forms on Whittaker modules. <i>Proceedings of the American Mathematical Society</i>. 2021;149(1):37-52. doi:<a href=\"https://doi.org/10.1090/proc/15205\">10.1090/proc/15205</a>","apa":"Brown, A., &#38; Romanov, A. (2021). Contravariant forms on Whittaker modules. <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/proc/15205\">https://doi.org/10.1090/proc/15205</a>","ista":"Brown A, Romanov A. 2021. Contravariant forms on Whittaker modules. Proceedings of the American Mathematical Society. 149(1), 37–52.","mla":"Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.” <i>Proceedings of the American Mathematical Society</i>, vol. 149, no. 1, American Mathematical Society, 2021, pp. 37–52, doi:<a href=\"https://doi.org/10.1090/proc/15205\">10.1090/proc/15205</a>.","chicago":"Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.” <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society, 2021. <a href=\"https://doi.org/10.1090/proc/15205\">https://doi.org/10.1090/proc/15205</a>."},"title":"Contravariant forms on Whittaker modules","isi":1,"scopus_import":"1","acknowledgement":"We would like to thank Peter Trapa for useful discussions, and Dragan Milicic and Arun Ram for valuable feedback on the structure of the paper. The first author acknowledges the support of the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie Grant Agreement No. 754411. The second author is\r\nsupported by the National Science Foundation Award No. 1803059.","oa_version":"Preprint"},{"ddc":["516"],"project":[{"name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425"}],"date_published":"2021-07-01T00:00:00Z","volume":66,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"date_updated":"2025-04-14T07:43:50Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","year":"2021","date_created":"2020-12-12T11:07:02Z","file_date_updated":"2021-08-06T09:52:29Z","file":[{"creator":"kschuh","file_size":983307,"file_name":"2021_DescreteCompGeopmetry_Boissonnat.pdf","date_updated":"2021-08-06T09:52:29Z","access_level":"open_access","relation":"main_file","file_id":"9795","checksum":"c848986091e56699dc12de85adb1e39c","success":1,"content_type":"application/pdf","date_created":"2021-08-06T09:52:29Z"}],"intvolume":"        66","publication":"Discrete & Computational Geometry","publication_status":"published","external_id":{"isi":["000597770300001"]},"department":[{"_id":"HeEd"}],"abstract":[{"text":"We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.","lang":"eng"}],"ec_funded":1,"doi":"10.1007/s00454-020-00250-8","article_processing_charge":"Yes (via OA deal)","article_type":"original","keyword":["Theoretical Computer Science","Computational Theory and Mathematics","Geometry and Topology","Discrete Mathematics and Combinatorics"],"_id":"8940","issue":"1","corr_author":"1","oa":1,"language":[{"iso":"eng"}],"author":[{"first_name":"Jean-Daniel","full_name":"Boissonnat, Jean-Daniel","last_name":"Boissonnat"},{"first_name":"Siargey","last_name":"Kachanovich","full_name":"Kachanovich, Siargey"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","first_name":"Mathijs"}],"type":"journal_article","day":"01","page":"386-434","status":"public","month":"07","quality_controlled":"1","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"publisher":"Springer Nature","citation":{"mla":"Boissonnat, Jean-Daniel, et al. “Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method.” <i>Discrete &#38; Computational Geometry</i>, vol. 66, no. 1, Springer Nature, 2021, pp. 386–434, doi:<a href=\"https://doi.org/10.1007/s00454-020-00250-8\">10.1007/s00454-020-00250-8</a>.","chicago":"Boissonnat, Jean-Daniel, Siargey Kachanovich, and Mathijs Wintraecken. “Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00250-8\">https://doi.org/10.1007/s00454-020-00250-8</a>.","short":"J.-D. Boissonnat, S. Kachanovich, M. Wintraecken, Discrete &#38; Computational Geometry 66 (2021) 386–434.","ama":"Boissonnat J-D, Kachanovich S, Wintraecken M. Triangulating submanifolds: An elementary and quantified version of Whitney’s method. <i>Discrete &#38; Computational Geometry</i>. 