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Let 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 [H.sub.n](F) and for the whole matrix space [M.sub.n](F). It is known that for n = 2, there are bijective linear maps [PHI] on [H.sub.n](F) and [M.sub.n](F) satisfying the condition per A = det [PHI](A). As an application of the obtained results, we show that if n [greater than or equal to] 3, then the situation is completely different and already for n = 3, there is no pair of maps ([PHI],[phi]), where [PHI] is an arbitrary bijective map on matrices and [phi]: F [right arrow] F is an arbitrary map such that per A = [phi](det [PHI](A)) for all matrices A from the spaces [H.sub.n](F) and [M.sub.n](F), respectively. Moreover, for the space [M.sub.n](F), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field F contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples., UDC 512.5 1. Introduction The permanent and the determinant of an (n x n)-matrix A = ([a.sub.ij]) with real entries [a.sub.ij] are defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [...]

We study lower bounds on the number of nonzero entries in (0, 1) matrices such that the permanent is always convertible to the determinant by placing [+ or -] signs on matrix entries., UDC 512.643 1. Introduction The permanent and the determinant of an n x n matrix A = ([a.sub.ij]) with real entries [a.sub.ij] are defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN [...]

We study nonlinear surjective mappings on Mn() and its subsets, which preserve the zeros of some fixed polynomials in noncommuting variables. [ABSTRACT FROM AUTHOR]

Additive mappings, which do not increase the minimal rank of alternate matrices, are completely classified. No condition is imposed on the underlying field. [ABSTRACT FROM AUTHOR]

Abstract: Additive bijections , which compress the spectrum between two unital, standard operator algebras, are characterized. Applications to local approximate (anti)multiplications are also given. [Copyright &y& Elsevier]

We study continuous maps on alternate matrices over complex field which preserve zeros of Lie product.