[{"date_created":"2023-02-12T23:00:59Z","date_updated":"2023-08-16T12:22:07Z","volume":63,"author":[{"full_name":"Koehl, Patrice","last_name":"Koehl","first_name":"Patrice"},{"full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X"},{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner"}],"publication_status":"published","department":[{"_id":"HeEd"}],"publisher":"American Chemical Society","acknowledgement":"P.K. acknowledges support from the University of California Multicampus Research Programs and Initiatives (Grant No. M21PR3267) and from the NSF (Grant No.1760485). H.E. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.\r\nOpen Access is funded by the Austrian Science Fund (FWF).","year":"2023","pmid":1,"license":"https://creativecommons.org/licenses/by/4.0/","file_date_updated":"2023-08-16T12:21:13Z","ec_funded":1,"language":[{"iso":"eng"}],"doi":"10.1021/acs.jcim.2c01346","isi":1,"quality_controlled":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"The Wittgenstein Prize","call_identifier":"FWF"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"external_id":{"pmid":["36638318"],"isi":["000920370700001"]},"month":"02","publication_identifier":{"issn":["1549-9596"],"eissn":["1549-960X"]},"file":[{"file_id":"14070","relation":"main_file","success":1,"checksum":"7d20562269edff1e31b9d6019d4983b0","date_updated":"2023-08-16T12:21:13Z","date_created":"2023-08-16T12:21:13Z","access_level":"open_access","file_name":"2023_JCIM_Koehl.pdf","creator":"dernst","file_size":8069223,"content_type":"application/pdf"}],"oa_version":"Published Version","title":"Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives","ddc":["510","540"],"status":"public","intvolume":" 63","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"12544","abstract":[{"text":"Geometry is crucial in our efforts to comprehend the structures and dynamics of biomolecules. For example, volume, surface area, and integrated mean and Gaussian curvature of the union of balls representing a molecule are used to quantify its interactions with the water surrounding it in the morphometric implicit solvent models. The Alpha Shape theory provides an accurate and reliable method for computing these geometric measures. In this paper, we derive homogeneous formulas for the expressions of these measures and their derivatives with respect to the atomic coordinates, and we provide algorithms that implement them into a new software package, AlphaMol. The only variables in these formulas are the interatomic distances, making them insensitive to translations and rotations. AlphaMol includes a sequential algorithm and a parallel algorithm. In the parallel version, we partition the atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented by a buffer zone to account for atoms whose ball representations may partially cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up compared to an independent serial implementation when using 32 processors. For instance, it takes 31 s to compute the geometric measures and derivatives of each atom in a viral capsid with more than 26 million atoms on 32 Intel processors running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant computations, which ultimately limit the impact of using multiple processors. AlphaMol is available as an OpenSource software.","lang":"eng"}],"issue":"3","type":"journal_article","date_published":"2023-02-13T00:00:00Z","article_type":"original","page":"973-985","publication":"Journal of Chemical Information and Modeling","citation":{"short":"P. Koehl, A. Akopyan, H. Edelsbrunner, Journal of Chemical Information and Modeling 63 (2023) 973–985.","mla":"Koehl, Patrice, et al. “Computing the Volume, Surface Area, Mean, and Gaussian Curvatures of Molecules and Their Derivatives.” Journal of Chemical Information and Modeling, vol. 63, no. 3, American Chemical Society, 2023, pp. 973–85, doi:10.1021/acs.jcim.2c01346.","chicago":"Koehl, Patrice, Arseniy Akopyan, and Herbert Edelsbrunner. “Computing the Volume, Surface Area, Mean, and Gaussian Curvatures of Molecules and Their Derivatives.” Journal of Chemical Information and Modeling. American Chemical Society, 2023. https://doi.org/10.1021/acs.jcim.2c01346.","ama":"Koehl P, Akopyan A, Edelsbrunner H. Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. Journal of Chemical Information and Modeling. 2023;63(3):973-985. doi:10.1021/acs.jcim.2c01346","apa":"Koehl, P., Akopyan, A., & Edelsbrunner, H. (2023). Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. Journal of Chemical Information and Modeling. American Chemical Society. https://doi.org/10.1021/acs.jcim.2c01346","ieee":"P. Koehl, A. Akopyan, and H. Edelsbrunner, “Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives,” Journal of Chemical Information and Modeling, vol. 63, no. 3. American Chemical Society, pp. 973–985, 2023.","ista":"Koehl P, Akopyan A, Edelsbrunner H. 2023. Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives. Journal of Chemical Information and Modeling. 63(3), 973–985."},"day":"13","has_accepted_license":"1","article_processing_charge":"No","scopus_import":"1"},{"title":"When different norms lead to same billiard trajectories?","status":"public","ddc":["510"],"intvolume":" 8","_id":"7791","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","file":[{"creator":"dernst","file_size":263926,"content_type":"application/pdf","access_level":"open_access","file_name":"2020_EuropMathematics_Akopyan.pdf","checksum":"f53e71fd03744075adcd0b8fc1b8423d","date_created":"2020-05-04T10:33:42Z","date_updated":"2020-07-14T12:48:03Z","file_id":"7796","relation":"main_file"}],"oa_version":"Published Version","type":"journal_article","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."}],"issue":"4","article_type":"original","page":"1309 - 1312","publication":"European Journal of Mathematics","citation":{"ieee":"A. Akopyan and R. Karasev, “When different norms lead to same billiard trajectories?,” European Journal of Mathematics, vol. 8, no. 4. Springer Nature, pp. 1309–1312, 2022.","apa":"Akopyan, A., & Karasev, R. (2022). When different norms lead to same billiard trajectories? European Journal of Mathematics. Springer Nature. https://doi.org/10.1007/s40879-020-00405-0","ista":"Akopyan A, Karasev R. 2022. When different norms lead to same billiard trajectories? European Journal of Mathematics. 8(4), 1309–1312.","ama":"Akopyan A, Karasev R. When different norms lead to same billiard trajectories? European Journal of Mathematics. 2022;8(4):1309-1312. doi:10.1007/s40879-020-00405-0","chicago":"Akopyan, Arseniy, and Roman Karasev. “When Different Norms Lead to Same Billiard Trajectories?” European Journal of Mathematics. Springer Nature, 2022. https://doi.org/10.1007/s40879-020-00405-0.","short":"A. Akopyan, R. Karasev, European Journal of Mathematics 8 (2022) 1309–1312.","mla":"Akopyan, Arseniy, and Roman Karasev. “When Different Norms Lead to Same Billiard Trajectories?” European Journal of Mathematics, vol. 8, no. 4, Springer Nature, 2022, pp. 1309–12, doi:10.1007/s40879-020-00405-0."},"date_published":"2022-12-01T00:00:00Z","scopus_import":"1","day":"01","has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","publication_status":"published","publisher":"Springer Nature","department":[{"_id":"HeEd"}],"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.","year":"2022","date_created":"2020-05-03T22:00:48Z","date_updated":"2024-02-22T15:58:42Z","volume":8,"author":[{"last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy"},{"first_name":"Roman","last_name":"Karasev","full_name":"Karasev, Roman"}],"file_date_updated":"2020-07-14T12:48:03Z","ec_funded":1,"quality_controlled":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"external_id":{"arxiv":["1912.12685"]},"language":[{"iso":"eng"}],"doi":"10.1007/s40879-020-00405-0","month":"12","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]}},{"file":[{"creator":"dernst","file_size":1966019,"content_type":"application/pdf","file_name":"2023_ExperimentalMath_Akopyan.pdf","access_level":"open_access","date_updated":"2023-08-14T11:55:10Z","date_created":"2023-08-14T11:55:10Z","success":1,"checksum":"3514382e3a1eb87fa6c61ad622874415","file_id":"14053","relation":"main_file"}],"oa_version":"Published Version","_id":"10222","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"title":"The beauty of random polytopes inscribed in the 2-sphere","status":"public","abstract":[{"text":"Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.","lang":"eng"}],"type":"journal_article","date_published":"2021-10-25T00:00:00Z","publication":"Experimental Mathematics","citation":{"short":"A. Akopyan, H. Edelsbrunner, A. Nikitenko, Experimental Mathematics (2021) 1–15.","mla":"Akopyan, Arseniy, et al. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” Experimental Mathematics, Taylor and Francis, 2021, pp. 1–15, doi:10.1080/10586458.2021.1980459.","chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, and Anton Nikitenko. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” Experimental Mathematics. Taylor and Francis, 2021. https://doi.org/10.1080/10586458.2021.1980459.","ama":"Akopyan A, Edelsbrunner H, Nikitenko A. The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics. 2021:1-15. doi:10.1080/10586458.2021.1980459","ieee":"A. Akopyan, H. Edelsbrunner, and A. Nikitenko, “The beauty of random polytopes inscribed in the 2-sphere,” Experimental Mathematics. Taylor and Francis, pp. 1–15, 2021.","apa":"Akopyan, A., Edelsbrunner, H., & Nikitenko, A. (2021). The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics. Taylor and Francis. https://doi.org/10.1080/10586458.2021.1980459","ista":"Akopyan A, Edelsbrunner H, Nikitenko A. 2021. The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics., 1–15."