2021;66(1):386-434. doi:<a href=\"https://doi.org/10.1007/s00454-020-00250-8\">10.1007/s00454-020-00250-8</a>","ieee":"J.-D. Boissonnat, S. Kachanovich, and M. Wintraecken, “Triangulating submanifolds: An elementary and quantified version of Whitney’s method,” <i>Discrete &#38; Computational Geometry</i>, vol. 66, no. 1. Springer Nature, pp. 386–434, 2021.","ista":"Boissonnat J-D, Kachanovich S, Wintraecken M. 2021. Triangulating submanifolds: An elementary and quantified version of Whitney’s method. Discrete &#38; Computational Geometry. 66(1), 386–434.","apa":"Boissonnat, J.-D., Kachanovich, S., &#38; Wintraecken, M. (2021). Triangulating submanifolds: An elementary and quantified version of Whitney’s method. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00250-8\">https://doi.org/10.1007/s00454-020-00250-8</a>"},"has_accepted_license":"1","title":"Triangulating submanifolds: An elementary and quantified version of Whitney’s method","scopus_import":"1","acknowledgement":"This work has been funded by the European Research Council under the European Union’s ERC Grant Agreement Number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). The third author also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. Open access funding provided by the Institute of Science and Technology (IST Austria).","oa_version":"Published Version","isi":1},{"date_published":"2021-02-01T00:00:00Z","ddc":["006","514","516"],"date_updated":"2026-04-08T07:01:30Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_created":"2021-02-02T14:11:06Z","year":"2021","file_date_updated":"2021-02-03T10:37:28Z","file":[{"creator":"patrickd","access_level":"closed","date_updated":"2021-02-03T10:37:28Z","file_size":13446994,"file_name":"thesis_source.zip","file_id":"9063","checksum":"bcf27986147cab0533b6abadd74e7629","relation":"source_file","date_created":"2021-02-02T14:09:25Z","content_type":"application/zip"},{"creator":"patrickd","access_level":"open_access","date_updated":"2021-02-02T14:09:18Z","file_size":5210329,"file_name":"thesis_pdfA2b.pdf","checksum":"9cc8af266579a464385bbe2aff6af606","file_id":"9064","relation":"main_file","date_created":"2021-02-02T14:09:18Z","success":1,"content_type":"application/pdf"}],"department":[{"_id":"HeEd"},{"_id":"GradSch"}],"publication_status":"published","doi":"10.15479/AT:ISTA:9056","degree_awarded":"PhD","alternative_title":["ISTA Thesis"],"abstract":[{"lang":"eng","text":"In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations. The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,\r\nand thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics. We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration\r\nfunction on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss the implications on persistence for periodic data sets."}],"_id":"9056","corr_author":"1","place":"Klosterneuburg","article_processing_charge":"No","author":[{"first_name":"Georg F","full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116","last_name":"Osang","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"}],"supervisor":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"oa":1,"language":[{"iso":"eng"}],"OA_place":"publisher","page":"134","status":"public","day":"01","type":"dissertation","publisher":"Institute of Science and Technology Austria","publication_identifier":{"issn":["2663-337X"]},"month":"02","has_accepted_license":"1","citation":{"mla":"Osang, Georg F. <i>Multi-Cover Persistence and Delaunay Mosaics</i>. Institute of Science and Technology Austria, 2021, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:9056\">10.15479/AT:ISTA:9056</a>.","chicago":"Osang, Georg F. “Multi-Cover Persistence and Delaunay Mosaics.” Institute of Science and Technology Austria, 2021. <a href=\"https://doi.org/10.15479/AT:ISTA:9056\">https://doi.org/10.15479/AT:ISTA:9056</a>.","ista":"Osang GF. 2021. Multi-cover persistence and Delaunay mosaics. Klosterneuburg: Institute of Science and Technology Austria.","apa":"Osang, G. F. (2021). <i>Multi-cover persistence and Delaunay mosaics</i>. Institute of Science and Technology Austria, Klosterneuburg. <a href=\"https://doi.org/10.15479/AT:ISTA:9056\">https://doi.org/10.15479/AT:ISTA:9056</a>","short":"G.F. Osang, Multi-Cover Persistence and Delaunay Mosaics, Institute of Science and Technology Austria, 2021.","ama":"Osang GF. Multi-cover persistence and Delaunay mosaics. 