Given a complex idempotent matrix A, we derive simple, sufficient and necessary conditions for X being nontrivial solution of the Yang-Baxter-like equation AXA = XAX, discriminating commuting solutions from non-commuting ones. On this basis, construct all nonlinear equation.

This paper is a survey on universal algorithms for solving the matrix Bellman equations over semirings and especially tropical and idempotent semirings. However, original algorithms are also presented. Some applications and software implementations are discussed.

Formal power series over non-commuting variables have been investigated as representations of the behavior of automata with multiplicities. Here we introduce and investigate the concepts of aperiodic and of star-free formal power series over semirings and partially commuting variables. We prove that if the semiring K is idempotent and commutative, or if K is idempotent and the variables are non...

Let A and B be 2 × non-zero complex matrices. P a linear combination of in the form P=c_1 A+c_2 where c_1,c_2 are nonzero scalar numbers. An idempotent matrix is which, when multiplied by itself, yields itself. In this study, we established entries according to given such that also an matrix. addition, result was obtained determined singular

An ideal I of a ring R is called right Baer-ideal if there exists an idempotent e 2 R such that r(I) = eR. We know that R is quasi-Baer if every ideal of R is a right Baer-ideal, R is n-generalized right quasi-Baer if for each I E R the ideal In is right Baer-ideal, and R is right principaly quasi-Baer if every principal right ideal of R is a right Baer-ideal. Therefore the concept of Baer idea...

In this paper we introduce extensions of Kleinberg’s Hubs & Authorities (HITS) algorithm to calculate the influence of nodes in a network whose adjacency matrix takes values over dioids, zerosumfree semirings with a natural order. We relate these extensions to both the Singular Value Problem and the Eigen Problem of matrices in these semirings. We show the original HITS algorithm to be a partic...

An unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition. For some particular cases, the minimum in the problem is known to be equal to the tropical spectral radius of the matrix. We examine the problem in the common setting of a general ...

Let D be a masa in B(H) where H is a separable Hilbert space. We find real numbers η0 < η1 < η2 < · · · < η6 so that for every bounded, normal D-bimodule map Φ on B(H), either ‖Φ‖ > η6 or ‖Φ‖ = ηk for some k ∈ {0, 1, 2, 3, 4, 5, 6}. When D is totally atomic, these maps are the idempotent Schur multipliers and we characterise those with norm ηk for 0 ≤ k ≤ 6. We also show that the Schur idempote...