},"article_type":"original","page":"1-15","day":"25","has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","scopus_import":"1","author":[{"full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","first_name":"Arseniy"},{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0659-3201","first_name":"Anton","last_name":"Nikitenko","full_name":"Nikitenko, Anton"}],"date_created":"2021-11-07T23:01:25Z","date_updated":"2023-08-14T11:57:07Z","year":"2021","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.\r\nWe are grateful to Dmitry Zaporozhets and Christoph Thäle for valuable comments and for directing us to relevant references. We also thank to Anton Mellit for a useful discussion on Bessel functions.","publication_status":"published","publisher":"Taylor and Francis","department":[{"_id":"HeEd"}],"file_date_updated":"2023-08-14T11:55:10Z","ec_funded":1,"doi":"10.1080/10586458.2021.1980459","language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"isi":["000710893500001"],"arxiv":["2007.07783"]},"oa":1,"quality_controlled":"1","isi":1,"project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"call_identifier":"FWF","name":"The Wittgenstein Prize","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"month":"10","publication_identifier":{"issn":["1058-6458"],"eissn":["1944-950X"]}},{"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."}],"type":"journal_article","oa_version":"Preprint","intvolume":" 66","status":"public","title":"On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs","_id":"8338","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","day":"01","scopus_import":"1","date_published":"2021-10-01T00:00:00Z","page":"938-976","article_type":"original","citation":{"apa":"Akopyan, A., Bobenko, A. I., Schief, W. K., & Techter, J. (2021). On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00240-w","ieee":"A. Akopyan, A. I. Bobenko, W. K. Schief, and J. Techter, “On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs,” Discrete and Computational Geometry, 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.","ama":"Akopyan A, Bobenko AI, Schief WK, Techter J. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. 2021;66:938-976. doi:10.1007/s00454-020-00240-w","chicago":"Akopyan, Arseniy, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” Discrete and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00240-w.","short":"A. Akopyan, A.I. Bobenko, W.K. Schief, J. Techter, Discrete and Computational Geometry 66 (2021) 938–976.","mla":"Akopyan, Arseniy, et al. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” Discrete and Computational Geometry, vol. 66, Springer Nature, 2021, pp. 938–76, doi:10.1007/s00454-020-00240-w."},"publication":"Discrete and Computational Geometry","ec_funded":1,"volume":66,"date_created":"2020-09-06T22:01:13Z","date_updated":"2024-03-07T14:51:11Z","author":[{"last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy"},{"last_name":"Bobenko","first_name":"Alexander I.","full_name":"Bobenko, Alexander I."},{"first_name":"Wolfgang K.","last_name":"Schief","full_name":"Schief, Wolfgang K."},{"full_name":"Techter, Jan","last_name":"Techter","first_name":"Jan"}],"publisher":"Springer Nature","department":[{"_id":"HeEd"}],"publication_status":"published","year":"2021","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).","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"month":"10","language":[{"iso":"eng"}],"doi":"10.1007/s00454-020-00240-w","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"}],"isi":1,"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1908.00856"}],"oa":1,"external_id":{"arxiv":["1908.00856"],"isi":["000564488500002"]}},{"month":"09","publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"language":[{"iso":"eng"}],"doi":"10.1007/s40879-020-00426-9","quality_controlled":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"}],"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/2001.02934","open_access":"1"}],"external_id":{"arxiv":["2001.02934"]},"ec_funded":1,"date_updated":"2021-12-02T15:10:17Z","date_created":"2020-09-20T22:01:38Z","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","last_name":"Akopyan","full_name":"Akopyan, Arseniy"},{"first_name":"Richard","last_name":"Schwartz","full_name":"Schwartz, Richard"},{"last_name":"Tabachnikov","first_name":"Serge","full_name":"Tabachnikov, Serge"}],"publication_status":"published","department":[{"_id":"HeEd"}],"publisher":"Springer Nature","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.","year":"2020","day":"09","article_processing_charge":"No","scopus_import":"1","date_published":"2020-09-09T00:00:00Z","article_type":"original","publication":"European Journal of Mathematics","citation":{"short":"A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics (2020).","mla":"Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” European Journal of Mathematics, Springer Nature, 2020, doi:10.1007/s40879-020-00426-9.","chicago":"Akopyan, Arseniy, Richard Schwartz, and Serge Tabachnikov. “Billiards in Ellipses Revisited.” European Journal of Mathematics. Springer Nature, 2020. https://doi.org/10.1007/s40879-020-00426-9.","ama":"Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. European Journal of Mathematics. 2020. doi:10.1007/s40879-020-00426-9","ieee":"A. Akopyan, R. Schwartz, and S. Tabachnikov, “Billiards in ellipses revisited,” European Journal of Mathematics. Springer Nature, 2020.","apa":"Akopyan, A., Schwartz, R., & Tabachnikov, S. (2020). Billiards in ellipses revisited. European Journal of Mathematics. Springer Nature. https://doi.org/10.1007/s40879-020-00426-9","ista":"Akopyan A, Schwartz R, Tabachnikov S. 2020. Billiards in ellipses revisited. European Journal of Mathematics."},"abstract":[{"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.","lang":"eng"}],"type":"journal_article","oa_version":"Preprint","title":"Billiards in ellipses revisited","status":"public","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","_id":"8538"},{"intvolume":" 2256","title":"Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures","status":"public","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"74","oa_version":"Preprint","type":"book_chapter","abstract":[{"text":"We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class\r\nof compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2.\r\nWe use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian\r\nmeasures.","lang":"eng"}],"page":"1-27","citation":{"chicago":"Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian Measures and Radial Non-Gaussian Measures.” In Geometric Aspects of Functional Analysis, edited by Bo’az Klartag and Emanuel Milman, 2256:1–27. LNM. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-36020-7_1.","mla":"Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian Measures and Radial Non-Gaussian Measures.” Geometric Aspects of Functional Analysis, edited by Bo’az Klartag and Emanuel Milman, vol. 2256, Springer Nature, 2020, pp. 1–27, doi:10.1007/978-3-030-36020-7_1.","short":"A. Akopyan, R. Karasev, in:, B. Klartag, E. Milman (Eds.), Geometric Aspects of Functional Analysis, Springer Nature, 2020, pp. 1–27.","ista":"Akopyan A, Karasev R. 2020.Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In: Geometric Aspects of Functional Analysis. vol. 2256, 1–27.","ieee":"A. Akopyan and R. Karasev, “Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures,” in Geometric Aspects of Functional Analysis, vol. 2256, B. Klartag and E. Milman, Eds. Springer Nature, 2020, pp. 1–27.","apa":"Akopyan, A., & Karasev, R. (2020). Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In B. Klartag & E. Milman (Eds.), Geometric Aspects of Functional Analysis (Vol. 2256, pp. 1–27). Springer Nature. https://doi.org/10.1007/978-3-030-36020-7_1","ama":"Akopyan A, Karasev R. Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In: Klartag B, Milman E, eds. Geometric Aspects of Functional Analysis. Vol 2256. LNM. Springer Nature; 2020:1-27. doi:10.1007/978-3-030-36020-7_1"},"publication":"Geometric Aspects of Functional Analysis","date_published":"2020-06-21T00:00:00Z","series_title":"LNM","scopus_import":"1","article_processing_charge":"No","day":"21","department":[{"_id":"HeEd"},{"_id":"JaMa"}],"publisher":"Springer Nature","editor":[{"full_name":"Klartag, Bo'az","first_name":"Bo'az","last_name":"Klartag"},{"full_name":"Milman, Emanuel","first_name":"Emanuel","last_name":"Milman"}],"publication_status":"published","year":"2020","volume":2256,"date_created":"2018-12-11T11:44:29Z","date_updated":"2023-08-17T13:48:31Z","author":[{"full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","last_name":"Akopyan"},{"full_name":"Karasev, Roman","last_name":"Karasev","first_name":"Roman"}],"ec_funded":1,"project":[{"call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117"}],"quality_controlled":"1","isi":1,"oa":1,"external_id":{"isi":["000557689300003"],"arxiv":["1808.07350"]},"main_file_link":[{"url":"https://arxiv.org/abs/1808.07350","open_access":"1"}],"language":[{"iso":"eng"}],"doi":"10.1007/978-3-030-36020-7_1","publication_identifier":{"isbn":["9783030360191"],"eissn":["16179692"],"eisbn":["9783030360207"],"issn":["00758434"]},"month":"06"},{"year":"2020","acknowledgement":" Supported by the Russian Foundation for Basic Research grant 18-01-00036.","publication_status":"published","department":[{"_id":"HeEd"}],"publisher":"Oxford University Press","author":[{"last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy"},{"first_name":"Roman","last_name":"Karasev","full_name":"Karasev, Roman"}],"date_updated":"2023-08-24T14:19:55Z","date_created":"2022-03-18T11:39:30Z","volume":2020,"month":"02","publication_identifier":{"issn":["1073-7928"],"eissn":["1687-0247"]},"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1702.07513","open_access":"1"}],"external_id":{"arxiv":["1702.07513"],"isi":["000522852700002"]},"isi":1,"quality_controlled":"1","doi":"10.1093/imrn/rny037","language":[{"iso":"eng"}],"type":"journal_article","abstract":[{"text":"In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces.","lang":"eng"}],"issue":"3","_id":"10867","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","title":"Waist of balls in hyperbolic and spherical spaces","status":"public","intvolume":" 2020","oa_version":"Preprint","scopus_import":"1","keyword":["General Mathematics"],"day":"01","article_processing_charge":"No","publication":"International Mathematics Research Notices","citation":{"short":"A. Akopyan, R. Karasev, International Mathematics Research Notices 2020 (2020) 669–697.","mla":"Akopyan, Arseniy, and Roman Karasev. “Waist of Balls in Hyperbolic and Spherical Spaces.” International Mathematics Research Notices, vol. 2020, no. 3, Oxford University Press, 2020, pp. 669–97, doi:10.1093/imrn/rny037.","chicago":"Akopyan, Arseniy, and Roman Karasev. “Waist of Balls in Hyperbolic and Spherical Spaces.” International Mathematics Research Notices. Oxford University Press, 2020. https://doi.org/10.1093/imrn/rny037.","ama":"Akopyan A, Karasev R. Waist of balls in hyperbolic and spherical spaces. International Mathematics Research Notices. 2020;2020(3):669-697. doi:10.1093/imrn/rny037","ieee":"A. Akopyan and R. Karasev, “Waist of balls in hyperbolic and spherical spaces,” International Mathematics Research Notices, vol. 2020, no. 3. Oxford University Press, pp. 669–697, 2020.","apa":"Akopyan, A., & Karasev, R. (2020). Waist of balls in hyperbolic and spherical spaces. International Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rny037","ista":"Akopyan A, Karasev R. 2020. Waist of balls in hyperbolic and spherical spaces. International Mathematics Research Notices. 2020(3), 669–697."},"article_type":"original","page":"669-697","date_published":"2020-02-01T00:00:00Z"},{"abstract":[{"text":"Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy.","lang":"eng"}],"issue":"1","type":"journal_article","file":[{"date_updated":"2021-02-19T13:56:24Z","date_created":"2021-02-19T13:56:24Z","checksum":"cea41de9937d07a3b927d71ee8b4e432","success":1,"relation":"main_file","file_id":"9171","content_type":"application/pdf","file_size":562359,"creator":"dernst","file_name":"2020_CompMathBiophysics_Akopyan2.pdf","access_level":"open_access"}],"oa_version":"Published Version","ddc":["510"],"status":"public","title":"The weighted mean curvature derivative of a space-filling diagram","intvolume":" 8","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"9157","day":"20","article_processing_charge":"No","has_accepted_license":"1","date_published":"2020-06-20T00:00:00Z","article_type":"original","page":"51-67","publication":"Computational and Mathematical Biophysics","citation":{"chicago":"Akopyan, Arseniy, and Herbert Edelsbrunner. “The Weighted Mean Curvature Derivative of a Space-Filling Diagram.” Computational and Mathematical Biophysics. De Gruyter, 2020. https://doi.org/10.1515/cmb-2020-0100.","mla":"Akopyan, Arseniy, and Herbert Edelsbrunner. “The Weighted Mean Curvature Derivative of a Space-Filling Diagram.” Computational and Mathematical Biophysics, vol. 8, no. 1, De Gruyter, 2020, pp. 51–67, doi:10.1515/cmb-2020-0100.","short":"A. Akopyan, H. Edelsbrunner, Computational and Mathematical Biophysics 8 (2020) 51–67.","ista":"Akopyan A, Edelsbrunner H. 2020. The weighted mean curvature derivative of a space-filling diagram. Computational and Mathematical Biophysics. 8(1), 51–67.","ieee":"A. Akopyan and H. Edelsbrunner, “The weighted mean curvature derivative of a space-filling diagram,” Computational and Mathematical Biophysics, vol. 8, no. 1. De Gruyter, pp. 51–67, 2020.","apa":"Akopyan, A., & Edelsbrunner, H. (2020). The weighted mean curvature derivative of a space-filling diagram. Computational and Mathematical Biophysics. De Gruyter. https://doi.org/10.1515/cmb-2020-0100","ama":"Akopyan A, Edelsbrunner H. The weighted mean curvature derivative of a space-filling diagram. Computational and Mathematical Biophysics. 2020;8(1):51-67. doi:10.1515/cmb-2020-0100"},"file_date_updated":"2021-02-19T13:56:24Z","ec_funded":1,"date_created":"2021-02-17T15:13:01Z","date_updated":"2023-10-17T12:34:51Z","volume":8,"author":[{"full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","first_name":"Arseniy"},{"first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"}],"publication_status":"published","department":[{"_id":"HeEd"}],"publisher":"De Gruyter","acknowledgement":"The authors of this paper thank Roland Roth for suggesting the analysis of the weighted\r\ncurvature derivatives for the purpose of improving molecular dynamics simulations and for his continued encouragement. They also thank Patrice Koehl for the implementation of the formulas and for his encouragement and advise along the road. Finally, they thank two anonymous reviewers for their constructive criticism.\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","year":"2020","month":"06","publication_identifier":{"issn":["2544-7297"]},"language":[{"iso":"eng"}],"doi":"10.1515/cmb-2020-0100","quality_controlled":"1","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"oa":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"}},{"issue":"1","abstract":[{"lang":"eng","text":"The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy."}],"type":"journal_article","oa_version":"Published Version","file":[{"file_id":"9170","relation":"main_file","date_updated":"2021-02-19T13:33:19Z","date_created":"2021-02-19T13:33:19Z","success":1,"checksum":"ca43a7440834eab6bbea29c59b56ef3a","file_name":"2020_CompMathBiophysics_Akopyan.pdf","access_level":"open_access","creator":"dernst","file_size":707452,"content_type":"application/pdf"}],"intvolume":" 8","ddc":["510"],"title":"The weighted Gaussian curvature derivative of a space-filling diagram","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"9156","article_processing_charge":"No","has_accepted_license":"1","day":"21","date_published":"2020-07-21T00:00:00Z","page":"74-88","article_type":"original","citation":{"ieee":"A. Akopyan and H. Edelsbrunner, “The weighted Gaussian curvature derivative of a space-filling diagram,” Computational and Mathematical Biophysics, vol. 8, no. 1. De Gruyter, pp. 74–88, 2020.","apa":"Akopyan, A., & Edelsbrunner, H. (2020). The weighted Gaussian curvature derivative of a space-filling diagram. Computational and Mathematical Biophysics. De Gruyter. https://doi.org/10.1515/cmb-2020-0101","ista":"Akopyan A, Edelsbrunner H. 2020. The weighted Gaussian curvature derivative of a space-filling diagram. Computational and Mathematical Biophysics. 8(1), 74–88.","ama":"Akopyan A, Edelsbrunner H. The weighted Gaussian curvature derivative of a space-filling diagram. Computational and Mathematical Biophysics. 2020;8(1):74-88. doi:10.1515/cmb-2020-0101","chicago":"Akopyan, Arseniy, and Herbert Edelsbrunner. “The Weighted Gaussian Curvature Derivative of a Space-Filling Diagram.” Computational and Mathematical Biophysics. De Gruyter, 2020. https://doi.org/10.1515/cmb-2020-0101.","short":"A. Akopyan, H. Edelsbrunner, Computational and Mathematical Biophysics 8 (2020) 74–88.","mla":"Akopyan, Arseniy, and Herbert Edelsbrunner. “The Weighted Gaussian Curvature Derivative of a Space-Filling Diagram.” Computational and Mathematical Biophysics, vol. 8, no. 1, De Gruyter, 2020, pp. 74–88, doi:10.1515/cmb-2020-0101."},"publication":"Computational and Mathematical Biophysics","ec_funded":1,"file_date_updated":"2021-02-19T13:33:19Z","volume":8,"date_updated":"2023-10-17T12:35:10Z","date_created":"2021-02-17T15:12:44Z","author":[{"orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","first_name":"Arseniy","full_name":"Akopyan, Arseniy"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"}],"publisher":"De Gruyter","department":[{"_id":"HeEd"}],"publication_status":"published","year":"2020","acknowledgement":"The authors of this paper thank Roland Roth for suggesting the analysis of theweighted\r\ncurvature derivatives for the purpose of improving molecular dynamics simulations. They also thank Patrice Koehl for the implementation of the formulas and for his encouragement and advise along the road. Finally, they thank two anonymous reviewers for their constructive criticism.