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Investigations of arithmetic properties of the hyper-geometric function values make it possible to single out two trends, namely, Siegel’s method and methods based on the effective construction of a linear approximating form. There are also methods combining both approaches mentioned. The Siegel’s method allows obtaining the most general results concerning the abovementioned problems. In many cases it was used to establish the algebraic independence of the values of corresponding functions. Although the effective methods do not allow obtaining propositions of such generality they have nevertheless some advantages. Among these advantages one can distinguish at least two: a higher precision of the quantitative results obtained by effective methods and a possibility to study the hyper-geometric functions with irrational parameters. In this paper we apply the effective construction to estimate a measure of the linear independence of the hyper-geometric function values over the imaginary quadratic field. The functions themselves were chosen by a special way so that it could be possible to demonstrate a new approach to the effective construction of a linear approximating form. This approach makes it possible also to extend the well-known effective construction methods of the linear approximating forms for poly-logarithms to the functions of more general type. To obtain the arithmetic result we had to establish a linear independence of the functions under consideration over the field of rational functions. It is apparently impossible to apply directly known theorems containing sufficient (and in some cases needful and sufficient) conditions for the system of functions appearing in the theorems mentioned. For this reason, a special technique has been developed to solve this problem. The paper presents the obtained arithmetic results concerning the values of integral functions, but, with appropriate alterations, the theorems proved can be adapted to the case of the hyper-geometric series with a finite radius of convergence. Subsequently, it will be possible to consider also a case of the irrational parameters, but there are still a number of difficulties, which are specific for the problems of such type and which are to be overcome.

One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and only if the corresponding parallel transport to the connection is a solution of the Laplace equation for the Levy Laplacian or of the d'Alembert equation for the Levy d'Alembertian respectively (see [5, 6]). There are two approaches to define Levy type operators, both of which date back to the original works of Levy (see [7]). The first is that the Levy Laplacian (or Levy d'Alembertian) is defined as an integral functional generated by a special form of the second derivative. This approach is used in the works [5, 6], as well as in the paper [8] of Leandre and Volovich, where stochastic Levy-Laplacian is discussed. Another approach to the Levy Laplacian is defining it as the Cesaro mean of second order derivatives along the family of vectors, which is an orthonormal basis in the Hilbert space. This definition of the Levy Laplacian is used for the description of solutions of the Yang-Mills equations in the paper [10].The present work shows that the definitions of the Levy Laplacian and the Levy d'Alembertian based on Cesaro averaging of the second order directional derivatives can be transferred to the stochastic case. In the article the values of these operators on a stochastic parallel transport associated with a connection (vector potential) are found. In this case, unlike the deterministic case and the stochastic case of Levy Laplacian from [8], these values are not equal to zero if the vector potential corresponding to the stochastic parallel transport is a solution of the Maxwell's equations. As a result, two approaches to definition of the Levy Laplacian in the stochastic case give different operators. This situation is different from the flat deterministic case, which is discussed in [11].It can be expected that the work can be summarized in the non-commutative case of the Yang-Mills theory.DOI: 10.7463/mathm.0615.0822138

In 1955 A.B. Shidlovski's general theorems were published. They allow us to reduce the problem of algebraic independence of the analytic function values, belonging to the specific class, to a simpler problem of algebraic independence of these functions. Since the abovementioned general theorems can be applied to the generalized hyper-geometric functions with rational parameters, there appeared many works in which the algebraic independence of such functions (and their derivatives) had been established. The A.B. Shidlovski's results generalize and develop a Siegel's method well known in the theory of transcendental numbers. Besides the Siegel's method to solve the problems concerning the arithmetic nature of the values of analytic functions one also applies methods based on the effective construction of linear approximating forms. Such methods enabled finding the most accurate estimates of linear forms and obtaining the numerous results concerning the arithmetic properties of the values of hyper-geometric functions with irrational parameters. This shows that effective methods are of some value for the development of the theory of transcendental numbers.Recently, in the context of studied arithmetic nature of the values of differentiated hypergeometric functions with respect to parameter, there was a need in results concerning the linear independence of such functions over the field of rational fractions. Similar investigations were also conducted earlier because of applications of A.B. Shidlovski's general theorems, but in that case a more difficult problem of algebraic independence had to be solved, and therefore only the simplest functions were considered. The paper studies the issue of linear independence of hypergeometric functions, differentiated with respect to parameter, and this parameter is included both in the numerator and in the denominator of the common member of the appropriate power series. The paper defines a condition (in some cases, it is necessary and sufficient) of linear independence of such functions, which is very convenient for checking in concrete cases. The paper results are obtained by calculating some determinants, which, naturally, arise from the problems under consideration. In the future, the theorems proved in this paper can be used to have the diverse statements concerning the arithmetic nature of the values of the appropriate functions.DOI: 10.7463/mathm.0415.0817328

