Related papers: On Computing the Kronecker Structure of Polynomial…
We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.
The complete characterization of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one is known, and it is stated in terms of very involved conditions. This paper is devoted to, loosing accuracy, better…
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…
Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we…
Translated into the language of representations of quivers, a challenge in matrix pencil theory is to find sufficient and necessary conditions for a Kronecker representation to be a subfactor of another Kronecker representation in terms of…
This paper demonstrates how certified computational tools can be used to address various problems in control theory. In particular, we introduce PACE.jl, a Julia package that implements symbolic elimination techniques, including (among…
We completely characterize the conformal radii of Siegel disks in the family $$P_\theta(z)=e^{2\pi i\theta}z+z^2,$$ corresponding to {\bf computable} parameters $\theta$. As a consequence, we constructively produce quadratic polynomials…
Matrix functions extend scalar function concepts to linear operators, offering a unified framework with broad applications in mathematics, science, and engineering. Classical definitions--via power series, spectral calculus, or Jordan…
The dynamics of all quadratic Newton maps of rational functions are completely described. The Julia set of such a map is found to be either a Jordan curve or totally disconnected. It is proved that no Newton map with degree at least three…
Block full rank pencils introduced in [Dopico et al., Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information…
In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping…
The asymptotic behaviour of the solutions of Poincar\'e's functional equation $f(\lambda z)=p(f(z))$ ($\lambda>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions of the complex plain. The constancy of an occurring…
Adopting the approach of [7] we study rational function carrying invariant line fields on the Julia set. In particular, we show that under certain weak conditions all possible measurable invariant line fields of a rational function on its…
We investigate the problem of determining the zeros of quaternionic polynomials using matrix method. In a recent paper, Dar et al. \cite{RD} proved that the zeros of a quaternionic polynomial and the left eigenvalues of the corresponding…
We study the row completion problem of polynomial and rational matrices with partial prescription of the structural data. The prescription of the complete structural data has been solved in Amparan et al., Lin. Alg. Appl. 720 (2025)…
It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
A Peano compactum is a compact metric space having locally connected components such that at most finitely many of them are of diameter greater than any fixed number C>0. Given a compactum K in the extended complex plane, it is known that…
Geometric computing with chain complexes allows for the computation of the whole chain of linear spaces and (co)boundary operators generated by a space decomposition into a cell complex. The space decomposition is stored and handled with…
It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has…