[{"issue":"4","publication":"European Journal of Mathematics","page":"1309 - 1312","file_date_updated":"2020-07-14T12:48:03Z","intvolume":"         8","status":"public","day":"01","type":"journal_article","date_created":"2020-05-03T22:00:48Z","file":[{"checksum":"f53e71fd03744075adcd0b8fc1b8423d","date_created":"2020-05-04T10:33:42Z","file_size":263926,"file_name":"2020_EuropMathematics_Akopyan.pdf","access_level":"open_access","date_updated":"2020-07-14T12:48:03Z","file_id":"7796","creator":"dernst","relation":"main_file","content_type":"application/pdf"}],"has_accepted_license":"1","department":[{"_id":"HeEd"}],"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"12","date_published":"2022-12-01T00:00:00Z","article_type":"original","_id":"7791","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"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.","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"quality_controlled":"1","arxiv":1,"volume":8,"date_updated":"2024-02-22T15:58:42Z","oa":1,"article_processing_charge":"Yes (via OA deal)","abstract":[{"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.","lang":"eng"}],"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy","last_name":"Akopyan","orcid":"0000-0002-2548-617X","first_name":"Arseniy"},{"first_name":"Roman","full_name":"Karasev, Roman","last_name":"Karasev"}],"publication_status":"published","citation":{"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>.","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>","ista":"Akopyan A, Karasev R. 2022. When different norms lead to same billiard trajectories? European Journal of Mathematics. 8(4), 1309–1312.","short":"A. Akopyan, R. Karasev, European Journal of Mathematics 8 (2022) 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.","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>","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>."},"ddc":["510"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"external_id":{"arxiv":["1912.12685"]},"title":"When different norms lead to same billiard trajectories?","year":"2022","doi":"10.1007/s40879-020-00405-0","ec_funded":1},{"status":"public","type":"journal_article","day":"30","file_date_updated":"2022-01-10T08:33:22Z","publication":"European Journal of Mathematics","language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","date_published":"2021-12-30T00:00:00Z","article_type":"original","month":"12","file":[{"success":1,"content_type":"application/pdf","relation":"main_file","creator":"cchlebak","file_id":"10610","file_name":"2021_EuJournalMath_Weighill.pdf","file_size":384908,"date_created":"2022-01-10T08:33:22Z","checksum":"c435dcfa1ad3aadc5cdd7366bc7f4e98","date_updated":"2022-01-10T08:33:22Z","access_level":"open_access"}],"date_created":"2022-01-09T23:01:27Z","department":[{"_id":"HeEd"}],"has_accepted_license":"1","author":[{"first_name":"Thomas","full_name":"Weighill, Thomas","last_name":"Weighill"},{"first_name":"Takamitsu","full_name":"Yamauchi, Takamitsu","last_name":"Yamauchi"},{"id":"c8b3499c-7a77-11eb-b046-aa368cbbf2ad","last_name":"Zava","full_name":"Zava, Nicolò","first_name":"Nicolò"}],"abstract":[{"text":"We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.","lang":"eng"}],"publication_status":"published","citation":{"ama":"Weighill T, Yamauchi T, Zava N. Coarse infinite-dimensionality of hyperspaces of finite subsets. <i>European Journal of Mathematics</i>. 2021. doi:<a href=\"https://doi.org/10.1007/s40879-021-00515-3\">10.1007/s40879-021-00515-3</a>","mla":"Weighill, Thomas, et al. “Coarse Infinite-Dimensionality of Hyperspaces of Finite Subsets.” <i>European Journal of Mathematics</i>, Springer Nature, 2021, doi:<a href=\"https://doi.org/10.1007/s40879-021-00515-3\">10.1007/s40879-021-00515-3</a>.","short":"T. Weighill, T. Yamauchi, N. Zava, European Journal of Mathematics (2021).","ista":"Weighill T, Yamauchi T, Zava N. 2021. Coarse infinite-dimensionality of hyperspaces of finite subsets. European Journal of Mathematics.","apa":"Weighill, T., Yamauchi, T., &#38; Zava, N. (2021). Coarse infinite-dimensionality of hyperspaces of finite subsets. <i>European Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40879-021-00515-3\">https://doi.org/10.1007/s40879-021-00515-3</a>","ieee":"T. Weighill, T. Yamauchi, and N. Zava, “Coarse infinite-dimensionality of hyperspaces of finite subsets,” <i>European Journal of Mathematics</i>. Springer Nature, 2021.","chicago":"Weighill, Thomas, Takamitsu Yamauchi, and Nicolò Zava. “Coarse Infinite-Dimensionality of Hyperspaces of Finite Subsets.” <i>European Journal of Mathematics</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s40879-021-00515-3\">https://doi.org/10.1007/s40879-021-00515-3</a>."