On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.Comment: 26 page

Reference values and repeatability of inspiratory capacity for men and women aged 65–85

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A Reilly formula and eigenvalue estimates for differential forms

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally we also obtain, as a by-product of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.Comment: 22 page

Superposed sedimentary and tectonic block-in-matrix fabrics in a subducted serpentinite m\ue9lange (High-pressure zermatt saas ophiolite, western alps)

The primary stratigraphic fabric of a chaotic rock unit in the Zermatt Saas ophiolite of the Western Alps was reworked by a polyphase Alpine tectonic deformation. Multiscalar structural criteria demonstrate that this unit was deformed by two ductile subduction-related phases followed by brittle-ductile then brittle deformation. Deformation partitioning operated at various scales, leaving relatively unstrained rock domains preserving internal texture, organization, and composition. During subduction, ductile deformation involved stretching, boudinage, and simultaneous folding of the primary stratigraphic succession. This deformation is particularly well-documented in alternating layers showing contrasting deformation style, such as carbonate-rich rocks and turbiditic serpentinite metasandstones. During collision and exhumation, deformation enhanced the boudinaged horizons and blocks, giving rise to spherical to lozenge-shaped blocks embedded in a carbonate-rich matrix. Structural criteria allow the recognition of two main domains within the chaotic rock unit, one attributable to original broken formations reflecting turbiditic sedimentation, the other ascribable to an original sedimentary m\ue9lange. The envisaged geodynamic setting for the formation of the protoliths is the Jurassic Ligurian-Piedmont ocean basin floored by mostly serpentinized peridotites, intensely tectonized by extensional faults that triggered mass transport processes and turbiditic sedimentation

Comprehensive genomic analysis reveals the prognostic role of LRRK2 copy-number variations in human malignancies

Genetic alterations of leucine-rich repeat kinase 2 (LRRK2), one of the most important contributors to familial Parkinson\u2019s disease (PD), have been hypothesized to play a role in cancer development due to demographical and preclinical data. Here, we sought to define the prevalence and prognostic significance of LRRK2 somatic mutations across all types of human malignancies by querying the publicly available online genomic database cBioPortal. Ninety-six different studies with 14,041 cases were included in the analysis, and 761/14,041 (5.4%) showed genetic alterations in LRRK2. Among these, 585 (76.9%) were point mutations, indels or fusions, 168 (22.1%) were copy number variations (CNVs), and 8 (1.0%) showed both types of alterations. One case showed the somatic mutation R1441C. A significant difference in terms of overall survival (OS) was noted between cases harboring somatic LRRK2 whole deletions, amplifications, and CNV-unaltered cases (median OS: 20.09, 57.40, and 106.57 months, respectively; p = 0.0008). These results suggest that both LRRK2 amplifications and whole gene deletions could play a role in cancer development, paving the way for future research in terms of potential treatment with LRRK2 small molecule inhibitors for LRRK2-amplified cases