Solvability of subprincipal type operators

In this paper we consider the solvability of pseudodifferential operators in the case when the principal symbol vanishes of order $k \ge 2$ at a nonradial involutive manifold $\Sigma_2$. We shall assume that the operator is of subprincipal type, which means that the $k$:th inhomogeneous blowup at $\Sigma_2$ of the refined principal symbol is of principal type with Hamilton vector field parallel to the base $\Sigma_2$, but transversal to the symplectic leaves of $\Sigma_2$ at the characteristics. When $k = \infty$ this blowup reduces to the subprincipal symbol. We also assume that the blowup is essentially constant on the leaves of $\Sigma_2$, and does not satisfying the Nirenberg-Treves condition (${\Psi}$). We also have conditions on the vanishing of the normal gradient and the Hessian of the blowup at the characteristics. Under these conditions, we show that $P$ is not solvable.Comment: Changed the formulation of Theorem 2.15, added an assuption. Corrected errors and clarified the arguments. Added reference

Molecular and Evolutionary Bases of Within-Patient Genotypic and Phenotypic Diversity in Escherichia coli Extraintestinal Infections

Although polymicrobial infections, caused by combinations of viruses, bacteria, fungi and parasites, are being recognised with increasing frequency, little is known about the occurrence of within-species diversity in bacterial infections and the molecular and evolutionary bases of this diversity. We used multiple approaches to study the genomic and phenotypic diversity among 226 Escherichia coli isolates from deep and closed visceral infections occurring in 19 patients. We observed genomic variability among isolates from the same site within 11 patients. This diversity was of two types, as patients were infected either by several distinct E. coli clones (4 patients) or by members of a single clone that exhibit micro-heterogeneity (11 patients); both types of diversity were present in 4 patients. A surprisingly wide continuum of antibiotic resistance, outer membrane permeability, growth rate, stress resistance, red dry and rough morphotype characteristics and virulence properties were present within the isolates of single clones in 8 of the 11 patients showing genomic micro-heterogeneity. Many of the observed phenotypic differences within clones affected the trade-off between self-preservation and nutritional competence (SPANC). We showed in 3 patients that this phenotypic variability was associated with distinct levels of RpoS in co-existing isolates. Genome mutational analysis and global proteomic comparisons in isolates from a patient revealed a star-like relationship of changes amongst clonally diverging isolates. A mathematical model demonstrated that multiple genotypes with distinct RpoS levels can co-exist as a result of the SPANC trade-off. In the cases involving infection by a single clone, we present several lines of evidence to suggest diversification during the infectious process rather than an infection by multiple isolates exhibiting a micro-heterogeneity. Our results suggest that bacteria are subject to trade-offs during an infectious process and that the observed diversity resembled results obtained in experimental evolution studies. Whatever the mechanisms leading to diversity, our results have strong medical implications in terms of the need for more extensive isolate testing before deciding on antibiotic therapies

Topological vector spaces, distributions and kernels

Topological vector spaces, distributions and kernel

Intrinsic stratifications of analytic varieties

By associating a Lie algebra of analytic vector elds to every point of an analytic variety and using the associated Nagano foliation, this work presents a coordinate-free stratication of analytic sets