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In this paper, stochastic differential equations (SDEs) with the first integral are considered. The exact solution of such SDEs belongs to a smooth manifold with probability 1. However, the numerical solution does not belong to the manifold, but it belongs to some of its neighborhood due to the numerical error. The main objective of the paper is to construct modified numerical methods for solving SDEs that preserve the first integral. In this study, exact solutions for three SDE systems with the first integral are obtained, and the proposed modification of numerical methods is tested on these systems.

The description of a method for modeling the motion of bodies in a viscous incompressible fluid with the use of counting technology on grids with overlapping («chimera» technology) is given. Equations describing the flow of a viscous incompressible fluid are approximated by the finite volume method on an arbitrary unstructured grid. Their iterative solution is implemented using the SIMPLE algorithm. The description of the basic equations in the case of grid motion is given in this paper. The features of realizing the conditions on the boundaries of the grid regions that are established during the construction of the interpolation template are described. A method for overcoming numerical instability in the use of a rigid body model is demonstrated. The feature of taking into account the forces of gravitation in the case of the presence of multiphase media is described. The results of solving the problem of the motion of a cylinder in a fluid, the problem of the drop of a sphere into a fluid, and the problem of the ship's model flooding are presented.

We present the approaches to solving a problem of shallow water oscillations in a parabolic basin (including an extra case of a horizontal plane). A series of assumptions about the form of solution and effects of Earth`s rotation and bottom friction are made. Then the resulting ODE systems are solved. The corresponding free surfaces have first or second order. The conditions of finiteness and localization of a flow are analyzed. The solutions are used in the verification of numerical algorithm of the large particles method, the efficiency of the carried out tests is discussed.

The presence of the nonlocal term in the nonlocal problems destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite element method and Newton-Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, this paper is devoted to the analysis of a linearized Theta-Galerkin finite element method for the time-dependent nonlocal problem with nonlinearity of Kirchhoff type. Hereby, we focus on time discretization based on θ -time stepping scheme with θ ∈ [1/2,1). Some a error estimates are derived for the standard Crank-Nicolson ( θ =1/2), the shifted Crank-Nicolson ( θ = 1/2 + δ, δ is the time-step) and the general case ( θ ≠ 1/2 + kδ, k = 0,1). Finally, numerical simulations that validate the theoretical findings are exhibited.

In this paper, we introduce the notion of controllability of a non-parametric statistical test and compare the powers of one new controllable non-parametric statistical test and the Wilcoxon-Mann-Whitney test for the cases with samples from exponential distribution.

The hierarchy of minimal mathematical models of the dynamics of p53-Mdm2-microRNA system has been developed. The models are based on the differential equations with time delay, hiding complex mechanisms of interaction in the signal system of the p53 protein. We consider the two types of interaction of p53 with microRNAs: the positive direct connection and the positive feedback. The feedback of microRNA-p53 is due to the negative effect of microRNA on the protein Mdm2, which itself is a negative regulator of p53. To approximate the direct positive effect of p53 on the microRNA, a linear function or the Goldbeter-Koshland type representation is used. The comparison of numerical solutions with medical and biological data of a number of specific p53-dependent microRNAs is made, which proves the adequacy of the models proposed and the results of numerical analysis. Special attention was given to the analysis of the positive feedback of p53 and microRNAs. The minimal models adopted have allowed us to consider the most general regularities of the p53-dependent microRNAs functioning. In the framework of these mathematical models it is shown that it is possible to neglect the connection Mdm2-miRNA for, at least, some of the most studied microRNAs associated with a direct positive connection with p53. However, those of the microRNAs, which are important negative regulator Mdm2, can have the most significant impact on the functioning of the entire system p53-Mdm2-microRNA. Conditions were obtained under which the regulatory function of microRNAs with respect to p53 is manifested. The results of numerical experiments indicate that such microRNAs can be considered to be a factor of the anticancer therapy.

This paper deals with the multi-parameter asymptotic description of the stress field near the crack tip of a finite crack in an infinite isotropic elastic plane medium subject to 1) tensile stress; 2) in-plane shear; 3) mixed mode loading for a wide range of mode-mixing situations (Mode I and Mode II). The multi-parameter series expansion of the stress tensor components containing higher order terms has been constructed. All the coefficients of the multi-parameter series expansion of the stress field are given. The main focus is on the discussion of the influence of considering the higher-order terms of the Williams expansion. Analysis of the higher order terms in the stress field is made. It is shown that the larger distance from the crack tip, the more terms are necessary to be kept in the asymptotic series expansion. Therefore, it can be concluded that several more higher-order terms of the Williams expansion must be used for the stress field description when the distance from the crack tip is not small enough. The crack propagation direction angle has been calculated. Two fracture criteria: maximum tangential stress criterion and the strain energy density criterion, are used. The multi-parameter form of two commonly used fracture criteria is introduced and tested. Thirty and more terms of the Williams expansion enable the angle to be calculated more precisely.

