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Дискретная математика и математическая кибернетика
Multidimensional threshold matrices and extremal matrices of order $2$
A. A. Taranenko Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
Аннотация:
The paper is devoted to multidimensional $(0,1)$-matrices extremal with respect to containing a polydiagonal (a fractional generalization of a diagonal). Every extremal matrix is a threshold matrix, i.e., an entry belongs to its support whenever a weighted sum of incident hyperplanes exceeds a given threshold.
Firstly, we prove that nonequivalent threshold matrices have different distributions of ones in hyperplanes. Next, we establish that extremal matrices of order $2$ are exactly selfdual threshold Boolean functions. Using this fact, we find the asymptotics of the number of extremal matrices of order $2$ and provide counterexamples to several conjectures on extremal matrices. Finally, we describe extremal matrices of order $2$ with a small diversity of hyperplanes.
Ключевые слова:
multidimensional matrix, extremal matrix, threshold matrix, selfdual Boolean function.
Поступила 3 апреля 2023 г., опубликована 14 ноября 2023 г.
Образец цитирования:
A. A. Taranenko, “Multidimensional threshold matrices and extremal matrices of order $2$”, Сиб. электрон. матем. изв., 20:2 (2023), 1052–1063
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/semr1628 https://www.mathnet.ru/rus/semr/v20/i2/p1052
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