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The problem discrete statement by the smoothed particle hydrodynamics method (SPH) include a discretization constants parameters set. Of them particular note is the model sound speed $c_0$, which relates the SPH-particle instantaneous density to the resulting pressure through the equation of state.

The paper describes an approach to the exact determination of the model sound speed required value. It is on the analysis based, how SPH-particle density changes with their relative shift. An example of the continuous medium motion taken the plane shear flow problem; the analysis object is the relative compaction function $\varepsilon_\rho$ in the SPH-particle. For various smoothing kernels was research the functions of $\varepsilon_\rho$, that allowed the pulsating nature of the pressures occurrence in particles to establish. Also the neighbors uniform distribution in the smoothing domain was determined, at which shaping the maximum of compaction in the particle.

Through comparison the function $\varepsilon_\rho$ with the SPH-approximation of motion equation is defined associate the discretization parameter $c_0$ with the smoothing kernel shape and other problem parameters. As a result, an equation is formulated that the necessary and sufficient model sound speed value provides finding. For such equation the expressions of root $c_0$ are given for three different smoothing kernels, that simplified from polynomials to numerical coefficients for the plane shear flow problem parameters.

Numerical methods of obtaining collective and one-particle states were used for the quantum description of two-nuclear systems behavior at the initial stage of near-barrier heavy nuclei fusion. The collective exited states in such systems represent concordant oscillations of surfaces of spherical nuclei. The one-particle states of the external neutrons are similar to the states of valence electrons of diatomic molecules.

The paper considers mathematical modeling of layered Benard–Marangoni convection of a viscous incompressible fluid. The fluid moves in an infinitely extended layer. The Oberbeck–Boussinesq system describing layered Benard–Marangoni convection is overdetermined, since the vertical velocity is zero identically. We have a system of five equations to calculate two components of the velocity vector, temperature and pressure (three equations of impulse conservation, the incompressibility equation and the heat equation). A class of exact solutions is proposed for the solvability of the Oberbeck–Boussinesq system. The structure of the proposed solution is such that the incompressibility equation is satisfied identically. Thus, it is possible to eliminate the «extra» equation. The emphasis is on the study of heat exchange on the free layer boundary, which is considered rigid. In the description of thermocapillary convective motion, heat exchange is set according to the Newton’s law of cooling. The application of this heat distribution law leads to the third-kind initial-boundary value problem. It is shown that within the presented class of exact solutions to the Oberbeck–Boussinesq equations the overdetermined initial-boundary value problem is reduced to the Sturm–Liouville problem. Consequently, the hydrodynamic fields are expressed using trigonometric functions (the Fourier basis). A transcendental equation is obtained to determine the eigenvalues of the problem. This equation is solved numerically. The numerical analysis of the solutions of the system of evolutionary and gradient equations describing fluid flow is executed. Hydrodynamic fields are analyzed by a computational experiment. The existence of counterflows in the fluid layer is shown in the study of the boundary value problem. The existence of counterflows is equivalent to the presence of stagnation points in the fluid, and this testifies to the existence of a local extremum of the kinetic energy of the fluid. It has been established that each velocity component cannot have more than one zero value. Thus, the fluid flow is separated into two zones. The tangential stresses have different signs in these zones. Moreover, there is a fluid layer thickness at which the tangential stresses at the liquid layer equal to zero on the lower boundary. This physical effect is possible only for Newtonian fluids. The temperature and pressure fields have the same properties as velocities. All the nonstationary solutions approach the steady state in this case.

Functional modeling of blood clotting and fibrin-polymer mesh formation is of a significant value for medical and biophysics applications. Despite the fact of some discrepancies present in simplified functional models their results are of the great interest for the experimental science as a handy tool of the analysis for research planning, data processing and verification. Under conditions of the good correspondence to the experiment functional models can be used as an element of the medical treatment methods and biophysical technologies. The aim of the paper in hand is a modeling of a point system of the fibrin-polymer formation as a multistage polymerization process with a sol-gel transition at the final stage. Complex-value Rosenbroke method of second order (CROS) used for computational experiments. The results of computational experiments are presented and discussed. It was shown that in the physiological range of the model coefficients there is a lag period of approximately 20 seconds between initiation of the reaction and fibrin gel appearance which fits well experimental observations of fibrin polymerization dynamics. The possibility of a number of the consequent $(n = 1–3)$ sol-gel transitions demonstrated as well. Such a specific behavior is a consequence of multistage nature of fibrin polymerization process. At the final stage the solution of fibrin oligomers of length 10 can reach a semidilute state, leading to an extremely fast gel formation controlled by oligomers’ rotational diffusion. Otherwise, if the semidilute state is not reached the gel formation is controlled by significantly slower process of translational diffusion. Such a duality in the sol-gel transition led authors to necessity of introduction of a switch-function in an equation for fibrin-polymer formation kinetics. Consequent polymerization events can correspond to experimental systems where fibrin mesh formed gets withdrawn from the volume by some physical process like precipitation. The sensitivity analysis of presented system shows that dependence on the first stage polymerization reaction constant is non-trivial.

