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### Seperable maximal pluriharmonic functions in two complex variables.

Mathematica Scandinavica

### The polynomial hull of unions of convex sets in ${ℂ}^{n}$

Colloquium Mathematicae

We prove that three pairwise disjoint, convex sets can be found, all congruent to a set of the form $\left({z}_{1},{z}_{2},{z}_{3}\right)\in {ℂ}^{3}:|{z}_{1}{|}^{2}+|{z}_{2}{|}^{2}+{|{z}_{3}|}^{2m}\le 1$, such that their union has a non-trivial polynomial convex hull. This shows that not all holomorphic functions on the interior of the union can be approximated by polynomials in the open-closed topology.

### Counterexamples to the Gleason problem

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### On the algebra of ${A}^{k}$-functions

Mathematica Bohemica

For a domain $\Omega \subset {ℂ}^{n}$ let $H\left(\Omega \right)$ be the holomorphic functions on $\Omega$ and for any $k\in ℕ$ let ${A}^{k}\left(\Omega \right)=H\left(\Omega \right)\cap {C}^{k}\left(\overline{\Omega }\right)$. Denote by ${𝒜}_{D}^{k}\left(\Omega \right)$ the set of functions $f\phantom{\rule{0.222222em}{0ex}}\Omega \to \left[0,\infty \right)$ with the property that there exists a sequence of functions ${f}_{j}\in {A}^{k}\left(\Omega \right)$ such that $\left\{|{f}_{j}|\right\}$ is a nonincreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }|{f}_{j}\left(z\right)|$. By ${𝒜}_{I}^{k}\left(\Omega \right)$ denote the set of functions $f\phantom{\rule{0.222222em}{0ex}}\Omega \to \left(0,\infty \right)$ with the property that there exists a sequence of functions ${f}_{j}\in {A}^{k}\left(\Omega \right)$ such that $\left\{|{f}_{j}|\right\}$ is a nondecreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }|{f}_{j}\left(z\right)|$. Let $k\in ℕ$ and let ${\Omega }_{1}$ and ${\Omega }_{2}$ be bounded ${A}^{k}$-domains of holomorphy in ${ℂ}^{{m}_{1}}$ and ${ℂ}^{{m}_{2}}$ respectively. Let ${g}_{1}\in {𝒜}_{D}^{k}\left({\Omega }_{1}\right)$, ${g}_{2}\in {𝒜}_{I}^{k}\left({\Omega }_{1}\right)$ and $h\in {𝒜}_{D}^{k}\left({\Omega }_{2}\right)\cap {𝒜}_{I}^{k}\left({\Omega }_{2}\right)$. We prove that the...

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