Vol 46, No 1 (2024)

Linear equations, i.e. Equations of the first degree, as well as systems of such equations, receive much attention both in algebra and in number theory. Of greatest interest is the case of such equations with integer coefficients, and in this case they need to be solved in integers. Such equations with the specified conditions are called linear Diophantine equations. Euler also considered ways to solve linear Diophantine equations with two unknowns, and one of these methods was based on the use of the Euclid algorithm. Another method for solving such equations, based on continued fractions, was also used by Lagrange. Euler’s method turned out to be more convenient and promising than the method of continued fractions. In this paper, we consider one new method for solving linear equations over a Euclidean ring, based on comparisons over suitable moduli. The previously known matrix method for solving such equations with an increasing number of unknowns is quite cumbersome due to the fact that it is associated with finding the inverses of unimodular integer matrices. Essential in our method of solving linear equations over a Euclidean ring is the use of the Euclidean algorithm and the linear GCD representation of elements in the Euclidean ring. The theorem proved in the work is applied to finding a solution to a linear equation in three unknowns over a ring of Gaussian integers, which, as is known, is a Euclidean ring. In conclusion, comments are made on possible ways of further development of the presented research.

In this work, the classical mathematical model of S.V. was investigated. Dubovsky to describe long waves N.D. Kondratiev (K-waves). This model describes the dynamics of free fluctuations in the efficiency of new technologies and the efficiency of capital productivity. From the point of view of mathematics, it is a system of nonlinear ordinary differential equations of the first order. The purpose of the research is to visualize the results of the solution using numerical modeling of a modification of the mathematical model of S.V. Dubovsky, which consists in taking into account the dependence of the accumulation rate on capital productivity and external inflow of investments and new technological models. It was also shown using the Bendixson test that the classical model of S.V. Dubovsky can generate closed phase trajectories, which indicates its use in describing economic crises and cycles. Similarly, it was shown that within the framework of the modified mathematical model S.V. Dubovsky can also have closed phase trajectories. It is shown using computer modeling that the dependence of the accumulation rate on capital productivity can influence the period of cyclical fluctuations, which is important when modeling real economic cycles and crises. Taking into account the external influx of investment and new technologies (managerial decisions) using harmonic functions significantly complicates the appearance of phase trajectories, however, closed phase trajectories are also possible here. These harmonic functions determine forced fluctuations in the efficiency of new technologies and the efficiency of capital productivity, and here resonance effects may occur, which were shown using computer modeling in this article. Computer simulation was carried out in the computer algebra environment Matlab.

The article presents the results of statistical processing of data from the earthquake catalog of the KBGSRAS for the period from 1 January 1962 to 31 December 2002 for the Kuril-Kamchatka island arc subduction zone (area – N, – E) within the framework of the earlier presented by the authors hereditarian criticality model. The compound power-law Poisson process in fractional time representation is considered as a model. The use of this model assumes quasi-stationary and quasi-homogeneous regime of the seismic process averaged over time and space during long-term observation. The study of the instability of this process over time is carried out using critical indices, which are determined by the numerical characteristics of the process and depend on the parameter  of the Gutenberg-Richter law.Based on the catalog data, the parameters of the seismic process were found by linear and nonlinear regression: the coefficient  and the exponent of the Caputo fractional derivative , by averaging over the magnitude interval in which the power law distribution of recurrence frequencies of events is performed. The significance of the obtained value of the Gutenberg-Richter law parameter  is estimated. Critical indices have been calculated, according to the values of which, and in comparison with the hereditarity parameter , the state of the seismic process in the period under consideration is determined.

The article presents a study of the computational efficiency of a parallel version of a numerical algorithm for solving the Riccati equation with a fractional variable order derivative of the Gerasimov-Caputo type. The numerical algorithm is a nonlocal implicit finite-difference scheme, which reduces to a system of nonlinear algebraic equations and is solved using a modified Newton method. The nonlocality of the numerical scheme creates a high computational load on computing resources, which creates the need to implement efficient parallel algorithms for solving them. The numerical algorithm studied for efficiency is implemented in the C language due to its versatility when working with memory. Parallelization was carried out using OpenMP technology. A series of computational experiments are being carried out on the NVIDIA DGX STATION computing server (Institute of Mathematics named after V.I. Romanovsky, Tashkent, Uzbekistan) and the HP Pavilion Gaming Laptop Z270X, where the Cauchy problem for the fractional Riccati equation with non-constant coefficients was solved. Based on the average computation time, the speedup, efficiency and cost of the algorithm are calculated. From the data analysis it is clear that the OpenMP parallel software implementation of the non-local implicit finite-difference scheme shows an acceleration of 9-12 times, depending on the number of CPU cores involved.

The work represents the results of analysis of electro-telluric potentials data obtained at the Yuzhno-Sakhalinsk test site (deployed in June 2023on the territory of the IMGG FEB RAS). Unexpectedly, a new kind of signals – series of quasiperiodic spikes (pulses) in night times were found in first few months after start of recording. Signals of 4-5 s length and of 130-150 s repetition period have a various waveform, which is derived from some primary quasi-sinusoidal signal. Such series were recorded from July 20 to September 11, 2023, and their average duration was nearly 8-9 hours. No episodes were found after September 12, up to December 20. The maximal intensity of the signals and the series as a whole was revealed in the period from 5 to 10 August. During this period the moderate earthquake M=3.8 occurred on 9 August, 2003 in the vicinity of measurement point (within a circular zone of 0.25 degrees radius around the test site) It was the strongest event from pair of that occurred in the given zone, the magnitudes were being M = 3.8 (08/09/2023) and M = 3.1 (09/19/2023). No similar series were observed before the second earthquake, being the weaker. Origination of quasiperiodic pulses series could be related to the preparedness of earthquake source – site. However extra surveys are required to proof this hypothesis.