\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","publication_identifier":{"issn":["2544-7297"]},"month":"07","language":[{"iso":"eng"}],"doi":"10.1515/cmb-2020-0101","project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","external_id":{"arxiv":["1908.06777"]},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1},{"status":"public","title":"Two circles and only a straightedge","intvolume":" 147","_id":"6050","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa_version":"Preprint","type":"journal_article","abstract":[{"text":"We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible. ","lang":"eng"}],"page":"91-102","publication":"Proceedings of the American Mathematical Society","citation":{"mla":"Akopyan, Arseniy, and Roman Fedorov. “Two Circles and Only a Straightedge.” Proceedings of the American Mathematical Society, vol. 147, AMS, 2019, pp. 91–102, doi:10.1090/proc/14240.","short":"A. Akopyan, R. Fedorov, Proceedings of the American Mathematical Society 147 (2019) 91–102.","chicago":"Akopyan, Arseniy, and Roman Fedorov. “Two Circles and Only a Straightedge.” Proceedings of the American Mathematical Society. AMS, 2019. https://doi.org/10.1090/proc/14240.","ama":"Akopyan A, Fedorov R. Two circles and only a straightedge. Proceedings of the American Mathematical Society. 2019;147:91-102. doi:10.1090/proc/14240","ista":"Akopyan A, Fedorov R. 2019. Two circles and only a straightedge. Proceedings of the American Mathematical Society. 147, 91–102.","ieee":"A. Akopyan and R. Fedorov, “Two circles and only a straightedge,” Proceedings of the American Mathematical Society, vol. 147. AMS, pp. 91–102, 2019.","apa":"Akopyan, A., & Fedorov, R. (2019). Two circles and only a straightedge. Proceedings of the American Mathematical Society. AMS. https://doi.org/10.1090/proc/14240"},"date_published":"2019-01-01T00:00:00Z","scopus_import":"1","day":"01","article_processing_charge":"No","publication_status":"published","publisher":"AMS","department":[{"_id":"HeEd"}],"year":"2019","date_updated":"2023-08-24T14:48:59Z","date_created":"2019-02-24T22:59:19Z","volume":147,"author":[{"orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","first_name":"Arseniy","full_name":"Akopyan, Arseniy"},{"full_name":"Fedorov, Roman","first_name":"Roman","last_name":"Fedorov"}],"quality_controlled":"1","isi":1,"external_id":{"arxiv":["1709.02562"],"isi":["000450363900008"]},"oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1709.02562"}],"language":[{"iso":"eng"}],"doi":"10.1090/proc/14240","month":"01"},{"intvolume":" 15","title":"An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape","status":"public","ddc":["570"],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"6419","oa_version":"Published Version","file":[{"checksum":"cf3889c8a8a16053dacf9c3776cbe217","date_updated":"2020-07-14T12:47:30Z","date_created":"2019-05-14T08:26:08Z","file_id":"6445","relation":"main_file","creator":"dernst","file_size":3726017,"content_type":"application/pdf","access_level":"open_access","file_name":"2019_PLOSGenetics_Pokusaeva.pdf"}],"type":"journal_article","issue":"4","abstract":[{"text":"Characterizing the fitness landscape, a representation of fitness for a large set of genotypes, is key to understanding how genetic information is interpreted to create functional organisms. Here we determined the evolutionarily-relevant segment of the fitness landscape of His3, a gene coding for an enzyme in the histidine synthesis pathway, focusing on combinations of amino acid states found at orthologous sites of extant species. Just 15% of amino acids found in yeast His3 orthologues were always neutral while the impact on fitness of the remaining 85% depended on the genetic background. Furthermore, at 67% of sites, amino acid replacements were under sign epistasis, having both strongly positive and negative effect in different genetic backgrounds. 46% of sites were under reciprocal sign epistasis. The fitness impact of amino acid replacements was influenced by only a few genetic backgrounds but involved interaction of multiple sites, shaping a rugged fitness landscape in which many of the shortest paths between highly fit genotypes are inaccessible.","lang":"eng"}],"citation":{"ama":"Pokusaeva V, Usmanova DR, Putintseva EV, et al. An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape. PLoS Genetics. 2019;15(4). doi:10.1371/journal.pgen.1008079","ista":"Pokusaeva V, Usmanova DR, Putintseva EV, Espinar L, Sarkisyan K, Mishin AS, Bogatyreva NS, Ivankov D, Akopyan A, Avvakumov S, Povolotskaya IS, Filion GJ, Carey LB, Kondrashov F. 2019. An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape. PLoS Genetics. 15(4), e1008079.","apa":"Pokusaeva, V., Usmanova, D. R., Putintseva, E. V., Espinar, L., Sarkisyan, K., Mishin, A. S., … Kondrashov, F. (2019). An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape. PLoS Genetics. Public Library of Science. https://doi.org/10.1371/journal.pgen.1008079","ieee":"V. 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Bogatyreva, et al. “An Experimental Assay of the Interactions of Amino Acids from Orthologous Sequences Shaping a Complex Fitness Landscape.” PLoS Genetics. 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The Regge symmetry, confocal conics, and the Schläfli formula. Bulletin of the London Mathematical Society. 2019;51(5):765-775. doi:10.1112/blms.12276"},"publication":"Bulletin of the London Mathematical Society","page":"765-775","article_type":"original","date_published":"2019-10-01T00:00:00Z","type":"journal_article","issue":"5","abstract":[{"lang":"eng","text":"The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here, we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry."}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"6793","intvolume":" 51","status":"public","title":"The Regge symmetry, confocal conics, and the Schläfli formula","oa_version":"Preprint","publication_identifier":{"issn":["00246093"],"eissn":["14692120"]},"month":"10","main_file_link":[{"url":"https://arxiv.org/abs/1903.04929","open_access":"1"}],"external_id":{"arxiv":["1903.04929"],"isi":["000478560200001"]},"oa":1,"project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","isi":1,"doi":"10.1112/blms.12276","language":[{"iso":"eng"}],"ec_funded":1,"year":"2019","department":[{"_id":"HeEd"}],"publisher":"London Mathematical Society","publication_status":"published","author":[{"full_name":"Akopyan, Arseniy","last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Izmestiev","first_name":"Ivan","full_name":"Izmestiev, Ivan"}],"volume":51,"date_updated":"2023-08-29T07:08:34Z","date_created":"2019-08-11T21:59:23Z"},{"month":"06","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme"}],"isi":1,"quality_controlled":"1","oa":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"isi":["000431418800004"]},"language":[{"iso":"eng"}],"doi":"10.1007/s10711-017-0265-6","publist_id":"7014","ec_funded":1,"file_date_updated":"2020-07-14T12:47:44Z","publisher":"Springer","department":[{"_id":"HeEd"}],"publication_status":"published","year":"2018","volume":194,"date_created":"2018-12-11T11:47:57Z","date_updated":"2023-09-08T11:40:29Z","author":[{"full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X"}],"scopus_import":"1","article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","day":"01","page":"55 - 64","article_type":"original","citation":{"ista":"Akopyan A. 2018. 3-Webs generated by confocal conics and circles. Geometriae Dedicata. 194(1), 55–64.","apa":"Akopyan, A. (2018). 3-Webs generated by confocal conics and circles. Geometriae Dedicata. Springer. https://doi.org/10.1007/s10711-017-0265-6","ieee":"A. Akopyan, “3-Webs generated by confocal conics and circles,” Geometriae Dedicata, vol. 194, no. 1. Springer, pp. 55–64, 2018.","ama":"Akopyan A. 3-Webs generated by confocal conics and circles. Geometriae Dedicata. 2018;194(1):55-64. doi:10.1007/s10711-017-0265-6","chicago":"Akopyan, Arseniy. “3-Webs Generated by Confocal Conics and Circles.” Geometriae Dedicata. Springer, 2018. https://doi.org/10.1007/s10711-017-0265-6.","mla":"Akopyan, Arseniy. “3-Webs Generated by Confocal Conics and Circles.” Geometriae Dedicata, vol. 194, no. 1, Springer, 2018, pp. 55–64, doi:10.1007/s10711-017-0265-6.","short":"A. Akopyan, Geometriae Dedicata 194 (2018) 55–64."},"publication":"Geometriae Dedicata","date_published":"2018-06-01T00:00:00Z","type":"journal_article","issue":"1","abstract":[{"lang":"eng","text":"We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them."}],"intvolume":" 194","ddc":["510"],"title":"3-Webs generated by confocal conics and circles","status":"public","_id":"692","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa_version":"Published Version","file":[{"file_id":"7222","relation":"main_file","checksum":"1febcfc1266486053a069e3425ea3713","date_created":"2020-01-03T11:35:08Z","date_updated":"2020-07-14T12:47:44Z","access_level":"open_access","file_name":"2018_Springer_Akopyan.pdf","creator":"kschuh","file_size":1140860,"content_type":"application/pdf"}]},{"article_processing_charge":"No","day":"06","scopus_import":"1","date_published":"2018-09-06T00:00:00Z","citation":{"ama":"Akopyan A, Segal Halevi E. Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. 2018;32(3):2242-2257. doi:10.1137/16M110407X","apa":"Akopyan, A., & Segal Halevi, E. (2018). Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics . https://doi.org/10.1137/16M110407X","ieee":"A. Akopyan and E. Segal Halevi, “Counting blanks in polygonal arrangements,” SIAM Journal on Discrete Mathematics, vol. 32, no. 3. Society for Industrial and Applied Mathematics , pp. 2242–2257, 2018.","ista":"Akopyan A, Segal Halevi E. 2018. Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. 