Uniqueness and topological properties of number representation

Let b be a complex number with |b| > 1 and let D be a finite subset of the complex plane C such that 0 ∊ D and card D ≥ 2. A number z is representable by the system (D, b) if z = Σajbj , where aj ∊ D. We denote by F the set of numbers which are representable by (D, b) with M = −1. The set W consists of numbers that are (D, b) representable with aj = 0 for all negative j. Let F1 be a set of numbers in F that can be uniquely represented by (D, b). It is shown that: The set of all extreme points of F is a subset of F1. If 0 ∊ F1, then W is discrete and closed. If b ∊ {z : |z| > 1}\D′, where D′ is a finite or countable set associated with D and W is discrete and closed, then 0 ∊ F1. For a real number system (D, b), F is homeomorphic to the Cantor set C iff F\F1 is nowhere dense subset of R

Wiman and Arima theorems for quasiregular mappings

Peer reviewe

Ahlfors Theorems for Differential Forms

Some counterparts of theorems of Phragmén-Lindelöf and of Ahlfors are proved for differential forms of -classes

Linear distortion of Hausdorff dimension and Cantor's function

Let $f$ be a mapping from a metric space $X$ to a metric space $Y$ , and let $\alpha$ be a positive real number. Write $dim(E)$ and $\mathcal{H}^s(E)$ for the Hausdorff dimension and the $s$-dimensional Hausdorff measure of a set $E$. We give sufficient conditions that the equality $dim(f(E)) = \alpha dim(E)$ holds for each $E \subseteq X$. The problem is studied also for the Cantor ternary function $G$. It is shown that there is a subset $M$ of the Cantor ternary set such that $\mathcal{H}^s(M) = 1$, with $s = \log 2/\log 3$ and $dim(G(E)) = (\log 3/log 2) dim(E)$, for every $E\subseteq M$

Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.Comment: 26 pages, 2 figures, includes several improvements in Sections 6-8 allowing to relax the assumptions in the main results. Final version published at http://link.springer.com/article/10.1007%2Fs11118-015-9463-

Fractional Sobolev-Poincaré inequalities in irregular domains

This paper is devoted to the study of fractional (q, p)-Sobolev-Poincaré in- equalities in irregular domains. In particular, the author establishes (essentially) sharp fractional (q, p)-Sobolev-Poincaré inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional (q, p)-Sobolev-Poincaré inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P., Sobolev-Poincaré implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out