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Malgorzata Stawiska

Publications and source records attributed to Malgorzata Stawiska.

10 recordsLinked to original sources
arXiv2025

Non-autonomous iteration of polynomials in the complex plane

We consider a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and $°p_1\geq 1$, $°p_n\geq 2$ for $n\geq 2$, satisfying certain asymptotic conditions. We prove that the function sequence $\left(\frac{1}{°p_n\cdot...\cdot °p_1}\log^+|p_n\circ...\circ p_1|\right)_{n=1}^\infty$ is uniformly convergent in $\mathbb{C}$. The non-autonomous filled Julia set $\mathcal{K}[(p_{n})_{n=1}^\infty]$ generated by the polynomial sequence $(p_{n})_{n=1}^\infty$ is defined and shown to be compact and regular with respect to the Green function. Our toy example is generated by $t_n=\frac{1}{2^{n-1}}T_n,\ n\in\{1,2,...\}$, where $T_n$ is the classical Chebyshev polynomial of degree $n$.

math.CV↗
arXiv2025

A generalized eigenvector-eigenvalue identity from the viewpoint of exterior algebra

We consider square matrices over $\mathbb{C}$ satisfying an identity relating their eigenvalues and the corresponding eigenvectors re-proved and discussed by Denton, Parker, Tao and Zhang, called the eigenvector-eigenvalue identity. We prove that for an eigenvalue $λ$ of a given matrix the identity holds if and only if the geometric multiplicity of $λ$ equals its algebraic multiplicity. We do not make any other assumptions on the matrix and allow the multiplicity of the eigenvalue to be greater than 1, which provides a substantial generalization of the identity. In the proof we use exterior algebra, particularly the properties of higher adjugates of a matrix.

math.RA↗
arXiv2020

Convex hulls of polynomial Julia sets

We prove P. Alexandersson's conjecture that for every complex polynomial $p$ of degree $d \geq 2$ the convex hull $H_p$ of the Julia set $J_p$ of $p$ satisfies $p^{-1}(H_p) \subset H_p$. We further prove that the equality $p^{-1}(H_p) = H_p$ is achieved only if $p$ is affinely conjugated to the Chebyshev polynomial $T_d$ of degree $d$, to $-T_d$ or a monomial $c z^d$ with $|c|=1$.

math.CV↗
arXiv2018

On Lagrange polynomials and the rate of approximation of planar sets by polynomial Julia sets

We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed by Lindsey and Younsi and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in $\mathbb{C}$. In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconformal arcs. Finally, for some classes of sets we estimate more precisely the rate of approximation by filled-in Julia sets in Hausdorff and Klimek metrics.

math.CV↗
arXiv2018

Polish Mathematicians and Mathematics in World War I

In this article we present diverse experiences of Polish mathematicians (in a broad sense) who during World War I fought for freedom of their homeland or conducted their research and teaching in difficult wartime circumstances. We first focus on those affiliated with Polish institutions of higher education: the existing Universities in Lwów in Kraków and the Lwów Polytechnics (Austro-Hungarian empire) as well as the reactivated University of Warsaw and the new Warsaw Polytechnics (the Polish Kingdom, formerly in the Russian empire). Then we consider the situations of Polish mathematicians in the Russian empire and other countries. We discuss not only individual fates, but also organizational efforts of many kinds (teaching at the academic level outside traditional institutions-- in Society for Scientific Courses in Warsaw and in Polish University College in Kiev; scientific societies in Kraków, Lwów, Moscow and Kiev; publishing activities) in order to illustrate the formation of modern Polish mathematical community.

math.HO↗
arXiv2016

Some approximation problems in semi-algebraic geometry

In this paper we deal with a best approximation of a vector with respect to a closed semi-algebraic set $C$ in the space $\mathbb{R}^n$ endowed with a semi-algebraic norm $ν$. Under additional assumptions on $ν$ we prove semi-algebraicity of the set of points of unique approximation and other sets associated with the distance to $C$. For $C$ irreducible algebraic we study the critical point correspondence and introduce the $ν$- distance degree, generalizing the notion appearing in \cite{DHOST} for the Euclidean norm. We discuss separately the case of the $\ell^p$ norm ($p>1$).

math.AG↗
arXiv2013

Best approximation on semi-algebraic sets and k-border rank approximation of symmetric tensors

In the first part of this paper we study a best approximation of a vector in Euclidean space R^n with respect to a closed semi-algebraic set C and a given semi-algebraic norm. Assuming that the given norm and its dual norm are differentiable we show that a best approximation is unique outside a hypersurface. We then study the case where C is an irreducible variety and the approximation is with respect to the Euclidean norm. We show that for a general point in x in R^n the number of critical points of the distance function of x to C is bounded above by a degree of a related dominant map. If C induces a smooth projective variety in V_P in P(C^n) then this degree is the top Chern number of a corresponding vector bundle on V_P. We then study the problem when a best k(> 1)-border rank approximation of a symmetric tensor is symmetric. We show that under certain dimensional conditions there exists an open semi-algebraic set of symmetric tensors for which a best k-border rank is unique and symmetric.

math.AG↗
arXiv2011

Lelong classes on toric varieties and a theorem of Siciak

We characterize Lelong classes on a toric manifold with an ample torus invariant line bundle, generalizing an approximation theorem due to Siciak. We include a counterexample to the theorem when the line bundle is globally generated, but not ample.

math.CV↗
arXiv2008

Weighted pluripotential theory on complex Kähler manifolds

We introduce a weighted version of the pluripotential theory on complex Kähler manifolds developed by Guedj and Zeriahi. We give the appropriate definition of a weighted pluricomplex Green function, its basic properties and consider its behaviour under holomorphic maps. We also establish a generalization of Siciak's H-principle.

math.CV↗