Extending Whitney's extension theorem: nonlinear function spaces

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains , with non-smooth boundary, in possibly non-compact manifolds. Assuming is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.publishedVersio

Biomarkers of disease differentiation: HCV recurrence versus acute cellular rejection

The wound-healing process induced by chronic hepatitis C virus (HCV) infection triggers liver damage characterized by fibrosis development and finally cirrhosis. Liver Transplantation (LT) is the optimal surgical treatment for HCV-cirrhotic patients at end-stage liver disease. However, acute cellular rejection (ACR) and HCV recurrence disease represent two devastating complications post-LT. The accurate differential diagnosis between both conditions is critical for treatment choice, and similar histological features represent a challenge for pathologists. Moreover, the HCV recurrence disease severity is highly variable post-LT. HCV recurrence disease progression is characterized by an accelerated fibrogenesis process, and almost 30% of those patients develop cirrhosis at 5-years of follow-up. Whole-genome gene expression (WGE) analyses through well-defined oligonucleotide microarray platforms represent a powerful tool for the molecular characterization of biological process. In the present manuscript, the utility of microarray technology is applied for the ACR and HCV-recurrence biological characterization in post-LT liver biopsy samples. Moreover, WGE analysis was performed to identify predictive biomarkers of HCV recurrence severity in formalin-fixed paraffin-embedded liver biopsies prospectively collected

Extending Whitney's extension theorem: nonlinear function spaces

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains , with non-smooth boundary, in possibly non-compact manifolds. Assuming is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator