Pluricomplex Green and Lempert functions for equally weighted poles

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.Comment: 25 page

Development of a Pacific oyster (Crassostrea gigas) 31,918-feature microarray: identification of reference genes and tissue-enriched expression patterns

Background: Research using the Pacific oyster Crassostrea gigas as a model organism has experienced rapid growth in recent years due to the development of high-throughput molecular technologies. As many as 56,268 EST sequences have been sequenced to date, representing a genome-wide resource that can be used for transcriptomic investigations. Results: In this paper, we developed a Pacific oyster microarray containing oligonucleotides representing 31,918 transcribed sequences selected from the publicly accessible GigasDatabase. This newly designed microarray was used to study the transcriptome of male and female gonads, mantle, gills, posterior adductor muscle, visceral ganglia, hemocytes, labial palps and digestive gland. Statistical analyses identified genes differentially expressed among tissues and clusters of tissue-enriched genes. These genes reflect major tissue-specific functions at the molecular level, such as tissue formation in the mantle, filtering in the gills and labial palps, and reproduction in the gonads. Hierarchical clustering predicted the involvement of unannotated genes in specific functional pathways such as the insulin/NPY pathway, an important pathway under study in our model species. Microarray data also accurately identified reference genes whose mRNA level appeared stable across all the analyzed tissues. Adp-ribosylation factor 1 9arf1) appeared to be the most robust reference for normalizing gene expression data across different tissues and is therefore proposed as a relevant reference gene for further gene expression analysis in the Pacific oyster. Conclusions: This study provides a new transcriptomic tool for studies of oyster biology, which will help in the annotation of its genome and which identifies candidate reference genes for gene expression analysis

Effects of nanoparticles on murine macrophages

Metallic nanoparticles are more and more widely used in an increasing number of applications. Consequently, they are more and more present in the environment, and the risk that they may represent for human health must be evaluated. This requires to increase our knowledge of the cellular responses to nanoparticles. In this context, macrophages appear as an attractive system. They play a major role in eliminating foreign matter, e.g. pathogens or infectious agents, by phagocytosis and inflammatory responses, and are thus highly likely to react to nanoparticles. We have decided to study their responses to nanoparticles by a combination of classical and wide-scope approaches such as proteomics. The long term goal of this study is the better understanding of the responses of macrophages to nanoparticles, and thus to help to assess their possible impact on human health. We chose as a model system bone marrow-derived macrophages and studied the effect of commonly used nanoparticles such as TiO2 and Cu. Classical responses of macrophage were characterized and proteomic approaches based on 2D gels of whole cell extracts were used. Preliminary proteomic data resulting from whole cell extracts showed different effects for TiO2-NPs and Cu-NPs. Modifications of the expression of several proteins involved in different pathways such as, for example, signal transduction, endosome-lysosome pathway, Krebs cycle, oxidative stress response have been underscored. These first results validate our proteomics approach and open a new wide field of investigation for NPs impact on macrophagesComment: Nanosafe2010: International Conference on Safe Production and Use of Nanomaterials 16-18 November 2010, Grenoble, France, Grenoble : France (2010

On Vector Bundles of Finite Order

We study growth of holomorphic vector bundles E over smooth affine manifolds. We define Finsler metrics of finite order on E by estimates on the holomorphic bisectional curvature. These estimates are very similar to the ones used by Griffiths and Cornalba to define Hermitian metrics of finite order. We then generalize the Vanishing Theorem of Griffiths and Cornalba to the Finsler context. We develop a value distribution theory for holomorphic maps from the projectivization of E to projective space. We show that the projectivization of E can be immersed into a projective space of sufficiently large dimension via a map of finite order.Comment: version 2 has some typos corrected; to appear in Manuscripta Mathematic

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball \B \sub \C^n with its relative logarithmic capacity in \C^n with respect to the same ball \B. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of \C^n is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at infinity on \C^n as well as the Cegrell class of \psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W \Sub \C^n. Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of \psh functions.Comment: 25 page

Approximation of conformal mappings using conformally equivalent triangular lattices

Consider discrete conformal maps defined on the basis of two conformally equivalent triangle meshes, that is edge lengths are related by scale factors associated to the vertices. Given a smooth conformal map $f$, we show that it can be approximated by such discrete conformal maps $f^\epsilon$. In particular, let $T$ be an infinite regular triangulation of the plane with congruent triangles and only acute angles (i.e.\ $<\pi/2$). We scale this tiling by $\epsilon>0$ and approximate a compact subset of the domain of $f$ with a portion of it. For $\epsilon$ small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by $\log|f'|$ on the boundary. Furthermore we show that the corresponding discrete conformal maps $f^\epsilon$ converge to $f$ uniformly in $C^1$ with error of order $\epsilon$.Comment: 14 pages, 3 figures; v2 typos corrected, revised introduction, some proofs extende

Convergence and multiplicities for the Lempert function

Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space Hol (\D,\Omega) of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.Comment: 24 pages; many typos corrected thanks to the referee of Arkiv for Matemati