Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\Delta u=4 e^{2u}$ and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence $\{z_j\}$ in the unit disk there is always a Blaschke product with $\{z_j\}$ as its set of critical points. Our work is closely related to the Berger-Nirenberg problem in differential geometry.Comment: 21 page

Expression of a cDNA encoding the glucose trimming enzyme glucosidase II in CHO cells and molecular characterization of the enzyme deficiency in a mutant mouse lymphoma cell line

Glucosidase II is an ER resident glycoprotein involved in the processing of N-linked glycans and probably a component of the ER quality control of glycoproteins. For cloning of glucosidase II cDNA, degenerate oligonucleotides based on amino acid sequences derived from proteolytic fragments of purified pig liver glucosidase II were used. An unamplified cDNA library from pig liver was screened with a 760 bp glucosidase II specific cDNA fragment obtained by RT-PCR. A 3.9 kb glucosidase II cDNA with an open reading frame of about 2.9 kb was obtained. The glucosidase II sequence did not contain known ER retention signals nor hydrophobic regions which could represent a transmem-brane domain; however, it contained a single N-glycosylation site close to the amino terminus. All studied pig and rat tissues exhibited an mRNA of approximately 4.4 kb with varying tissue expression levels. The authenticity of the identified cDNA with that coding for glucosidase II was proven by overexpression in CHO cells. Mouse lymphoma PHAR 2.7 cells, deficient in glucosidase II activity, were shown to be devoid of transcript

Strict Wick-type deformation quantization on Riemann surfaces: Rigidity and Obstructions

Let $X$ be a hyperbolic Riemann surface. We study a convergent Wick-type star product $\star_X$ on $X$ which is induced by the canonical convergent star product $\star_{\mathbb{D}}$ on the unit disk $\mathbb{D}$ via Uniformization Theory. While by construction, the resulting Fr\'echet algebras $(\mathcal{A}(X),\star_X)$ are strongly isomorphic for conformally equivalent Riemann surfaces, our work exhibits additional severe topological obstructions. In particular, we show that the Fr\'echet algebra $(\mathcal{A}(X),\star_X)$ degenerates if and only if the connectivity of $X$ is at least $3$, and $(\mathcal{A}(X),\star_X)$ is noncommutative if and only if $X$ is simply connected. We also explicitly determine the algebra $\mathcal{A}_X$ and the star product $\star_X$ for the intermediate case of doubly connected Riemann surfaces $X$. As a perhaps surprinsing result, we deduce that two such Fr\'echet algebras are strongly isomorphic if and only if either both Riemann surfaces are conformally equivalent to an (not neccesarily the same) annulus or both are conformally equivalent to a punctured disk.Comment: References update