2021. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:9056\">10.15479/AT:ISTA:9056</a>","ieee":"G. F. Osang, “Multi-cover persistence and Delaunay mosaics,” Institute of Science and Technology Austria, Klosterneuburg, 2021."},"related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"187"},{"id":"8703","relation":"part_of_dissertation","status":"public"}]},"title":"Multi-cover persistence and Delaunay mosaics","oa_version":"Published Version"},{"acknowledgement":"AB was supported in part by the European Union’s Horizon 2020 research and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No. 754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation, Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like to thank the Institute for Mathematics and its Applications for hosting a workshop titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen Access funding provided by Institute of Science and Technology (IST Austria).","oa_version":"Published Version","scopus_import":"1","title":"Probabilistic convergence and stability of random mapper graphs","arxiv":1,"citation":{"chicago":"Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic Convergence and Stability of Random Mapper Graphs.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s41468-020-00063-x\">https://doi.org/10.1007/s41468-020-00063-x</a>.","mla":"Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper Graphs.” <i>Journal of Applied and Computational Topology</i>, vol. 5, no. 1, Springer Nature, 2021, pp. 99–140, doi:<a href=\"https://doi.org/10.1007/s41468-020-00063-x\">10.1007/s41468-020-00063-x</a>.","ieee":"A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence and stability of random mapper graphs,” <i>Journal of Applied and Computational Topology</i>, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.","short":"A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational Topology 5 (2021) 99–140.","ama":"Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. 2021;5(1):99-140. doi:<a href=\"https://doi.org/10.1007/s41468-020-00063-x\">10.1007/s41468-020-00063-x</a>","ista":"Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 5(1), 99–140.","apa":"Brown, A., Bobrowski, O., Munch, E., &#38; Wang, B. (2021). Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-020-00063-x\">https://doi.org/10.1007/s41468-020-00063-x</a>"},"has_accepted_license":"1","month":"03","quality_controlled":"1","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"publisher":"Springer Nature","day":"01","type":"journal_article","page":"99-140","status":"public","oa":1,"language":[{"iso":"eng"}],"author":[{"first_name":"Adam","full_name":"Brown, Adam","last_name":"Brown","id":"70B7FDF6-608D-11E9-9333-8535E6697425"},{"full_name":"Bobrowski, Omer","last_name":"Bobrowski","first_name":"Omer"},{"first_name":"Elizabeth","full_name":"Munch, Elizabeth","last_name":"Munch"},{"full_name":"Wang, Bei","last_name":"Wang","first_name":"Bei"}],"article_processing_charge":"Yes (via OA deal)","article_type":"original","issue":"1","_id":"9111","ec_funded":1,"abstract":[{"lang":"eng","text":"We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data."