On the problem in max algebra: every system of intervals is a spectrum
We consider the two-sided eigenproblem over max algebra. It is shown that any finite system of real intervals and points can be represented as spectrum of this eigenproblem.
On hyperplanes and semispaces in max–min convex geometry
The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.
Characterizing matrices with -simple image eigenspace in max-min semiring
A matrix is said to have -simple image eigenspace if any eigenvector belonging to the interval is the unique solution of the system in . The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that...
Minimizing maximum lateness in two-stage projects by tropical optimization
We are considering a two-stage optimal scheduling problem, which involves two similar projects with the same starting times for workers and the same deadlines for tasks. It is required that the starting times for workers and deadlines for tasks should be optimal for the first-stage project and, under this condition, also for the second-stage project. Optimality is measured with respect to the maximal lateness (or maximal delay) of tasks, which has to be minimized. We represent this problem as a...
Modifying the tropical version of Stickel's key exchange protocol
A tropical version of Stickel's key exchange protocol was suggested by Grigoriev and Shpilrain (2014) and successfully attacked by Kotov and Ushakov (2018). We suggest some modifications of this scheme that use commuting matrices in tropical algebra and discuss some possibilities of attacks on these new modifications. We suggest some simple heuristic attacks on one of our new protocols, and then we generalize the Kotov and Ushakov attack on tropical Stickel's protocol and discuss the application...
Eigenspace of a three-dimensional max-Łukasiewicz fuzzy matrix
Eigenvectors of a fuzzy matrix correspond to stable states of a complex discrete-events system, characterized by a given transition matrix and fuzzy state vectors. Description of the eigenspace (set of all eigenvectors) for matrices in max-min or max-drast fuzzy algebra was presented in previous papers. In this paper the eigenspace of a three-dimensional fuzzy matrix in max-Łukasiewicz algebra is investigated. Necessary and sufficient conditions are shown under which the eigenspace restricted to...
Generalized Kotov-Ushakov attack on tropical Stickel protocol based on modified tropical circulant matrices
After the Kotov-Ushakov attack on the tropical implementation of Stickel protocol, various attempts have been made to create a secure variant of such implementation. Some of these attempts used a special class of commuting matrices resembling tropical circulants, and they have been proposed with claims of resilience against the Kotov-Ushakov attack, and even being potential post-quantum candidates. This paper, however, reveals that a form of the Kotov-Ushakov attack remains applicable and, moreover,...
A bound for the rank-one transient of inhomogeneous matrix products in special case
We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length are rank-one, as it was shown in [6] (Shue, Anderson, Dey 1998). We establish a bound on the transient after which any product of matrices whose length exceeds that bound becomes rank-one.
Quantum relativistic Toda chain at root of unity: isospectrality, modified -operator, and functional Bethe ansatz.
Quantum relativistic Toda chain.
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