Lions-type compactness and Rubik actions on the Heisenberg group

In this paper we prove a Lions-type compactness embedding result for symmetric unbounded domains of the Heisenberg group. The natural group action on the Heisenberg group TeX is provided by the unitary group U(n) × {1} and its appropriate subgroups, which will be used to construct subspaces with specific symmetry and compactness properties in the Folland-Stein’s horizontal Sobolev space TeX. As an application, we study the multiplicity of solutions for a singular subelliptic problem by exploiting a technique of solving the Rubik-cube applied to subgroups of U(n) × {1}. In our approach we employ concentration compactness, group-theoretical arguments, and variational methods.

Modulus method and radial stretch map in the Heisenberg group

We develop a modulus method for surface families inside a domain in the Heisenberg group and we prove that the stretch map between two Heisenberg spherical rings is a minimiser for the mean distortion among the class of contact quasiconformal maps between these rings which satisfy certain boundary conditions.

The effect of projections on dimension in the Heisenberg group

We prove analogs of classical almost sure dimension theorems for Euclidean projection mappings in the first Heisenberg group, equipped with a sub-Riemannian metric.

Highly pathogenic adapted HIV-1 strains limit host immunity and dictate rapid disease progression.

OBJECTIVE: The study of HIV-1 rapid progressors has been limited to specific case reports. Nevertheless, identification and characterization of the viral and host factors involved in rapid progression are crucial when attempting to uncover the correlates of rapid disease outcome. DESIGN: We carried out comparative functional analyses in rapid progressors (n = 46) and standard progressors (n = 46) early after HIV-1 seroconversion (≤1 year). The viral traits tested were viral replicative capacity, co-receptor usage, and genomic variation. Host CD8 T-cell responses, humoral activity, and HLA immunogenetic markers were also determined. RESULTS: Our data demonstrate an unusual convergence of highly pathogenic HIV-1 strains in rapid progressors. Compared with standard progressors, rapid progressor viral strains show higher in-vitro replicative capacity (81.5 vs. 67.9%; P = 0.025) and greater X4/DM co-receptor usage (26.3 vs. 2.8%; P = 0.006) in early infection. Limited or absent functional HIV-1 CD8 T-cell responses and neutralizing activity were measured in rapid progressors. Moreover, the increase in common HLA allele-restricted CD8 T-cell escape mutations in rapid progressors acts as a signature of uncontrolled HIV-1 replication and early impairment of adaptive cellular responses. CONCLUSION: Our data support a dominant role for viral factors in rapid progressors. Robust HIV-1 replication and intrinsic viral properties limit host adaptive immune responses, thus driving rapid disease progression.