A weighted Hardy inequality and nonexistence of positive solutions

In this article, we prove that the following weighted Hardy inequality $$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$ holds with optimal Hardy constant ${\big(\frac{|d-p|}{p}\big)^{p}}$ for all ${u \in W^{1,p}_{\mu,0}(\Omega)}$ if the dimension d ≥ 2, 1 < p < d, and for all ${u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}$ if p > d ≥ 1. Here we assume that Ω is an open subset of ${\mathbb{R}^{d}}$ with ${0 \in \Omega}$ , A is a real d × d-symmetric positive definite matrix, c > 0, and $$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$ If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on ${W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}$ . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p <  + ∞, and when Ω =  ]0, + ∞[.     

Polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory

We consider one-dimensional problem for the thermoelastic diffusion theory and we obtain polynomial decay estimates. Then we show that the solution decays exponentially to zero as time goes to infinity; that is, denoting by E(t) the first-order energy of the system, we show that positive constants C ₀ and c ₀ exist which satisfy E(t) ≤ C ₀ E(0)e ^{?c ₀ t} .     

Poor rich demarcation of Matthew effect on scale-free systems and its application

In a scale-free network,only a minority of nodes are connected very often,while the majority of nodes are connected rarely. However,what is the ratio of minority nodes to majority nodes resulting from the Matthew effect In this paper,based on a simple preferential random model,the poor-rich demarcation points are found to vary in a limited range,and form a poor-rich demarcation interval that approximates to k/m ∈ [3,4]. As a result,the (cumulative) degree distribution of a scale-free network can be divided into three intervals: the poor interval,the demarcation interval and the rich interval. The inequality of the degree distribution in each interval is measured. Finally,the Matthew effect is applied to the ABC analysis of project management.     

Pointwise Efficiency in Multiobjective Stochastic Linear Programming

In most of the approaches to the Multiobjective Stochastic Linear Programming problem that have been proposed in the literature, the notion of quality of a solution is not adequately defined. We reconsider this problem from a decision point of view, in contexts where either the decision maker's preference structure cannot be described by a utility function, or where this structure is expressed by an unknown non-decreasing utility function and the probability distribution of the random parameters is unknown. We define a fundamental set of ‘pointwise admissible’ solutions, as well as several subsets of particular interest. We discuss the relevance of these various pointwise efficient sets, their interrelations, and their practical identification.     

Positivity and Stability for a Population Equation with Diffusion on 1

Using positivity, abstract extrapolation spaces, and a generalization of the Jörgens–Vidav-Voigt perturbation theorem for the essential spectral radius of strongly continuous semigroups given in [19], we study the asymptotic behavior of a linear age-dependent population problem with spatial diffusion where the birth and the death rates depend on the spatial variable.     

[M(CO)4PPh3] radicals (M = Cr, Mo, W): DFT and single crystal EPR investigations

[M(CO)₄PPh₃]⁻ (M = Mo, W) were trapped at 77 K in X-irradiated single crystals of M(CO)₅PPh₃ and studied by EPR. Structures of [M(CO)₄PPh₃]⁻ (M = Cr, Mo, W) were optimized by DFT; predicted g and ³¹P-hyperfine tensors agree with experiments for M = Mo, W. The anions adopt a slightly distorted pyramidal structure with PPh₃ in basal position and the spin mostly delocalized in a metal-d_z² orbital and carbon-p_z orbitals of carbonyls. The EPR tensors are slightly modified by annealing, they suggest that new constraints in the matrix distort the structure of [M(CO)₄PPh₃]⁻ (M = Cr, Mo, W).     

Radical Reactions in an X-Irradiated Phosphaalkene: A single-crystal ESR study

Single crystals of 1-[2,4,6-tri(tert-butyl)phenyl]-2-phenylphosphaethene ( PPPE ) and of ²D- and ¹³C-enriched PPPE were studied by ESR after X-ray irradiation. Two phosphorus-centered radicals were trapped in the crystals. The first one was characterized by its ³¹P, ¹H-, and ¹³C-hyperfine tensors, the second one exhibited coupling with ³¹P only. Comparison of these parameters with those predicted by ab initio calculations on some phosphinyl species indicates that these two radicals probably result, on the one hand, from an addition of a H-atom to the C-atom of the P?C bond and, on the other, from a cyclization of the parent molecule. The proposed mechanisms are consistent with the mutual orientations of the hyperfine eigenvectors and bond directions in the undamaged molecule. A C-centered radical which results from an addition of a H-atom to the P-atom of the phosphaethylene bond is also detected.