A population biological model with a singular nonlinearity

summary:We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$\begin {cases} -{\rm div}(|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u)=|x|^{-(\alpha +1)p+\beta } \Big (a u^{p-1}-f(u)-\dfrac {c}{u^{\gamma }}\Big ), \quad x\in \Omega ,\\ u=0, \quad x\in \partial \Omega , \end {cases}$$ where $\Omega$ is a bounded smooth domain of ${\mathbb R}^N$ with $0\in \Omega$, $1<p<N$, $0\leq \alpha < {(N-p)}/{p}$, $\gamma \in (0,1)$, and $a$, $\beta$, $c$ and $\lambda$ are positive parameters. Here $f\colon [0,\infty )\to {\mathbb R}$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions to establish our results

Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray-Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis

Long-term values in Markov decision processes, (Co)algebraically

This paper studies Markov decision processes (MDPs) from the categorical perspective of coalgebra and algebra. Probabilistic systems, similar to MDPs but without rewards, have been extensively studied, also coalgebraically, from the perspective of program semantics. In this paper, we focus on the role of MDPs as models in optimal planning, where the reward structure is central. The main contributions of this paper are (i) to give a coinductive explanation of policy improvement using a new proof principle, based on Banach’s Fixpoint Theorem, that we call contraction coinduction, and (ii) to show that the long-term value function of a policy with respect to discounted sums can be obtained via a generalized notion of corecursive algebra, which is designed to take boundedness into account. We also explore boundedness features of the Kantorovich lifting of the distribution monad to metric spaces

Lattice-theoretic progress measures and coalgebraic model checking

In the context of formal verification in general and model checking in particular, parity games serve as a mighty vehicle: many problems are encoded as parity games, which are then solved by the seminal algorithm by Jurdzinski. In this paper we identify the essence of this workflow to be the notion of progress measure, and formalize it in general, possibly infinitary, lattice-theoretic terms. Our view on progress measures is that they are to nested/alternating fixed points what invariants are to safety/greatest fixed points, and what ranking functions are to liveness/least fixed points. That is, progress measures are combination of the latter two notions (invariant and ranking function) that have been extensively studied in the context of (program) verification. We then apply our theory of progress measures to a general model-checking framework, where systems are categorically presented as coalgebras. The framework's theoretical robustness is witnessed by a smooth transfer from the branching-time setting to the linear-time one. Although the framework can be used to derive some decision procedures for finite settings, we also expect the proposed framework to form a basis for sound proof methods for some undecidable/infinitary problems

Coalgebraic Trace Semantics via Forgetful Logics

Abstract. We use modal logic as a framework for coalgebraic trace semantics, and show the flexibility of the approach with concrete ex-amples such as the language semantics of weighted, alternating and tree automata. We provide a sufficient condition under which a logical seman-tics coincides with the trace semantics obtained via a given determiniza-tion construction. Finally, we consider a condition that guarantees the existence of a canonical determinization procedure that is correct with respect to a given logical semantics. That procedure is closely related to Brzozowski’s minimization algorithm.

Tribological behavior of arcing contact materials based on copper infiltrated tungsten composites

Tungsten copper composites with 70±3 wt.% W, maximum 1.5 wt.% Ni, and balance Cu were achieved as disks (diameter × height of 50×6 mm) by copper infiltration process of tungsten skeletons. Elemental analysis was assessed by WDXRF spectroscopy. Hydrostatic density was evaluated in ethanol. Vickers hardness and Young’s modulus were determined in ambient air by instrumented indentation technique and Oliver&Pharr computation method. Tribological behavior was investigated under 30 N up to 400 m sliding distance and naphthenic mineral oil lubricant with a standard tribometer of ball-on-disk type. The results yielded highly dense materials with relative density over 96%, Vickers hardness (HVIT) of 244…323, Young’s modulus (EIT) of 156…185 GPa, mean coefficient of friction of 0.11…0.22 and specific wear rate up to 8×10 ˉ ⁶mm3/(Nm). The developed composites with low coefficient of friction and high wear resistance for use as arcing contacts in oil circuit breakers will endow high performance in service