Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality

We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow to draw some consequences about second order differential operators modeled on these nonsmooth Hormander's vector fields.Comment: 60 pages, LaTeX; Section 6 added and Section 7 (6 in the previous version) changed. Some references adde

Levi umbilical surfaces in complex space

We define a complex connection on a real hypersurface of \C^{n+1} which is naturally inherited from the ambient space. Using a system of Codazzi-type equations, we classify connected real hypersurfaces in \C^{n+1}, $n\ge 2$, which are Levi umbilical and have non zero constant Levi curvature. It turns out that such surfaces are contained either in a sphere or in the boundary of a complex tube domain with spherical section.Comment: 18 page

Sub-Riemannian Fast Marching in SE(2)

We propose a Fast Marching based implementation for computing sub-Riemanninan (SR) geodesics in the roto-translation group SE(2), with a metric depending on a cost induced by the image data. The key ingredient is a Riemannian approximation of the SR-metric. Then, a state of the art Fast Marching solver that is able to deal with extreme anisotropies is used to compute a SR-distance map as the solution of a corresponding eikonal equation. Subsequent backtracking on the distance map gives the geodesics. To validate the method, we consider the uniform cost case in which exact formulas for SR-geodesics are known and we show remarkable accuracy of the numerically computed SR-spheres. We also show a dramatic decrease in computational time with respect to a previous PDE-based iterative approach. Regarding image analysis applications, we show the potential of considering these data adaptive geodesics for a fully automated retinal vessel tree segmentation.Comment: CIARP 201

First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds

We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator for non-singular C^2 surfaces and another one for C2 (eventually singular) surfaces. Then we can obtain a necessary condition for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we classify complete stable surfaces in the roto-traslation group RT .Comment: 36 pages. Misprints corrected. Statement of Proposition 9.8 slightly changed and Remark 9.9 adde

Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields

We consider a family of vector fields and we assume a horizontal regularity on their derivatives. We discuss the notion of commutator showing that different definitions agree. We apply our results to the proof of a ball-box theorem and Poincar\'e inequality for nonsmooth H\"ormander vector fields.Comment: arXiv admin note: material from arXiv:1106.2410v1, now three separate articles arXiv:1106.2410v2, arXiv:1201.5228, arXiv:1201.520

The role of fundamental solution in Potential and Regularity Theory for subelliptic PDE

In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formulas on the level sets of the fundamental solution, which are the starting point to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results, namely estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem

Mycoplasma genitalium: an efficient strategy to generate genetic variation from a minimal genome

Mycoplasma genitalium, a human pathogen associated with sexually transmitted diseases, is unique in that it has smallest genome of any known free-living organism. The goal of this study was to investigate if and how M. genitalium uses a minimal genome to generate genetic variations. We analysed the sequence variability of the third gene (MG192 or mgpC) of the M. genitalium MgPa adhesion operon, demonstrated that the MG192 gene is highly variable among and within M. genitalium strains in vitro and in vivo, and identified MG192 sequence shifts in the course of in vitro passage of the G37 type strain and in sequential specimens from an M. genitalium-infected patient. In order to establish the origin of the MG192 variants, we examined nine genomic loci containing partial copies of the MgPa operon, known as MgPar sequences. Our analysis suggests that the MG192 sequence variation is achieved by recombination between the MG192 expression site and MgPar sequences via gene cross-over and, possibly, also by gene conversion. It appears plausible that M. genitalium has the ability to generate unlimited variants from its minimized genome, which presumably allows the organism to adapt to diverse environments and/or to evade host defences by antigenic variation

Genome-Scale Analysis of Mycoplasma agalactiae Loci Involved in Interaction with Host Cells

Mycoplasma agalactiae is an important pathogen of small ruminants, in which it causes contagious agalactia. It belongs to a large group of “minimal bacteria” with a small genome and reduced metabolic capacities that are dependent on their host for nutrients. Mycoplasma survival thus relies on intimate contact with host cells, but little is known about the factors involved in these interactions or in the more general infectious process. To address this issue, an assay based on goat epithelial and fibroblastic cells was used to screen a M. agalactiae knockout mutant library. Mutants with reduced growth capacities in cell culture were selected and 62 genomic loci were identified as contributing to this phenotype. As expected for minimal bacteria, “transport and metabolism” was the functional category most commonly implicated in this phenotype, but 50% of the selected mutants were disrupted in coding sequences (CDSs) with unknown functions, with surface lipoproteins being most commonly represented in this category. Since mycoplasmas lack a cell wall, lipoproteins are likely to be important in interactions with the host. A few intergenic regions were also identified that may act as regulatory sequences under co-culture conditions. Interestingly, some mutants mapped to gene clusters that are highly conserved across mycoplasma species but located in different positions. One of these clusters was found in a transcriptionally active region of the M. agalactiae chromosome, downstream of a cryptic promoter. A possible scenario for the evolution of these loci is discussed. Finally, several CDSs identified here are conserved in other important pathogenic mycoplasmas, and some were involved in horizontal gene transfer with phylogenetically distant species. These results provide a basis for further deciphering functions mediating mycoplasma-host interactions

The Role of the Multiple Banded Antigen of Ureaplasma parvum in Intra-Amniotic Infection: Major Virulence Factor or Decoy?

The multiple banded antigen (MBA) is a predicted virulence factor of Ureaplasma species. Antigenic variation of the MBA is a potential mechanism by which ureaplasmas avoid immune recognition and cause chronic infections of the upper genital tract of pregnant women. We tested whether the MBA is involved in the pathogenesis of intra-amniotic infection and chorioamnionitis by injecting virulent or avirulent-derived ureaplasma clones (expressing single MBA variants) into the amniotic fluid of pregnant sheep. At 55 days of gestation pregnant ewes (n = 20) received intra-amniotic injections of virulent-derived or avirulent-derived U. parvum serovar 6 strains (2×104 CFU), or 10B medium (n = 5). Amniotic fluid was collected every two weeks post-infection and fetal tissues were collected at the time of surgical delivery of the fetus (140 days of gestation). Whilst chronic colonisation was established in the amniotic fluid of animals infected with avirulent-derived and virulent-derived ureaplasmas, the severity of chorioamnionitis and fetal inflammation was not different between these groups (p>0.05). MBA size variants (32–170 kDa) were generated in vivo in amniotic fluid samples from both the avirulent and virulent groups, whereas in vitro antibody selection experiments led to the emergence of MBA-negative escape variants in both strains. Anti-ureaplasma IgG antibodies were detected in the maternal serum of animals from the avirulent (40%) and virulent (55%) groups, and these antibodies correlated with increased IL-1β, IL-6 and IL-8 expression in chorioamnion tissue (p<0.05). We demonstrate that ureaplasmas are capable of MBA phase variation in vitro; however, ureaplasmas undergo MBA size variation in vivo, to potentially prevent eradication by the immune response. Size variation of the MBA did not correlate with the severity of chorioamnionitis. Nonetheless, the correlation between a maternal humoral response and the expression of chorioamnion cytokines is a novel finding. This host response may be important in the pathogenesis of inflammation-mediated adverse pregnancy outcomes