Polynomial convolutions in max-plus algebra

Rosenmann, Amnon; Lehner, Franz; Peperko, Aljoša

doi:10.1016/j.laa.2019.05.020

Cited by 19 publications

(14 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The characteristic maxpolynomial of A (see e.g. [3,14,15]) is a max polynomial where I denotes the identity matrix. We call its tropical roots (the points of nondifferentiability of A (x) considered as a function on [0, ∞) ) algebraic max eigenvalues (or also tropical eigenvalues) of A.…”

Section: Theorem 1 If a ∈ M N (ℝ + ) Thenmentioning

confidence: 99%

“…The set of all algebraic max eigenvalues is denoted by trop (A) . For ∈ trop (A) its multiplicity as a tropical root of A (x) (see e.g [3,14,15]) is called an algebraic multiplicity of . It is known that max (A) ⊂ trop (A) [15,Remark 2.3] and that (A) = max{ ∶ ∈ trop (A)} , but in general, the sets max (A) and trop (A) may differ.…”

Section: Theorem 1 If a ∈ M N (ℝ + ) Thenmentioning

confidence: 99%

See 1 more Smart Citation

On the numerical ranges of matrices in max algebra

Thaghizadeh

Zahraei

Peperko

et al. 2020

Banach J. Math. Anal.

View full text Add to dashboard Cite

Let $$M_{n}({\mathbb {R}}_{+})$$ M n ( R + ) be the set of all $$n \times n$$ n × n nonnegative matrices. Recently, in Tavakolipour and Shakeri (Linear Multilinear Algebra 67, 2019, https://doi.org/10.1080/03081087.2018.1478946), the concept of the numerical range in tropical algebra was introduced and an explicit formula describing it was obtained. We study the isomorphic notion of the numerical range of nonnegative matrices in max algebra and give a short proof of the known formula. Moreover, we study several generalizations of the numerical range in max algebra. Let $$1 \le k \le n$$ 1 ≤ k ≤ n be a positive integer and $$C \in M_{ n}({\mathbb {R}}_{+}).$$ C ∈ M n ( R + ) . We introduce the notions of max $$k-$$ k - numerical range and max $$C-$$ C - numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range $$W_\text {max}(\varSigma )$$ W max ( Σ ) of a bounded set $$\varSigma$$ Σ of $$n \times n$$ n × n nonnegative matrices is introduced and some of its properties are also investigated.

show abstract

Section: Theorem 1 If a ∈ M N (ℝ + ) Thenmentioning

confidence: 99%

Section: Theorem 1 If a ∈ M N (ℝ + ) Thenmentioning

confidence: 99%

On the numerical ranges of matrices in max algebra

Thaghizadeh

Zahraei

Peperko

et al. 2020

Banach J. Math. Anal.

View full text Add to dashboard Cite

show abstract

“…For more information on tropical algebra we refer to the monograph of Butkovič [17]. Min-plus algebra is isomorphic to max-plus algebra, which is the semifield R ∪ {−∞}, where addition is replaced by maximum and multiplication by addition (see e.g., [17,18] and the references there), and also to max-times algebra R + , where addition is replaced by maximum and multiplication is the same as in standard arithmetic (see e.g., [19] and the references there). Tropical algebra is a part of a broader branch of mathematics, called "idempotent mathematics", which was developed mainly by Maslov and his collaborators (see e.g., [20,21]).…”

Section: Tropical Algebramentioning

confidence: 99%

Independent Rainbow Domination Numbers of Generalized Petersen Graphs P(n,2) and P(n,3)

2020

Self Cite

View full text Add to dashboard Cite

We obtain new results on independent 2- and 3-rainbow domination numbers of generalized Petersen graphs P ( n , k ) for certain values of n , k ∈ N . By suitably adjusting and applying a well established technique of tropical algebra (path algebra) we obtain exact 2-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) and P ( n , 3 ) thus confirming a conjecture proposed by Shao et al. In addition, we compute exact 3-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) . The method used here is developed for rainbow domination and for Petersen graphs. However, with some natural modifications, the method used can be applied to other domination type invariants, and to many other classes of graphs including grids and tori.

show abstract

“…Izhakian et al [29,30] showed that systems of tropical polynomials formed from univariate monomials define subsemigroups with respect to tropical addition (maximum). Rosenmann et al [31] created an exact and simple description of all roots of convolutions in terms of the roots of involved maxpolynomials. Wang and Tao [32] introduced the matrix representation of formal polynomials over max-plus algebra to factorize polynomial functions.…”

Section: Introductionmentioning

confidence: 99%

Ordered Structures of Polynomials over Max-Plus Algebra

2021

View full text Add to dashboard Cite

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.

show abstract

Polynomial convolutions in max-plus algebra

Cited by 19 publications

References 21 publications

On the numerical ranges of matrices in max algebra

On the numerical ranges of matrices in max algebra

Independent Rainbow Domination Numbers of Generalized Petersen Graphs P(n,2) and P(n,3)

Ordered Structures of Polynomials over Max-Plus Algebra

Contact Info

Product

Resources

About