It is known that the dual representation of problems (through solutions of the main and the conjugate in the Lagrange sense equations) allows one to formulate the perturbation theory serving as basement for the successive approximation method in the inverse problems theory. If, according to preliminary predictions, the solution of an inverse problem (for example, the structure of the medium of interest) belongs to a certain set A, then selecting a suitable (trial, reference) element a ₀ as an unperturbed one and applying the perturbation theory, one can approximately describe the behavior of the solution of the direct problem in this domain and find a subset A₀ that best matches the measurement data. However, as the accuracy requirements increase, the domain A₀ of the first approximation is rapidly narrowing, expanding it by adding higher terms of the expansion complicates the decision procedure. For this reason, a number of works have been devoted to the search for unperturbed approaches. Among them is the method of variational interpolation (VI-method), in the frame of which not one, but several problems a₁, a₂, …, a_n are used in order to compose from their solutions the desired one. The functional of interest is represented in the stationary form, and the coefficients of the expansion are determined from the condition of stationarity of the bilinear form. This paper demonstrates the application of VI-method to solving inverse problems in the frame of simple model situations associated with cosmic rays astrophysics.

The article considers consequences of Spinoza’s recognition of the fact that the universe, which he names also God, is the sole and single totality of an infinite multitude of solitary modes of its universal attributes, to which human also belongs. Special attention is paid to the specificity of Spinoza's picture of the universe caused by the philosopher's recognition of the strict necessity of the «nature» of everything that exists and at the same time to his efforts to reject that which by the end of his life was called «quietism». Meanwhile, in Spinozas's view, the «last goal» of human is only an adequate perception of the world and himself. According to Spinoza, only intellectual intuition, as he says, may attain it in an undoubtedly adequate way. Naturally, the article studies thoroughly and evaluates the component of Spinoza’s teaching which deals with intellectual intuition. It is summarized not just in his statement that it is only intellectual intuition which comprehends the individually unique and the singular, i.e. that the totality of which Spinoza thinks to be the only really existing components of the universe. Its equally important feature, according to Spinoza, is the fact that clear and distinct things in that what we cognize we perceive in no other way but «through themselves». The article shows the main direction of change in the object orientation of Spinoza's research attention in his Ethics which takes place as the philosopher increases his efforts to reveal a level of importance of a person’s reliance on intellectual intuition in his cognitive activity and spiritual quest as a whole. The author analyzes how Spinoza substantiates both the absolute truth of the result of the functioning of intellectual intuition and the perpetual state of soul beatitude which feels human possessing this result.

The distribution of the Cayander larch Larix cajanderi Mayr within the northern boundary, on the right bank of the Lena river in Ust’-Lenskiy nature reserve in Yakutia is discussed in the article. In the studied area, normal, curved, curtain, semi-shelter and elfin forms of growth of the Cajander larch were recorded, and the geographical coordinates of their locations are given. Larch communities are distributed over 6 ecological-topological areas of the terrain: along the banks of the Lena river, between the Tit-Ary island and the mouth of the Chinke river; in the interfluve of the Chinke and Sobol’-Yuryage rivers; below the mouth of the Sobol’-Yuryage river; on the left bank of the Chinke river valley; on the right bank of the valleys of the Chinke and Chinke Salata rivers. On the right bank of the Lena river between the Tit-Ary island and upstream from the mouth of the Chinke river, larch forests are common with small gaps. Their last locations are found in the mouth part of the Chinke river. In the interfluve between the Chinke and Sobol’-Yuryage rivers there is a small number of semi-elfin forest growth forms of larch trees. The last reliably registered larch communities were found at a distance of 2.1 km below the mouth of the Sobol’-Yuryage river. On the left bank of the Chinke river valley, on the slopes of the northern exposure, 5 locations of larch were found. All of them are located below the confluence of the Chinke-Yuryage and the Chinke-Salata rivers, at a distance from the Lena river up to 1.4 km. On the right side of the valley of the Chinke and the Chinke-Yuryage rivers, on the slopes of the southern exposition, larch communities are more common and were observed at a distance of 3 km from the Lena river to (thirteen locations). In the valley of the Chinke-Salata river larch is found only in one place, on the slope of the southern exposure, at a distance of 0.5 km from the confluence with the Chinke-Yuryage river. The conservation regime of the territory eliminates the anthropogenic impact on the natural process of larch moving to the north, which makes it possible to organize and conduct long-term monitoring observations of the dynamics of forest vegetation in the far north.