}],"doi":"10.1007/s41468-020-00063-x","publication_status":"published","department":[{"_id":"HeEd"}],"external_id":{"arxiv":["1909.03488"]},"intvolume":"         5","file":[{"creator":"dernst","date_updated":"2021-02-11T14:43:59Z","file_size":2090265,"file_name":"2020_JourApplCompTopology_Brown.pdf","access_level":"open_access","relation":"main_file","checksum":"3f02e9d47c428484733da0f588a3c069","file_id":"9112","success":1,"content_type":"application/pdf","date_created":"2021-02-11T14:43:59Z"}],"publication":"Journal of Applied and Computational Topology","file_date_updated":"2021-02-11T14:43:59Z","year":"2021","date_created":"2021-02-11T14:41:02Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_updated":"2025-04-14T07:43:51Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"volume":5,"date_published":"2021-03-01T00:00:00Z","ddc":["510"],"project":[{"name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"}]},{"language":[{"iso":"eng"}],"oa":1,"author":[{"last_name":"Heiler","full_name":"Heiler, Georg","first_name":"Georg"},{"first_name":"Tobias","full_name":"Reisch, Tobias","last_name":"Reisch"},{"full_name":"Hurt, Jan","last_name":"Hurt","first_name":"Jan"},{"full_name":"Forghani, Mohammad","last_name":"Forghani","first_name":"Mohammad"},{"first_name":"Aida","full_name":"Omani, Aida","last_name":"Omani"},{"first_name":"Allan","last_name":"Hanbury","full_name":"Hanbury, Allan"},{"full_name":"Karimipour, Farid","orcid":"0000-0001-6746-4174","last_name":"Karimipour","id":"2A2BCDC4-CF62-11E9-BE5E-3B1EE6697425","first_name":"Farid"}],"article_processing_charge":"No","_id":"9253","quality_controlled":"1","month":"03","publication_identifier":{"isbn":["9781728162515"]},"publisher":"IEEE","type":"conference","day":"19","page":"3123-3132","status":"public","citation":{"ista":"Heiler G, Reisch T, Hurt J, Forghani M, Omani A, Hanbury A, Karimipour F. 2021. Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. 2020 IEEE International Conference on Big Data. Big Data: International Conference on Big Data, 3123–3132.","apa":"Heiler, G., Reisch, T., Hurt, J., Forghani, M., Omani, A., Hanbury, A., &#38; Karimipour, F. (2021). Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. In <i>2020 IEEE International Conference on Big Data</i> (pp. 3123–3132). Atlanta, GA, United States: IEEE. <a href=\"https://doi.org/10.1109/bigdata50022.2020.9378374\">https://doi.org/10.1109/bigdata50022.2020.9378374</a>","ieee":"G. Heiler <i>et al.</i>, “Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic,” in <i>2020 IEEE International Conference on Big Data</i>, Atlanta, GA, United States, 2021, pp. 3123–3132.","short":"G. Heiler, T. Reisch, J. Hurt, M. Forghani, A. Omani, A. Hanbury, F. Karimipour, in:, 2020 IEEE International Conference on Big Data, IEEE, 2021, pp. 3123–3132.","ama":"Heiler G, Reisch T, Hurt J, et al. Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. In: <i>2020 IEEE International Conference on Big Data</i>. IEEE; 2021:3123-3132. doi:<a href=\"https://doi.org/10.1109/bigdata50022.2020.9378374\">10.1109/bigdata50022.2020.9378374</a>","mla":"Heiler, Georg, et al. “Country-Wide Mobility Changes Observed Using Mobile Phone Data during COVID-19 Pandemic.” <i>2020 IEEE International Conference on Big Data</i>, IEEE, 2021, pp. 3123–32, doi:<a href=\"https://doi.org/10.1109/bigdata50022.2020.9378374\">10.1109/bigdata50022.2020.9378374</a>.","chicago":"Heiler, Georg, Tobias Reisch, Jan Hurt, Mohammad Forghani, Aida Omani, Allan Hanbury, and Farid Karimipour. “Country-Wide Mobility Changes Observed Using Mobile Phone Data during COVID-19 Pandemic.” In <i>2020 IEEE International Conference on Big Data</i>, 3123–32. IEEE, 2021. <a href=\"https://doi.org/10.1109/bigdata50022.2020.9378374\">https://doi.org/10.1109/bigdata50022.2020.9378374</a>."