32(3), 2242–2257.","short":"A. Akopyan, E. Segal Halevi, SIAM Journal on Discrete Mathematics 32 (2018) 2242–2257.","mla":"Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” SIAM Journal on Discrete Mathematics, vol. 32, no. 3, Society for Industrial and Applied Mathematics , 2018, pp. 2242–57, doi:10.1137/16M110407X.","chicago":"Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics , 2018. https://doi.org/10.1137/16M110407X."},"publication":"SIAM Journal on Discrete Mathematics","page":"2242 - 2257","issue":"3","abstract":[{"text":"Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks.","lang":"eng"}],"type":"journal_article","oa_version":"Preprint","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"58","intvolume":" 32","title":"Counting blanks in polygonal arrangements","status":"public","month":"09","doi":"10.1137/16M110407X","language":[{"iso":"eng"}],"oa":1,"external_id":{"arxiv":["1604.00960"],"isi":["000450810500036"]},"main_file_link":[{"url":"https://arxiv.org/abs/1604.00960","open_access":"1"}],"project":[{"grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7"}],"isi":1,"quality_controlled":"1","ec_funded":1,"publist_id":"7996","author":[{"first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy"},{"full_name":"Segal Halevi, Erel","last_name":"Segal Halevi","first_name":"Erel"}],"volume":32,"date_created":"2018-12-11T11:44:24Z","date_updated":"2023-09-11T12:48:39Z","year":"2018","department":[{"_id":"HeEd"}],"publisher":"Society for Industrial and Applied Mathematics ","publication_status":"published"},{"month":"04","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme"}],"quality_controlled":"1","isi":1,"external_id":{"isi":["000423197800019"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1602.04637"}],"oa":1,"language":[{"iso":"eng"}],"doi":"10.1090/tran/7292","ec_funded":1,"publist_id":"7363","department":[{"_id":"HeEd"}],"publisher":"American Mathematical Society","publication_status":"published","year":"2018","acknowledgement":"DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”; People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) REA grant agreement n◦[291734]","volume":370,"date_updated":"2023-09-11T14:19:12Z","date_created":"2018-12-11T11:46:35Z","author":[{"first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy"},{"last_name":"Bobenko","first_name":"Alexander","full_name":"Bobenko, Alexander"}],"scopus_import":"1","article_processing_charge":"No","day":"01","page":"2825 - 2854","citation":{"ista":"Akopyan A, Bobenko A. 2018. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 370(4), 2825–2854.","apa":"Akopyan, A., & Bobenko, A. (2018). Incircular nets and confocal conics. Transactions of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/tran/7292","ieee":"A. Akopyan and A. Bobenko, “Incircular nets and confocal conics,” Transactions of the American Mathematical Society, vol. 370, no. 4. American Mathematical Society, pp. 2825–2854, 2018.","ama":"Akopyan A, Bobenko A. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 2018;370(4):2825-2854. doi:10.1090/tran/7292","chicago":"Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” Transactions of the American Mathematical Society. American Mathematical Society, 2018. https://doi.org/10.1090/tran/7292.","mla":"Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” Transactions of the American Mathematical Society, vol. 370, no. 4, American Mathematical Society, 2018, pp. 2825–54, doi:10.1090/tran/7292.","short":"A. Akopyan, A. Bobenko, Transactions of the American Mathematical Society 370 (2018) 2825–2854."},"publication":"Transactions of the American Mathematical Society","date_published":"2018-04-01T00:00:00Z","type":"journal_article","issue":"4","abstract":[{"lang":"eng","text":"We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the plane. These are congruences of straight lines in the plane with the combinatorics of the square grid, combinatorially colored as a checkerboard, such that all black coordinate quadrilaterals possess inscribed circles. We show how this larger class of IC-nets appears quite naturally in Laguerre geometry of oriented planes and spheres and leads to new remarkable incidence theorems. Most of our results are valid in hyperbolic and spherical geometries as well. We present also generalizations in spaces of higher dimension, called checkerboard IS-nets. The construction of these nets is based on a new 9 inspheres incidence theorem."}],"intvolume":" 370","title":"Incircular nets and confocal conics","status":"public","_id":"458","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa_version":"Preprint"},{"date_published":"2018-09-01T00:00:00Z","citation":{"chicago":"Akopyan, Arseniy, and Anton Petrunin. “Long Geodesics on Convex Surfaces.” Mathematical Intelligencer. Springer, 2018. https://doi.org/10.1007/s00283-018-9795-5.","short":"A. Akopyan, A. Petrunin, Mathematical Intelligencer 40 (2018) 26–31.","mla":"Akopyan, Arseniy, and Anton Petrunin. “Long Geodesics on Convex Surfaces.” Mathematical Intelligencer, vol. 40, no. 3, Springer, 2018, pp. 26–31, doi:10.1007/s00283-018-9795-5.","apa":"Akopyan, A., & Petrunin, A. (2018). Long geodesics on convex surfaces. Mathematical Intelligencer. Springer. https://doi.org/10.1007/s00283-018-9795-5","ieee":"A. Akopyan and A. Petrunin, “Long geodesics on convex surfaces,” Mathematical Intelligencer, vol. 40, no. 3. Springer, pp. 26–31, 2018.","ista":"Akopyan A, Petrunin A. 2018. Long geodesics on convex surfaces. Mathematical Intelligencer. 40(3), 26–31.","ama":"Akopyan A, Petrunin A. Long geodesics on convex surfaces. Mathematical Intelligencer. 2018;40(3):26-31. doi:10.1007/s00283-018-9795-5"},"publication":"Mathematical Intelligencer","page":"26 - 31","article_processing_charge":"No","day":"01","scopus_import":"1","oa_version":"Preprint","_id":"106","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","intvolume":" 40","title":"Long geodesics on convex surfaces","status":"public","issue":"3","abstract":[{"text":"The goal of this article is to introduce the reader to the theory of intrinsic geometry of convex surfaces. We illustrate the power of the tools by proving a theorem on convex surfaces containing an arbitrarily long closed simple geodesic. Let us remind ourselves that a curve in a surface is called geodesic if every sufficiently short arc of the curve is length minimizing; if, in addition, it has no self-intersections, we call it simple geodesic. A tetrahedron with equal opposite edges is called isosceles. The axiomatic method of Alexandrov geometry allows us to work with the metrics of convex surfaces directly, without approximating it first by a smooth or polyhedral metric. Such approximations destroy the closed geodesics on the surface; therefore it is difficult (if at all possible) to apply approximations in the proof of our theorem. On the other hand, a proof in the smooth or polyhedral case usually admits a translation into Alexandrov’s language; such translation makes the result more general. In fact, our proof resembles a translation of the proof given by Protasov. Note that the main theorem implies in particular that a smooth convex surface does not have arbitrarily long simple closed geodesics. However we do not know a proof of this corollary that is essentially simpler than the one presented below.","lang":"eng"}],"type":"journal_article","doi":"10.1007/s00283-018-9795-5","language":[{"iso":"eng"}],"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1702.05172","open_access":"1"}],"external_id":{"isi":["000444141200005"],"arxiv":["1702.05172"]},"quality_controlled":"1","isi":1,"month":"09","author":[{"full_name":"Akopyan, Arseniy","first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X"},{"full_name":"Petrunin, Anton","first_name":"Anton","last_name":"Petrunin"}],"volume":40,"date_created":"2018-12-11T11:44:40Z","date_updated":"2023-09-13T08:49:16Z","year":"2018","publisher":"Springer","department":[{"_id":"HeEd"}],"publication_status":"published","publist_id":"7948"},{"date_published":"2018-04-01T00:00:00Z","publication":"Comptes Rendus Mathematique","citation":{"chicago":"Akopyan, Arseniy. “On the Number of Non-Hexagons in a Planar Tiling.” Comptes Rendus Mathematique. Elsevier, 2018. https://doi.org/10.1016/j.crma.2018.03.005.","short":"A. Akopyan, Comptes Rendus Mathematique 356 (2018) 412–414.","mla":"Akopyan, Arseniy. “On the Number of Non-Hexagons in a Planar Tiling.” Comptes Rendus Mathematique, vol. 356, no. 4, Elsevier, 2018, pp. 412–14, doi:10.1016/j.crma.2018.03.005.","apa":"Akopyan, A. (2018). On the number of non-hexagons in a planar tiling. Comptes Rendus Mathematique. Elsevier. https://doi.org/10.1016/j.crma.2018.03.005","ieee":"A. Akopyan, “On the number of non-hexagons in a planar tiling,” Comptes Rendus Mathematique, vol. 356, no. 4. Elsevier, pp. 412–414, 2018.","ista":"Akopyan A. 2018. On the number of non-hexagons in a planar tiling. Comptes Rendus Mathematique. 356(4), 412–414.","ama":"Akopyan A. On the number of non-hexagons in a planar tiling. Comptes Rendus Mathematique. 