The first nontrivial eigenvalue for a system of p−p-Laplacians with Neumann and Dirichlet boundary conditions

We deal with the first eigenvalue for a system of two $p-$Laplacians with Dirichlet and Neumann boundary conditions. If \Delta_{p}w=\mbox{div}(|\nabla w|^{p-2}w) stands for the $p-$Laplacian and $\frac{\alpha}{p}+\frac{\beta}{q}=1,$ we consider $$\begin{cases} -\Delta_pu= \lambda \alpha |u|^{\alpha-2} u|v|^{\beta} &\text{ in }\Omega,\\ -\Delta_q v= \lambda \beta |u|^{\alpha}|v|^{\beta-2}v &\text{ in }\Omega,\\ \end{cases}$$ with mixed boundary conditions $$u=0, \qquad |\nabla v|^{q-2}\dfrac{\partial v}{\partial \nu }=0, \qquad \text{on }\partial \Omega.$$ We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem $$\lambda_{p,q}^{\alpha,\beta} = \min \left\{\dfrac{\displaystyle\int_{\Omega}\dfrac{|\nabla u|^p}{p}\, dx +\int_{\Omega}\dfrac{|\nabla v|^q}{q}\, dx} {\displaystyle\int_{\Omega} |u|^\alpha|v|^{\beta}\, dx} \colon (u,v)\in \mathcal{A}_{p,q}^{\alpha,\beta}\right\},$$ where $$\mathcal{A}_{p,q}^{\alpha,\beta}=\left\{(u,v)\in W^{1,p}_0(\Omega)\times W^{1,q}(\Omega)\colon uv\not\equiv0\text{ and }\int_{\Omega}|u|^{\alpha}|v|^{\beta-2}v \, dx=0\right\}.$$ We also study the limit of $\lambda_{p,q}^{\alpha,\beta}$ as $p,q\to \infty$ assuming that $\frac{\alpha}{p} \to \Gamma \in (0,1)$, and $\frac{q}{p} \to Q \in (0,\infty)$ as $p,q\to \infty.$ We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take $Q=1$ and the limits $\Gamma \to 1$ and $\Gamma \to 0$.Comment: 21 pages, 1 figur

Afterglow lightcurves, viewing angle and the jet structure of gamma-ray bursts

Gamma ray bursts are often modelled as jet-like outflows directed towards the observer; the cone angle of the jet is then commonly inferred from the time at which there is a steepening in the power-law decay of the afterglow. We consider an alternative model in which the jet has a beam pattern where the luminosity per unit solid angle (and perhaps also the initial Lorentz factor) decreases smoothly away from the axis, rather than having a well-defined cone angle within which the flow is uniform. We show that the break in the afterglow light curve then occurs at a time that depends on the viewing angle. Instead of implying a range of intrinsically different jets - some very narrow, and others with similar power spread over a wider cone - the data on afterglow breaks could be consistent with a standardized jet, viewed from different angles. We discuss the implication of this model for the luminosity function.Comment: Corrected typo in Eq. 1

Exact ground state Monte Carlo method for Bosons without importance sampling

Generally ``exact'' Quantum Monte Carlo computations for the ground state of many Bosons make use of importance sampling. The importance sampling is based, either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of Path Integral Ground State (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a non zero overlap with the ground state, that gives rise to a discrete path which is sampled via a Metropolis like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid 4He with PIGS by considering as initial wave function a constant, i.e. the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e. there is no importance sampling. For both the phases we obtain results converging to those obtained by considering the best available variational wave function (the Shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function and the static structure factor, but also for off-diagonal ones, such as the one--body density matrix. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature Path Integral Monte Carlo.Comment: 11 pages, 10 figure

Computation of microdosimetric distributions for small sites

Object of this study is the computation of microdosimetric functions for sites which are too small to permit experimental determination of the distributions by Rossi-counters. The calculations are performed on simulated tracks generated by Monte-Carlo techniques. The first part of the article deals with the computational procedure. The second part presents numerical results for protons of energies 0.5, 5, 20 MeV and for site diameters of 5, 10, 100 nm

Mycotoxins nivalenol and deoxynivalenol differently modulate cytokine mRNA expression in Jurkat T cells.

Deoxynivalenol (DON) and its hydroxylated form nivalenol (NIV) are Fusarium mycotoxins that occur in cereal grains alone or in combination. Several studies have shown that these metabolites affect lymphocyte functions. However, the molecular mechanisms underlying their activities are still partially known. To address this issue, we examined the influence of NIV and DON in modulating IFNc, IL-2 and IL-8 mRNA levels in Jurkat T cells. In PMA/ionomycin stimulated cells, pre-incubated with increasing concentrations of NIV, transcription was induced in the range 0.06–2 lM; higher concentrations of NIV were found non-stimulating (4 lM) or inhibitory (8 lM) for IFNc and IL-2 whereas IL-8 was still induced. DON administration elicited a similar profile for IL-8 and IFNc, whilst IL-2 mRNA was induced in a broader range of concentrations. Combination of NIV and DON at 1:1 and 1:10 ratios essentially restored the cytokine transcriptional pattern observed with NIV alone but the level of transcripts, with the exception of IL-8, peaked at lower concentrations suggesting interactive effects. Moreover both mycotoxins caused inhibition of cell proliferation, mediated by induction of apoptosis, confirming previous results and highlighting the usefulness of Jurkat as a T-cell model to study the effects of mycotoxins on the immune functions in humans