Optimal Szeg\"o-Weinberger type inequalities

Denote with $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ the first nontrivial eigenvalue of the Neumann problem \begin{equation*} \left\{\begin{array}{lll} -\text{div}\left(e^{h\left(|x|\right)}\nabla u\right) =\mu e^{h\left(|x|\right)}u & \text{in} & \Omega & & \frac{\partial u}{\partial \nu}=0 & \text{on} & \partial \Omega , \end{array} \right. \end{equation*} where $\Omega$ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. Under suitable assumption on $h$ we prove that the ball centered at the origin is the unique set maximizing $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ among all Lipschitz bounded domains $\Omega$ of $\mathbb{R}^{N}$ of prescribed $e^{h\left(|x|\right)}dx$-measure and symmetric about the origin. Moreover, an example in the model case $h\left(|x|\right) =|x|^{2},$ shows that, in general, the assumption on the symmetry of the domain cannot be dropped. In the one-dimensional case, i.e. when $\Omega$ reduces to an interval $(a,b),$ we consider a wide class of weights (including both Gaussian and anti-Gaussian). We then describe the behavior of the eigenvalue as the interval $(a,b)$ slides along the $x$-axis keeping fixed its weighted length

Uniformly bounded representations and completely bounded multipliers of SL(2,R)

We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2, R) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal

Modern meson--exchange potential and superfluid neutron star crust matter

In this work we study properties of neutron star crusts, where matter is expected to consist of nuclei surrounded by superfluid neutrons and a homogeneous background of relativistic electrons. The nuclei are disposed in a Coulomb lattice, and it is believed that the structure of the lattice influences considerably the specific heat of the neutronic matter inside the crust of a neutron star. Using a modern meson--exchange potential in the framework of a local--density approximation we calculate the neutronic specific heat accounting for various shapes of the Coulomb lattice, from spherical to non--spherical nuclear shapes. We find that a realistic nucleon--nucleon potential leads to a significant increase in the neutronic specific heat with respect to that obtained assuming a uniform neutron distribution. The increase is largest for the non--spherical phase of the crust. These results may have consequences for the thermal history of young neutron stars.Comment: Revtex, 5 pages, 4 figures included as uuencoded p

R-modes in Neutron Stars with Crusts: Turbulent Saturation, Spin-down, and Crust Melting

Rossby waves (r-modes) have been suggested as a means to regulate the spin periods of young or accreting neutron stars, and also to produce observable gravitational wave radiation. R-modes involve primarily transverse, incompressive motions of the star's fluid core. However, neutron stars gain crusts early in their lives: therefore, r-modes also imply shear in the fluid beneath the crust. We examine the criterion for this shear layer to become turbulent, and derive the rate of dissipation in the turbulent regime. Unlike dissipation from a viscous boundary layer, turbulent energy loss is nonlinear in mode energy and can therefore cause the mode to saturate at amplitudes typically much less than unity. This energy loss also reappears as heat below the crust. We study the possibility of crust melting as well as its implications for the spin evolution of low-mass X-ray binaries. Lastly, we identify some universal features of the spin evolution that may have observational consequences.Comment: 12 pages, 4 figures, submitted to Ap

HASH(0x563d44119a80)

HASH(0x563d43e27db0)HASH(0x563d44003f78

Microscopic structure of a vortex line in superfluid neutron star matter

The microscopic structure of an isolated vortex line in superfluid neutron star matter is studied by solving the Bogoliubov-de Gennes equations. Our calculation, which is the starting point for a microscopic calculation of pinning forces in neutron stars, shows that the size of the vortex core varies differently with density, and is in general smaller than assumed in some earlier calculations of vortex pinning in neutron star crusts. The implications of this result are discussedComment: 5 pages, 2 figure

In human retinoblastoma Y79 cells okadaic acid\u2013parthenolide co-treatment induces synergistic apoptotic effects, with PTEN as a key player.

Retinoblastoma is the most common intraocular malignancy of childhood. In developing countries, treatment is limited, long-term survival rates are low and current chemotherapy causes significant morbidity to pediatric patients and significantly limits dosing. Therefore there is an urgent need to identify new therapeutic strategies to improve the clinical outcome of patients with retinoblastoma. here, we investigated the effects of two natural compounds okadaic acid (OKa) and parthenolide (PN) on human retinoblastoma Y79 cells. For the first time we showed that OKa/PN combination at subtoxic doses induces potent synergistic apoptotic effects accompanied by lowering in p-akt levels, increasing in the stabilized forms of p53 and potent decrease in ps166-Mdm2. We also showed the key involvement of PTeN which, after OKa/PN treatment, potently increased before p53, thus suggesting that p53 activation was under PTeN action. Moreover, after PTEN-knockdown p-akt/ ps166Mdm2 increased over basal levels and p53 significantly lowered, while OKa/PN treatment failed both to lower p-akt and ps166-Mdm2 and to increase p53 below/over their basal levels respectively. OKa/PN treatment potently increased ROs levels whereas decreased those of Gsh. Reducing cellular Gsh by l-butathionine-[s,R]-sulfoximine treatment significantly anticipated the cytotoxic effect exerted by OKa/ PN. Furthermore, the effects of OKa/PN treatment on both Gsh content and cell viability were less pronounced in PTeN silenced cells than in control cells. The results provide strong suggestion for combining a treatment approach that targets the PTeN/akt/Mdm2/p53 pathway