An Efficient Algorithm for Eigenvalue Problem of Latin Squares in a Bipartite Min-Max-Plus System

Umer, Mubasher; Hayat, Umar; Abbas, Fazal; Agarwal, Alekh; Kitanov, Petko

doi:10.3390/sym12020311

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…Let ρ(A) denote the spectral radius of A, i.e., the largest eigenvalue. In Tropical Algebra, the eigenvalue defines the cycle time in a TEG and is still active research topic, with recent results, such as involving nontrivial eigenvectors [19,20]. The symbol is because a b is the analog of a division in Max-Plus algebra.…”

Section: Tropical Algebramentioning

confidence: 99%

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

2020

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Tropical Algebra is used to model the dynamics of Timed Event Graphs (TEG), a particular class of Timed Discrete-Event System (TDES) in which we are interested only in synchronization and delay phenomena. Whenever this TEG has control inputs, we can use them to control the synchronization of the system to achieve some objective. Thus, this paper formulates a framework based on tropical algebra and lexicographic optimization to synchronize a TEG when dealing with many synchronization objectives that are ranked in previous priority order. We call this kind of problem the Tropical Lexicographic Synchronization Optimization (TLSO). This work develops a solution to this problem, based on Tropical Fractional Linear Programming (TFLP) and lexicographic optimization concepts. In this way, the basics of tropical algebra are determined, including essential terms to this paper, such as left and right residuations, and the following stages of the solution to the TLSO problem are explained. Therefore, this work presents a general framework based on structured algebraic models with application to TEG synchronization. By synchronization, we mean balancing and organizing events chronologically in order to achieve the desired goal. So, we are dealing with concepts closely related to symmetry ones. An illustrative numerical example is presented, which demonstrates the implementation of the proposed algorithms. The acquired results confirm the efficiency of the proposed methodology. Codes used for implementing the algorithms are listed in the appendix section of the article.

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Section: Tropical Algebramentioning

confidence: 99%

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

2020

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“…These nonlinear systems can be described by a max-plus linear time-invariant model, which is called the max-plus linear system. The matrix theory in max-plus algebra has been developed, including the computation for eigenvalues and eigenvectors [12][13][14][15][16][17], The Cayley-Hamilton theorem [18,19], QR decomposition [20] and solvability of linear equations [21][22][23]. Meanwhile, the polynomial theory in max-plus algebra has also been studied.…”

Section: Introductionmentioning

confidence: 99%

Ordered Structures of Polynomials over Max-Plus Algebra

2021

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The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element.

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Trivial and Nontrivial Eigenvectors for Latin Squares in Max-Plus Algebra

et al. 2022

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A square array whose all rows and columns are different permutations of the same length over the same symbol set is known as a Latin square. A Latin square may or may not be symmetric. For classification and enumeration purposes, symmetric, non-symmetric, conjugate symmetric, and totally symmetric Latin squares play vital roles. This article discusses the Eigenproblem of non-symmetric Latin squares in well known max-plus algebra. By defining a certain vector corresponding to each cycle of a permutation of the Latin square, we characterize and find the Eigenvalue as well as the possible Eigenvectors.

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An Efficient Algorithm for Eigenvalue Problem of Latin Squares in a Bipartite Min-Max-Plus System

Cited by 3 publications

References 16 publications

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

Ordered Structures of Polynomials over Max-Plus Algebra

Trivial and Nontrivial Eigenvectors for Latin Squares in Max-Plus Algebra

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