Painlevé analysis and exact solutions of the Korteweg–de Vries equation with a source

We consider the Korteweg–de Vries equation with a source. The source depends on the solution as polynomials with constant coefficients. Using the Painlevé test we show that the generalized Korteweg–de Vries equation is not integrable by the inverse scattering transform. However there are some exact solutions of the generalized Korteweg–de Vries equation for two forms of the source. We present these exact solutions.     

Dynamics and optimal control in a spatially structured economic growth model with pollution diffusion and environmental taxation

In the first part of this paper we present a spatially structured dynamic economic growth model which takes into account the level of pollution and a possible taxation based on the amount of produced pollution. In the second part we analyze an optimal harvesting control problem with an objective function composed of three terms, namely the intertemporal utility of the decision maker, the space–time average of the level of pollution in the habitat, and the disutility due to the imposition of taxation.     

Exact Finite-Dimensional Filter for Exponential Functionals of the State of Beneš Systems

The purpose of this article is to derive finite-dimensional filters for exponential functionals of the state of nonlinear Bene? type filtering systems with correlated noises. These new filters can be applied in the state reconstruction of doubly–stochastic auto–regressive processes.     

The structural and catalytic properties of nanoparticles@MOF composites: A case study of Au@ZIF-8 hybrid crystals

Strain analysis of the MOF and its composites using high-resolution X-ray diffractionmeter (XRD) was carried out and the presence of non-uniform, depth-related strain in the MOF crystals was confirmed. Further analysis showed that the magnitude and distribution of strain in MOF crystals can be tuned with the incorporation of nanoparticles (NPs). Moreover, the spatial controlled structures can also optimize functionalities of the NPs@MOF, which was exemplified by their applications on the catalytic reduction of nitroarenes. It is anticipated that the investigation of MOF structure evolution through controlling the architectures of the MOF/NPs hybrid materials will shed a light on the study of optimizing the mechanical and chemical properties of MOF composites.