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The paper is a study of the mathematical model and kinematics of a parallel spherical manipulator. This type of manipulator was proposed back in the 80s of the last century and has since found application in exoskeletons and rehabilitation robots due to its structure, which allows imitating natural joint movements of the human body.

The Parallel Spherical Manipulator is a robot with three legs and two platforms, a base platform and a mobile platform. Its legs consist of two support links that are arc-shaped. Mathematically, the manipulator can be described using two virtual pyramids that are placed on top of each other.

The paper considers two types of manipulator configurations: classical and asymmetric, and solves basic kinematic problems for each. The study shows that the asymmetric design of the manipulator has the maximum workspace, especially when the motors are mounted at the joints of the manipulator’s links inside legs.

To optimize the parameters of the parallel spherical manipulator, we introduced a metric of usable workspace volume. This metric represents the volume of the sector of the sphere in which the robot does not experience internal collisions or singular states. There are three types of singular states possible within a parallel spherical manipulator — serial, parallel, and mixed singularity. We used all three types of singularities to calculate the useful volume. In our research work, we solved the problem related to maximizing the usable volume of the workspace.

Through our research work, we found that the asymmetric configuration of the spherical manipulator maximizes the workspace when the motors are located at the articulation point of the robot leg support arms. At the same time, the parameter $\beta_1$ must be zero degrees to maximize the workspace. This allowed us to create a prototype robot in which we eliminated the use of lower links in legs in favor of a radiused rail along which the motors run. This allowed us to reduce the linear dimensions of the robot itself and gain on the stiffness of the structure.

The results obtained can be used to optimize the parameters of the parallel spherical manipulator in various industrial and scientific applications, as well as for further research of other types of parallel robots and manipulators.

An analytical solution is obtained for seismic wave fields in a spherically symmetric Earth. In the case of an arbitrary layered medium, the solution, which includes Bessel functions, is constructed by means of a differential sweep method. Asymptotic of Bessel functions is used for stable calculation of wave fields. It is shown that the classical asymptotic in the case of a sphere of large (in wavelengths) dimensions gives an error in the solution. The new asymptotic is used for efficient calculation of a solution without errors with high detail. A program has been created that makes it possible to carry out calculations for high-frequency (1 hertz and higher) teleseismic wave fields in a discrete (layered) sphere of planetary dimensions. Calculations can be carried even out on personal computers with OpenMP parallelization.

In the works of Burmin (2019) proposed a spherically symmetric model of the Earth. It is characterized by the fact that in it the outer core has a viscosity and, therefore, an effective shear modulus other than zero. For this model of the Earth, a highly detailed calculation was carried out with a carrier frequency of 1 hertz. As a result of the analytical calculation, it was found that highfrequency oscillations of small amplitude, the so-called “precursors”, appear ahead of the PKP waves. An analytical calculation showed that the theoretical seismograms for this model of the Earth are in many respects similar to the experimental data. This confirms the correctness of the ideas underlying its construction.

This paper discusses the modeling of the deformation of polymer nanocomposites containing "hard" and "soft" inclusions, using cellular automata and parallel computing. The paper describes an algorithm based on the model, a comparison with experimental data is shown, software for the numerical experiment is described.

The paper proposes a new mathematical model to study the interaction dynamics of the longitudinal wall of a narrow channel with its end seal. The end seal was considered as the edge wall on a spring, i.e. spring-mass system. These walls interaction occurs via a viscous liquid filling the narrow channel; thus required the formulation and solution of the hydroelasticity problem. However, this problem has not been previously studied. The problem consists of the Navier–Stokes equations, the continuity equation, the edge wall dynamics equation, and the corresponding boundary conditions. Two cases of fluid motion in a narrow channel with parallel walls were studied. In the first case, we assumed the liquid motion as the creeping one, and in the second case as the laminar, taking into account the motion inertia. The hydroelasticty problem solution made it possible to determine the distribution laws of velocities and pressure in the liquid layer, as well as the motion law of the edge wall. It is shown that during creeping flow, the liquid physical properties and the channel geometric dimensions completely determine the damping in the considered oscillatory system. Both the end wall velocity and the longitudinal wall velocity affect the damping properties of the liquid layer. If the fluid motion inertia forces were taken into account, their influence on the edge wall vibrations was revealed, which manifested itself in the form of two added masses in the equation of its motion. The added masses and damping coefficients of the liquid layer due to the joint consideration of the liquid layer inertia and its viscosity were determined. The frequency and phase responses of the edge wall were constructed for the regime of steady-state harmonic oscillations. The simulation showed that taking into account the fluid layer inertia and its damping properties leads to a shift in the resonant frequencies to the low-frequency region and an increase in the oscillation amplitudes of the edge wall.

