Cyclic classes and attraction cones in max algebra

In max algebra it is well-known that the sequence A^k, with A an irreducible square matrix, becomes periodic at sufficiently large k. This raises a number of questions on the periodic regime of A^k and A^k x, for a given vector x. Also, this leads to the concept of attraction cones in max algebra, by which we mean sets of vectors whose ultimate orbit period does not exceed a given number. This paper shows that some of these questions can be solved by matrix squaring (A,A^2,A^4, ...), analogously to recent findings concerning the orbit period in max-min algebra. Hence the computational complexity of such problems is of the order O(n^3 log n). The main idea is to apply an appropriate diagonal similarity scaling A -> X^{-1}AX, called visualization scaling, and to study the role of cyclic classes of the critical graph. For powers of a visualized matrix in the periodic regime, we observe remarkable symmetry described by circulants and their rectangular generalizations. We exploit this symmetry to derive a concise system of equations for attraction cpne, and we present an algorithm which computes the coefficients of the system.Comment: 38 page

APPLICATION OF SYSTEM MAX-PLUS LINEAR EQUATIONS ON SERIAL MANUFACTURING MACHINE WITH STORAGE UNIT

The set  together with the operation maximum (max) denoted as  and addition (+) denoted as  is called max-plus algebra. Max-plus algebra may be used to apply algebraically a few programs of Discrete Event Systems (DES), certainly one of the examples in the production system. In this study, the application of max-plus algebra in a serial manufacturing machine with a storage unit is discussed. The results of this are the generalization system max-plus-linear equations on a production system that is, in addition, noted the max-plus-linear time-invariant system. From the max-plus-linear time-invariant system, it can be obtained the equation  which is then used to determine the beginning time of a production system so the manufacturing machine work periodically. The eigenvector and eigenvalue of the matrix  are then used to find the beginning time and the period time of the manufacturing machine. Furthermore, the time when the product leaves the manufacturing machine with the time while the raw material enters the manufacturing machine is given and vice versa are obtained from the max-plus-linear time-invariant system that is can be formed in the equation

Sifat Periodik Jaringan Antrian Seri Tertutup Dengan Pendekatan Aljabar Max-plus

. This article discussed about the properties of closed periodic queuing network series susing max-plus algebra. The result showed that the properties of closed periodic dinamic queuing network series can be determined by using the concept of eigen values ​​and eigen vectors of max-plus matrix in the network model. Through the max-plus eigen vector fundamental, can be determined faster early time departure of customers of departure to the next customer periodically, with a large period of max-plus eigenvalu

On the max-algebraic core of a nonnegative matrix

The max-algebraic core of a nonnegative matrix is the intersection of column spans of all max-algebraic matrix powers. Here we investigate the action of a matrix on its core. Being closely related to ultimate periodicity of matrix powers, this study leads us to new modifications and geometric characterizations of robust, orbit periodic and weakly stable matrices.Comment: 27 page

Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.Comment: 27 page

A Max-Plus Model of Asynchronous Cellular Automata

This paper presents a new framework for asynchrony. This has its origins in our attempts to better harness the internal decision making process of cellular automata (CA). Thus, we show that a max-plus algebraic model of asynchrony arises naturally from the CA requirement that a cell receives the state of each neighbour before updating. The significant result is the existence of a bijective mapping between the asynchronous system and the synchronous system classically used to update cellular automata. Consequently, although the CA outputs look qualitatively different, when surveyed on "contours" of real time, the asynchronous CA replicates the synchronous CA. Moreover, this type of asynchrony is simple - it is characterised by the underlying network structure of the cells, and long-term behaviour is deterministic and periodic due to the linearity of max-plus algebra. The findings lead us to proffer max-plus algebra as: (i) a more accurate and efficient underlying timing mechanism for models of patterns seen in nature, and (ii) a foundation for promising extensions and applications.Comment: in Complex Systems (Complex Systems Publications Inc), Volume 23, Issue 4, 201

Generalizations of Bounds on the Index of Convergence to Weighted Digraphs

We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure

CSR expansions of matrix powers in max algebra

We study the behavior of max-algebraic powers of a reducible nonnegative n by n matrix A. We show that for t>3n^2, the powers A^t can be expanded in max-algebraic powers of the form CS^tR, where C and R are extracted from columns and rows of certain Kleene stars and S is diadonally similar to a Boolean matrix. We study the properties of individual terms and show that all terms, for a given t>3n^2, can be found in O(n^4 log n) operations. We show that the powers have a well-defined ultimate behavior, where certain terms are totally or partially suppressed, thus leading to ultimate CS^tR terms and the corresponding ultimate expansion. We apply this expansion to the question whether {A^ty, t>0} is ultimately linear periodic for each starting vector y, showing that this question can be also answered in O(n^4 log n) time. We give examples illustrating our main results.Comment: 25 pages, minor corrections, added 3 illustration

An Overview of Transience Bounds in Max-Plus Algebra

We survey and discuss upper bounds on the length of the transient phase of max-plus linear systems and sequences of max-plus matrix powers. In particular, we explain how to extend a result by Nachtigall to yield a new approach for proving such bounds and we state an asymptotic tightness result by using an example given by Hartmann and Arguelles.Comment: 13 pages, 2 figure