Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system

In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system: \[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$ , $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$ , respectively.

Monotonicity properties and solvability of dominated best approximation problem in Orlicz spaces equipped with s-norms

In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the s-norms defined by the formula $$\left\| x\right\| _{\Phi ,s}=\inf _{k>0}\frac{1}{k}s\left( \int _T \Phi (kx)d\mu \right) $$ x Φ , s = inf k > 0 1 k s ∫ T Φ ( k x ) d μ , where s and $$\Phi $$ Φ are an outer and Orlicz function respectively and x is a measurable real-valued function over a $$\sigma $$ σ -finite measure space $$(T,\Sigma ,\mu )$$ ( T , Σ , μ ) . In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the s-norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function s is strictly increasing or the measure $$\mu $$ μ is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

Can we trust the standardized mortality ratio? A formal analysis and evaluation based on axiomatic requirements

Background The standardized mortality ratio (SMR) is often used to assess and compare hospital performance. While it has been recognized that hospitals may differ in their SMRs due to differences in patient composition, there is a lack of rigorous analysis of this and other—largely unrecognized—properties of the SMR. Methods This paper proposes five axiomatic requirements for adequate standardized mortality measures: strict monotonicity (monotone relation to actual mortality rates), case-mix insensitivity (independence of patient composition), scale insensitivity (independence of hospital size), equivalence principle (equal rating of hospitals with equal actual mortality rates in all patient groups), and dominance principle (better rating of unambiguously better performing hospitals). Given these axiomatic requirements, effects of variations in patient composition, hospital size, and actual and expected mortality rates on the SMR were examined using basic algebra and calculus. In this regard, we distinguished between standardization using expected mortality rates derived from a different dataset (external standardization) and standardization based on a dataset including the considered hospitals (internal standardization). The results were illustrated by hypothetical examples. Results Under external standardization, the SMR fulfills the axiomatic requirements of strict monotonicity and scale insensitivity but violates the requirement of case-mix insensitivity, the equivalence principle, and the dominance principle. All axiomatic requirements not fulfilled under external standardization are also not fulfilled under internal standardization. In addition, the SMR under internal standardization is scale sensitive and violates the axiomatic requirement of strict monotonicity. Conclusions The SMR fulfills only two (none) out of the five proposed axiomatic requirements under external (internal) standardization. Generally, the SMRs of hospitals are differently affected by variations in case mix and actual and expected mortality rates unless the hospitals are identical in these characteristics. These properties hamper valid assessment and comparison of hospital performance based on the SMR.

The Hypervolume Indicator

The hypervolume indicator is one of the most used set-quality indicators for the assessment of stochastic multiobjective optimizers, as well as for selection in evolutionary multiobjective optimization algorithms. Its theoretical properties justify its wide acceptance, particularly the strict monotonicity with respect to set dominance, which is still unique of hypervolume-based indicators. This article discusses the computation of hypervolume-related problems, highlighting the relations between them, providing an overview of the paradigms and techniques used, a description of the main algorithms for each problem, and a rundown of the fastest algorithms regarding asymptotic complexity and runtime. By providing a complete overview of the computational problems associated to the hypervolume indicator, this article serves as the starting point for the development of new algorithms and supports users in the identification of the most appropriate implementations available for each problem.

Picone’s identity for -Laplace operator and its applications

UDC 517.9We prove a nonlinear analogue of Picone's identity for -Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system.

Quantity, quality, equality: introducing a new measure of social welfare

In this essay I propose a new measure of social welfare. It captures the intuitive idea that quantity, quality, and equality of individual welfare all matter for social welfare. More precisely, it satisfies six conditions: Equivalence, Dominance, Quality, Strict Monotonicity, Equality and Asymmetry. These state that (i) populations equivalent in individual welfare are equal in social welfare; (ii) a population that dominates another in individual welfare is better; (iii) a population that has a higher average welfare than another population is better, other things being equal; (iv) the addition of a well-faring individual makes a population better, whereas the addition of an ill-faring individual makes a population worse; (v) a population that has a higher degree of equality than another population is better, other things being equal; and (vi) individual illfare matters more for social welfare than individual welfare. By satisfying the six conditions, the measure improves on previously proposed measures, such as the utilitarian Total and Average measures, as well as different kinds of Prioritarian measures.

On the balancing property of Matkowski means

Let $$I\subseteq \mathbb {R}$$ I ⊆ R be a nonempty open subinterval. We say that a two-variable mean $$M:I\times I\rightarrow \mathbb {R}$$ M : I × I → R enjoys the balancing property if, for all $$x,y\in I$$ x , y ∈ I , the equality $$\begin{aligned} {M\big (M(x,M(x,y)),M(M(x,y),y)\big )=M(x,y)} \end{aligned}$$ M ( M ( x , M ( x , y ) ) , M ( M ( x , y ) , y ) ) = M ( x , y ) holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, that M is analytic, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes regular quasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question are continuously differentiable. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class of Matkowski means.

Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

This paper is concerned with nonlocal diffusion systems of three species with delays. By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. In order to obtain more information of the asymptotic behavior of such solutions at positive infinity, for the special kernels, we discuss the asymptotic behavior of such solutions of such system without delays, via the stable manifold theorem. In addition, by using the sliding method, the strict monotonicity and uniqueness of traveling wave fronts are also obtained.