The paper is aimed at determining the marginal distributions of the estimates of model parameters for a nonlinear dynamic model of the cell population dynamics. The cell population system evolves in the laboratory (in vitro) and comprises two types of cells.Basic estimates of model parameters are obtained on a single limited sample of experimental data. The value of the number of populations of each cell type is obtained from the experiment at equal intervals of time. This paper proposes a technique for determining the marginal distributions of the parameter estimates using numerical modeling.This technique includes the identification of parameters of nonlinear models and test of the obtained model adequacy with base parameter estimates, the identification of the initial data and finding the reference trajectory. The initial data for the trajectory are found using the least squares method, while minimizing the deviation from the experimental trajectory. Data sets on measured densities of the populations at specific points in time are generated using the reference trajectory and the normally distributed random numbers generator. The problem of obtaining estimates of system parameters is solved for each data set. Tests of hypotheses about the marginal distribution of each parameter are carried out based on the calculated set of estimated parameters. To prove hypothesis, the Kolmogorov test is used. The description of a numerical example is included. The obtained marginal distributions of the parameter estimates can be further used to evaluate the probabilities of different scenarios of the population system development.DOI: 10.7463/mathm.0415.0812686

The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.

It is well known that the Dirichlet problem for the Laplace equation in a ball has a unique polynomial solution (harmonic polynomial) in the case if the given boundary value is the trace of an arbitrary polynomial on the sphere. S.M.Nikol'skii generalized this result in the case of a boundary value problem of the first kind for a linear differential self-adjoint operator of the order 2l with constant coefficients (in particular polyharmonic) and for a domain that is an ellipsoid in R n . For a polyharmonic equation in a ball (homogeneous and inhomogeneous), V.V. Karachik proposed the Almansi formula-based algorithm to construct a polynomial solution of the Dirichlet problem. The paper considers the Poisson equation with the polynomial right-hand side in a multidimensional infinite layer bounded by two hyper-planes. Shows that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value problem with polynomial boundary conditions have a unique solution in the class of functions of polynomial growth, and this solution is a polynomial. Gives an algorithm for constructing this polynomial solution and considers examples. In particular, presents formulas to give exact values of certain integrals (including multi-dimensional ones) and sums of trigonometric series.

Development and operation of modern machinery and latest technology require reliable estimates of the strength characteristics of the critical elements of structures and technological equipment under the impact of high-intensity thermomechanical loading, accompanied, as a rule, by complex contact interaction. Mathematical modeling of stress-strain state of such parts and components in the contact area, based on adequate mathematical models, modern numerical methods and efficient algorithms that implement the direct determination of displacement fields, strains and stresses, is the main tool that allows fast acquisition of data required for the calculations of strength and durability. The paper considers an algorithm for constructing the numerical solution of the contact problem of elasticity theory in relation to the body, which has an obvious one-sided discrete contact interaction with an elastic half-space. The proposed algorithm is specially designed to have a correction of the tangential forces at discrete contact points, allowing us to achieve sufficiently accurate implementation of the adopted law of friction. The algorithm is embedded in a general finite element technology, with which the application code is generated. Numerical study of discrete unilateral contact interaction of an elastic plate and a rigid half-space showed a high efficiency of the developed algorithm and the application code that implements it.DOI: 10.7463/mathm.0415.0801840