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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 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]

If the cosquares of complex matrices and are similar and there is a unimodular number in the spectrum of the cosquares, then and are not necessarily congruent. Assume that such an eigenvalue is unique. In this case, so far, one could verify the congruence of and by using a rational algorithm only in two situations: (1) the eigenvalue is simple or semi-simple; (2)there is only one Jordan block associated with in the Jordan form of the cosquares. We propose a rational algorithm for checking congruence in the case where two Jordan blocks of the same order are associated with . [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]

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]

We consider a class of generalized derivations that arise in connection with the problem of adjoining unity to an algebra with generalized derivation, and of searching envelopes for Novikov–Poisson algebras. Conditions for the existence of the localization of an algebra with ternary derivation are specified, as well as conditions under which given an algebra with ternary derivation, we can construct a Novikov–Poisson algebra and a Jordan superalgebra. Finally, we show how the simplicity of an algebra with Brešar generalized derivation is connected with simplicity of the appropriate Novikov algebra. [ABSTRACT FROM AUTHOR]

Second-order cone linear complementarity problems (SOCLCPs) have wide applications in real world, and the latest modulus method is proved to be an efficient solver. Here, inspired by the state-of-the-art modulus method and Anderson acceleration (AA), we construct the Anderson accelerating preconditioned modulus (AA+PMS) approach. Theoretically, in the first stage, we utilize the Fr e ́ chet-differentiability of the absolute value function in Jordan algebra to explore its new properties. On this basis, we establish the convergence theory for the PMS approach different from the previous analysis, and further discuss the selection strategy of parameters involved. In the second stage, we demonstrate the strong semi-smoothness of the absolute value function in Jordan algebra and, thus, establish the local convergence theory for the AA+PMS approach. Finally, we conduct rich numerical experiments with application to some well-structured examples, the second-order cone programming, the Signorini problem of the Laplacian and the three-dimensional frictional contact problem to verify the robustness and effectiveness of the AA+PMS approach. [ABSTRACT FROM AUTHOR]

We reconsider the dynamical systems approach to analyze inflationary universe in the Jordan frame models of scalar field nonminimally coupled to curvature. The adopted set of variables allows us to clearly distinguish between different asymptotic states in the phase space, including the kinetic and inflationary regimes. Inflation is realized as a heteroclinic trajectory originating either at infinity from a nonhyperbolic asymptotic de Sitter point or from a regular saddle de Sitter point. We also present a comprehensive picture of possible initial conditions leading to sufficient inflationary expansion and show their extent on the phase diagrams. In addition we comment on the slow roll conditions applicable in the Jordan frame and show how they approximate the leading inflationary "attractor solution". As particular examples we portrait quadratic and quartic potential models and note that increasing the nonminimal coupling diminishes the range of good initial conditions in the quadratic case, but enlarges is in the quartic case. [ABSTRACT FROM AUTHOR]

Given two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible. Establishing whether two quantum channels are compatible is a fundamental problem in quantum information. Here, the authors prove its equivalence to the quantum state marginal problem, introduce an efficient way to solve both, and draw further connection to the measurement compatibility problem. [ABSTRACT FROM AUTHOR]

New complete invariants for Jordan parts of von Neumann algebras are presented. We shall prove that the poset of all finite dimensional abelian von Neumann subalgebras ordered by set theoretic inclusion is a complete Jordan invariant for von Neumann algebras. On the other hand, we exhibit an example showing that not any order isomorphism on this structure is derived from a Jordan isomorphism. We apply our results to the Choquet order of orthogonal measures on state spaces of von Neumann algebras. Among others we show that the poset of decompositions of a fixed faithful normal state on a von Neumann algebra endowed with the Choquet order is a complete Jordan invariant for σ-finite von Neumann algebras. [ABSTRACT FROM AUTHOR]

For a class of second-order cone nonlinear complementarity problems, abbreviated as SOCNCPs, we establish the modulus-based matrix splitting relaxation iteration methods, which are obtained by reformulating equivalently SOCNCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The global convergence theorems are given under suitable choices of the involved splitting matrix and parameter matrices. When the splitting matrix is symmetric positive definite, the strategy choice of the parameters is discussed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SOCNCPs. [ABSTRACT FROM AUTHOR]