2018;356(4):412-414. doi:10.1016/j.crma.2018.03.005"},"article_type":"original","page":"412-414","day":"01","article_processing_charge":"No","scopus_import":"1","oa_version":"Preprint","_id":"409","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","title":"On the number of non-hexagons in a planar tiling","status":"public","intvolume":" 356","abstract":[{"text":"We give a simple proof of T. Stehling's result [4], whereby in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except a finite number are hexagons.","lang":"eng"}],"issue":"4","type":"journal_article","doi":"10.1016/j.crma.2018.03.005","language":[{"iso":"eng"}],"external_id":{"isi":["000430402700009"],"arxiv":["1805.01652"]},"oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1805.01652"}],"quality_controlled":"1","isi":1,"month":"04","publication_identifier":{"issn":["1631073X"]},"author":[{"full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","last_name":"Akopyan"}],"date_created":"2018-12-11T11:46:19Z","date_updated":"2023-09-13T09:34:12Z","volume":356,"year":"2018","publication_status":"published","department":[{"_id":"HeEd"}],"publisher":"Elsevier","publist_id":"7420"},{"file":[{"file_name":"2018_ForumMahtematics_Akopyan.pdf","access_level":"open_access","creator":"dernst","content_type":"application/pdf","file_size":249246,"file_id":"6356","relation":"main_file","date_created":"2019-04-30T06:14:58Z","date_updated":"2020-07-14T12:47:28Z","checksum":"5a71b24ba712a3eb2e46165a38fbc30a"}],"oa_version":"Published Version","_id":"6355","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","intvolume":" 6","title":"Any cyclic quadrilateral can be inscribed in any closed convex smooth curve","status":"public","ddc":["510"],"abstract":[{"text":"We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.","lang":"eng"}],"type":"journal_article","date_published":"2018-05-31T00:00:00Z","citation":{"chicago":"Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be Inscribed in Any Closed Convex Smooth Curve.” Forum of Mathematics, Sigma. Cambridge University Press, 2018. https://doi.org/10.1017/fms.2018.7.","short":"A. Akopyan, S. Avvakumov, Forum of Mathematics, Sigma 6 (2018).","mla":"Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be Inscribed in Any Closed Convex Smooth Curve.” Forum of Mathematics, Sigma, vol. 6, e7, Cambridge University Press, 2018, doi:10.1017/fms.2018.7.","ieee":"A. Akopyan and S. Avvakumov, “Any cyclic quadrilateral can be inscribed in any closed convex smooth curve,” Forum of Mathematics, Sigma, vol. 6. Cambridge University Press, 2018.","apa":"Akopyan, A., & Avvakumov, S. (2018). Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. Forum of Mathematics, Sigma. Cambridge University Press. https://doi.org/10.1017/fms.2018.7","ista":"Akopyan A, Avvakumov S. 2018. Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. Forum of Mathematics, Sigma. 6, e7.","ama":"Akopyan A, Avvakumov S. Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. Forum of Mathematics, Sigma. 2018;6. doi:10.1017/fms.2018.7"},"publication":"Forum of Mathematics, Sigma","has_accepted_license":"1","article_processing_charge":"No","day":"31","related_material":{"record":[{"id":"8156","relation":"dissertation_contains","status":"public"}]},"author":[{"last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy"},{"full_name":"Avvakumov, Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","first_name":"Sergey","last_name":"Avvakumov"}],"volume":6,"date_created":"2019-04-30T06:09:57Z","date_updated":"2023-09-19T14:50:12Z","year":"2018","publisher":"Cambridge University Press","department":[{"_id":"UlWa"},{"_id":"HeEd"},{"_id":"JaMa"}],"publication_status":"published","ec_funded":1,"file_date_updated":"2020-07-14T12:47:28Z","article_number":"e7","doi":"10.1017/fms.2018.7","language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"arxiv":["1712.10205"],"isi":["000433915500001"]},"oa":1,"project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics"}],"isi":1,"quality_controlled":"1","publication_identifier":{"issn":["2050-5094"]},"month":"05"},{"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","day":"01","scopus_import":"1","date_published":"2018-06-01T00:00:00Z","citation":{"ista":"Akopyan A, Balitskiy A, Grigorev M. 2018. On the circle covering theorem by A.W. Goodman and R.E. Goodman. Discrete & Computational Geometry. 59(4), 1001–1009.","ieee":"A. Akopyan, A. Balitskiy, and M. Grigorev, “On the circle covering theorem by A.W. Goodman and R.E. Goodman,” Discrete & Computational Geometry, vol. 59, no. 4. Springer, pp. 1001–1009, 2018.","apa":"Akopyan, A., Balitskiy, A., & Grigorev, M. (2018). On the circle covering theorem by A.W. Goodman and R.E. Goodman. Discrete & Computational Geometry. Springer. https://doi.org/10.1007/s00454-017-9883-x","ama":"Akopyan A, Balitskiy A, Grigorev M. On the circle covering theorem by A.W. Goodman and R.E. Goodman. Discrete & Computational Geometry. 2018;59(4):1001-1009. doi:10.1007/s00454-017-9883-x","chicago":"Akopyan, Arseniy, Alexey Balitskiy, and Mikhail Grigorev. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” Discrete & Computational Geometry. Springer, 2018. https://doi.org/10.1007/s00454-017-9883-x.","mla":"Akopyan, Arseniy, et al. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” Discrete & Computational Geometry, vol. 59, no. 4, Springer, 2018, pp. 1001–09, doi:10.1007/s00454-017-9883-x.","short":"A. Akopyan, A. Balitskiy, M. Grigorev, Discrete & Computational Geometry 59 (2018) 1001–1009."},"publication":"Discrete & Computational Geometry","page":"1001-1009","article_type":"original","issue":"4","abstract":[{"lang":"eng","text":"In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, … , rn in the plane, it is always possible to cover them by a disk of radius R= ∑ ri, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K⊂ Rd with homothety coefficients τ1, … , τn> 0 , it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets."}],"type":"journal_article","file":[{"creator":"dernst","content_type":"application/pdf","file_size":482518,"access_level":"open_access","file_name":"2018_DiscreteComp_Akopyan.pdf","success":1,"date_updated":"2019-01-18T09:27:36Z","date_created":"2019-01-18T09:27:36Z","file_id":"5844","relation":"main_file"}],"oa_version":"Published Version","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"1064","intvolume":" 59","title":"On the circle covering theorem by A.W. Goodman and R.E. Goodman","status":"public","ddc":["516","000"],"publication_identifier":{"eissn":["14320444"],"issn":["01795376"]},"month":"06","doi":"10.1007/s00454-017-9883-x","language":[{"iso":"eng"}],"oa":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"isi":["000432205500011"]},"project":[{"call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"isi":1,"quality_controlled":"1","publist_id":"6324","ec_funded":1,"file_date_updated":"2019-01-18T09:27:36Z","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","last_name":"Akopyan","full_name":"Akopyan, Arseniy"},{"last_name":"Balitskiy","first_name":"Alexey","full_name":"Balitskiy, Alexey"},{"first_name":"Mikhail","last_name":"Grigorev","full_name":"Grigorev, Mikhail"}],"volume":59,"date_updated":"2023-09-20T12:08:51Z","date_created":"2018-12-11T11:49:57Z","year":"2018","publisher":"Springer","department":[{"_id":"HeEd"}],"publication_status":"published"},{"publisher":"arXiv","department":[{"_id":"HeEd"},{"_id":"JaMa"}],"status":"public","publication_status":"published","title":"Convex fair partitions into arbitrary number of pieces","year":"2018","_id":"75","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","date_updated":"2023-12-18T10:51:02Z","date_created":"2018-12-11T11:44:30Z","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"8156"}]},"author":[{"full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","first_name":"Arseniy"},{"full_name":"Avvakumov, Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","last_name":"Avvakumov","first_name":"Sergey"},{"last_name":"Karasev","first_name":"Roman","full_name":"Karasev, Roman"}],"type":"preprint","article_number":"1804.03057","ec_funded":1,"abstract":[{"text":"We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization.","lang":"eng"}],"project":[{"grant_number":"716117","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020"}],"oa":1,"external_id":{"arxiv":["1804.03057"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1804.03057"}],"citation":{"apa":"Akopyan, A., Avvakumov, S., & Karasev, R. (2018). Convex fair partitions into arbitrary number of pieces. arXiv. https://doi.org/10.48550/arXiv.1804.03057","ieee":"A. Akopyan, S. Avvakumov, and R. Karasev, “Convex fair partitions into arbitrary number of pieces.” arXiv, 2018.","ista":"Akopyan A, Avvakumov S, Karasev R. 2018. Convex fair partitions into arbitrary number of pieces. 1804.03057.","ama":"Akopyan A, Avvakumov S, Karasev R. Convex fair partitions into arbitrary number of pieces. 2018. doi:10.48550/arXiv.1804.03057","chicago":"Akopyan, Arseniy, Sergey Avvakumov, and Roman Karasev. “Convex Fair Partitions into Arbitrary Number of Pieces.” arXiv, 2018. https://doi.org/10.48550/arXiv.1804.03057.","short":"A. Akopyan, S. Avvakumov, R. Karasev, (2018).","mla":"Akopyan, Arseniy, et al. Convex Fair Partitions into Arbitrary Number of Pieces. 1804.03057, arXiv, 2018, doi:10.48550/arXiv.1804.03057."},"language":[{"iso":"eng"}],"doi":"10.48550/arXiv.1804.03057","date_published":"2018-09-13T00:00:00Z","article_processing_charge":"No","month":"09","day":"13"},{"type":"journal_article","issue":"4","abstract":[{"lang":"eng","text":"We answer a question of M. Gromov on the waist of the unit ball."}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","_id":"707","intvolume":" 49","title":"A tight estimate for the waist of the ball ","status":"public","oa_version":"Preprint","scopus_import":1,"day":"01","citation":{"apa":"Akopyan, A., & Karasev, R. (2017). A tight estimate for the waist of the ball . Bulletin of the London Mathematical Society. Wiley-Blackwell. https://doi.org/10.1112/blms.12062","ieee":"A. Akopyan and R. Karasev, “A tight estimate for the waist of the ball ,” Bulletin of the London Mathematical Society, vol. 49, no. 4. Wiley-Blackwell, pp. 690–693, 2017.","ista":"Akopyan A, Karasev R. 2017. A tight estimate for the waist of the ball . Bulletin of the London Mathematical Society. 