In this paper, the system of equations of magnetic hydrodynamics (MHD) is considered. The exact solutions found describe fluid flows in a porous medium and are related to the development of a core simulator and are aimed at creating a domestic technology «digital deposit» and the tasks of controlling the parameters of incompressible fluid. The central problem associated with the use of computer technology is large-dimensional grid approximations and high-performance supercomputers with a large number of parallel microprocessors. Kinetic methods for solving differential equations and methods for «gluing» exact solutions on coarse grids are being developed as possible alternatives to large-dimensional grid approximations. A comparative analysis of the efficiency of computing systems allows us to conclude that it is necessary to develop the organization of calculations based on integer arithmetic in combination with universal approximate methods. A class of exact solutions of the Navier – Stokes system is proposed, describing three-dimensional flows for an incompressible fluid, as well as exact solutions of nonstationary three-dimensional magnetic hydrodynamics. These solutions are important for practical problems of controlled dynamics of mineralized fluids, as well as for creating test libraries for verification of approximate methods. A number of phenomena associated with the formation of macroscopic structures due to the high intensity of interaction of elements of spatially homogeneous systems, as well as their occurrence due to linear spatial transfer in spatially inhomogeneous systems, are highlighted. It is fundamental that the emergence of structures is a consequence of the discontinuity of operators in the norms of conservation laws. The most developed and universal is the theory of computational methods for linear problems. Therefore, from this point of view, the procedures of «immersion» of nonlinear problems into general linear classes by changing the initial dimension of the description and expanding the functional spaces are important. Identification of functional solutions with functions makes it possible to calculate integral averages of an unknown, but at the same time its nonlinear superpositions, generally speaking, are not weak limits of nonlinear superpositions of approximations of the method, i.e. there are functional solutions that are not generalized in the sense of S. L. Sobolev.

Laser damage to transparent solids is a major limiting factor output power of laser systems. For laser rangefinders, the most likely destruction cause of elements of the optical system (lenses, mirrors) actually, as a rule, somewhat dusty, is not an optical breakdown as a result of avalanche, but such a thermal effect on the dust speck deposited on an element of the optical system (EOS), which leads to its ignition. It is the ignition of a speck of dust that initiates the process of EOS damage.

The corresponding model of this process leading to the ignition of a speck of dust takes into account the nonlinear Stefan –Boltzmann law of thermal radiation and the infinite thermal effect of periodic radiation on the EOS and the speck of dust. This model is described by a nonlinear system of differential equations for two functions: the EOS temperature and the dust particle temperature. It is proved that due to the accumulating effect of periodic thermal action, the process of reaching the dust speck ignition temperature occurs almost at any a priori possible changes in this process of the thermophysical parameters of the EOS and the dust speck, as well as the heat exchange coefficients between them and the surrounding air. Averaging these parameters over the variables related to both the volume and the surfaces of the dust speck and the EOS is correct under the natural constraints specified in the paper. The entire really significant spectrum of thermophysical parameters is covered thanks to the use of dimensionless units in the problem (including numerical results).

A thorough mathematical study of the corresponding nonlinear system of differential equations made it possible for the first time for the general case of thermophysical parameters and characteristics of the thermal effect of periodic laser radiation to find a formula for the value of the permissible radiation intensity that does not lead to the destruction of the EOS as a result of the ignition of a speck of dust deposited on the EOS. The theoretical value of the permissible intensity found in the general case in the special case of the data from the Grasse laser ranging station (south of France) almost matches that experimentally observed in the observatory.

In parallel with the solution of the main problem, we derive a formula for the power absorption coefficient of laser radiation by an EOS expressed in terms of four dimensionless parameters: the relative intensity of laser radiation, the relative illumination of the EOS, the relative heat transfer coefficient from the EOS to the surrounding air, and the relative steady-state temperature of the EOS.