},"arxiv":1,"scopus_import":"1","oa_version":"Preprint","isi":1,"title":"Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic","date_updated":"2023-08-07T14:00:13Z","conference":{"end_date":"2020-12-13","name":"Big Data: International Conference on Big Data","start_date":"2020-12-10","location":"Atlanta, GA, United States"},"date_published":"2021-03-19T00:00:00Z","year":"2021","main_file_link":[{"url":"https://arxiv.org/abs/2008.10064","open_access":"1"}],"date_created":"2021-03-21T11:34:07Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publication":"2020 IEEE International Conference on Big Data","abstract":[{"text":"In March 2020, the Austrian government introduced a widespread lock-down in response to the COVID-19 pandemic. Based on subjective impressions and anecdotal evidence, Austrian public and private life came to a sudden halt. Here we assess the effect of the lock-down quantitatively for all regions in Austria and present an analysis of daily changes of human mobility throughout Austria using near-real-time anonymized mobile phone data. We describe an efficient data aggregation pipeline and analyze the mobility by quantifying mobile-phone traffic at specific point of interests (POIs), analyzing individual trajectories and investigating the cluster structure of the origin-destination graph. We found a reduction of commuters at Viennese metro stations of over 80% and the number of devices with a radius of gyration of less than 500 m almost doubled. The results of studying crowd-movement behavior highlight considerable changes in the structure of mobility networks, revealed by a higher modularity and an increase from 12 to 20 detected communities. We demonstrate the relevance of mobility data for epidemiological studies by showing a significant correlation of the outflow from the town of Ischgl (an early COVID-19 hotspot) and the reported COVID-19 cases with an 8-day time lag. This research indicates that mobile phone usage data permits the moment-by-moment quantification of mobility behavior for a whole country. We emphasize the need to improve the availability of such data in anonymized form to empower rapid response to combat COVID-19 and future pandemics.","lang":"eng"}],"doi":"10.1109/bigdata50022.2020.9378374","publication_status":"published","external_id":{"arxiv":["2008.10064"],"isi":["000662554703032"]},"department":[{"_id":"HeEd"}]},{"intvolume":"     12635","publication":"15th International Conference on Algorithms and Computation","publication_status":"published","department":[{"_id":"UlWa"},{"_id":"HeEd"},{"_id":"KrCh"}],"external_id":{"isi":["001435069600018"],"arxiv":["2101.03928"]},"abstract":[{"lang":"eng","text":" matching is compatible to two or more labeled point sets of size n with labels   {1,…,n}  if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with   ⌊2n−−√⌋  edges. More generally, for any   ℓ  labeled point sets we construct compatible matchings of size   Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any   ℓ  given sets of n points there exists a labeling of each set such that the largest compatible matching has   O(n2/(ℓ+1))  edges. Finally, we show that   Θ(logn)  copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge."}],"ec_funded":1,"doi":"10.1007/978-3-030-68211-8_18","alternative_title":["LNCS"],"date_published":"2021-02-16T00:00:00Z","project":[{"call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"name":"Quantitative Graph Games: Theory and Applications","call_identifier":"FP7","grant_number":"279307","_id":"2581B60A-B435-11E9-9278-68D0E5697425"},{"name":"Modern Graph Algorithmic Techniques in Formal Verification","call_identifier":"FWF","_id":"2584A770-B435-11E9-9278-68D0E5697425","grant_number":"P 23499-N23"},{"grant_number":"S11407","_id":"25863FF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Game Theory"}],"date_updated":"2026-04-16T09:18:21Z","volume":12635,"conference":{"name":"WALCOM: Algorithms and Computation","end_date":"2021-03-02","location":"Yangon, Myanmar","start_date":"2021-02-28"},"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","year":"2021","date_created":"2021-03-28T22:01:41Z","main_file_link":[{"url":"https://arxiv.org/abs/2101.03928","open_access":"1"}],"arxiv":1,"citation":{"ieee":"O. Aichholzer <i>et al.