Chimera state in coupled map lattice of matrices

In recent years, a lot of research has focused on understanding the behavior of when synchronous and asynchronous phases occur, that is, the existence of chimera states in various networks. Chimera states have wide-range applications in many disciplines including biology, chemistry, physics, or engineering. The object of research in this paper is a coupled map lattice of matrices when each node is described by an iterative map of matrices of order two. A regular topology network of iterative maps of matrices was formed by replacing the scalar iterative map with the iterative map of matrices in each node. The coupled map of matrices is special in a way where we can observe the effect of divergence. This effect can be observed when the matrix of initial conditions is a nilpotent matrix. Also, the evolution of the derived network is investigated. It is found that the network of the supplementary variable $\mu$ can evolve into three different modes: the quiet state, the state of divergence, and the formation of divergence chimeras. The space of parameters of node coupling including coupling strength $\varepsilon$ and coupling range $r$ is also analyzed in this study. Image entropy is applied in order to identify chimera state parameter zones.

On $m$-th roots of nilpotent matrices

A new necessary and sufficient condition for the existence of an $m$-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the $m$-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an $m$-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an $m$-th root is corrected and generalized. Moreover, for a singular matrix having an $m$-th root with a pair of nilpotent Jordan blocks of sizes $s$ and $l$, a new $m$-th root is constructed by replacing that pair by another one of sizes $s+i$ and $l-i$, for special $s,l,i$. This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix $A$ over an arbitrary field that is a sum of two commuting matrices, several results for the existence of $m$-th roots of $A^k$ are obtained.

On the Generalized Strongly Nil-Clean Property of Matrix Rings

Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] is expressible as a sum [Formula: see text] of (commuting) idempotent matrices [Formula: see text] and a nilpotent matrix [Formula: see text].

On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

We study when every square matrix over an algebraically closed field or over a finite field is decomposable into a sum of a potent matrix and a nilpotent matrix of order 2. This can be related to our recent paper, published in Linear & Multilinear Algebra (2022). We also completely address the question when each square matrix over an infinite field can be decomposed into a periodic matrix and a nilpotent matrix of order 2

Certain Properties of Square Matrices over Fields with Applications to Rings

We prove that any square nilpotent matrix over a field is a difference of two idempotent matrices as well as that any square matrix over an algebraically closed field is a sum of a nilpotent square-zero matrix and a diagonalizable matrix. We further apply these two assertions to a variation of π-regular rings. These results somewhat improve on establishments due to Breaz from Linear Algebra & amp; Appl. (2018) and Abyzov from Siberian Math. J. (2019) as well as they also refine two recent achievements due to the present author, published in Vest. St. Petersburg Univ. - Ser. Math., Mech. & amp; Astr. (2019) and Chebyshevskii Sb. (2019), respectively.

Linear dynamical systems of dimension two over the ring of integers modulo pt

In this paper, we investigate the linear dynamical system [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text] ([Formula: see text] is a prime). In order to facilitate the visualization of this system, we associate a graph [Formula: see text] on it, whose nodes are the points of [Formula: see text], and for which there is an arrow from [Formula: see text] to [Formula: see text], when [Formula: see text] for a fixed [Formula: see text] matrix [Formula: see text]. In this paper, the in-degree of each node in [Formula: see text] is obtained, and a complete description of [Formula: see text] is given, when [Formula: see text] is an idempotent matrix, or a nilpotent matrix, or a diagonal matrix. The results in this paper generalize Elspas’ [1959] and Toledo’s [2005].

All Solutions of the Yang–Baxter-Like Matrix Equation for Nilpotent Matrices of Index Two

Let A be a nilpotent matrix of index two, and consider the Yang–Baxter-like matrix equation AXA=XAX. We first obtain a system of matrix equations of smaller sizes to find all the solutions of the original matrix equation. When A is a nilpotent matrix with rank 1 and rank 2, we get all solutions of the Yang–Baxter-like matrix equation.

REGULARIZATION OF NON-NORMAL MATRICES BY GAUSSIAN NOISE—THE BANDED TOEPLITZ AND TWISTED TOEPLITZ CASES

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let$M_{N}$be a deterministic$N\times N$matrix, and let$G_{N}$be a complex Ginibre matrix. We consider the matrix${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where$\unicode[STIX]{x1D6FE}>1/2$. With$L_{N}$the empirical measure of eigenvalues of${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties$L_{N}$to the singular values of$z-M_{N}$, with$z\in \mathbb{C}$. We then compute the limit of$L_{N}$when$M_{N}$is an upper-triangular Toeplitz matrix of finite symbol: if$M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$where$\mathfrak{d}$is fixed,$a_{i}\in \mathbb{C}$are deterministic scalars and$J$is the nilpotent matrix$J(i,j)=\mathbf{1}_{j=i+1}$, then$L_{N}$converges, as$N\rightarrow \infty$, to the law of$\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$where$U$is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when$\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in$M_{N}$.

Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.