The dynamics of public attention to COVID-19 epidemic is studied. The level of public attention is described by the daily number of search requests in Google made by users from a given country. In the empirical part of the work, data on the number of requests and the number of infected cases for a number of countries are considered. It is shown that in all cases the maximum of public attention occurs earlier than the maximum daily number of newly infected individuals. Thus, for a certain period of time, the growth of the epidemics occurs in parallel with the decline in public attention to it. It is also shown that the decline in the number of requests is described by an exponential function of time. In order to describe the revealed empirical pattern, a mathematical model is proposed, which is a modification of the model of the decline in attention after a one-time political event. The model develops the approach that considers decision-making by an individual as a member of the society in which the information process takes place. This approach assumes that an individual’s decision about whether or not to make a request on a given day about COVID is based on two factors. One of them is an attitude that reflects the individual’s long-term interest in a given topic and accumulates the individual’s previous experience, cultural preferences, social and economic status. The second is the dynamic factor of public attention to the epidemic, which changes during the process under consideration under the influence of informational stimuli. With regard to the subject under consideration, information stimuli are related to epidemic dynamics. The behavioral hypothesis is that if on some day the sum of the attitude and the dynamic factor exceeds a certain threshold value, then on that day the individual in question makes a search request on the topic of COVID. The general logic is that the higher the rate of infection growth, the higher the information stimulus, the slower decreases public attention to the pandemic. Thus, the constructed model made it possible to correlate the rate of exponential decrease in the number of requests with the rate of growth in the number of cases. The regularity found with the help of the model was tested on empirical data. It was found that the Student’s statistic is 4.56, which allows us to reject the hypothesis of the absence of a correlation with a significance level of 0.01.

We consider a new set of tests which assigns to detection of human capability for parallel calculation. The new tests support the homogenous statistical distribution of results in distinction to the tests discussed in our previous works. This feature simplifies the analysis of test results and decreases the estimate of statistical error. The new experimental data is close to results obtained in previous experiments.

One of problems in developing the gas condensate fields lies on the fact that the condensed hydrocarbons in the gas-bearing layer can get stuck in the pores of the formation and hence cannot be extracted. In this regard, research is underway to increase the recoverability of hydrocarbons in such fields. This research includes a wide range of studies on mathematical simulations of the passage of gas condensate mixtures through a porous medium under various conditions.

In the present work, within the classical approach based on the Darcy law and the law of continuity of flows, we formulate an initial-boundary value problem for a system of nonlinear differential equations that describes a depletion of a multicomponent gas-condensate mixture in porous reservoir. A computational scheme is developed on the basis of the finite-difference approximation and the fourth order Runge .Kutta method. The scheme can be used for simulations both in the spatially one-dimensional case, corresponding to the conditions of the laboratory experiment, and in the two-dimensional case, when it comes to modeling a flat gas-bearing formation with circular symmetry.

The computer implementation is based on the combination of C++ and Maple tools, using the MPI parallel programming technique to speed up the calculations. The calculations were performed on the HybriLIT cluster of the Multifunctional Information and Computing Complex of the Laboratory of Information Technologies of the Joint Institute for Nuclear Research.

Numerical results are compared with the experimental data on the pressure dependence of output of a ninecomponent hydrocarbon mixture obtained at a laboratory facility (VNIIGAZ, Ukhta). The calculations were performed for two types of porous filler in the laboratory model of the formation: terrigenous filler at 25 .„R and carbonate one at 60 .„R. It is shown that the approach developed ensures an agreement of the numerical results with experimental data. By fitting of numerical results to experimental data on the depletion of the laboratory reservoir, we obtained the values of the parameters that determine the inter-phase transition coefficient for the simulated system. Using the same parameters, a computer simulation of the depletion of a thin gas-bearing layer in the circular symmetry approximation was carried out.

This paper targets programmers and developers interested in utilizing parallel programming techniques to enhance application performance. The Oracle Solaris Studio software provides state-of-the-art optimizing and parallelizing compilers for C, C++ and Fortran, an advanced debugger, and optimized mathematical and performance libraries. Also included are an extremely powerful performance analysis tool for profiling serial and parallel applications, a thread analysis tool to detect data races and deadlock in memory parallel programs, and an Integrated Development Environment (IDE). The Oracle Message Passing Toolkit software provides the high-performance MPI libraries and associated run-time environment needed for message passing applications that can run on a single system or across multiple compute systems connected with high performance networking, including Gigabit Ethernet, 10 Gigabit Ethernet, InfiniBand and Myrinet. Examples of OpenMP and MPI are provided throughout the paper, including their usage via the Oracle Solaris Studio and Oracle Message Passing Toolkit products for development and deployment of both serial and parallel applications on SPARC and x86/x64 based systems. Throughout this paper it is demonstrated how to develop and deploy an application parallelized with OpenMP and/or MPI.