</i>, “On compatible matchings,” in <i>15th International Conference on Algorithms and Computation</i>, Yangon, Myanmar, 2021, vol. 12635, pp. 221–233.","short":"O. Aichholzer, A.M. Arroyo Guevara, Z. Masárová, I. Parada, D. Perz, A. Pilz, J. Tkadlec, B. Vogtenhuber, in:, 15th International Conference on Algorithms and Computation, Springer Nature, 2021, pp. 221–233.","ama":"Aichholzer O, Arroyo Guevara AM, Masárová Z, et al. On compatible matchings. In: <i>15th International Conference on Algorithms and Computation</i>. Vol 12635. Springer Nature; 2021:221-233. doi:<a href=\"https://doi.org/10.1007/978-3-030-68211-8_18\">10.1007/978-3-030-68211-8_18</a>","apa":"Aichholzer, O., Arroyo Guevara, A. M., Masárová, Z., Parada, I., Perz, D., Pilz, A., … Vogtenhuber, B. (2021). On compatible matchings. In <i>15th International Conference on Algorithms and Computation</i> (Vol. 12635, pp. 221–233). Yangon, Myanmar: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-68211-8_18\">https://doi.org/10.1007/978-3-030-68211-8_18</a>","ista":"Aichholzer O, Arroyo Guevara AM, Masárová Z, Parada I, Perz D, Pilz A, Tkadlec J, Vogtenhuber B. 2021. On compatible matchings. 15th International Conference on Algorithms and Computation. WALCOM: Algorithms and Computation, LNCS, vol. 12635, 221–233.","chicago":"Aichholzer, Oswin, Alan M Arroyo Guevara, Zuzana Masárová, Irene Parada, Daniel Perz, Alexander Pilz, Josef Tkadlec, and Birgit Vogtenhuber. “On Compatible Matchings.” In <i>15th International Conference on Algorithms and Computation</i>, 12635:221–33. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/978-3-030-68211-8_18\">https://doi.org/10.1007/978-3-030-68211-8_18</a>.","mla":"Aichholzer, Oswin, et al. “On Compatible Matchings.” <i>15th International Conference on Algorithms and Computation</i>, vol. 12635, Springer Nature, 2021, pp. 221–33, doi:<a href=\"https://doi.org/10.1007/978-3-030-68211-8_18\">10.1007/978-3-030-68211-8_18</a>."},"related_material":{"record":[{"relation":"later_version","status":"public","id":"11938"}]},"title":"On compatible matchings","oa_version":"Preprint","acknowledgement":"A.A. funded by the Marie Skłodowska-Curie grant agreement No. 754411. Z.M. partially funded by Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31. I.P., D.P., and B.V. partially supported by FWF within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35. A.P. supported by a Schrödinger fellowship of the FWF: J-3847-N35. J.T. partially supported by ERC Start grant no. (279307: Graph Games), FWF grant no. P23499-N23 and S11407-N23 (RiSE).","scopus_import":"1","isi":1,"article_processing_charge":"No","_id":"9296","oa":1,"language":[{"iso":"eng"}],"author":[{"last_name":"Aichholzer","full_name":"Aichholzer, Oswin","first_name":"Oswin"},{"first_name":"Alan M","full_name":"Arroyo Guevara, Alan M","orcid":"0000-0003-2401-8670","last_name":"Arroyo Guevara","id":"3207FDC6-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Masárová, Zuzana","orcid":"0000-0002-6660-1322","last_name":"Masárová","id":"45CFE238-F248-11E8-B48F-1D18A9856A87","first_name":"Zuzana"},{"first_name":"Irene","last_name":"Parada","full_name":"Parada, Irene"},{"first_name":"Daniel","last_name":"Perz","full_name":"Perz, Daniel"},{"full_name":"Pilz, Alexander","last_name":"Pilz","first_name":"Alexander"},{"id":"3F24CCC8-F248-11E8-B48F-1D18A9856A87","last_name":"Tkadlec","orcid":"0000-0002-1097-9684","full_name":"Tkadlec, Josef","first_name":"Josef"},{"first_name":"Birgit","last_name":"Vogtenhuber","full_name":"Vogtenhuber, Birgit"}],"day":"16","type":"conference","status":"public","page":"221-233","quality_controlled":"1","month":"02","publisher":"Springer Nature","publication_identifier":{"isbn":["9783030682101"],"eisbn":["9783030682118"],"issn":["0302-9743"],"eissn":["1611-3349"]}},{"title":"The multi-cover persistence of Euclidean balls","related_material":{"record":[{"id":"187","relation":"earlier_version","status":"public"}]},"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).","oa_version":"Published Version","scopus_import":"1","isi":1,"pmid":1,"citation":{"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>.","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>.","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>","ista":"Edelsbrunner H, Osang GF. 2021. The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. 