In the paper, it is represented a linearization method for the stress-strain curves of nonlinearly deformable beams and circular plates in order to generalize the pure bending vibration equations. It is considered composite, on average isotropic prismatic beams of a constant rectangular cross-section and circular plates of a constant thickness made of nonlinearly elastic materials. The technique consists in determining the approximate Young’s moduli from the initial stress-strain state of beam and plate subjected to the action of the bending moment.

The paper proposes two criteria for linearization: the equality of the specific potential energy of deformation and the minimization of the standard deviation in the state equation approximation. The method allows obtaining in the closed form the estimated value of the natural frequencies of layered and structurally heterogeneous, on average isotropic nonlinearly elastic beams and circular plates. This makes it possible to significantly reduce the resources in the vibration analysis and modeling of these structural elements. In addition, the paper shows that the proposed linearization criteria allow to estimate the natural frequencies with the same accuracy.

Since in the general case even isotropic materials exhibit different resistance to tension and compression, it is considered the piecewise-linear Prandtl’s diagrams with proportionality limits and tangential Young’s moduli that differ under tension and compression as the stress-strain curves of the composite material components. As parameters of the stress-strain curve, it is considered the effective Voigt’s characteristics (under the hypothesis of strain homogeneity) for a longitudinally layered material structure; the effective Reuss’ characteristics (under the hypothesis of strain homogeneity) for a transversely layered beam and an axially laminated plate. In addition, the effective Young’s moduli and the proportionality limits, obtained by the author’s homogenization method, are given for a structurally heterogeneous, on average isotropic material. As an example, it is calculated the natural frequencies of two-phase beams depending on the component concentrations.

In the framework of modern non-equilibrium thermodynamics (macroscopic approach of description and mathematical modeling of the dynamics of real physical and chemical processes), the authors developed a potential- flow method for describing and mathematical modeling of real physical and chemical processes applicable in the general case of real macroscopic physicochemical systems. In accordance with the potential-flow method, the description and mathematical modeling of these processes consists in determining through the interaction potentials of the thermodynamic forces driving these processes and the kinetic matrix determined by the kinetic properties of the system in question, which in turn determine the dynamics of the course of physicochemical processes in this system under the influence of the thermodynamic forces in it. Knowing the thermodynamic forces and the kinetic matrix of the system, the rates of the flow of physicochemical processes in the system are determined, and according to these conservation laws the rates of change of its state coordinates are determined. It turns out in this way a closed system of equations of physical and chemical processes in the system. Knowing the interaction potentials in the system, the kinetic matrices of its simple subsystems (individual processes that are conjugate to each other and not conjugate with other processes), the coefficients entering into the conservation laws, the initial state of the system under consideration, external flows into the system, one can obtain a complete dynamics of physicochemical processes in the system. However, in the case of a complex physico-chemical system in which a large number of physicochemical processes take place, the dimension of the system of equations for these processes becomes appropriate. Hence, the problem arises of automating the formation of the described system of equations of the dynamics of physical and chemical processes in the system under consideration. In this article, we develop a library of software data types that implement a user-defined physicochemical system at the level of its design scheme (coordinates of the state of the system, energy degrees of freedom, physico-chemical processes, flowing, external flows and the relationship between these listed components) and algorithms references in these types of data, as well as calculation of the described system parameters. This library includes both program types of the calculation scheme of the user-defined physicochemical system, and program data types of the components of this design scheme (coordinates of the system state, energy degrees of freedom, physicochemical processes, flowing, external flows). The relationship between these components is carried out by reference (index) addressing. This significantly speeds up the calculation of the system characteristics, because faster access to data.

An isolated system, which possesses a discrete set of microscopic states, is considered. The system performs spontaneous random transitions between the microstates. Kinetic equations for the probabilities of the system staying in various microstates are formulated. A general dimensionless expression for entropy of such a system, which depends on the probability distribution, is considered. Two problems are stated: 1) to study the effect of possible unequal probabilities of different microstates, in particular, when the system is in its internal equilibrium, on the system entropy value, and 2) to study the kinetics of microstate probability distribution and entropy evolution of the system in nonequilibrium states. The kinetics for the rates of transitions between the microstates is assumed to be first-order. Two variants of the effects of possible nonequiprobability of the microstates are considered: i) the microstates form two subgroups the probabilities of which are similar within each subgroup but differ between the subgroups, and ii) the microstate probabilities vary arbitrarily around the point at which they are all equal. It is found that, under a fixed total number of microstates, the deviations of entropy from the value corresponding to the equiprobable microstate distribution are extremely small. The latter is a rigorous substantiation of the known hypothesis about the equiprobability of microstates under the thermodynamic equilibrium. On the other hand, based on several characteristic examples, it is shown that the structure of random transitions between the microstates exerts a considerable effect on the rate and mode of the establishment of the system internal equilibrium, on entropy time dependence and expression of the entropy production rate. Under definite schemes of these transitions, there are possibilities of fast and slow components in the transients and of the existence of transients in the form of damped oscillations. The condition of universality and stability of equilibrium microstate distribution is that for any pair of microstates, a sequence of transitions should exist, which provides the passage from one microstate to next, and, consequently, any microstate traps should be absent.

