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The article is devoted to modeling the shallow water hydrodynamic processes using the example of the Azov Sea. The article presents a mathematical model of the hydrodynamics of a shallow water body, which allows one to calculate three-dimensional fields of the velocity vector of movement of the aquatic environment. Application of regularizers according to B.N.Chetverushkin in the continuity equation led to a change in the method of calculating the pressure field, based on solving the wave equation. A discrete finite-difference scheme has been constructed for calculating pressure in an area whose linear vertical dimensions are significantly smaller than those in horizontal coordinate directions, which is typical for the geometry of shallow water bodies. The method and algorithm for solving grid equations with a tridiagonal preconditioner are described. The proposed method is used to solve grid equations that arise when calculating pressure for the three-dimensional problem of hydrodynamics of the Azov Sea. It is shown that the proposed method converges faster than the modified alternating triangular method. A parallel implementation of the proposed method for solving grid equations is presented and theoretical and practical estimates of the acceleration of the algorithm are carried out taking into account the latency time of the computing system. The results of computational experiments for solving problems of hydrodynamics of the Sea of Azov using the hybrid MPI + OpenMP technology are presented. The developed models and algorithms were used to reconstruct the environmental disaster that occurred in the Sea of Azov in 2001 and to solve the problem of the movement of the aquatic environment in estuary areas. Numerical experiments were carried out on the K-60 hybrid computing cluster of the Keldysh Institute of Applied Mathematics of Russian Academy of Sciences.

The article considers a mathematical model of decomposition of a complex product into assembly units. This is an important engineering problem, which affects the organization of discrete production and its operational management. A review of modern approaches to mathematical modeling and automated computer-aided of decompositions is given. In them, graphs, networks, matrices, etc. serve as mathematical models of structures of technical systems. These models describe the mechanical structure as a binary relation on a set of system elements. The geometrical coordination and integrity of machines and mechanical devices during the manufacturing process is achieved by means of basing. In general, basing can be performed on several elements simultaneously. Therefore, it represents a variable arity relation, which can not be correctly described in terms of binary mathematical structures. A new hypergraph model of mechanical structure of technical system is described. This model allows to give an adequate formalization of assembly operations and processes. Assembly operations which are carried out by two working bodies and consist in realization of mechanical connections are considered. Such operations are called coherent and sequential. This is the prevailing type of operations in modern industrial practice. It is shown that the mathematical description of such operation is normal contraction of an edge of the hypergraph. A sequence of contractions transforming the hypergraph into a point is a mathematical model of the assembly process. Two important theorems on the properties of contractible hypergraphs and their subgraphs proved by the author are presented. The concept of $s$-hypergraphs is introduced. $S$-hypergraphs are the correct mathematical models of mechanical structures of any assembled technical systems. Decomposition of a product into assembly units is defined as cutting of an $s$-hypergraph into $s$-subgraphs. The cutting problem is described in terms of discrete mathematical programming. Mathematical models of structural, topological and technological constraints are obtained. The objective functions are proposed that formalize the optimal choice of design solutions in various situations. The developed mathematical model of product decomposition is flexible and open. It allows for extensions that take into account the characteristics of the product and its production.

The article proposes a method of joint analysis of multidimensional financial time series based on the evaluation of the set of properties of stock quotes in a sliding time window and the subsequent averaging of property values for all analyzed companies. The main purpose of the analysis is to construct measures of joint behavior of time series reacting to the occurrence of a synchronous or coherent component. The coherence of the behavior of the characteristics of a complex system is an important feature that makes it possible to evaluate the approach of the system to sharp changes in its state. The basis for the search for precursors of sharp changes is the general idea of increasing the correlation of random fluctuations of the system parameters as it approaches the critical state. The increments in time series of stock values have a pronounced chaotic character and have a large amplitude of individual noises, against which a weak common signal can be detected only on the basis of its correlation in different scalar components of a multidimensional time series. It is known that classical methods of analysis based on the use of correlations between neighboring samples are ineffective in the processing of financial time series, since from the point of view of the correlation theory of random processes, increments in the value of shares formally have all the attributes of white noise (in particular, the “flat spectrum” and “delta-shaped” autocorrelation function). In connection with this, it is proposed to go from analyzing the initial signals to examining the sequences of their nonlinear properties calculated in time fragments of small length. As such properties, the entropy of the wavelet coefficients is used in the decomposition into the Daubechies basis, the multifractal parameters and the autoregressive measure of signal nonstationarity. Measures of synchronous behavior of time series properties in a sliding time window are constructed using the principal component method, moduli values of all pairwise correlation coefficients, and a multiple spectral coherence measure that is a generalization of the quadratic coherence spectrum between two signals. The shares of 16 large Russian companies from the beginning of 2010 to the end of 2016 were studied. Using the proposed method, two synchronization time intervals of the Russian stock market were identified: from mid-December 2013 to mid- March 2014 and from mid-October 2014 to mid-January 2016.

