A geometric study of Wasserstein spaces: Hadamard spaces

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space W(X). In this paper we investigate the geometry of W(X) when X is a Hadamard space, by which we mean that $X$ has globally non-positive sectional curvature and is locally compact. Although it is known that -except in the case of the line- W(X) is not non-positively curved, our results show that W(X) have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for W(X) that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in W(X).Comment: This second version contains only the first part of the preceeding one. The visibility properties of W(X) and the isometric rigidity have been split off to other articles after a referee's commen

A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces

A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a naturel set of spaces generalizing the usual Hilbert cube. In a second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space, with uniform bound and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d.Comment: v2 Largely expanded version, as reflected by the change of title; all part I on generalized Hausdorff dimension is new, as well as the embedding of Hilbert cubes into Wasserstein spaces. v3 modified according to the referee final remarks ; to appear in Journal of Topology and Analysi

Symmetric spaces of higher rank do not admit differentiable compactifications

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answer negatively.Comment: 13 pages, to appear in Mathematische Annale

Expanding e-MERLIN with the Goonhilly Earth Station

A consortium of universities has recently been formed with the goal of using the decommissioned telecommunications infrastructure at the Goonhilly Earth Station in Cornwall, UK, for astronomical purposes. One particular goal is the introduction of one or more of the ~30-metre parabolic antennas into the existing e-MERLIN radio interferometer. This article introduces this scheme and presents some simulations which quantify the improvements that would be brought to the e-MERLIN system. These include an approximate doubling of the spatial resolution of the array, an increase in its N-S extent with strong implications for imaging the most well-studied equatorial fields, accessible to ESO facilities including ALMA. It also increases the overlap between the e-MERLIN array and the European VLBI Network. We also discuss briefly some niche science areas in which an e-MERLIN array which included a receptor at Goonhilly would be potentially world-leading, in addition to enhancing the existing potential of e-MERLIN in its role as a Square Kilometer Array pathfinder instrument.Comment: 7 pages, 3 figures, to appear in the proceedings of "Astronomy with megastructures: Joint science with the E-ELT and SKA", 10-14 May 2010, Crete, Greece (Eds: Isobel Hook, Dimitra Rigopoulou, Steve Rawlings and Aris Karastergiou

Goonhilly: a new site for e-MERLIN and the EVN

The benefits for the e-MERLIN and EVN arrays of using antennae at the satellite communication station at Goonhilly in Cornwall are discussed. The location of this site - new to astronomy - will provide an almost equal distribution of long baselines in the east-west- and north-south directions, and opens up the possibility to get significantly improved observations of equatorial fields with e-MERLIN. These additional baselines will improve the sensitivity on a set of critical spatial scales and will increase the angular resolution of e-MERLIN by a factor of two. e-MERLIN observations, including many allocated under the e-MERLIN Legacy programme, will benefit from the enhanced angular resolution and imaging capability especially for sources close to or below the celestial equator (where ESO facilities such as ALMA will operate) of including the Goonhilly telescopes. Furthermore, the baselines formed between Goonhilly and the existing stations will close the gap between the baselines of e-MERLIN and those of the European VLBI Network (EVN) and therefore enhance the legacy value of e-MERLIN datasets.Comment: 10 pages, 2 figue

Epidermal Growth Factor–PEG Functionalized PAMAM-Pentaethylenehexamine Dendron for Targeted Gene Delivery Produced by Click Chemistry

Aim of this study was the site-specific conjugation of an epidermal growth factor (EGF)-polyethylene glycol (PEG) chain by click chemistry onto a poly(amido amine) (PAMAM) dendron, as a key step toward defined multifunctional carriers for targeted gene delivery. For this purpose, at first propargyl amine cored PAMAM dendrons with ester ends were synthesized. The chain terminal ester groups were then modified by oligoamines with different secondary amino densities. The oligoamine-modified PAMAM dendrons were well biocompatible, as demonstrated in cytotoxicity assays. Among the different oligoamine-modified dendrons, PAMAM-pentaethylenehexamine (PEHA) dendron polyplexes displayed the best gene transfer ability. Conjugation of PAMAM-PEHA dendron with PEG spacer was conducted via click reaction, which was performed before amidation with PEHA. The resultant PEG-PAMAM-PEHA copolymer was then coupled with EGF ligand. pDNA transfections in HuH-7 hepatocellular carcinoma cells showed a 10-fold higher efficiency with the polyplexes containing conjugated EGF as compared to the ligand-free ones, demonstrating the concept of ligand targeting. Overall gene transfer efficiencies, however, were moderate, suggesting that additional measures for overcoming subsequent intracellular bottlenecks in delivery have to be taken

A semi-empirical simulation of the extragalactic radio continuum sky for next generation radio telescopes

We have developed a semi-empirical simulation of the extragalactic radio continuum sky suitable for aiding the design of next generation radio interferometers such as the Square Kilometre Array (SKA). The emphasis is on modelling the large-scale cosmological distribution of radio sources rather than the internal structure of individual galaxies. Here we provide a description of the simulation to accompany the online release of a catalogue of similar or equal to 320 million simulated radio sources. The simulation covers a sky area of 20x20 deg^2 - a plausible upper limit to the instantaneous field of view attainable with future (e.g. SKA) aperture array technologies - out to a cosmological redshift of z=20, and down to flux density limits of 10 nJy at 151, 610 MHz, 1.4, 4.86 and 18 GHz. Five distinct source types are included: radio-quiet active galactic nuclei (AGN), radio-loud AGN of the Fanaroff-Riley type I (FR I) and FR II structural classes, and star-forming galaxies, the latter split into Populations of quiescent and starbursting galaxies

Functional Analysis of the Cytoskeleton Protein MreB from Chlamydophila pneumoniae

In rod-shaped bacteria, the bacterial actin ortholog MreB is considered to organize the incorporation of cell wall precursors into the side-wall, whereas the tubulin homologue FtsZ is known to tether incorporation of cell wall building blocks at the developing septum. For intracellular bacteria, there is no need to compensate osmotic pressure by means of a cell wall, and peptidoglycan has not been reliably detected in Chlamydiaceae. Surprisingly, a nearly complete pathway for the biosynthesis of the cell wall building block lipid II has been found in the genomes of Chlamydiaceae. In a previous study, we discussed the hypothesis that conservation of lipid II biosynthesis in cell wall-lacking bacteria may reflect the intimate molecular linkage of cell wall biosynthesis and cell division and thus an essential role of the precursor in cell division. Here, we investigate why spherical-shaped chlamydiae harbor MreB which is almost exclusively found in elongated bacteria (i.e. rods, vibrios, spirilla) whereas they lack the otherwise essential division protein FtsZ. We demonstrate that chlamydial MreB polymerizes in vitro and that polymerization is not inhibited by the blocking agent A22. As observed for MreB from Bacillus subtilis, chlamydial MreB does not require ATP for polymerization but is capable of ATP hydrolysis in phosphate release assays. Co-pelleting and bacterial two-hybrid experiments indicate that MreB from Chlamydophila (Chlamydia) pneumoniae interacts with MurF, MraY and MurG, three key components in lipid II biosynthesis. In addition, MreB polymerization is improved in the presence of MurF. Our findings suggest that MreB is involved in tethering biosynthesis of lipid II and as such may be necessary for maintaining a functional divisome machinery in Chlamydiaceae