[{"oa":1,"_id":"13128","arxiv":1,"issue":"4","type":"journal_article","author":[{"full_name":"Mohammadi, Ali","last_name":"Mohammadi","first_name":"Ali"},{"first_name":"Thang","full_name":"Pham, Thang","last_name":"Pham"},{"id":"1917d194-076e-11ed-97cd-837255f88785","full_name":"Wang, Yiting","last_name":"Wang","orcid":"0000-0002-2856-767X","first_name":"Yiting"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","status":"public","date_created":"2023-06-11T22:00:40Z","language":[{"iso":"eng"}],"publication":"Canadian Mathematical Bulletin","page":"1280-1295","intvolume":"        66","citation":{"ieee":"A. Mohammadi, T. Pham, and Y. Wang, “An energy decomposition theorem for matrices and related questions,” <i>Canadian Mathematical Bulletin</i>, vol. 66, no. 4. Cambridge University Press, pp. 1280–1295, 2023.","mla":"Mohammadi, Ali, et al. “An Energy Decomposition Theorem for Matrices and Related Questions.” <i>Canadian Mathematical Bulletin</i>, vol. 66, no. 4, Cambridge University Press, 2023, pp. 1280–95, doi:<a href=\"https://doi.org/10.4153/S000843952300036X\">10.4153/S000843952300036X</a>.","ista":"Mohammadi A, Pham T, Wang Y. 2023. An energy decomposition theorem for matrices and related questions. Canadian Mathematical Bulletin. 66(4), 1280–1295.","ama":"Mohammadi A, Pham T, Wang Y. An energy decomposition theorem for matrices and related questions. <i>Canadian Mathematical Bulletin</i>. 2023;66(4):1280-1295. doi:<a href=\"https://doi.org/10.4153/S000843952300036X\">10.4153/S000843952300036X</a>","chicago":"Mohammadi, Ali, Thang Pham, and Yiting Wang. “An Energy Decomposition Theorem for Matrices and Related Questions.” <i>Canadian Mathematical Bulletin</i>. Cambridge University Press, 2023. <a href=\"https://doi.org/10.4153/S000843952300036X\">https://doi.org/10.4153/S000843952300036X</a>.","short":"A. Mohammadi, T. Pham, Y. Wang, Canadian Mathematical Bulletin 66 (2023) 1280–1295.","apa":"Mohammadi, A., Pham, T., &#38; Wang, Y. (2023). An energy decomposition theorem for matrices and related questions. <i>Canadian Mathematical Bulletin</i>. Cambridge University Press. <a href=\"https://doi.org/10.4153/S000843952300036X\">https://doi.org/10.4153/S000843952300036X</a>"},"year":"2023","date_published":"2023-12-01T00:00:00Z","publication_status":"published","doi":"10.4153/S000843952300036X","publication_identifier":{"eissn":["1496-4287"],"issn":["0008-4395"]},"article_processing_charge":"No","publisher":"Cambridge University Press","department":[{"_id":"GradSch"}],"month":"12","date_updated":"2026-04-08T13:04:49Z","title":"An energy decomposition theorem for matrices and related questions","isi":1,"day":"01","oa_version":"Preprint","volume":66,"external_id":{"arxiv":["2106.07328"],"isi":["001011963000001"]},"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2106.07328"}],"abstract":[{"lang":"eng","text":"Given 𝐴 ⊆𝐺⁡𝐿2⁡(𝔽𝑞), we prove that there exist disjoint subsets 𝐵,𝐶 ⊆𝐴 such that 𝐴 =𝐵 ⊔𝐶 and their additive and multiplicative energies satisfying\r\nmax⁡{𝐸+⁡(𝐵),𝐸×⁡(𝐶)}≪|𝐴|3/𝑀⁡(|𝐴|), where\r\n𝑀⁡(|𝐴|)=min⁡{𝑞4/3/|𝐴|1/3⁢(log⁡|𝐴|)2/3, |𝐴|4/5/𝑞13/5⁢(log⁡|𝐴|)27/10}.\r\n \r\nWe also study some related questions on moderate expanders over matrix rings, namely, for 𝐴,𝐵,𝐶 ⊆𝐺⁡𝐿2⁡(𝔽𝑞), we have\r\n|𝐴⁢𝐵+𝐶|, |(𝐴+𝐵)⁢𝐶|≫𝑞4,\r\n whenever |𝐴|⁢|𝐵|⁢|𝐶| ≫𝑞10+1/2. These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh ([2019], Expanding phenomena over matrix rings, 𝐹⁡𝑜⁢𝑟⁢𝑢⁢𝑚⁢𝑀⁢𝑎⁢𝑡⁢ℎ., 31, 951–970)."