We consider quantum graphs providing reflectionless wave transmission at the vertices. Imposing Kuska’s version of so-called absorbing boundary conditions we derive the constraints, which make usual continuity and Kirchhoff conditions equivalent to transparent boundary conditions.

We consider the family of 3 х 3 operator matrices H(K), K ϵ T^d := (-π; π]^d arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus T^d. We obtain an analog of the Faddeev equation for the eigenfunctions of H(K). An analytic description of the essential spectrum of H(K) is established. Further, it is shown that the essential spectrum of H(K) consists the union of at most three bounded closed intervals.

The effect of an external magnetic field on the binding energy of a hole in an impurity complex A⁺ +e in a spherically symmetric quantum dot, as well as frequency dependence of the spectral intensity of recombination radiation of the quasi-zero-dimensional structure with impurity complexes A⁺ + e have been investigated. It is shown that in an external magnetic field there is a spatial anisotropy for the binding energy of A⁺-state due to hybrid quantization in the quantum dot radial plane and dimensional quantization in the direction of an external magnetic field. It is shown that in an external magnetic field the spectral intensity curve of the recombination radiation shifts to the short-wavelength region of the spectrum and probability of the radiative transition of an electron to the level of A⁺-center increases, which is caused by increase in the overlap integral of the envelope wave functions of a hole bound at the A⁺-center and of an electron localized in the ground state of quantum dot.

The heat capacity of two interacting electrons confined in a quantum dot presented in a magnetic field has been calculated by solving the Hamiltonian using the exact diagonalization method. The statistical average energies for confined and interacting electrons have been computed for various values of magnetic fields, confining frequency and temperature. We had investigated the dependence of the heat capacity on quantum dot Hamiltonian’s parameters and temperature. The singlettriplet transitions in the ground state of the quantum dot spectra and the corresponding jumps in the heat capacity curves had been shown. The comparisons show that our results are in very good agreement with theoretical reported works.

The magnetic moment of a phosphorene monolayer is calculated in the presence of a perpendicular magnetic field B. Landau levels are strongly anisotropic with different anisotropic effective masses. The oscillatory character of the magnetic moment dependences on the magnetic field is investigated.

The zero-range potential method has been generalized to the case of two-electron impurity centers with an effective nuclear charge equal to zero in a spherically symmetric quantum dot (QD), and on the basis of this method the first ionization potential has been calculated by variational method. It is shown that as the radius of QD decreases, the threshold value of the second ionization potential also decreases, beginning with which the existence of the two-electron bound state is possible due to an increase in the size-quantization energy accompanied by suppression of mutual electron repulsion. The light impurity absorption coefficient has been calculated using the dipole approximation for double ionization of the twoelectron impurity center by a single photon in a quasi-zero-dimensional structure, which is the transparent dielectric matrix with semiconductor QDs synthesized in it. It is shown that characteristic feature of the double photoionization spectrum is a two-humped profile of the spectral curve due to electron correlations.

In the present paper, we discuss the kinetic equations for the evolution of particles of a nanodispersed substance, distinguishing by properties (sizes, velocities, positions, etc.). The aim of the present investigation is to determine the coefficients for the equations by the distribution functions, which are obtained experimentally. The experiment is characterized by the time interval, which is needed for the measurement of the distribution function. However, the nanodispersed substance is obtained from a highly supersaturated solution or vapor and this time interval is large, thus, one is able to measure distribution functions only when the processes of the integration and the fragmentation of the particles become rather slow. So it is advisable to reconstruct the kinetics for the formation of a nanodispersed substance by the experimental distribution functions measured when the processes are rather slow. The first problem that arises is the obtaining of correct equations, and, hence, the derivation of the equations from each other. From the discrete system of equations for the evolution of discrete distribution functions of particles of a nanodispersed substance, we obtain the continuum equation of the Fokker-Planck type, or of the Einstein-Kolmogorov type, or of the diffuse approximation on the distribution function of nanoparticles distinguishing by the numbers of molecules forming them. We consider the distribution functions, which approximate the experimental data. We determine the coefficients for the equation of the Fokker-Planck type by the stationary and non-stationary distribution functions of a nanodispersed substance. Due to unity of the kinetic approach, the present work may be useful for specialists of various areas, who study the evolution of structures (not only with nanosize) with differing properties.