In this paper we present Rayleigh formulas obtained from Kirchhoff integral formulas, which can later be used to obtain migration images. The relevance of the studies conducted in the work is due to the widespread use of migration in the interests of seismic oil and gas seismic exploration. A special feature of the work is the use of an elastic approximation to describe the dynamic behaviour of a geological environment, in contrast to the widespread acoustic approximation. The proposed approach will significantly improve the quality of seismic exploration in complex cases, such as permafrost and shelf zones of the southern and northern seas. The complexity of applying a system of equations describing the state of a linear-elastic medium to obtain Rayleigh formulas and algorithms based on them is a significant increase in the number of computations, the mathematical and analytical complexity of the resulting algorithms in comparison with the case of an acoustic medium. Therefore in industrial seismic surveys migration algorithms for the case of elastic waves are not currently used, which creates certain difficulties, since the acoustic approximation describes only longitudinal seismic waves in geological environments. This article presents the final analytical expressions that can be used to develop software systems using the description of elastic seismic waves: longitudinal and transverse, thereby covering the entire range of seismic waves: longitudinal reflected PP-waves, longitudinal reflected SP-waves, transverse reflected PS-waves and transverse reflected SS-waves. Also, the results of comparison of numerical solutions obtained on the basis of Rayleigh formulas with numerical solutions obtained by the grid-characteristic method are presented. The value of this comparison is due to the fact that the method based on Rayleigh integrals is based on analytical expressions, while the grid-characteristic method is a method of numerical integration of solutions based on a calculated grid. In the comparison, different types of sources were considered: a point source model widely used in marine and terrestrial seismic surveying and a flat wave model, which is also sometimes used in field studies.

The repressor is the first genetic regulatory network in synthetic biology, which was artificially constructed in 2000. It is a closed network of three genetic elements — $lacI$, $\lambda cI$ and $tetR$, — which have a natural origin, but are not found in nature in such a combination. The promoter of each of the three genes controls the next cistron via the negative feedback, suppressing the expression of the neighboring gene. In this paper, the nonlinear dynamics of a modified repressilator, which has time delays in all parts of the regulatory network, has been studied for the first time. Delay can be both natural, i.e. arises during the transcription/translation of genes due to the multistage nature of these processes, and artificial, i.e. specially to be introduced into the work of the regulatory network using synthetic biology technologies. It is assumed that the regulation is carried out by proteins being in a dimeric form. The considered repressilator has two more important modifications: the location on the same plasmid of the gene $gfp$, which codes for the fluorescent protein, and also the presence in the system of a DNA sponge. In the paper, the nonlinear dynamics has been considered within the framework of the deterministic description. By applying the method of decomposition into fast and slow motions, the set of nonlinear differential equations with delay on a slow manifold has been obtained. It is shown that there exists a single equilibrium state which loses its stability in an oscillatory manner at certain values of the control parameters. For a symmetric repressilator, in which all three genes are identical, an analytical solution for the neutral Andronov–Hopf bifurcation curve has been obtained. For the general case of an asymmetric repressilator, neutral curves are found numerically. It is shown that the asymmetric repressor generally is more stable, since the system is oriented to the behavior of the most stable element in the network. Nonlinear dynamic regimes arising in a repressilator with increase of the parameters are studied in detail. It was found that there exists a limit cycle corresponding to relaxation oscillations of protein concentrations. In addition to the limit cycle, we found the slow manifold not associated with above cycle. This is the long-lived transitional regime, which reflects the process of long-term synchronization of pulsations in the work of individual genes. The obtained results are compared with the experimental data known from the literature. The place of the model proposed in the present work among other theoretical models of the repressilator is discussed.

A cluster method of mathematical modeling of interval-stochastic thermal processes in complex electronic systems (ES), is developed. In the cluster method, the construction of a complex ES is represented in the form of a thermal model, which is a system of clusters, each of which contains a core that combines the heat-generating elements falling into a given cluster, the cluster shell and a medium flow through the cluster. The state of the thermal process in each cluster and every moment of time is characterized by three interval-stochastic state variables, namely, the temperatures of the core, shell, and medium flow. The elements of each cluster, namely, the core, shell, and medium flow, are in thermal interaction between themselves and elements of neighboring clusters. In contrast to existing methods, the cluster method allows you to simulate thermal processes in complex ESs, taking into account the uneven distribution of temperature in the medium flow pumped into the ES, the conjugate nature of heat exchange between the medium flow in the ES, core and shells of clusters, and the intervalstochastic nature of thermal processes in the ES, caused by statistical technological variation in the manufacture and installation of electronic elements in ES and random fluctuations in the thermal parameters of the environment. The mathematical model describing the state of thermal processes in a cluster thermal model is a system of interval-stochastic matrix-block equations with matrix and vector blocks corresponding to the clusters of the thermal model. The solution to the interval-stochastic equations are statistical measures of the state variables of thermal processes in clusters - mathematical expectations, covariances between state variables and variance. The methodology for applying the cluster method is shown on the example of a real ES.