Ordering properties of radial ground states and singular ground states of quasilinear elliptic equations

none2noIn this paper we discuss the ordering properties of positive radial solutions of the equation Δpu(x)+k|x|δuq-1(x)=0where x∈ Rn, n> p> 1 , k> 0 , δ> - p, q> p. We are interested both in regular ground states u (GS), defined and positive in the whole of Rn, and in singular ground states v (SGS), defined and positive in Rn { 0 } and such that lim |x|→v(x) = + ∞. A key role in this analysis is played by two bifurcation parameters pJL(δ) and pjl(δ) , such that pJL(δ) > p∗(δ) > pjl(δ) > p: pJL(δ) generalizes the classical Joseph–Lundgren exponent, and pjl(δ) its dual. We show that GS are well ordered, i.e. they cannot cross each other if and only if q≥ pJL(δ) ; this way we extend to the p> 1 case the result proved in Miyamoto (Nonlinear Differ Equ Appl 23(2):24, 2016), Miyamoto and Takahashi (Arch Math Basel 108(1):71–83, 2017) for the p≥ 2 case. Analogously we show that SGS are well ordered, if and only if q≤ pjl(δ) ; this latter result seems to be known just in the classical p= 2 and δ= 0 case, and also the expression of pjl(δ) has not appeared in literature previously.openColucci R.; Franca M.Colucci, R.; Franca, M

On the non-autonomous hopf bifurcation problem: Systems with rapidly varying coefficients

We consider the Cauchy-problem for the parabolic equation $$u_t = Delta u+ f(u,|x|),$$ where $x in mathbb R^n$, n >2, and $f(u,|x|)$ is either critical or supercritical with respect to the Joseph-Lundgren exponent. In particular, we improve and generalize some known results concerning stability and weak asymptotic stability of positive ground states

Dust from AGBs: relevant factors and modelling uncertainties

The dust formation process in the winds of Asymptotic Giant Branch stars is discussed, based on full evolutionary models of stars with mass in the range $1$M$_{\odot} \leq$M$\leq 8$M$_{\odot}$, and metallicities $0.001 < Z <0.008$. Dust grains are assumed to form in an isotropically expanding wind, by growth of pre--existing seed nuclei. Convection, for what concerns the treatment of convective borders and the efficiency of the schematization adopted, turns out to be the physical ingredient used to calculate the evolutionary sequences with the highest impact on the results obtained. Low--mass stars with M$\leq 3$M$_{\odot}$ produce carbon type dust with also traces of silicon carbide. The mass of solid carbon formed, fairly independently of metallicity, ranges from a few $10^{-4}$M$_{\odot}$, for stars of initial mass $1-1.5$M$_{\odot}$, to $\sim 10^{-2}$M$_{\odot}$ for M$\sim 2-2.5$M$_{\odot}$; the size of dust particles is in the range $0.1 \mu$m$\leq a_C \leq 0.2\mu$m. On the contrary, the production of silicon carbide (SiC) depends on metallicity. For $10^{-3} \leq Z \leq 8\times 10^{-3}$ the size of SiC grains varies in the range $0.05 \mu {\rm m} < {\rm a_{SiC}} < 0.1 \mu$m, while the mass of SiC formed is $10^{-5}{\rm M}_{\odot} < {\rm M_{SiC}} < 10^{-3}{\rm M}_{\odot}$. Models of higher mass experience Hot Bottom Burning, which prevents the formation of carbon stars, and favours the formation of silicates and corundum. In this case the results scale with metallicity, owing to the larger silicon and aluminium contained in higher--Z models. At Z=$8\times 10^{-3}$ we find that the most massive stars produce dust masses $m_d \sim 0.01$M$_{\odot}$, whereas models of smaller mass produce a dust mass ten times smaller. The main component of dust are silicates, although corundum is also formed, in not negligible quantities ($\sim 10-20\%$).Comment: Paper accepted for publication in Monthly Notices of the Royal Astronomical Society Main Journal (2014 January 4

Work and Quantum Phase Transitions: Is there Quantum Latency?

We study the physics of quantum phase transitions from the perspective of non-equilibrium thermodynamics. For first order quantum phase transitions, we find that the average work done per quench in crossing the critical point is discontinuous. This leads us to introduce the quantum latent work in analogy with the classical latent heat of first order classical phase transitions. For second order quantum phase transitions the irreversible work is closely related to the fidelity susceptibility for weak sudden quenches of the system Hamiltonian. We demonstrate our ideas with numerical simulations of first, second, and infinite order phase transitions in various spin chain models.Comment: accepted in PR

A survey of UV-excess AGNs in the South Galactic Pole

Spectra, position, magnitudes and colors are presented for 485 faint (B<20.5) emission line objects selected with the ultraviolet-excess (UVX) criterion on a area of 24.6 sq. deg in the South Galactic Pole. The objects were selected from the analysis of pixel-to-pixel stacking of COSMOS scans of UKST U, J and R plates. The candidates were observed with the Meudon-ESO Fiber Optics System (MEFOS) at the ESO 3.6m telescope. 429 type 1 AGNs have been identified (373 in the redshift range 0.3<z<2.2). This sample has allowed the measure of a difference on the QSO clustering evolution in comparison with that found for galaxies (La Franca et al 1998). The region is part of the ESO Imaging Survey (EIS) and of the 2dF QSO redshift survey.Comment: 32 pages, 19 figures. To appear on A&AS, revised after minor comment

AGB and SAGB stars: modelling dust production at solar metallicity

We present dust yields for asymptotic giant branch (AGB) and super--asymptotic giant branch (SAGB) stars of solar metallicity. Stars with initial mass $1.5~M_{\odot} \leq M_{\rm ini} \leq 3~M_{\odot}$ reach the carbon star stage during the AGB phase and produce mainly solid carbon and SiC. The size and the amount of the carbon particles formed follows a positive trend with themass of the star; the carbon grains with the largest size ($a_{\rm C} \sim 0.2\mu$m) are produced by AGB stars with $M_{\rm ini} = 2.5-3~M_{\odot}$, as these stars are those achieving the largest enrichment of carbon in the surface regions. The size of SiC grains, being sensitive to the surface silicon abundance, keeps around $a_{\rm SiC} \sim 0.1\mu$m. The mass of carbonaceous dust formed is in the range $10^{-4} - 5\times 10^{-3}~M_{\odot}$, whereas the amount of SiC produced is $2\times 10^{-4} - 10^{-3}~M_{\odot}$. Massive AGB/SAGB stars with $M_{\rm ini} > 3~M_{\odot}$ experience HBB, that inhibits formation of carbon stars. The most relevant dust species formed in these stars are silicates and alumina dust, with grain sizes in the range $0.1\mu m < a_{\rm ol} < 0.15\mu$m and $a_{\rm Al_2O_3} \sim 0.07\mu$m, respectively. The mass of silicates produced spans the interval $3.4\times 10^{-3}~M_{\odot} \leq M_{\rm dust} \leq 1.1\times 10^{-2}~M_{\odot}$ and increases with the initial mass of the star.Comment: Accepted for publication in MNRA