49(4), 690–693.","ama":"Akopyan A, Karasev R. A tight estimate for the waist of the ball . Bulletin of the London Mathematical Society. 2017;49(4):690-693. doi:10.1112/blms.12062","chicago":"Akopyan, Arseniy, and Roman Karasev. “A Tight Estimate for the Waist of the Ball .” Bulletin of the London Mathematical Society. Wiley-Blackwell, 2017. https://doi.org/10.1112/blms.12062.","short":"A. Akopyan, R. Karasev, Bulletin of the London Mathematical Society 49 (2017) 690–693.","mla":"Akopyan, Arseniy, and Roman Karasev. “A Tight Estimate for the Waist of the Ball .” Bulletin of the London Mathematical Society, vol. 49, no. 4, Wiley-Blackwell, 2017, pp. 690–93, doi:10.1112/blms.12062."},"publication":"Bulletin of the London Mathematical Society","page":"690 - 693","date_published":"2017-08-01T00:00:00Z","ec_funded":1,"publist_id":"6982","year":"2017","publisher":"Wiley-Blackwell","department":[{"_id":"HeEd"}],"publication_status":"published","author":[{"full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","last_name":"Akopyan"},{"full_name":"Karasev, Roman","last_name":"Karasev","first_name":"Roman"}],"volume":49,"date_updated":"2021-01-12T08:11:41Z","date_created":"2018-12-11T11:48:02Z","publication_identifier":{"issn":["00246093"]},"month":"08","main_file_link":[{"url":"https://arxiv.org/abs/1608.06279","open_access":"1"}],"oa":1,"project":[{"call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734"}],"quality_controlled":"1","doi":"10.1112/blms.12062","language":[{"iso":"eng"}]},{"ec_funded":1,"publist_id":"6173","volume":308,"date_updated":"2023-09-20T11:21:27Z","date_created":"2018-12-11T11:50:34Z","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","last_name":"Akopyan","full_name":"Akopyan, Arseniy"},{"first_name":"Imre","last_name":"Bárány","full_name":"Bárány, Imre"},{"first_name":"Sinai","last_name":"Robins","full_name":"Robins, Sinai"}],"department":[{"_id":"HeEd"}],"publisher":"Academic Press","publication_status":"published","year":"2017","publication_identifier":{"issn":["00018708"]},"month":"02","language":[{"iso":"eng"}],"doi":"10.1016/j.aim.2016.12.026","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7"}],"isi":1,"quality_controlled":"1","external_id":{"isi":["000409292900015"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1508.07594"}],"oa":1,"abstract":[{"lang":"eng","text":"In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform."}],"type":"journal_article","oa_version":"Submitted Version","intvolume":" 308","status":"public","title":"Algebraic vertices of non-convex polyhedra","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"1180","article_processing_charge":"No","day":"21","scopus_import":"1","date_published":"2017-02-21T00:00:00Z","page":"627 - 644","citation":{"ama":"Akopyan A, Bárány I, Robins S. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 2017;308:627-644. doi:10.1016/j.aim.2016.12.026","ista":"Akopyan A, Bárány I, Robins S. 2017. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 308, 627–644.","apa":"Akopyan, A., Bárány, I., & Robins, S. (2017). Algebraic vertices of non-convex polyhedra. Advances in Mathematics. Academic Press. https://doi.org/10.1016/j.aim.2016.12.026","ieee":"A. Akopyan, I. Bárány, and S. Robins, “Algebraic vertices of non-convex polyhedra,” Advances in Mathematics, vol. 308. Academic Press, pp. 627–644, 2017.","mla":"Akopyan, Arseniy, et al. “Algebraic Vertices of Non-Convex Polyhedra.” Advances in Mathematics, vol. 308, Academic Press, 2017, pp. 627–44, doi:10.1016/j.aim.2016.12.026.","short":"A. Akopyan, I. Bárány, S. Robins, Advances in Mathematics 308 (2017) 627–644.","chicago":"Akopyan, Arseniy, Imre Bárány, and Sinai Robins. “Algebraic Vertices of Non-Convex Polyhedra.” Advances in Mathematics. Academic Press, 2017. https://doi.org/10.1016/j.aim.2016.12.026."},"publication":"Advances in Mathematics"},{"scopus_import":"1","day":"01","article_processing_charge":"No","publication":"The American Mathematical Monthly","citation":{"short":"A. Akopyan, V. Vysotsky, The American Mathematical Monthly 124 (2017) 588–596.","mla":"Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” The American Mathematical Monthly, vol. 124, no. 7, Mathematical Association of America, 2017, pp. 588–96, doi:10.4169/amer.math.monthly.124.7.588.","chicago":"Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” The American Mathematical Monthly. Mathematical Association of America, 2017. https://doi.org/10.4169/amer.math.monthly.124.7.588.","ama":"Akopyan A, Vysotsky V. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 2017;124(7):588-596. doi:10.4169/amer.math.monthly.124.7.588","ieee":"A. Akopyan and V. Vysotsky, “On the lengths of curves passing through boundary points of a planar convex shape,” The American Mathematical Monthly, vol. 124, no. 7. Mathematical Association of America, pp. 588–596, 2017.","apa":"Akopyan, A., & Vysotsky, V. (2017). On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. Mathematical Association of America. https://doi.org/10.4169/amer.math.monthly.124.7.588","ista":"Akopyan A, Vysotsky V. 2017. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 124(7), 588–596."},"article_type":"original","page":"588 - 596","date_published":"2017-01-01T00:00:00Z","type":"journal_article","abstract":[{"text":"We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor ½ cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape.","lang":"eng"}],"issue":"7","_id":"909","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","title":"On the lengths of curves passing through boundary points of a planar convex shape","intvolume":" 124","oa_version":"Submitted Version","month":"01","publication_identifier":{"issn":["00029890"]},"oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1605.07997"}],"external_id":{"isi":["000413947300002"],"arxiv":["1605.07997"]},"isi":1,"quality_controlled":"1","project":[{"name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"doi":"10.4169/amer.math.monthly.124.7.588","language":[{"iso":"eng"}],"ec_funded":1,"publist_id":"6534","year":"2017","publication_status":"published","department":[{"_id":"HeEd"}],"publisher":"Mathematical Association of America","author":[{"last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy"},{"full_name":"Vysotsky, Vladislav","last_name":"Vysotsky","first_name":"Vladislav"}],"date_created":"2018-12-11T11:49:09Z","date_updated":"2023-10-17T11:24:57Z","volume":124},{"_id":"1330","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","title":"Billiards in convex bodies with acute angles","status":"public","intvolume":" 216","oa_version":"Preprint","type":"journal_article","abstract":[{"text":"In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ⊂ Rd has the property that the tangent cone of every non-smooth point q ∉ ∂K is acute (in a certain sense), then there is a closed billiard trajectory in K.","lang":"eng"}],"issue":"2","publication":"Israel Journal of Mathematics","citation":{"chicago":"Akopyan, Arseniy, and Alexey Balitskiy. “Billiards in Convex Bodies with Acute Angles.” Israel Journal of Mathematics. Springer, 2016. https://doi.org/10.1007/s11856-016-1429-z.","short":"A. Akopyan, A. Balitskiy, Israel Journal of Mathematics 216 (2016) 833–845.","mla":"Akopyan, Arseniy, and Alexey Balitskiy. “Billiards in Convex Bodies with Acute Angles.” Israel Journal of Mathematics, vol. 216, no. 2, Springer, 2016, pp. 833–45, doi:10.1007/s11856-016-1429-z.","apa":"Akopyan, A., & Balitskiy, A. (2016). Billiards in convex bodies with acute angles. Israel Journal of Mathematics. Springer. https://doi.org/10.1007/s11856-016-1429-z","ieee":"A. Akopyan and A. Balitskiy, “Billiards in convex bodies with acute angles,” Israel Journal of Mathematics, vol. 216, no. 2. Springer, pp. 833–845, 2016.","ista":"Akopyan A, Balitskiy A. 2016. Billiards in convex bodies with acute angles. Israel Journal of Mathematics. 216(2), 833–845.","ama":"Akopyan A, Balitskiy A. Billiards in convex bodies with acute angles. Israel Journal of Mathematics. 