arXiv subjects

Yoshihide Watanabe

Publications and source records attributed to Yoshihide Watanabe.

6 recordsLinked to original sources

arXiv2026

Tropical linearization and stability analysis of discrete dynamical systems at the tropical origin }

The tropical semiring is a semiring of extended real numbers, where the operations of `max' and `+' replace the usual addition and multiplication, respectively. Difference equations obtained from the ultradiscrete limit of discrete dynamical systems are described in terms of the tropical semiring. We propose a tropical linearization approach for the stability analysis of difference equations, including those describing ulradiscrete dynamical systems. We show that the fixed point at the tropical origin is asymptotically stable if the maximum eigenvalue of the tropical Jacobian matrix is negative. On the other hand, it is unstable if the maximum eigenvalue of the tropical Jacobian matrix is positive. Since $0$ is the tropical multiplicative identity, these results are analogous to those in the usual linearization process.

math.DS↗

arXiv2024

On max-plus two-sided linear systems whose solution sets are min-plus linear

The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is defined in terms of a combination of the following two operations: addition, $a \oplus b := \max(a,b)$, and multiplication, $a \otimes b := a + b$. In this study, we propose a new method to characterize the set of all solutions of a max-plus two-sided linear system $A \otimes x = B \otimes x$. We demonstrate that the minimum ``min-plus'' linear subspace containing the ``max-plus'' solution space can be computed by applying the alternating method algorithm, which is a well-known method to compute single solutions of two-sided systems. Further, we derive a sufficient condition for the ``min-plus'' and ``max-plus'' subspaces to be identical. The computational complexity of the method presented in this study is pseudo-polynomial.

math.CO↗

arXiv2021

Independence and orthogonality of algebraic eigenvectors over the max-plus algebra

The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is a semiring with the two operations: addition $a \oplus b := \max(a,b)$ and multiplication $a \otimes b := a + b$. Roots of the characteristic polynomial of a max-plus matrix are called algebraic eigenvalues. Recently, algebraic eigenvectors with respect to algebraic eigenvalues were introduced as a generalized concept of eigenvectors. In this paper, we present properties of algebraic eigenvectors analogous to those of eigenvectors in the conventional linear algebra. First, we prove that for generic matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are linearly independent. We further prove that for symmetric matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are orthogonal to each other.

math.CO↗

arXiv2020

$q$-VFCA: $q$-state Vector-valued Fuzzy Cellular Automata

Elementary fuzzy Cellular Automata (CA) are known as continuous counterpart of elementary CA, which are 2-state CA, via the polynomial representation of local rules. In this paper, we first develop a new fuzzification methodology for $q$-state CA. It is based on the vector representation of $q$-state CA, that is, the $q$-states are assigned to the standard basis vectors of the $q$-dimensional real space and the local rule can be expressed by a tuple of $q$ polynomials. Then, the $q$-state vector-valued fuzzy CA are defined by expanding the set of the states to the convex hull of the standard basis vectors in the $q$-dimensional real space. The vector representation of states enables us to enumerate the number-conserving rules of 3-state vector-valued fuzzy CA in a systematic way.

nlin.CG↗

arXiv2017

Network Simplex Algorithm associated with the Maximum Flow Problem

In the present paper, we apply the network simplex algorithm for solving the minimum cost flow problem, to the maximum flow problem. Then we prove that the cycling phenomenon which causes the infinite loop in the algorithm, does not occur in the network simplex algorithm associated with the maximum flow problem.

math.CO↗

arXiv2017

Two characteristic polynomials corresponding to graphical networks over min-plus algebra

In this paper, we investigate characteristic polynomials of matrices in min-plus algebra. Eigenvalues of min-plus matrices are known to be the minimum roots of the characteristic polynomials based on tropical determinants which are designed from emulating standard determinants. Moreover, minimum roots of characteristic polynomials have a close relationship to graphs associated with min-plus matrices consisting of vertices and directed edges with weights. The literature has yet to focus on the other roots of min-plus characteristic polynomials. Thus, here we consider how to relate the 2nd, 3rd,... minimum roots of min-plus characteristic polynomials to graphical features. We then define new characteristic polynomials of min-plus matrices by considering an analogue of the Faddeev-LeVerrier algorithm that generates the characteristic polynomials of linear matrices. We conclusively show that minimum roots of the proposed characteristic polynomials coincide with min-plus eigenvalues, and observe the other roots as in the study of the already known characteristic polynomials. We also give an example to illustrate the difference between the already known and proposed characteristic polynomials.

math.CO↗