65, 1296–1313.","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>","short":"H. Edelsbrunner, G.F. Osang, Discrete and Computational Geometry 65 (2021) 1296–1313.","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."},"has_accepted_license":"1","type":"journal_article","day":"31","page":"1296–1313","status":"public","quality_controlled":"1","month":"03","publisher":"Springer Nature","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"article_processing_charge":"Yes (via OA deal)","article_type":"original","_id":"9317","corr_author":"1","oa":1,"language":[{"iso":"eng"}],"author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","full_name":"Osang, Georg F","orcid":"0000-0002-8882-5116","last_name":"Osang"}],"publication_status":"published","external_id":{"pmid":["34720303"],"isi":["000635460400001"]},"department":[{"_id":"HeEd"}],"ec_funded":1,"abstract":[{"lang":"eng","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."}],"doi":"10.1007/s00454-021-00281-9","file_date_updated":"2021-12-01T10:56:53Z","file":[{"relation":"main_file","checksum":"59b4e1e827e494209bcb4aae22e1d347","file_id":"10394","success":1,"content_type":"application/pdf","date_created":"2021-12-01T10:56:53Z","creator":"cchlebak","file_name":"2021_DisCompGeo_Edelsbrunner_Osang.pdf","file_size":677704,"date_updated":"2021-12-01T10:56:53Z","access_level":"open_access"}],"intvolume":"        65","publication":"Discrete and Computational Geometry","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2021","date_created":"2021-04-11T22:01:15Z","ddc":["516"],"project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"}],"date_published":"2021-03-31T00:00:00Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"volume":65,"date_updated":"2025-06-12T06:36:54Z"},{"citation":{"ista":"Edelsbrunner H, Nikitenko A, Osang GF. 2021. A step in the Delaunay mosaic of order k. Journal of Geometry. 112(1), 15.","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>","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>","short":"H. Edelsbrunner, A. Nikitenko, G.F. Osang, Journal of Geometry 112 (2021).","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.","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>.","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>."},"has_accepted_license":"1","title":"A step in the Delaunay mosaic of order k","scopus_import":"1","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","article_type":"original","_id":"9465","issue":"1","corr_author":"1","language":[{"iso":"eng"}],"oa":1,"author":[{"first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","last_name":"Nikitenko","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201"},{"first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","orcid":"0000-0002-8882-5116","full_name":"Osang, Georg F"}],"type":"journal_article","day":"01","status":"public","month":"04","quality_controlled":"1","publication_identifier":{"eissn":["1420-8997"],"issn":["0047-2468"]},"article_number":"15","publisher":"Springer Nature","file_date_updated":"2021-06-11T13:16:26Z","publication":"Journal of Geometry","intvolume":"       112","file":[{"creator":"kschuh","file_size":694706,"date_updated":"2021-06-11T13:16:26Z","file_name":"2021_Geometry_Edelsbrunner.pdf","access_level":"open_access","relation":"main_file","file_id":"9544","checksum":"e52a832f1def52a2b23d21bcc09e646f","success":1,"content_type":"application/pdf","date_created":"2021-06-11T13:16:26Z"}],"publication_status":"published","department":[{"_id":"HeEd"}],"abstract":[{"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.","lang":"eng"}],"doi":"10.1007/s00022-021-00577-4","ddc":["510"],"date_published":"2021-04-01T00:00:00Z","volume":112,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"date_updated":"2025-07-10T12:01:46Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2021","date_created":"2021-06-06T22:01:29Z"},{"title":"Erdős-Hajnal-type