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.     

Energy structure and absorption spectra of colloidal CdS nanocrystals in gelatin matrix

We investigate the energy structure of colloidal CdS nanocrystals by measuring the UV–vis absorption spectra. Nanocrystals were synthesized by sol–gel method in a gelatin matrix in the size range from 2.5 to 3.9 (±0.2) nm. In order to interpret the UV–vis absorption spectra we calculate the energy spectrum of electron quasi-stationary states using the model of open nanocrystal as well as the hole stationary spectrum in a two-band approach. It is shown that the main contribution to the absorption spectrum is made by interband transitions 1S_3/2→1S_e and 1P_3/2→1P_e, and its shape is determined by the size distribution of nanoparticles. For this system the estimated values of the effective masses of the heavy hole and light hole are 1.44m₀ and 0.28m₀, respectively.     

Characterization of the channel walls roughness in photonic crystal fibers

We present electron microscope (FEI NanoSEM) and atomic force microscopy measurements of surface roughness in nanochannels in photonic crystal fibers (PCF). A method was invented to cleave the PCF along the axis without damaging the surface structure in the nanochannels allowing us to characterize the morphology of the nanochannels in the PCF. A multi-wall carbon nanotube mounted onto commercial AFM probes and super sharp silicon non-contact mode AFM probes were used to characterize the wall roughness in the nanochannels. The roughness is shown to have a Gaussian distribution, and has an amplitude smaller than 0.5 nm. The height–height correlation function is an exponential correlation function with an autocorrelation length of 13 nm, and 27 nm corresponding with scan sizes of 200×100 nm², and 1600×200 nm², respectively.     

Control of electron spin–orbit anisotropy in pyramidal InAs quantum dots

We investigate the electron spin–orbit interaction anisotropy of pyramidal InAs quantum dots using a fully three-dimensional Hamiltonian. The dependence of the spin–orbit interaction strength on the orientation of externally applied in-plane magnetic fields is consistent with recent experiments, and it can be explained from the interplay between Rashba and Dresselhaus spin–orbit terms in dots with asymmetric confinement. Based on this, we propose manipulating the dot composition and height as efficient means for controlling the spin–orbit anisotropy.