Nanocrystalline cerium dioxide is known as a unique redox active nanomaterial. Cerium dioxide is considered as the basis for future biomedical preparations, including radioprotectors. In the framework of this study, we synthesized citrate-stabilized CeO₂ nanoparticles and carried out a comprehensive in vitro assessment of their radioprotective properties on a NCTC L929 murine fibroblast culture. It was shown that CeO₂ nanoparticles ensure the survival of murine fibroblasts, even after high-dose X-ray irradiation, reducing the number of dead cells in the culture and modulating the mRNA level of the key antioxidant enzymes – superoxide dismutase 1 (SOD1) and superoxide dismutase 2 (SOD2). The results obtained confirm the potential for studying the properties of CeO₂ nanoparticles as basic materials for designing new efficient and safe preparations for protection against ionizing radiation.

In this work, a facile and cost-effective layer by layer method was proposed to synthesize novel high stable and effective electrode material based on the Co-doped Cu(OH)₂ nanocrystals. The crystals have orthorhombic structure and a rod-like morphology with a 23_2 nm in width and 43_4 nm in length. The composition of the nanocrystals corresponds to the 1 % Co-doped Cu(OH)2 by EDX with no noticeable impurities as it was found by FTIR spectroscopy. It was shown that nickel electrode modified with nanorods is characterized by an overvoltage value of -347 mV at 10 mA/cm2, which is 250 mV lower than that of an initial pure nickel electrode. The value of Tafel slope that reaches 138 mV/dec, high stability of the Co-doped Cu(OH)₂ nanorods in chronopotentiometric (10 hours) and cyclic volamperometric (500 cycles) tests allows us to consider them as a prospective basis of electrode materials for the hydrogen evolution from renewable water-alcohol sources.

A globally acknowledged green synthesis of reduced graphene oxide (rGO) from graphene oxide (GO) is presented in this paper. The graphene oxide powder was synthesized from Graphite powder by a modification of Hummer’s method. The GO is exposed to focused sunlight to obtain reduced graphene oxide (rGO). The reduction of GO under solar light is an eco-friendly method to conventional method of rGO preparation. The mechanism of the reduction of GO by sunlight imperative to exfoliation was seen to be well defined. The rGO powder was characterized by Xray Diffraction (XRD), Field-Emission Scanning Electron Microscopy (FESEM), Raman spectroscopy, Fourier-Transform Infrared Spectroscopy (FTIR) and High-Resolution Transmission Electron Microscopy (HRTEM). This eco-friendly method of synthesizing of rGO paves way for an alternative method of rGO nanosheets preparation and it can be effectively used for fabrication of various electronic devices.

Luminescent hydrophobic composite films based on nanocrystalline (CNC) and nanofibrillated (CNF) cellulose matrix with up-conversion MF₂:Ho (M = Ca, Sr) particles and acrylic resin (ACR) as a coating have been synthesized. Flexible, translucent composite films were obtained by molding from the CNC/CNF suspensions with up-conversion particles. ACR coating was applied to the composite film by spraying. Studies have shown that ACR coating with a layer thickness of 7 – 10 γm provides hydrophobic properties for the films, increasing the water contact angle up to 100 ± 2° with a simultaneous improvement in the luminescent properties. Transparency of CNC/CNF/MF₂:Ho-ACR films in the visible and near IR region improves by 20 – 25 % without compromising the flexibility and thermal stability. The manufactured water-resistant composite films can be utilized as potential photonics materials, in particular for visualization of near-IR laser radiation and luminescent labels.