New Estimates of Numerical Values Related to a Simplex
Let \(n\in {\mathbb N}\) and \(Q_n=[0,1]^n\). For a nondegenerate simplex \(S\subset {\mathbb R}^n\), by \(\sigma S\) we denote the homothetic copy of~\(S\) with center of homothety in the center of gravity of \(S\) and ratio of~homothety \(\sigma\). By \(\xi(S)\) we mean the minimal \(\sigma>0\...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Yaroslavl State University
2017-02-01
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| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/428 |
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| Summary: | Let \(n\in {\mathbb N}\) and \(Q_n=[0,1]^n\). For a nondegenerate simplex \(S\subset {\mathbb R}^n\), by \(\sigma S\) we denote the homothetic copy of~\(S\) with center of homothety in the center of gravity of \(S\) and ratio of~homothety \(\sigma\). By \(\xi(S)\) we mean the minimal \(\sigma>0\) such that \(Q_n\subset \sigma S\). By \(\alpha(S)\) denote the minimal \(\sigma>0\) such that \(Q_n\) is~contained in a translate of~\(\sigma S\). By \(d_i(S)\) we denote the \(i\)th axial diameter of \(S\), i.\,e. the maximum length of~the segment contained in \(S\) and parallel to the \(i\)th coordinate axis. Formulae for~\(\xi(S)\), \(\alpha(S)\), \(d_i(S)\) were proved earlier by the first author. Define \(\xi_n=\min\{ \xi(S): S\subset Q_n\}. \) We always have \(\xi_n\geq n.\) We discuss some conjectures formulated in the previous papers. One of~these conjectures is the following. For~every \(n\), there exists \(\gamma>0\), not depending on \(S\subset Q_n\), such that an~inequality \(\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n)\) holds. Denote by \(\varkappa_n\) the minimal \(\gamma\) with such a~property. We prove that \(\varkappa_1=\frac{1}{2}\); for \(n>1\), we obtain \(\varkappa_n\geq 1\). If \(n>1\) and \(\xi_n=n,\) then \(\varkappa_n=1\). The equality \(\xi_n=n\) holds if \(n+1\) is an Hadamard number, i.\,e. there exists an Hadamard matrix of~order \(n+1\). This proposition is known; we give one more proof with the direct use of Hadamard matrices. We prove that \(\xi_5=5\). Therefore, there exists \(n\) such that \(n+1\) is not an Hadamard number and nevertheless \(\xi_n=n\). The~minimal \(n\) with such a property is equal to \(5\). This involves \(\varkappa_5=1\) and also disproves the following previous conjecture of the first author concerning the characterization of Hadamard numbers in terms of~homothety of simplices: \(n+1\) is an Hadamard number if and only if \(\xi_n=n.\) This statement is valid only in one direction. There exists a simplex \(S\subset Q_5\) such that the boundary of the simplex \(5S\) contains all the vertices of the cube \(Q_5\). We describe a one-parameter family of simplices contained in \(Q_5\) with the property \(\alpha(S)=\xi(S)=5.\) These simplices were found with the use of numerical and symbolic computations. %Numerical experiments allow to discover Another new result is an inequality \(\xi_6\ |
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| ISSN: | 1818-1015 2313-5417 |