Displaying similar documents to “On the problem A x = λ B x in max algebra: every system of intervals is a spectrum”

On an algorithm for testing T4 solvability of max-plus interval systems

Helena Myšková (2012)

Kybernetika

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In this paper, we shall deal with the solvability of interval systems of linear equations in max-plus algebra. Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗ , where a ⊕ b = max { a , b } , a ⊗ b = a + b . The notation 𝔸 ⊗ x = 𝕓 represents an interval system of linear equations, where 𝔸 = [ b ¯ , A ¯ ] and 𝕓 = [ b ̲ , b ¯ ] are given interval matrix and interval vector, respectively. We can define several types of solvability of interval systems. In this paper, we define the T4 solvability and...

Max-min interval systems of linear equations with bounded solution

Helena Myšková (2012)

Kybernetika

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Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗ , where a ⊕ b = max { a , b } , a ⊗ b = min { a , b } . The notation 𝐀 ⊗ 𝐱 = 𝐛 represents an interval system of linear equations, where 𝐀 = [ A ̲ , A ¯ ] , 𝐛 = [ b ̲ , b ¯ ] are given interval matrix and interval vector, respectively, and a solution is from a given interval vector 𝐱 = [ x ̲ , x ¯ ] . We define six types of solvability of max-min interval systems with bounded solution and give necessary and sufficient conditions for them.

On ergodicity of some cylinder flows

Krzysztof Frączek (2000)

Fundamenta Mathematicae

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We study ergodicity of cylinder flows of the form    T f : T × ℝ → T × ℝ , T f ( x , y ) = ( x + α , y + f ( x ) ) , where f : T → ℝ is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that D k f is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of D k f have some good properties, then T f is ergodic. Moreover, there exists ε f > 0 such that if v : T → ℝ is a function with zero integral such that D k v is of bounded...

On tropical Kleene star matrices and alcoved polytopes

María Jesús de la Puente (2013)

Kybernetika

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In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero-diagonal real matrix A is characterized by A being a Kleene star. We give applications to alcoved polytopes, using normal idempotent matrices (which form a subclass of Kleene stars). For a normal matrix we define a norm and show that this is the radius of a hyperplane section of its tropical span.