---
_id: '13128'
abstract:
- lang: eng
  text: "Given \U0001D434 ⊆\U0001D43A⁡\U0001D43F2⁡(\U0001D53D\U0001D45E), we prove
    that there exist disjoint subsets \U0001D435,\U0001D436 ⊆\U0001D434 such that
    \U0001D434 =\U0001D435 ⊔\U0001D436 and their additive and multiplicative energies
    satisfying\r\nmax⁡{\U0001D438+⁡(\U0001D435),\U0001D438×⁡(\U0001D436)}≪|\U0001D434|3/\U0001D440⁡(|\U0001D434|),
    where\r\n\U0001D440⁡(|\U0001D434|)=min⁡{\U0001D45E4/3/|\U0001D434|1/3⁢(log⁡|\U0001D434|)2/3,
    |\U0001D434|4/5/\U0001D45E13/5⁢(log⁡|\U0001D434|)27/10}.\r\n \r\nWe also study
    some related questions on moderate expanders over matrix rings, namely, for \U0001D434,\U0001D435,\U0001D436
    ⊆\U0001D43A⁡\U0001D43F2⁡(\U0001D53D\U0001D45E), we have\r\n|\U0001D434⁢\U0001D435+\U0001D436|,
    |(\U0001D434+\U0001D435)⁢\U0001D436|≫\U0001D45E4,\r\n whenever |\U0001D434|⁢|\U0001D435|⁢|\U0001D436|
    ≫\U0001D45E10+1/2. These improve earlier results due to Karabulut, Koh, Pham,
    Shen, and Vinh ([2019], Expanding phenomena over matrix rings, \U0001D439⁡\U0001D45C⁢\U0001D45F⁢\U0001D462⁢\U0001D45A⁢\U0001D440⁢\U0001D44E⁢\U0001D461⁢ℎ.,
    31, 951–970)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Ali
  full_name: Mohammadi, Ali
  last_name: Mohammadi
- first_name: Thang
  full_name: Pham, Thang
  last_name: Pham
- first_name: Yiting
  full_name: Wang, Yiting
  id: 1917d194-076e-11ed-97cd-837255f88785
  last_name: Wang
  orcid: 0000-0002-2856-767X
citation:
  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>
  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.
  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.
date_created: 2023-06-11T22:00:40Z
date_published: 2023-12-01T00:00:00Z
date_updated: 2026-04-08T13:04:49Z
day: '01'
department:
- _id: GradSch
doi: 10.4153/S000843952300036X
external_id:
  arxiv:
  - '2106.07328'
  isi:
  - '001011963000001'
intvolume: '        66'
isi: 1
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2106.07328
month: '12'
oa: 1
oa_version: Preprint
page: 1280-1295
publication: Canadian Mathematical Bulletin
publication_identifier:
  eissn:
  - 1496-4287
  issn:
  - 0008-4395
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: An energy decomposition theorem for matrices and related questions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 66
year: '2023'
...
---
_id: '10860'
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.
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.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Grigory
  full_name: Ivanov, Grigory
  id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
  last_name: Ivanov
citation:
  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>
  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.
  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.
corr_author: '1'
date_created: 2022-03-18T09:55:59Z
date_published: 2021-12-18T00:00:00Z
date_updated: 2024-10-09T21:01:50Z
day: '18'
department:
- _id: UlWa
doi: 10.4153/s000843952000096x
external_id:
  arxiv:
  - '1804.10055'
  isi:
  - '000730165300021'
intvolume: '        64'
isi: 1
issue: '4'
keyword:
- General Mathematics
- Tight frame
- Grassmannian
- zonotope
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1804.10055
month: '12'
oa: 1
oa_version: Preprint
page: 942-963
publication: Canadian Mathematical Bulletin
publication_identifier:
  eissn:
  - 1496-4287
  issn:
  - 0008-4395
publication_status: published
publisher: Canadian Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tight frames and related geometric problems
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 64
year: '2021'
...