2016;216(2):833-845. doi:10.1007/s11856-016-1429-z"},"page":"833 - 845","date_published":"2016-10-15T00:00:00Z","scopus_import":1,"day":"15","year":"2016","acknowledgement":"Supported by People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n°[291734]. Supported by the Russian Foundation for Basic Research grant 15-31-20403 (mol a ved), by the Russian Foundation for Basic Research grant 15-01-99563 A, in part by the Moebius Contest Foundation for Young Scientists, and in part by the Simons Foundation.","publication_status":"published","publisher":"Springer","department":[{"_id":"HeEd"}],"author":[{"orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","first_name":"Arseniy","full_name":"Akopyan, Arseniy"},{"last_name":"Balitskiy","first_name":"Alexey","full_name":"Balitskiy, Alexey"}],"date_created":"2018-12-11T11:51:24Z","date_updated":"2021-01-12T06:49:56Z","volume":216,"ec_funded":1,"publist_id":"5938","main_file_link":[{"url":"https://arxiv.org/abs/1506.06014","open_access":"1"}],"oa":1,"quality_controlled":"1","project":[{"name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"doi":"10.1007/s11856-016-1429-z","language":[{"iso":"eng"}],"month":"10"},{"date_created":"2018-12-11T11:51:34Z","date_updated":"2021-01-12T06:50:09Z","volume":144,"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","first_name":"Arseniy","last_name":"Akopyan","full_name":"Akopyan, Arseniy"},{"full_name":"Balitskiy, Alexey","first_name":"Alexey","last_name":"Balitskiy"},{"last_name":"Karasev","first_name":"Roman","full_name":"Karasev, Roman"},{"last_name":"Sharipova","first_name":"Anastasia","full_name":"Sharipova, Anastasia"}],"publication_status":"published","publisher":"American Mathematical Society","department":[{"_id":"HeEd"}],"year":"2016","acknowledgement":"The first and third authors were supported by the Dynasty Foundation. The first, second and third authors were supported by the Russian Foundation for Basic Re- search grant 15-31-20403 (mol a ved).","publist_id":"5885","ec_funded":1,"language":[{"iso":"eng"}],"doi":"10.1090/proc/13062","quality_controlled":"1","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7"}],"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1401.0442","open_access":"1"}],"month":"10","oa_version":"Preprint","status":"public","title":"Elementary approach to closed billiard trajectories in asymmetric normed spaces","intvolume":" 144","_id":"1360","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","abstract":[{"lang":"eng","text":"We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results. "}],"issue":"10","type":"journal_article","date_published":"2016-10-01T00:00:00Z","page":"4501 - 4513","publication":"Proceedings of the American Mathematical Society","citation":{"ama":"Akopyan A, Balitskiy A, Karasev R, Sharipova A. Elementary approach to closed billiard trajectories in asymmetric normed spaces. Proceedings of the American Mathematical Society. 2016;144(10):4501-4513. doi:10.1090/proc/13062","ista":"Akopyan A, Balitskiy A, Karasev R, Sharipova A. 2016. Elementary approach to closed billiard trajectories in asymmetric normed spaces. Proceedings of the American Mathematical Society. 144(10), 4501–4513.","ieee":"A. Akopyan, A. Balitskiy, R. Karasev, and A. Sharipova, “Elementary approach to closed billiard trajectories in asymmetric normed spaces,” Proceedings of the American Mathematical Society, vol. 144, no. 10. American Mathematical Society, pp. 4501–4513, 2016.","apa":"Akopyan, A., Balitskiy, A., Karasev, R., & Sharipova, A. (2016). Elementary approach to closed billiard trajectories in asymmetric normed spaces. Proceedings of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/proc/13062","mla":"Akopyan, Arseniy, et al. “Elementary Approach to Closed Billiard Trajectories in Asymmetric Normed Spaces.” Proceedings of the American Mathematical Society, vol. 144, no. 10, American Mathematical Society, 2016, pp. 4501–13, doi:10.1090/proc/13062.","short":"A. Akopyan, A. Balitskiy, R. Karasev, A. Sharipova, Proceedings of the American Mathematical Society 144 (2016) 4501–4513.","chicago":"Akopyan, Arseniy, Alexey Balitskiy, Roman Karasev, and Anastasia Sharipova. “Elementary Approach to Closed Billiard Trajectories in Asymmetric Normed Spaces.” Proceedings of the American Mathematical Society. American Mathematical Society, 2016. https://doi.org/10.1090/proc/13062."},"day":"01","article_processing_charge":"No","scopus_import":1},{"language":[{"iso":"eng"}],"doi":"10.1137/140993843","quality_controlled":"1","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme"}],"main_file_link":[{"url":"http://arxiv.org/abs/1410.3736","open_access":"1"}],"oa":1,"month":"07","date_updated":"2021-01-12T06:52:41Z","date_created":"2018-12-11T11:53:36Z","volume":47,"author":[{"last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy"},{"last_name":"Plakhov","first_name":"Alexander","full_name":"Plakhov, Alexander"}],"publication_status":"published","department":[{"_id":"HeEd"}],"publisher":"SIAM","year":"2015","publist_id":"5423","ec_funded":1,"date_published":"2015-07-14T00:00:00Z","page":"2754 - 2769","publication":"Society for Industrial and Applied Mathematics","citation":{"chicago":"Akopyan, Arseniy, and Alexander Plakhov. “Minimal Resistance of Curves under the Single Impact Assumption.” Society for Industrial and Applied Mathematics. SIAM, 2015. https://doi.org/10.1137/140993843.","short":"A. Akopyan, A. Plakhov, Society for Industrial and Applied Mathematics 47 (2015) 2754–2769.","mla":"Akopyan, Arseniy, and Alexander Plakhov. “Minimal Resistance of Curves under the Single Impact Assumption.” Society for Industrial and Applied Mathematics, vol. 47, no. 4, SIAM, 2015, pp. 2754–69, doi:10.1137/140993843.","ieee":"A. Akopyan and A. Plakhov, “Minimal resistance of curves under the single impact assumption,” Society for Industrial and Applied Mathematics, vol. 47, no. 4. SIAM, pp. 2754–2769, 2015.","apa":"Akopyan, A., & Plakhov, A. (2015). Minimal resistance of curves under the single impact assumption. Society for Industrial and Applied Mathematics. SIAM. https://doi.org/10.1137/140993843","ista":"Akopyan A, Plakhov A. 2015. Minimal resistance of curves under the single impact assumption. Society for Industrial and Applied Mathematics. 47(4), 2754–2769.","ama":"Akopyan A, Plakhov A. Minimal resistance of curves under the single impact assumption. Society for Industrial and Applied Mathematics. 2015;47(4):2754-2769. doi:10.1137/140993843"},"day":"14","scopus_import":1,"oa_version":"Preprint","status":"public","title":"Minimal resistance of curves under the single impact assumption","intvolume":" 47","_id":"1710","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"We consider the hollow on the half-plane {(x, y) : y ≤ 0} ⊂ ℝ2 defined by a function u : (-1, 1) → ℝ, u(x) < 0, and a vertical flow of point particles incident on the hollow. It is assumed that u satisfies the so-called single impact condition (SIC): each incident particle is elastically reflected by graph(u) and goes away without hitting the graph of u anymore. We solve the problem: find the function u minimizing the force of resistance created by the flow. We show that the graph of the minimizer is formed by two arcs of parabolas symmetric to each other with respect to the y-axis. Assuming that the resistance of u ≡ 0 equals 1, we show that the minimal resistance equals π/2 - 2arctan(1/2) ≈ 0.6435. This result completes the previously obtained result [SIAM J. Math. Anal., 46 (2014), pp. 2730-2742] stating in particular that the minimal resistance of a hollow in higher dimensions equals 0.5. We additionally consider a similar problem of minimal resistance, where the hollow in the half-space {(x1,...,xd,y) : y ≤ 0} ⊂ ℝd+1 is defined by a radial function U satisfying the SIC, U(x) = u(|x|), with x = (x1,...,xd), u(ξ) < 0 for 0 ≤ ξ < 1, and u(ξ) = 0 for ξ ≥ 1, and the flow is parallel to the y-axis. The minimal resistance is greater than 0.5 (and coincides with 0.6435 when d = 1) and converges to 0.5 as d → ∞.","lang":"eng"}],"issue":"4","type":"journal_article"},{"ec_funded":1,"publist_id":"5276","year":"2015","publication_status":"published","publisher":"Springer","department":[{"_id":"HeEd"}],"author":[{"first_name":"Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy"},{"first_name":"Sergey","last_name":"Pirogov","full_name":"Pirogov, Sergey"},{"full_name":"Rybko, Aleksandr","first_name":"Aleksandr","last_name":"Rybko"}],"date_updated":"2021-01-12T06:53:28Z","date_created":"2018-12-11T11:54:14Z","volume":160,"month":"07","main_file_link":[{"open_access":"1","url":"arxiv.org/abs/1406.5313"}],"oa":1,"quality_controlled":"1","project":[{"name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"doi":"10.1007/s10955-015-1238-5","language":[{"iso":"eng"}],"type":"journal_article","abstract":[{"text":"We construct a non-linear Markov process connected with a biological model of a bacterial genome recombination. The description of invariant measures of this process gives us the solution of one problem in elementary probability theory.","lang":"eng"}],"issue":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"1828","title":"Invariant measures of genetic recombination process","status":"public","intvolume":" 160","oa_version":"Preprint","scopus_import":1,"day":"01","article_processing_charge":"No","publication":"Journal of Statistical Physics","citation":{"chicago":"Akopyan, Arseniy, Sergey Pirogov, and Aleksandr Rybko. “Invariant Measures of Genetic Recombination Process.” Journal of Statistical Physics. Springer, 2015. https://doi.org/10.1007/s10955-015-1238-5.","mla":"Akopyan, Arseniy, et al. “Invariant Measures of Genetic Recombination Process.” Journal of Statistical Physics, vol. 160, no. 1, Springer, 2015, pp. 163–67, doi:10.1007/s10955-015-1238-5.","short":"A. Akopyan, S. Pirogov, A. Rybko, Journal of Statistical Physics 160 (2015) 163–167.","ista":"Akopyan A, Pirogov S, Rybko A. 2015. Invariant measures of genetic recombination process. Journal of Statistical Physics. 160(1), 163–167.","ieee":"A. Akopyan, S. Pirogov, and A. Rybko, “Invariant measures of genetic recombination process,” Journal of Statistical Physics, vol. 160, no. 1. Springer, pp. 163–167, 2015.","apa":"Akopyan, A., Pirogov, S., & Rybko, A. (2015). Invariant measures of genetic recombination process. Journal of Statistical Physics. Springer. https://doi.org/10.1007/s10955-015-1238-5","ama":"Akopyan A, Pirogov S, Rybko A. Invariant measures of genetic recombination process. Journal of Statistical Physics. 2015;160(1):163-167. doi:10.1007/s10955-015-1238-5"},"page":"163 - 167","date_published":"2015-07-01T00:00:00Z"}]