},"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","acknowledgement":"We would like to thank the referees for their careful reading and the comments that improved our work. The third named author would like to thank the Division of Mathematics, Physics and Earth Sciences of the Graduate School of Science and Engineering of Ehime University and the second named author for hosting his visit in June 2018. Open access funding provided by Institute of Science and Technology (IST Austria).","oa_version":"Published Version","quality_controlled":"1","_id":"10608","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"oa":1,"date_updated":"2022-01-10T08:36:55Z","article_processing_charge":"Yes (via OA deal)","title":"Coarse infinite-dimensionality of hyperspaces of finite subsets","doi":"10.1007/s40879-021-00515-3","year":"2021","ddc":["500"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"}},{"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"09","article_type":"original","date_published":"2020-09-09T00:00:00Z","date_created":"2020-09-20T22:01:38Z","department":[{"_id":"HeEd"}],"status":"public","day":"09","type":"journal_article","publication":"European Journal of Mathematics","title":"Billiards in ellipses revisited","external_id":{"arxiv":["2001.02934"]},"doi":"10.1007/s40879-020-00426-9","year":"2020","ec_funded":1,"main_file_link":[{"url":"https://arxiv.org/abs/2001.02934","open_access":"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."}],"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan"},{"first_name":"Richard","full_name":"Schwartz, Richard","last_name":"Schwartz"},{"first_name":"Serge","last_name":"Tabachnikov","full_name":"Tabachnikov, Serge"}],"publication_status":"published","citation":{"ieee":"A. Akopyan, R. Schwartz, and S. Tabachnikov, “Billiards in ellipses revisited,” <i>European Journal of Mathematics</i>. Springer Nature, 2020.","apa":"Akopyan, A., Schwartz, R., &#38; Tabachnikov, S. (2020). 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>","chicago":"Akopyan, Arseniy, Richard Schwartz, and Serge Tabachnikov. “Billiards in Ellipses Revisited.” <i>European Journal of Mathematics</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s40879-020-00426-9\">https://doi.org/10.1007/s40879-020-00426-9</a>.","ama":"Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. <i>European Journal of Mathematics</i>. 2020. doi:<a href=\"https://doi.org/10.1007/s40879-020-00426-9\">10.1007/s40879-020-00426-9</a>","mla":"Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” <i>European Journal of Mathematics</i>, Springer Nature, 2020, doi:<a href=\"https://doi.org/10.1007/s40879-020-00426-9\">10.1007/s40879-020-00426-9</a>.","ista":"Akopyan A, Schwartz R, Tabachnikov S. 2020. Billiards in ellipses revisited. European Journal of Mathematics.","short":"A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics (2020)."},"_id":"8538","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","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.","oa_version":"Preprint","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"}],"quality_controlled":"1","arxiv":1,"oa":1,"date_updated":"2021-12-02T15:10:17Z","article_processing_charge":"No"},{"page":"909–928","publication":"European Journal of Mathematics","issue":"3","status":"public","intvolume":"         5","type":"journal_article","day":"15","date_created":"2018-12-11T11:46:29Z","department":[{"_id":"TaHa"}],"language":[{"iso":"eng"}],"scopus_import":1,"publisher":"Springer Nature","date_published":"2019-09-15T00:00:00Z","article_type":"original","month":"09","project":[{"grant_number":"291734","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","oa_version":"Preprint","user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","publication_identifier":{"issn":["2199-675X"],"eissn":["2199-6768"]},"_id":"441","article_processing_charge":"No","oa":1,"date_updated":"2021-01-12T07:56:46Z","volume":5,"publist_id":"7382","arxiv":1,"author":[{"first_name":"Nikita","full_name":"Kalinin, Nikita","last_name":"Kalinin"},{"id":"35084A62-F248-11E8-B48F-1D18A9856A87","first_name":"Mikhail","full_name":"Shkolnikov, Mikhail","last_name":"Shkolnikov","orcid":"0000-0002-4310-178X"}],"citation":{"ieee":"N. Kalinin and M. Shkolnikov, “Tropical formulae for summation over a part of SL(2,Z),” <i>European Journal of Mathematics</i>, vol. 5, no. 3. Springer Nature, pp. 909–928, 2019.","apa":"Kalinin, N., &#38; Shkolnikov, M. (2019). Tropical formulae for summation over a part of SL(2,Z). <i>European Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40879-018-0218-0\">https://doi.org/10.1007/s40879-018-0218-0</a>","chicago":"Kalinin, Nikita, and Mikhail Shkolnikov. “Tropical Formulae for Summation over a Part of SL(2,Z).” <i>European Journal of Mathematics</i>. Springer Nature, 2019. <a href=\"https://doi.org/10.1007/s40879-018-0218-0\">https://doi.org/10.1007/s40879-018-0218-0</a>.","mla":"Kalinin, Nikita, and Mikhail Shkolnikov. “Tropical Formulae for Summation over a Part of SL(2,Z).” <i>European Journal of Mathematics</i>, vol. 5, no. 3, Springer Nature, 2019, pp. 909–928, doi:<a href=\"https://doi.org/10.1007/s40879-018-0218-0\">10.1007/s40879-018-0218-0</a>.","ama":"Kalinin N, Shkolnikov M. Tropical formulae for summation over a part of SL(2,Z). <i>European Journal of Mathematics</i>. 2019;5(3):909–928. doi:<a href=\"https://doi.org/10.1007/s40879-018-0218-0\">10.1007/s40879-018-0218-0</a>","ista":"Kalinin N, Shkolnikov M. 2019. Tropical formulae for summation over a part of SL(2,Z). European Journal of Mathematics. 5(3), 909–928.","short":"N. Kalinin, M. Shkolnikov, European Journal of Mathematics 5 (2019) 909–928."},"publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1711.02089"}],"title":"Tropical formulae for summation over a part of SL(2,Z)","external_id":{"arxiv":["1711.02089"]},"ec_funded":1,"doi":"10.1007/s40879-018-0218-0","year":"2019"}]