results for monotone paths","acknowledgement":"We would like to thank the anonymous referees for their useful comments and suggestions. János Pach is partially supported by Austrian Science Fund (FWF) grant Z 342-N31 and by ERC Advanced grant “GeoScape.” István Tomon is partially supported by Swiss National Science Foundation grant no. 200021_196965, and thanks the support of MIPT Moscow. Both authors are partially supported by The Russian Government in the framework of MegaGrant no. 075-15-2019-1926.","scopus_import":"1","oa_version":"Published Version","isi":1,"citation":{"ista":"Pach J, Tomon I. 2021. Erdős-Hajnal-type results for monotone paths. Journal of Combinatorial Theory. Series B. 151, 21–37.","apa":"Pach, J., &#38; Tomon, I. (2021). Erdős-Hajnal-type results for monotone paths. <i>Journal of Combinatorial Theory. Series B</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jctb.2021.05.004\">https://doi.org/10.1016/j.jctb.2021.05.004</a>","ama":"Pach J, Tomon I. Erdős-Hajnal-type results for monotone paths. <i>Journal of Combinatorial Theory Series B</i>. 2021;151:21-37. doi:<a href=\"https://doi.org/10.1016/j.jctb.2021.05.004\">10.1016/j.jctb.2021.05.004</a>","short":"J. Pach, I. Tomon, Journal of Combinatorial Theory. Series B 151 (2021) 21–37.","ieee":"J. Pach and I. Tomon, “Erdős-Hajnal-type results for monotone paths,” <i>Journal of Combinatorial Theory. Series B</i>, vol. 151. Elsevier, pp. 21–37, 2021.","chicago":"Pach, János, and István Tomon. “Erdős-Hajnal-Type Results for Monotone Paths.” <i>Journal of Combinatorial Theory. Series B</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.jctb.2021.05.004\">https://doi.org/10.1016/j.jctb.2021.05.004</a>.","mla":"Pach, János, and István Tomon. “Erdős-Hajnal-Type Results for Monotone Paths.” <i>Journal of Combinatorial Theory. Series B</i>, vol. 151, Elsevier, 2021, pp. 21–37, doi:<a href=\"https://doi.org/10.1016/j.jctb.2021.05.004\">10.1016/j.jctb.2021.05.004</a>."},"has_accepted_license":"1","day":"09","type":"journal_article","status":"public","page":"21-37","quality_controlled":"1","month":"06","publisher":"Elsevier","publication_identifier":{"issn":["0095-8956"]},"article_type":"original","article_processing_charge":"No","_id":"9602","corr_author":"1","language":[{"iso":"eng"}],"oa":1,"author":[{"first_name":"János","full_name":"Pach, János","last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4"},{"full_name":"Tomon, István","last_name":"Tomon","first_name":"István"}],"publication_status":"published","department":[{"_id":"HeEd"}],"external_id":{"isi":["000702280800002"]},"abstract":[{"lang":"eng","text":"An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck > 0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least n^ck . This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs.\r\nA key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k) > 0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k ≥ 3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees."}],"doi":"10.1016/j.jctb.2021.05.004","file_date_updated":"2021-06-28T13:33:23Z","file":[{"creator":"asandaue","file_name":"2021_JournalOfCombinatorialTheory_Pach.pdf","file_size":418168,"date_updated":"2021-06-28T13:33:23Z","access_level":"open_access","relation":"main_file","checksum":"15fbc9064cd9d1c777ac0043b78c8f12","file_id":"9612","content_type":"application/pdf","success":1,"date_created":"2021-06-28T13:33:23Z"}],"intvolume":"       151","publication":"Journal of Combinatorial Theory. Series B","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","year":"2021","date_created":"2021-06-27T22:01:47Z","date_published":"2021-06-09T00:00:00Z","ddc":["510"],"project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science"}],"date_updated":"2025-04-15T07:16:52Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"volume":151}]