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Let n ≥ 1 be a fixed integer and R be a ring with involution ' ∗ '. For any two elements x and y in R, the n-skew Lie product and n-skew Jordan product are given by ▿ [ x , y ] n = ▿ [ x , ▿ [ x , y ] n - 1 ] with ▿ [ x , y ] 0 = y , ▿ [ x , y ] 1 = ▿ [ x , y ] = x y - y x ∗ , ▿ [ x , y ] 2 = x 2 y - 2 x y x ∗ + y (x ∗) 2 and x ⋄ n y = x ⋄ (x ⋄ n - 1 y) with x ⋄ 0 y = y , x ⋄ 1 y = x ⋄ y = x y + y x ∗ , x ⋄ 2 y = x 2 y + 2 x y x ∗ + y (x ∗) 2 . The purpose of this paper is to characterize Jordan left ∗ -centralizers satisfying certain functional identities involving skew Lie product and skew Jordan products. In particular, it is proved that if R is a 2-torsion free prime ring with involution of the second kind admits a non-zero Jordan left ∗ -centralizer T such that ▿ [ x , T (x) ] n ∈ Z (R) (n = 1 , 2) for all x ∈ R , then T (x) = λ x ∗ for all x ∈ R , where λ ∈ C , the extended centroid of R. We also characterize Jordan left ∗ -centralizers of prime rings with involution via skew Jordan product. [ABSTRACT FROM AUTHOR]

Let H and K be complex Hilbert spaces.Denote by B(H) and B(K) the algebras of all bounded linear operators on H and K, respectively. In this paper, we characterize that nonlinear map φ : B(H) → B(K) that preserves the higher-dimensional numerical ranges of Jordan *-products. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space with dim H ≥ 3, B s (H) be the Jordan algebra of all bounded self-adjoint operators on H, and let F : B (H) → [ d , ∞ ] with d ≥ 0 be a radial unitary similarity invariant function. In this paper, a characterization is obtained for maps ϕ on B s (H) satisfying F (ϕ (A) ϕ (B) ϕ (A)) = F (A B A) (A , B ∈ B s (H)) . As applications, a general form of the maps on B s (H) preserving the p-norm, the maps preserving the pseudo-spectral radius and the maps preserving the pseudo-spectrum are also described. [ABSTRACT FROM AUTHOR]

The existence of complementarity constraints causes the difficulties for studying mathematical programs with second-order cone complementarity constraints, since the standard constraint qualification, such as Robinson's constraint qualification, is invalid. Therefore, various stationary conditions including strong, Mordukhovich and Clarke stationary conditions have been proposed, according to different reformulations of the second-order cone complementarity constraints. In this paper, we present a new reformulation of this problem by taking into consideration the Jordan algebra associated with the second-order cone. It ensures that the classical Karush–Kuhn–Tucker condition coincides with the strong stationary condition of the original problem. Furthermore, we propose a class of approximation methods to solve mathematical programs with second-order cone complementarity constraints. Any accumulation point of the iterative sequences, generated by the approximation method, is Clarke stationary under the corresponding linear independence constraint qualification. This stationarity can be enhanced to strong stationarity with an extra strict complementarity condition. Preliminary numerical experiments indicate that the proposed method is effective. [ABSTRACT FROM AUTHOR]

Inspired by the pioneer work of H.L. Resnikoff, which is described in full detail in the first part of this two-part paper, we give a quantum description of the space P of perceived colors. We show that P is the effect space of a rebit, a real quantum qubit, whose state space is isometric to Klein's hyperbolic disk. This chromatic state space of perceived colors can be represented as a Bloch disk of real dimension 2 that coincides with Hering's disk given by the color opponency mechanism. Attributes of perceived colors, hue and saturation, are defined in terms of Von Neumann entropy. [ABSTRACT FROM AUTHOR]