}],"article_type":"original"},{"date_created":"2022-03-18T09:55:59Z","scopus_import":"1","keyword":["General Mathematics","Tight frame","Grassmannian","zonotope"],"status":"public","publication":"Canadian Mathematical Bulletin","language":[{"iso":"eng"}],"intvolume":"        64","page":"942-963","oa":1,"arxiv":1,"_id":"10860","corr_author":"1","issue":"4","author":[{"last_name":"Ivanov","full_name":"Ivanov, Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","first_name":"Grigory"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","acknowledgement":"The author was supported by the Swiss National Science Foundation grant 200021_179133. The author acknowledges the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no. 075-15-2019-1926.","type":"journal_article","oa_version":"Preprint","day":"18","volume":64,"quality_controlled":"1","external_id":{"isi":["000730165300021"],"arxiv":["1804.10055"]},"article_type":"original","abstract":[{"text":"A tight frame is the orthogonal projection of some orthonormal basis of Rn onto Rk. We show that a set of vectors is a tight frame if and only if the set of all cross products of these vectors is a tight frame. We reformulate a range of problems on the volume of projections (or sections) of regular polytopes in terms of tight frames and write a first-order necessary condition for local extrema of these problems. As applications, we prove new results for the problem of maximization of the volume of zonotopes.","lang":"eng"}],"main_file_link":[{"url":"https://arxiv.org/abs/1804.10055","open_access":"1"}],"year":"2021","date_published":"2021-12-18T00:00:00Z","citation":{"chicago":"Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” <i>Canadian Mathematical Bulletin</i>. Canadian Mathematical Society, 2021. <a href=\"https://doi.org/10.4153/s000843952000096x\">https://doi.org/10.4153/s000843952000096x</a>.","ama":"Ivanov G. Tight frames and related geometric problems. <i>Canadian Mathematical Bulletin</i>. 2021;64(4):942-963. doi:<a href=\"https://doi.org/10.4153/s000843952000096x\">10.4153/s000843952000096x</a>","apa":"Ivanov, G. (2021). Tight frames and related geometric problems. <i>Canadian Mathematical Bulletin</i>. Canadian Mathematical Society. <a href=\"https://doi.org/10.4153/s000843952000096x\">https://doi.org/10.4153/s000843952000096x</a>","short":"G. Ivanov, Canadian Mathematical Bulletin 64 (2021) 942–963.","mla":"Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” <i>Canadian Mathematical Bulletin</i>, vol. 64, no. 4, Canadian Mathematical Society, 2021, pp. 942–63, doi:<a href=\"https://doi.org/10.4153/s000843952000096x\">10.4153/s000843952000096x</a>.","ieee":"G. Ivanov, “Tight frames and related geometric problems,” <i>Canadian Mathematical Bulletin</i>, vol. 64, no. 4. Canadian Mathematical Society, pp. 942–963, 2021.","ista":"Ivanov G. 2021. Tight frames and related geometric problems. Canadian Mathematical Bulletin. 64(4), 942–963."},"article_processing_charge":"No","publisher":"Canadian Mathematical Society","publication_identifier":{"issn":["0008-4395"],"eissn":["1496-4287"]},"doi":"10.4153/s000843952000096x","publication_status":"published","date_updated":"2024-10-09T21:01:50Z","department":[{"_id":"UlWa"}],"month":"12","isi":1,"title":"Tight frames and related geometric problems"}]