Entire curves avoiding given sets in C^n

Let $F\subset\Bbb C^n$ be a proper closed subset of $\Bbb C^n$ and $A\subset\Bbb C^n\setminus F$ at most countable ($n\geq 2$). We give conditions of $F$ and $A$, under which there exists a holomorphic immersion (or a proper holomorphic embedding) $\phi:\Bbb C\to\Bbb C^n$ with $A\subset\phi(\Bbb C)\subset\Bbb C^n\setminus F$.Comment: 10 page

A Carleman type theorem for proper holomorphic embeddings

In 1927, Carleman showed that a continuous, complex-valued function on the real line can be approximated in the Whitney topology by an entire function restricted to the real line. In this paper, we prove a similar result for proper holomorphic embeddings. Namely, we show that a proper \cC^r embedding of the real line into \C^n can be approximated in the strong \cC^r topology by a proper holomorphic embedding of \C into \C^n

Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for H\"olderian almost complex structures

If $X$ is an almost complex manifold, with an almost complex structure $J$ of class \CC^\alpha, for some $\alpha >0$, for every point $p\in X$ and every tangent vector $V$ at $p$, there exists a germ of $J$-holomorphic disc through $p$ with this prescribed tangent vector. This existence result goes back to Nijenhuis-Woolf. All the $J$ holomorphic curves are of class \CC^{1,\alpha} in this case. Then, exactly as for complex manifolds one can define the Royden-Kobayashi pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is an upper semi-continuous function on the tangent bundle. For complex manifolds it is the crucial point in Royden's proof of the equivalence of the two standard definitions of the Kobayashi pseudo-metric. The upper semi-continuity of the Royden-Kobayashi pseudo-norm has been established by Kruglikov for structures that are smooth enough. In [I-R], it is shown that \CC^{1,\alpha} regularity of $J$ is enough. Here we show the following: Theorem. There exists an almost complex structure $J$ of class \CC^{1\over 2} on the unit bidisc \D^2\subset \C^2, such that the Royden-Kobayashi seudo-norm is not an upper semi-continuous function on the tangent bundle.Comment: 5 page

A biogenic amine and a neuropeptide act identically: tyramine signals through calcium in drosophila tubule stellate cells

Insect osmoregulation is subject to highly sophisticated endocrine control. In Drosophila, both Drosophila kinin and tyramine act on the Malpighian (renal) tubule stellate cell to activate chloride shunt conductance, and so increase the fluid production rate. Drosophila kinin is known to act through intracellular calcium, but the mode of action of tyramine is not known. Here, we used a transgenically encoded GFP::apoaequorin translational fusion, targeted to either principal or stellate cells under GAL4/UAS control, to demonstrate that tyramine indeed acts to raise calcium in stellate, but not principal cells. Furthermore, the EC(50) tyramine concentration for half-maximal activation of the intracellular calcium signal is the same as that calculated from previously published data on tyramine-induced increase in chloride flux. In addition, tyramine signalling to calcium is markedly reduced in mutants of NorpA (a phospholipase C) and itpr, the inositol trisphosphate receptor gene, which we have previously shown to be necessary for Drosophila kinin signalling. Therefore, tyramine and Drosophila kinin signals converge on phospholipase C, and thence on intracellular calcium; and both act to increase chloride shunt conductance by signalling through itpr. To test this model, we co-applied tyramine and Drosophila kinin, and showed that the calcium signals were neither additive nor synergistic. The two signalling pathways thus represent parallel, independent mechanisms for distinct tissues (nervous and epithelial) to control the same aspect of renal function

Generalization of a theorem of Gonchar

Let $X, Y$ be two complex manifolds, let $D\subset X,$ $G\subset Y$ be two nonempty open sets, let $A$ (resp. $B$) be an open subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Under a geometric condition on the boundary sets $A$ and $B,$ we show that every function locally bounded, separately continuous on $W,$ continuous on $A\times B,$ and separately holomorphic on $(A\times G) \cup (D\times B)$ "extends" to a function continuous on a "domain of holomorphy" $\hat{W}$ and holomorphic on the interior of $\hat{W}.$Comment: 14 pages, to appear in Arkiv for Matemati

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,...,x_k)$ where $p \in C[x_1,...,x_k],$ acts $m-$transitively on the smooth part reg$X$ of $X$ for any $m \in N.$ We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.Comment: 39 Pages, LaTeX; a revised version with minor changes and correction

Abstract basins of attraction

Abstract basins appear naturally in different areas of several complex variables. In this survey we want to describe three different topics in which they play an important role, leading to interesting open problems