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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 $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and for the whole matrix space M( $$ \mathbb{F} $$). It is known that for n = 2, there are bijective linear maps Φ on $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and M( $$ \mathbb{F} $$) satisfying the condition per A = det Φ( A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ , ϕ), where Φ is an arbitrary bijective map on matrices and ϕ : $$ \mathbb{F} $$ → $$ \mathbb{F} $$ is an arbitrary map such that per A = ϕ(det Φ( A)) for all matrices A from the spaces $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and M( $$ \mathbb{F} $$), respectively. Moreover, for the space M( $$ \mathbb{F} $$), 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. [ABSTRACT FROM AUTHOR]

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 ± signs on matrix entries. [ABSTRACT FROM AUTHOR]

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

Suppose [image omitted] is a field with proper involution and of arbitrary characteristic. Additive maps, which do not increase rank-one on hermitian matrices with entries from [image omitted] are classified. The result is then used to classify additive maps that do not increase minimal rank on the set of symmetric elements, relative to general involution on a matrix algebra. [ABSTRACT FROM AUTHOR]

A complete characterization of additive mappings Φ : ℬ( $${\fancyscript X}$$ ) → ℬ( $${\fancyscript Y}$$ ), which map rank–one idempotents to themselves or to zero, is given. [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]

Abstract: An operator T on the separable infinite-dimensional Hilbert space is constructed so that the commutant of every operator which is not a scalar multiple of the identity operator and commutes with T coincides with the commutant of T. On the other hand, it is shown that for several classes of operators it is possible to construct a finite sequence of operators, starting at a given operator from the class and ending in a rank-one projection such that each operator in the sequence commutes with its predecessor. The classes which we study are: finite-rank operators, normal operators, partial isometries, and contractions. It is also shown that for any given set of yes/no conditions between points in some finite set, there always exist operators on a finite-dimensional Hilbert space such that their commutativity relations exactly satisfy those conditions. [Copyright &y& Elsevier]