Том 58, № 7 (2018)

This work is one of many that are devoted to the further investigation of local interpolating polynomial splines of the fifth order approximation. Here, new polynomial and trigonometrical basic splines are presented. The main features of these splines are the following: the approximation is constructed separately for each grid interval (or elementary rectangular), the approximation constructed as the sum of products of the basic splines and the values of function in nodes and/or the values of its derivatives and/or the values of integrals of this function over subintervals. Basic splines are determined by using a solving system of equations which are provided by the set of functions. It is known that when integrals of the function over the intervals is equal to the integrals of the approximation of the function over the intervals then the approximation has some physical parallel. The splines which are constructed here satisfy the property of the fifth order approximation. Here, the one-dimensional polynomial and trigonometrical basic splines of the fifth order approximation are constructed when the values of the function are known in each point of interpolation. For the construction of the spline, we use the discrete analogues of the first derivative and quadrature with the appropriate order of approximation. We compare the properties of these splines with splines which are constructed when the values of the first derivative of the function are known in each point of interpolation and the values of integral over each grid interval are given. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented.

For the minimization problem for a differentiable function on a set defined by inequality constraints, a simple proof of the Kuhn–Tucker theorem in the Fritz John form is presented. Only an elementary property of the projection of a point onto a convex closed set is used. The approach proposed by the authors is applied to prove Farkas’ theorem. All results are extended to the case of Banach spaces.

The relation between the classical theory of Pfaffian systems and the modern theory of controlled systems is discussed. It is shown that this relation helps solve classification problems and terminal control problems for controlled systems.

The problem of reconstructing unknown inputs in a first-order quasilinear stochastic differential equation is studied by applying dynamic inversion theory. The disturbances in the deterministic and stochastic terms of the equation are simultaneously reconstructed using discrete information on some realizations of the stochastic process. The problem is reduced to an inverse one for ordinary differential equations satisfied by the expectation and variance of the original process. A finite-step software implementable solution algorithm is proposed, and its accuracy with respect to the number of measured realizations is estimated. An illustrative example is given.

An algorithm for solving a stationary nonlinear problem of a nonisothermal flow of an incompressible viscoelastic polymer fluid between two coaxial cylinders is developed on the basis of Chebyshev approximations and the collocation method. In test calculations, the absence of saturation of the algorithm is shown. A posteriori estimates of two error components in the numerical solution—the error of approximation method and the round-off error—are obtained. The behavior of these components as a function of the number of nodes in the spatial grid of the algorithm and the radius of the inner cylinder is analyzed. The calculations show exponential convergence, stability to rounding errors, and high time efficiency of the algorithm developed.

Assuming that the fluid viscosity is an exponential-power function of temperature, a boundary value problem for the Navier–Stokes equations linearized with respect to velocity is solved and the uniqueness of the solution is proved. The problem of a nonuniformly heated spherical solid particle settling in fluid is considered as an application.

Two two-dimensional plates with bending described by Sophie Germain’s equation with the biharmonic operator are joined in the form of a cross with clamped ends, but with free lateral sides outside the cross core. Asymptotics of the deflection of the junction with respect to the relative width of the plates regarded as a small parameter is constructed and justified. The variational-asymptotic model includes a system of two ordinary differential equations of the fourth and second orders with Dirichlet conditions at the endpoints of the one-dimensional cross and the Kirchhoff transmission conditions at its center. They are derived by analyzing the boundary layer near the crossing of the plates and mean that the deflection and the angles of rotation at the central point are continuous and that the total bending force and the principal torques vanish. Possible generalizations of the asymptotic analysis are discussed.

This paper continues the series of studies of Boolean polynomials (Zhegalkin polynomials or algebraic normal forms (ANFs)). The investigation of properties of Boolean polynomials is an important topic of discrete mathematics and combinatorial analysis. Theoretical results in this field have wide practical applications. For example, a number of popular public-key cryptosystems are based on Reed–Muller codes, and the representation of these codes, as well as encryption and decryption algorithms are based on Boolean polynomials. The spectral properties of such codes are determined by the number of zeros of Boolean polynomials and are analyzed using the randomization lemma. Since the problem of determining the number of zeros Z_g of the Boolean polynomial g(x) is NP-hard, the algorithms taking into account the “combinatorial structure of the polynomial” (even though they are search algorithms) are of practical interest. In this paper, such an algorithm based on the properties of the monomial matrix is proposed. For various types of polynomials, exact formulas for the number of roots and the expectation of the number of roots are obtained. A subclass of Boolean polynomials consisting of polynomials with separating variables is considered. A result that can be considered as a generalization of the randomization lemma is proved. The theoretical results provide a basis for estimating the applicability of polynomials for solving various practical problems.