[{"publication":"Canadian Mathematical Bulletin","oa_version":"Preprint","month":"12","language":[{"iso":"eng"}],"type":"journal_article","date_published":"2023-12-01T00:00:00Z","publication_identifier":{"issn":["0008-4395"],"eissn":["1496-4287"]},"oa":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2106.07328","open_access":"1"}],"status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","_id":"13128","issue":"4","author":[{"first_name":"Ali","last_name":"Mohammadi","full_name":"Mohammadi, Ali"},{"full_name":"Pham, Thang","last_name":"Pham","first_name":"Thang"},{"last_name":"Wang","first_name":"Yiting","full_name":"Wang, Yiting","orcid":"0000-0002-2856-767X","id":"1917d194-076e-11ed-97cd-837255f88785"}],"date_created":"2023-06-11T22:00:40Z","article_processing_charge":"No","department":[{"_id":"GradSch"}],"publication_status":"published","intvolume":"        66","title":"An energy decomposition theorem for matrices and related questions","quality_controlled":"1","page":"1280-1295","publisher":"Cambridge University Press","article_type":"original","year":"2023","citation":{"ista":"Mohammadi A, Pham T, Wang Y. 2023. An energy decomposition theorem for matrices and related questions. Canadian Mathematical Bulletin. 66(4), 1280–1295.","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>.","short":"A. Mohammadi, T. Pham, Y. Wang, Canadian Mathematical Bulletin 66 (2023) 1280–1295.","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>.","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.","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>","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>"},"date_updated":"2024-01-29T11:00:46Z","external_id":{"arxiv":["2106.07328"],"isi":["001011963000001"]},"isi":1,"day":"01","arxiv":1,"doi":"10.4153/S000843952300036X","abstract":[{"text":"Given  A⊆GL2(Fq), we prove that there exist disjoint subsets  B,C⊆A such that  A=B⊔C and their additive and multiplicative energies satisfying max{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 We also study some related questions on moderate expanders over matrix rings, namely, for  A,B,C⊆GL2(Fq), we have |AB+C|, |(A+B)C|≫q4, 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,  ForumMath., 31, 951–970).\r\n","lang":"eng"}],"volume":66},{"year":"2021","citation":{"ista":"Ivanov G. 2021. Tight frames and related geometric problems. Canadian Mathematical Bulletin. 64(4), 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>.","short":"G. Ivanov, Canadian Mathematical Bulletin 64 (2021) 942–963.","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>.","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.","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>","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>"},"date_updated":"2023-09-05T12:43:09Z","external_id":{"isi":["000730165300021"],"arxiv":["1804.10055"]},"isi":1,"day":"18","doi":"10.4153/s000843952000096x","arxiv":1,"abstract":[{"lang":"eng","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."}],"volume":64,"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.","scopus_import":"1","_id":"10860","issue":"4","author":[{"full_name":"Ivanov, Grigory","last_name":"Ivanov","first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E"}],"department":[{"_id":"UlWa"}],"article_processing_charge":"No","date_created":"2022-03-18T09:55:59Z","publication_status":"published","intvolume":"        64","title":"Tight frames and related geometric problems","quality_controlled":"1","page":"942-963","publisher":"Canadian Mathematical Society","article_type":"original","type":"journal_article","date_published":"2021-12-18T00:00:00Z","publication_identifier":{"eissn":["1496-4287"],"issn":["0008-4395"]},"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1804.10055","open_access":"1"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","status":"public","publication":"Canadian Mathematical Bulletin","oa_version":"Preprint","month":"12","keyword":["General Mathematics","Tight frame","Grassmannian","zonotope"],"language":[{"iso":"eng"}]}]