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31.
Permanent versus determinant over a finite field
Gregor Dolinar, Aleksandr Èmilevič Guterman, Bojan Kuzma, Marko Orel, 2013, published scientific conference contribution

Abstract: Let ▫$\mathbb{F}$▫ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices ▫$\mathcal{H}_n(\mathbb{F})$▫ and for the whole matrix space ▫$M_n(\mathbb{F})$▫. It is known that for ▫$n = 2$▫, there are bijective linear maps ▫$\Phi$▫ on ▫$\mathcal{H}_n(\mathbb{F})$▫ and ▫$M_n(\mathbb{F})$▫ satisfying the condition per ▫$A = \det \Phi(A)$▫. As an application of the obtained results, we show that if ▫$n \ge 3$▫, then the situation is completely different and already for ▫$n = 3$▫, there is no pair ofmaps ▫$(\Phi, \phi)$▫, where ▫$\Phi$▫ is an arbitrary bijective map on matrices and ▫$\phi \colon \mathbb{F} \to \mathbb{F}$▫ is an arbitrary map such that per ▫$A = \phi(\det \Phi(A))$▫ for all matrices ▫$A$▫ from the spaces ▫$\mathcal{H}_n(\mathbb{F})$▫ and ▫$M_n(\mathbb{F})$▫, respectively. Moreover, for the space ▫$M_n(\mathbb{F})$▫, we show that such a pair of transformations does not exist also for an arbitrary ▫$n > 3$▫ if the field ▫$\mathbb{F}$▫ contains sufficiently many elements (depending on ▫$n$▫). Our results are illustrated by a number of examples.
Keywords: mathematics, linear algebra, matrix theory, permanent, determinant
Published in RUP: 03.04.2017; Views: 2030; Downloads: 123
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32.
Ohranjevalci na Banachovih algebrah
Ajda Fošner, 2009, scientific monograph

Keywords: Banachova algebra, ohranjevalci obrnljivosti
Published in RUP: 21.12.2015; Views: 1813; Downloads: 62
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Prime and semiprime rings with symmetric skew 3-derivations
Ajda Fošner, 2013, original scientific article

Abstract: In this paper we introduce the notion of symmetric skew 3-derivations of prime or semiprime rings and prove that under certain conditions a prime ring with a nonzero symmetric skew 3-derivation has to be commutative.
Keywords: algebra, prime ring, semiprime ring, symmetric skew 3-derivation, centralizing mapping, commuting mapping
Published in RUP: 15.10.2013; Views: 3727; Downloads: 138
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Noninvertibility preservers on Banach algebras
Bojan Kuzma, 2006, short scientific article

Abstract: It is proved that a linear surjection ▫$\Phi: \mathcal{A} \to \mathcal{B}$▫, which preserves noninvertibility between semisimple, unital, complex Banach algebras, is automatically injective.
Keywords: mathematics, functional analysis, linear preserver, noninvertible element, semisimple Banach algebra, socle
Published in RUP: 15.10.2013; Views: 3028; Downloads: 126
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