Related papers: Bounded critical Fatou components are Jordan domai…

Weighted fractional Poincar\'e-type inequalities are proved on John domains whenever the weights defined on the domain are depending on the distance to the boundary and to an arbitrary compact set in the boundary of the domain.

For the study of the 2-dimensional space of cubic polynomials, J. Milnor considers the complex 1-dimensional slice S_n of the cubic polynomials which have a super-attracting orbit of period n. He gives in [M4] a detailed conjectural picture…

A conjecture for the dimension and the character of the homogenous components of the free Jordan algebras is proposed. As a support of the conjecture, some numerical evidences are generated by a computer and some new theoretical results are…

For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…

We consider the zero distribution of random polynomials of the form $P_n(z) = \sum_{k=0}^n a_k B_k(z)$, where $\{a_k\}_{k=0}^{\infty}$ are non-trivial i.i.d. complex random variables with mean $0$ and finite variance. Polynomials…

Theorem converse to Jordan's curve theorem says that {\it if a compact set $K$ has two complementary domains in $R^{2}$, from each of which it is at every point accessible, it is a simple closed curve}. We show that the requirement of this…

A Jordan H\"older theorem is established for derived module categories of piecewise hereditary algebras. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module…

On a compact subset of the plane with connected complement, is it possible to uniformly approximate a continuous function, holomorphic and non-vanishing on the interior, with polynomials non-vanishing on the entire compact set? In this…

We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel.…

We compute the exact values of the Jordan constants of abelian surfaces over finite fields.

Using a definition of Jordan curve similar to that of Dieudonn\'e, we prove that our notion is equivalent to that used by Berg et al. in their constructive proof of the Jordan Curve Theorem. We then establish a number of properties of…

In this paper, we prove that for any post-critically finite rational map $f$ on the Riemann sphere $\overline{\mathbb{C}}$, and for each sufficiently large integer $n$, there exists a finite and connected graph $G$ in the Julia set of $f$…

We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks 2 and 3.

The w-asymmetry induced by transfinite partitions makes it impossible for Jordan curves to have an infinite length.

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over…

Let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Given a square matrix $A \in \mathbb{F}^{n \times n}$ and a polynomial $f \in \mathbb{F}[w]$, we determine the Jordan canonical form of the formal Fr\'{e}chet…

We mainly generalize the notion of abelian transcendental semigroup to nearly abelian transcendental semigroup. We prove that Fatou set, Julia set and escaping set of nearly abelian transcendental semigroup are completely invariant. We…

We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.

We prove that if two subsets ${A}$ and ${B}$ of the plane are connected, ${A}$ is bounded, and the Euclidean distance $\rho({A},{B})$ between ${A}$ and ${B}$ is greater than zero, then for every positive $\varepsilon<\rho({A},{B})$, the…

We describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the possible eventual connectivity of Fatou…