Related papers: Exponential iteration and Borel sets
The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that…
In these notes, uniform convergence on compacta is studied on the space of functions taking values in the set of finite Borel measures. Related limit theorems, including L\'evy's continuity theorem and functional limit theorems for…
Let $f$ be Fatou's function, that is, $f(z)= z+1+e^{-z}$. We prove that the escaping set of $f$ has the structure of a `spider's web' and we show that this result implies that the non-escaping endpoints of the Julia set of $f$ together with…
In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a…
Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…
A result of the author shows that the behavior of Gowers norms on bounded exponent abelian groups is connected to finite nilspaces. Motivated by this, we investigate the structure of finite nilspaces. As an application we prove inverse…
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…
We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…
The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the…
Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the…
The problem of classifying tuples of nilpotent matrices over a field under simultaneous conjugation is considered "hopeless". However, for any given matrix order over a finite field, the number of concerned orbits is always finite. This…
We study the lattice of all Borel clones on $2 = \{0,1\}$: classes of Borel functions $f : 2^n \to 2$, $n \le \omega$, which are closed under composition and include all projections. This is a natural extension to countable arities of…
The goal of this note is to prove the following theorem: Let $p_a(z) = z^2+a$ be a quadratic polynomial which has no irrational indifferent periodic points, and is not infinitely renormalizable. Then the Lebesgue measure of the Julia set…
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…
In this paper we study the higher-order Euler numbers and polynomials and we introduce the mutiple zeta functions which interpolate higher-order Euler polynomials and numbers at negative integers
We introduce the concept of escaping set for semigroups of transcendental entire functions using Fatou-Julia theory. Several results of the escaping set associated with the iteration of one transcendental entire function have been extended…
This paper, being the sequel of [An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators], studies a class of linear ordinary differential operators with polynomial coefficients called \emph{exactly solvable};…
We introduce the notion of \pi-extension of the semigroup \mathbb{Z}_+ and study the extensions of the Toeplitz algebras by isometric operators. We show that when the action of the Toeplitz algebra is irreducible all such extensions…
In this paper we investigate the level sets of extremal Sobolev functions for subcritical exponents p. We conjecture that as p increases the corresponding extremal functions become more peaked, which we can measure by comparing their…
We describe in terms of the j-invariant all elliptic surfaces pi: X -> C with a section, such that h^{1,1}(X)=rank NS(X) and the Mordell-Weil group of pi is finite. We use this to give a complete solution to infinitesimal Torelli for…