Vol 29, No 1 (2023)

This article discusses a model for obtaining a sparse representation of a signal vector in ${ℝ}^k$, based on a system of linear equations with an orthogonal matrix. Such a representation minimizes a target function that combines the deviation from the exact solution and a chosen functional $J$. The functionals chosen are the Euclidean norm, the norm $|\cdot {|}_1$, and the quasi-norm $|\cdot {|}_0$. The Euclidean norm only allows for the exact solution, while the other two allow for a balance between the residual and the parameter $\lambda$ in the functional, resulting in sparser solutions. Graphs are plotted showing the dependence between the coordinates of the optimal vector and the parameter , and examples are provided.

The article describes the processing of the results of a series of experiments performed by the interference-optical method of holographic interferometry (holographic photoelasticity) aimed at computing the amplitude coefficients of the M. Williams series constituting the stress and displacement fields at the crack tip for several cracked configurations. The main objective of this study is the experimental and computational determination of the coefficients of the M. Williams series for the stress, strain and displacement fields associated with the crack tip in an isotropic linearly elastic medium taking into account regular (non-singular) terms in the multiparameter Williams series. These coefficients are named generalized stress intensity factors. The method of holographic interferometry is shown to be a convenient and efficacious tool for reconstructing the stress field near the tip of the crack, because during the experiment it is possible to obtain two families of interference fringe patterns: absolute retardation fringes (isodromics) for vertical and horizontal polarizations. Experimental outcomes were thoroughly processed using the developed digital application allowing us to accumulate the isodromics orders and coordinates of points belonging to absolute retardations. In this work, absolute retardation fringes in a plate with a central horizontal crack and a crack inclined at different angles are obtained. For each type of experimental sample, the coefficients of the Williams series were calculated taking into account non-singular terms (in the representation of M. Williams, fifteen terms were preserved). A procedure for linearization of nonlinear algebraic equations following from the relations of Favre’s law is proposed. By solving the obtained overdeterministic system of linear algebraic equations, the generalized stress intensity factors (coefficients of the M. Williams series) are estimated. Conjointly, the finite element analysis of the specimens with same geometry was effectuated. The experimentally determined values of the Williams series are compared with the results of the finite element calculation of the stress-strain state performed in the SIMULIA Abaqus software.The results of the numerical and experimental studies were found to be quite consistent. It is lucidly shown that it is imperative to keep the higher order terms in the Williams series expansions for the fields associated with the crack tip.

Due to the fact that many wave emitters that a person is able to perceive belong to low-frequency ones, studies of sound fields created by such emitters are relevant. Thanks to the results obtained, it becomes possible to understand in which directions and with what power the sound field created by them will propagate, form practical recommendations for choosing the zone most suitable for observing them, solve inverse problems to determine their location. As a result of the analysis of existing models used to describe acoustic emitters, it was found that the most adequate models are those that take into account the directionality of sound sources. Among them, the parametric model proposed by G.N. Kuznetsov and A.N. Stepanov deserves special attention, which was used in the work. As a model representation of the atmosphere, a system of homogeneous layers was chosen, in one of which the source is located. For the selected models of the source and medium, the boundary value problem of finding the potential of the field created by the source is set, exact and approximate relations are obtained that can be used to solve direct and inverse problems associated with a multipole emitter. The influence of such factors as the height and frequency of the source, as well as the horizontal distance between the source and receiver on the amplitude component of the field has been studied. 

In this paper, we investigated the dynamics of entanglement of pairs of qubits in a system of three identical qubits that interact non-resonantly with the selected mode of a microwave resonator without loss with the Kerr medium by means of single-photon transitions. We have found solutions to the quantum time Schrodinger equation for the total wave function of the system for the initial separable, biseparable and true entangled states of qubits and the Fock initial state of the resonator field. Based on these solutions, the criterion of entanglement of qubit pairs — negativity is calculated. The results of numerical simulation of the negativity of qubit pairs have shown that the presence of disorder and Kerr nonlinearity in the case of an initial non-entangled state of a pair of qubits can lead to a significant increase in the degree of their entanglement. In the case of an initial entangled state of a pair of qubits, the disorder and the Kerr medium can lead to a significant stabilization of the initial entanglement.