Related papers: Boundary Gauss--Lucas type theorems on the disk

A concept of boundedness of the $\mathbf{L}$-index in joint variables (see in Bandura A. I., Bordulyak M. T., Skaskiv O. B. "Sufficient conditions of boundedness of L-index in joint variables", Mat. Stud. 45 (2016), 12--26.…

Let $M$ be a complete simply connected manifold which is in addition Gromov hyperbolic, coercive and roughly starlike. For a given harmonic function on $M$, a local Fatou Theorem and a pointwise criteria of non-tangential convergence coming…

In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii) Geometry of Polynomials Level Sets, and iii) Shape…

We study the partition function and entropy of U(1) gauge theories with multiple boundaries on the black holes background. The nontrivial boundary conditions allow residual zero longitudinal momentum modes and Wilson lines stretched between…

The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carath\'eodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence…

Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according…

It is shown that under certain boundary conditions the virial theorem has to be modified. We analyze the origin of the extra term and compute it in particular examples. The Coulomb and harmonic oscillator with point interaction have been…

We study hyperbolic Gaussian analytic functions in the unit polydisk of $\mathbb C^n$. Following the scheme previously used in the unit ball we first study the asymptotics of fluctuations of linear statistics as the directional intensities…

A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…

We study the boundaries of non-univalent simply connected Baker domains of transcendental maps (both entire and meromorphic), of hyperbolic and simply parabolic type. We prove non-ergodicity and non-recurrence for the boundary map, and…

Every function in the Dirichlet space on the unit disc has an inner/outer factorization. We study which inner functions occur in this way. For Blaschke products, this is the well known question of which subsets of the disc are zero sets for…

Let f be a transcendental map, and let U be an attracting or parabolic basin, or a doubly parabolic Baker domain. Assume U is simply connected. Then, we prove that periodic points are dense in the boundary of U, under certain hypothesis on…

Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called…

We present a systematic study on a class of nonlocal integral functionals for functions defined on a bounded domain and the naturally induced function spaces. The function spaces are equipped with a seminorm depending on finite differences…

The Katz-Sarnak Density Conjecture states that zeros of families of $L$-functions are well-modeled by eigenvalues of random matrix ensembles. For suitably restricted test functions, this correspondence yields upper bounds for the families'…

The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary…

Analytic self-maps of the unit disc whose hyperbolic derivative is uniformly bounded by a constant smaller than one, are called contractive. We describe these maps in terms of their Aleksandrov-Clark measures and in terms of their…

In this paper, we develop a theory on the degenerations of Blaschke products $\mathcal{B}_d$ to study the boundaries of hyperbolic components. We give a combinatorial classification of geometrically finite polynomials on the boundary of the…