Estimates for the dilatation of σ\sigma-harmonic mappings

We consider planar $\sigma$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u^i)=0$, for $i=1,2$. We investigate whether a locally invertible $\sigma$-harmonic mapping $U$ is also quasiconformal. Under mild regularity assumptions, only involving $\det \sigma$ and the antisymmetric part of $\sigma$, we prove quantitative bounds which imply quasiconformality.Comment: 8 pages, to appear on Rendiconti di Matematica e delle sue applicazion

Quantitative estimates on Jacobians for hybrid inverse problems

We consider $\sigma$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u_i)=0$, for $i=1,\ldots,n$. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.Comment: 15 pages, submitte

Invertible harmonic mappings, beyond Kneser

We prove necessary and sufficient criteria of invertibility for planar harmonic mappings which generalize a classical result of H. Kneser, also known as the Rad\'{o}-Kneser-Choquet theorem.Comment: One section added. 15 page

Noise Estimate of Pendular Fabry-Perot through Reflectivity Change

A key issue in developing pendular Fabry-Perot interferometers as very accurate displacement measurement devices, is the noise level. The Fabry-Perot pendulums are the most promising device to detect gravitational waves, and therefore the background and the internal noise should be accurately measured and reduced. In fact terminal masses generates additional internal noise mainly due to thermal fluctuations and vibrations. We propose to exploit the reflectivity change, that occurs in some special points, to monitor the pendulums free oscillations and possibly estimate the noise level. We find that in spite of long transients, it is an effective method for noise estimate. We also prove that to only retain the sequence of escapes, rather than the whole time dependent dynamics, entails the main characteristics of the phenomenon. Escape times could also be relevant for future gravitational wave detector developments.Comment: PREPRINT Metrology for Aerospace (MetroAeroSpace), 2014 IEEE Publication Year: 2014, Page(s): 468 - 47

Spin connection as Lorentz gauge field: propagating torsion

We propose a modified gravitational action containing besides the Einstein-Cartan term some quadratic contributions resembling the Yang-Mills lagrangian for the Lorentz spin connections. We outline how a propagating torsion arises and we solve explicitly the linearised equations of motion on a Minkowski background. We identify among torsion components six degrees of freedom: one is carried by a pseudo-scalar particle, five by a tachyon field. By adding spinor fields and neglecting backreaction on the geometry, we point out how only the pseudo-scalar particle couples directly with fermions, but the resulting coupling constant is suppressed by the ratio between fermion and Planck masses. Including backreaction, we demonstrate how the tachyon field provides causality violation in the matter sector, via an interaction mediated by gravitational waves.Comment: 7 pages, no figures, new section adde

Pathophysiological role of extrasynaptic GABAA receptors in typical absence epilepsy

GABA is the principal inhibitory neurotransmitter in the mammalian CNS. It acts via two classes of receptors, the GABAA, a ligand gated ion channel (ionotropic receptor) and the metabotropic G-protein coupled GABAB receptor. While synaptic GABAA receptors underlie classical ‘phasic’ GABAA receptor-mediated inhibition, extrasynaptic GABAA receptors (eGABAAR) mediate a new form of inhibition, termed ‘tonic’ GABAA inhibition. The subunit composition of eGABAARs differs from those present at the synapse, resulting in pharmacologically and functionally distinct properties. In this mini-review the findings presented at the 2nd Neuroscience Day meeting held last July in Malta will be summarised. Particular emphasis will be given to the important pathophysiological role of eGABAAR within thalamocortical circuits as a major player in nonconvulsive absence epilepsy. The new findings presented at the conference suggest that enhanced tonic inhibition is a common cause of seizures in several animal models of absence epilepsy and may provide new targets for therapeutic intervention.peer-reviewe

Nonlinear equations involving the square root of the Laplacian

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega. \end{array}\right. \end{equation*} The existence of at least two non-trivial $L^{\infty}$-bounded weak solutions is established for large value of the parameter $\lambda$ requiring that the nonlinear term $f$ is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method