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.

Metal hydrides are an interesting class of chemical compounds that can reversibly bind a large amount of hydrogen and are, therefore, of interest for energy applications. Understanding the factors affecting the kinetics of hydride formation and decomposition is especially important. Features of the material, experimental setup and conditions affect the mathematical description of the processes, which can undergo significant changes during the processing of experimental data. The article proposes a general approach to numerical modeling of the formation and decomposition of metal hydrides and solving inverse problems of estimating material parameters from measurement data. The models are divided into two classes: diffusive ones, that take into account the gradient of hydrogen concentration in the metal lattice, and models with fast diffusion. The former are more complex and take the form of non-classical boundary value problems of parabolic type. A rather general approach to the grid solution of such problems is described. The second ones are solved relatively simply, but can change greatly when model assumptions change. Our experience in processing experimental data shows that a flexible software tool is needed; a tool that allows, on the one hand, building models from standard blocks, freely changing them if necessary, and, on the other hand, avoiding the implementation of routine algorithms. It also should be adapted for high-performance systems of different paradigms. These conditions are satisfied by the HIMICOS library presented in the paper, which has been tested on a large number of experimental data. It allows simulating the kinetics of formation and decomposition of metal hydrides, as well as related tasks, at three levels of abstraction. At the low level, the user defines the interface procedures, such as calculating the time layer based on the previous layer or the entire history, calculating the observed value and the independent variable from the task variables, comparing the curve with the reference. Special algorithms can be used for solving quite general parabolic-type boundary value problems with free boundaries and with various quasilinear (i.e., linear with respect to the derivative only) boundary conditions, as well as calculating the distance between the curves in different metric spaces and with different normalization. This is the middle level of abstraction. At the high level, it is enough to choose a ready tested model for a particular material and modify it in relation to the experimental conditions.

Instrument of numerical-analytical modeling of characteristics of propagation of electromagnetic waves in chaotic space plasma with taking into account effects of gravitation is developed for interpretation of data of measurements of astrophysical precision instruments of new education. The task of propagation of waves in curved (Riemann’s) space is solved in Euclid’s space by introducing of the effective index of refraction of vacuum. The gravitational potential can be calculated for various model of distribution of mass of astrophysical objects and at solution of Poisson’s equation. As a result the effective index of refraction of vacuum can be evaluated. Approximate model of the effective index of refraction is suggested with condition that various objects additively contribute in total gravitational field. Calculation of the characteristics of electromagnetic waves in the gravitational field of astrophysical objects is performed by the approximation of geometrical optics with condition that spatial scales of index of refraction a lot more wavelength. Light differential equations in Euler’s form are formed the basis of numerical-analytical instrument of modeling of trajectory characteristic of waves. Chaotic inhomogeneities of space plasma are introduced by model of spatial correlation function of index of refraction. Calculations of refraction scattering of waves are performed by the approximation of geometrical optics. Integral equations for statistic moments of lateral deviations of beams in picture plane of observer are obtained. Integrals for moments are reduced to system of ordinary differential equations the firsts order with using analytical transformations for cooperative numerical calculation of arrange and meansquare deviations of light. Results of numerical-analytical modeling of trajectory picture of propagation of electromagnetic waves in interstellar space with taking into account impact of gravitational fields of space objects and refractive scattering of waves on inhomogeneities of index of refraction of surrounding plasma are shown. Based on the results of modeling quantitative estimation of conditions of stochastic blurring of the effect of gravitational lensing of electromagnetic waves at various frequency ranges is performed. It’s shown that operating frequencies of meter range of wavelengths represent conditional low-frequency limit for observational of the effect of gravitational lensing in stochastic space plasma. The offered instrument of numerical-analytical modeling can be used for analyze of structure of electromagnetic radiation of quasar propagating through group of galactic.

