We construct and analyze a strongly consistent second-order finite difference scheme for the steady two-dimensional Stokes flow. The pressure Poisson equation is explicitly incorporated into the scheme. Our approach suggested by the first two authors is based on a combination of the finite volume method, difference elimination, and numerical integration. We make use of the techniques of the differential and difference Janet/Gröbner bases. In order to prove strong consistency of the generated scheme we correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. Additionally, we compute the modified differential system of the obtained scheme and analyze the scheme's accuracy and strong consistency by considering this system. An evaluation of our scheme against the established marker-andcell method is carried out.

PT symmetry is a unique platform for light manipulation and versatile use in unidirectional invisibility, lasing, sensing, etc. Broken and unbroken PT -symmetric states in non-Hermitian open systems are described by scattering matrices. A multilayer structure, as a simplest example of the open system, has no certain definition of the scattering matrix, since the output ports can be permuted. The uncertainty in definition of the exceptional points bordering PT -symmetric and PT -symmetry-broken states poses an important problem, because the exceptional points are indispensable in applications as sensing and mode discrimination. Here we derive the proper scattering matrix from the unambiguous relation between the PT -symmetric Hamiltonian and scattering matrix. We reveal that the exceptional points of the scattering matrix with permuted output ports are not related to the PT symmetry breaking. Nevertheless, they can be employed for finding a lasing onset as demonstrated in our time-domain calculations and scattering-matrix pole analysis. Our results are important for various applications of the non-Hermitian systems including encircling exceptional points, coherent perfect absorption, PT -symmetric plasmonics, etc.

For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.

Indium-substituted strontium hexaferrites were prepared by the conventional solid-phase reaction method. Neutron diffraction patterns were obtained at room temperature and analyzed using the Rietveld methods. A linear dependence of the unit cell parameters is found. In3+ cations are located mainly in octahedral positions of 4fVI and 12 k. The average crystallite size varies within 0.84–0.65 μm. With increasing substitution, the TC Curie temperature decreases monotonically down to ~ 520 K. ZFC and FC measurements showed a frustrated state. Upon substitution, the average and maximum sizes of ferrimagnetic clusters change in the opposite direction. The Mr remanent magnetization decreases down to ~ 20.2 emu/g at room temperature. The Ms spontaneous magnetization and the keff effective magnetocrystalline anisotropy constant are determined. With increasing substitution, the maximum of the ε/ real part of permittivity decreases in magnitude from ~ 3.3 to ~ 1.9 and shifts towards low frequencies from ~ 45.5 GHz to ~ 37.4 GHz. The maximum of the tg(α) dielectric loss tangent decreases from ~ 1.0 to ~ 0.7 and shifts towards low frequencies from ~ 40.6 GHz to ~ 37.3 GHz. The low-frequency maximum of the μ/ real part of permeability decreases from ~ 1.8 to ~ 0.9 and slightly shifts towards high frequencies up to ~ 34.7 GHz. The maximum of the tg(δ) magnetic loss tangent decreases from ~ 0.7 to ~ 0.5 and shifts slightly towards low frequencies from ~ 40.5 GHz to ~ 37.7 GHz. The discussion of microwave properties is based on the saturation magnetization, natural ferromagnetic resonance and dielectric polarization types.

Bi nanocrystalline films were formed from perchlorate electrolyte (PE) on Cu substrate via electrochemical deposition with different duration and current densities. The microstructural, morphological properties, and elemental composition were studied using scanning electron microscopy (SEM), atomic force microscopy (AFM), and energy-dispersive X-ray microanalysis (EDX). The optimal range of current densities for Bi electrodeposition in PE using polarization measurements was demonstrated. For the first time, it was shown and explained why, with a deposition duration of 1 s, co-deposition of Pb and Bi occurs. The correlation between synthesis conditions and chemical composition and microstructure for Bi films was discussed. The analysis of the microstructure evolution revealed the changing mechanism of the films’ growth from pillar-like (for Pb-rich phase) to layered granular form (for Bi) with deposition duration rising. This abnormal behavior is explained by the appearance of a strong Bi growth texture and coalescence effects. The investigations of porosity showed that Bi films have a closely-packed microstructure. The main stages and the growth mechanism of Bi films in the galvanostatic regime in PE with a deposition duration of 1–30 s are proposed.