### ...- Invarianten bei verallgemeinerten Carlesonmengen.

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The 3x+k function ${T}_{k}\left(n\right)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map ${T}_{k}\left(\xb7\right)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of ${T}_{k}\left(\xb7\right)$. We consider the generating functions ${f}_{k,m}\left(z\right)={\sum}_{n>0,n\to m}{z}^{n}$, which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions ${f}_{k,m}\left(z\right)$ to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold for all m...

Solutions to Beltrami differential equation with prescribed boundary correspondence in some plane domains are given.

Using a distributional approach to integration in superspace, we investigate a Cauchy-Pompeiu integral formula in super Dunkl-Clifford analysis and several related results, such as Stokes formula, Morera's theorem and Painlevé theorem for super Dunkl-monogenic functions. These results are nice generalizations of well-known facts in complex analysis.

Let $G\subset \mathrm{\mathbf{S}\mathbf{U}}(2,1)$ be a non-elementary complex hyperbolic Kleinian group. If $G$ preserves a complex line, then $G$ is $\u2102$-Fuchsian; if $G$ preserves a Lagrangian plane, then $G$ is $\mathbb{R}$-Fuchsian; $G$ is Fuchsian if $G$ is either $\u2102$-Fuchsian or $\mathbb{R}$-Fuchsian. In this paper, we prove that if the traces of all elements in $G$ are real, then $G$ is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application...