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Abstract. In this paper, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn × mn Hermitian matrices such that φ(A ⊗ B) and A ⊗ B have the same spectrum for any m × m Hermitian A and n × n Hermitian B. Such a map has the form A ⊗ B → U (ϕ1(A) ⊗ ϕ2(B))U * for mn × mn Hermitian matrices in tensor form A ⊗ B, where U is a unitary matrix, and for j ∈ {1, 2}, ϕj is the identity map X → X or the transposition map X → X t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A ⊗ B is also obtained. The results are connected bipartite (quantum) systems and are extended to multipartite systems.2010 Math. Subj. Class.: 15A69, 15A86, 15B57, 15A18.

Motivated by the powerful and elegant works of Miers (1971Miers ( , 1973Miers ( , 1978 we mainly study nonlinear Lie-type derivations of von Neumann algebras. Let A be a von Neumann algebra without abelian central summands of type I1. It is shown that every nonlinear Lie n-derivation of A has the standard form, that is, can be expressed as a sum of an additive derivation and a central-valued mapping which annihilates each (n − 1)th commutator of A. Several potential research topics related to our work are also presented.

We consider an optimization problem given by a discrete inclusion, whose trajectories are constrained to closed sets. Necessary optimality conditions in the form of the maximum principle and in terms of local generalized derived cones of constraints are obtained.

Abstract. The aim of this paper is to define generalized (m, n)-Jordan centralizers and to prove that on a prime ring with nonzero center and char(R) = 6mn(m+n)(m+2n) every generalized (m, n)-Jordan centralizer is a two-sided centralizer.Throughout, R will represent an associative ring with a center Z(R). Let n ≥ 2 be an integer. A ring R is said to be n-torsion free if for x ∈ R, nx = 0 implies x = 0. Recall that R is prime if aRb = {0} implies a = 0 oris fulfilled for all x ∈ R. One can easily prove that every derivation is a Jordan derivation, but converse is in general not true. A classical result due to Herstein [5, Theorem 3.3] asserts that a Jordan derivation on a prime ring of characteristic different from two is a derivation. A brief proof of Herstein's result can be found in [3]. This result was extended to 2-torsion free semiprime rings by Cusack [4] (see [2] for an alternative proof).An additive mapping T : R → R is called a left (right) centralizer if T (xy) = T (x)y (T (xy) = xT (y)) holds for all pairs x, y ∈ R. If R has the identity element, then T : R → R is a left centralizer if and only if T is of the form T (x) = ax for all x ∈ R, where a ∈ R is a fixed element. For a semiprime ring R, left centralizers are of the form T (x) = qx for all x ∈ R, where q is a fixed element of a Martindale right ring of quotients Q r (see, for example, Chapter 2 in [1]). An additive mapping T : R → R is called a left (right) Jordan centralizer if T (x 2 ) = T (x)x (T (x 2 ) = xT (x)) holds for all x ∈ R. We call an additive mapping T : R → R a two-sided centralizer 2010 Mathematics Subject Classification: 16N60, 39B05.