Calcium is a major nutrient regulating metabolism in a plant. Deficiency of calcium results in a growth decline of plant tissues. Ca may be lost from forest soils due to acidic atmospheric deposition and tree harvesting. Plant-available calcium compounds are in the soil cation exchange complex and soil waters. Model of soil calcium dynamics linking it with the model of soil organic matter dynamics ROMUL in forest ecosystems is developed. ROMUL describes the mineralization and humification of the fraction of fresh litter which is further transformed into complex of partially humified substance (CHS) and then to stable humus (H) in dependence on temperature, soil moisture and chemical composition of the fraction (nitrogen, lignin and ash contents, pH). Rates of decomposition and humification being coefficients in the system of ordinary differential equations are evaluated using laboratory experiments and verified on a set of field experiments. Model of soil calcium dynamics describes calcium flows between pools of soil organic matter. Outputs are plant nutrition, leaching, synthesis of secondary minerals. The model describes transformation and mineralization of forest floor in detail. Experimental data for calibration model was used from spruсe forest of Bulgaria.

Modern mathematical models in the dynamics system “soil–plant” are considered. The components of this system are: agricultural plant, microorganisms of the rhizosphere (root zone of plants), the mineral nutrition elements of plants in their mobile and immobile forms. The model of submitted system based on the analysis of the adopted provisions was developed. The construction of system elements allows to display the coordinated dynamics of these elements among themselves. In particular, the dynamics of mineral nutrition elements in plants and the dynamics of their biomass are determined by the current contents in the rhizosphere of mineral fertilizers and organic origin substances (plant roots, leaves, etc.). The immobility of plants spatial distribution and the mobile spatial nature of microorganisms are assumed. This mechanism is determined by diffusion. Mutual relationships between weeds and pests are suggested. The dynamics of the mineral nutrition elements is determined by the peculiarity of sorption in the soil solution, environmental conditions, organic decomposition and fertilizer application. An analytical study for a system where each of the components is represented by only one species (fertilizer, the association of microorganisms and plants) was performed. An adaptation of the wave propagation model in the “resource–consumer” system (Kolmogorov–Petrovsky–Piskunov waves) has been developed for annual agricultural crops. The developed model has been adapted for the growth of Krasnoufimskaya-100 spring wheat in a vessel on peat lowland soil, where nitrogen, phosphorus, and potassium fertilizers were added variably. Sample distributions are plants biomass and the content of mineral nutrition elements in them. The parametric identification of the model and its adequacy was performed. An assessment of the model adequacy showed a good agreement between the model and experimental data.

The grid-characteristic method is successfully used for solving hyperbolic systems of partial differential equations (for example, transport / acoustic / elastic equations). It allows to construct correctly algorithms on contact boundaries and boundaries of the integration domain, to a certain extent to take into account the physics of the problem (propagation of discontinuities along characteristic curves), and has the property of monotonicity, which is important for considered problems. In the cases of two-dimensional and three-dimensional problems the method makes use of a coordinate splitting technique, which enables us to solve the original equations by solving several one-dimensional ones consecutively. It is common to use up to 3-rd order one-dimensional schemes with simple splitting techniques which do not allow for the convergence order to be higher than two (with respect to time). Significant achievements in the operator splitting theory were done, the existence of higher-order schemes was proved. Its peculiarity is the need to perform a step in the opposite direction in time, which gives rise to difficulties, for example, for parabolic problems.

In this work coordinate splitting of the 3-rd and 4-th order were used for the two-dimensional hyperbolic problem of the linear elasticity. This made it possible to increase the final convergence order of the computational algorithm. The paper empirically estimates the convergence in L1 and L∞ norms using analytical solutions of the system with the sufficient degree of smoothness. To obtain objective results, we considered the cases of longitudinal and transverse plane waves propagating both along the diagonal of the computational cell and not along it. Numerical experiments demonstrated the improved accuracy and convergence order of constructed schemes. These improvements are achieved with the cost of three- or fourfold increase of the computational time (for the 3-rd and 4-th order respectively) and no additional memory requirements. The proposed improvement of the computational algorithm preserves the simplicity of its parallel implementation based on the spatial decomposition of the computational grid.

Although each task of recreating artifacts is truly unique, the modeling process for façades, foundations and building elements can be parametrized. This paper is focused on a complex of the existing programming libraries and solutions that need to be united into a single computer system to solve such a task. An algorithm of generating 3D filling of objects under reconstruction is presented. The solution architecture necessary for the system's adaptation for a cloud environment is studied.