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We extend the concept of a G-Drazin inverse from the set M n of all n × n complex matrices to the set R D of all Drazin invertible elements in a ring R with identity. We also generalize a partial order induced by G-Drazin inverses from M n to the set of all regular elements in R D , study its properties, compare it to known partial orders, and generalize some known results. [ABSTRACT FROM AUTHOR]

We classify maps which preserve orthogonality on the Cayley projective plane over octonions. In addition, we also classify orthogonality preserving maps on finite dimensional projective spaces over reals, complexes, or quaternions. Unlike similar results which extend Uhlhorns's theorem we assume neither injectivity/surjectivity nor that orthogonality is preserved in both directions. [ABSTRACT FROM AUTHOR]

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 the maps that transform one immanant into another. No surjectivity or linearity is imposed; the rudiments of the former are weakly embedded into the functional equation via {ie874-01}. We show that this property alone implies that Φ is linear and bijective. [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]

We study non-linear maps on the set of n × n matrices over a field, which preserve the zeros of a fixed homogeneous multilinear polynomial. This solves Kaplansky-Watkins problem from 1976 about the structure of such transformations. [ABSTRACT FROM AUTHOR]