Undecidable properties of self-affine sets and multi-tape automata

We study the decidability of the topological properties of some objects coming from fractal geometry. We prove that having empty interior is undecidable for the sets defined by two-dimensional graph-directed iterated function systems. These results are obtained by studying a particular class of self-affine sets associated with multi-tape automata. We first establish the undecidability of some language-theoretical properties of such automata, which then translate into undecidability results about their associated self-affine sets.Comment: 10 pages, v2 includes some corrections to match the published versio

Direct three-dimensional microfabrication of hydrogels via two-photon lithography in aqueous solution

Hydrogels capable of releasing growth factors from very precise 3D structures to be placed on neural electrodes were microfabricated by two-photon lithography in aqueous solution. Hydrophobic chromophores with a large two-photon absorption cross section were solubilized using a nonionic surfactant and the resulting complex were applied with an oil soluble photoradical initiator to fabricate hydrogel microstructures. The results show that hydrogel microstructures are formed when the initiator 2,2-dimethoxy-2-phenyl acetophenone (Irgacure 651) and the 2-photon sensitive chromophore are present in the polymerizable mixture. The presence of water in the prepolymer solution helps in controlling important properties of the resulting structure such as modulus and the pore size which can be fine-tuned for its intended application

Elevational gradients of species diversity, breeding system and floral traits of orchid species on Reunion Island

Aim: This research investigates changes in orchid species composition and diversity, plant breeding system and floral traits along an elevational gradient spanning 2300 m (200-2500 m). Location: The study was conducted on Réunion Island (Mascarene Islands, Indian Ocean). Methods: Data on the distribution of 135 orchid species from 35 genera were gathered from 121 localities situated between 200 and 2500 m a.s.l. For each locality, 500-m transects were walked and each orchid species was recorded. Measures of species diversity (species richness, a modified Shannon diversity index and the modified Shannon equitability index) were related to altitude using ordinary least-squares regression. Species turnover and elevational gradients in species composition were determined by: (1) relating scores of detrended correspondence analysis to altitude using ordinary least-squares regression, and (2) relating Sørensen similarity indices to differences in altitude using Mantel tests. Finally, the average proportion of species displaying similar floral traits or showing the same breeding system were compared among altitudinal zones. Results: Species richness per transect ranged from 1 to 36 species (mean 14.3) and decreased significantly with increasing altitude. Similarly, species evenness decreased significantly with increasing altitude. Around 50% of all orchid species were rare (occurred in fewer than 5% of all localities), and only a few occurred in more than 50% of all localities. Orchid species composition changed continuously with altitude, indicating turnover of species with increasing altitude. Analogously, orchid breeding systems and floral traits also changed with altitude. Relatively more auto-pollinating species were found at high altitudes compared with mid- and low-altitude sites where animal-pollinated species were most abundant. Species characterized by a cleistogamous pollination system were found almost exclusively in high-altitude sites, whereas the proportion of species displaying floral traits related to pollination by long-tongued moths (sphinx) and flies sharply decreased with increasing altitude. Main conclusions: Environmental conditions associated with altitude exert a large influence on orchid species composition and the distribution of orchid breeding systems. Our results revealed a high proportion of auto-pollinating species, and confirm earlier findings that auto-pollinating species are more frequent in high-altitude sites. © Blackwell Publishing Ltd