We give simple algebraic proofs of results on generators and bases of max cones, some of which are known. We show that every generating set S for a cone in max algebra can be partitioned into two parts: the independent set of extremals E in the cone and a set F every member of which is redundant in S. We exploit the result that extremals are minimal… (More)
The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A, a diagonal matrix X such that all… (More)
Matrix elements of quantum intertwiner as well as the modified Q-operator for the quantum relativistic Toda chain at root of unity are constructed explicitly. Modified Q-operators make isospectrality transformations of quantum transfer matrices so that the classical counterparts of Q-operators correspond to the Bäcklund isospectrality transformations of the… (More)
Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring: it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a… (More)
(2012): Z-matrix equations in max-algebra, nonnegative linear algebra and other semirings, Linear and Multilinear Algebra, This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is… (More)
The Analytic Hierarchy Process (AHP) is widely used for decision making involving multiple criteria. Elsner and van den Driessche [10, 11] introduced a max-algebraic approach to the single criterion AHP. We extend this to the multi-criteria AHP, by considering multi-objective generalisations of the single objective optimisation problem solved in these… (More)
This paper is a survey on universal algorithms for solving the matrix Bell-man equations over semirings and especially tropical and idempotent semirings. However, original algorithms are also presented. Some applications and software implementations are discussed.
This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments , semispaces, hyperplanes and an account of separation and non-separation results based on semispaces. There are some new results. In particular, we give new " colorful " extensions of the max-min Carathéodory… (More)
This paper presents two universal algorithms for generalized discrete matrix Bellman equations with symmetric Toeplitz matrix. The algorithms are semiring extensions of two well-known methods solving Toeplitz systems in the ordinary linear algebra.
The subject of this paper is the consecutive procedure of discretization and quantization of two similar classical integrable systems in three-dimensional space-time: the standard three-wave equations and less known modified three-wave equations. The quantized systems in discrete space-time may be understood as the regularized integrable quantum field… (More)