Approximation of classes of smooth functions of several variables

The paper deals with imbedding theorems and harmonic approximation for classes of periodic functions of several variables. Necessary and sufficient conditions for imbedding are found for various classes of smooth functions. The paper contains asymptotic estimates for fractional order derivatives of multidimensional Dirichlet and Favard kernels, for Fourier best approximation and for the Kolmogorov diameter of smooth function classes.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 207–226, 1984.     

Spaces of Functions of Generalized Bounded Variation. II: Questions of Uniform Convergence of Fourier Series

Tests are given for uniform convergence of Fourier series for spaces of functions of generalized bounded variation; along with the well-known tests (of Salem–Oskolkov–Young, Chanturiya, and Waterman) we suggest new tests. We show that the Waterman test for uniform convergence of Fourier series is strongest and unimprovable. We present a theorem on exact estimates for the Fourier coefficients for spaces of functions of bounded variation which contains classical results, improves several well-known results, and gives some new results.     

Cubature Formulas of a Nonalgebraic Degree of Precision

We deal with cubature formulas that are exact for polynomials and also for polynomials multiplied by r, where r is the Euclidean distance to the origin. A general lower bound for the number of nodes for a specified degree of precision is given. This bound is improved for centrally symmetric integrals. A set of constraints (consistency conditions) is introduced for the construction of fully symmetric formulas. For one dimension and arbitrary degree, it is shown that the lower bound is sharp for centrally symmetric integrals. For higher dimensions, as an illustration, cubature formulas are only constructed for low degrees. March 6, 2000. Date revised: April 30, 2001. Date accepted: May 31, 2001.     

On the coinvariants of modular representations of cyclic groups of prime order

We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derksen and Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an incidental result, we identify the coefficients of the monomials appearing in the orbit product of a terminal variable for the three-dimensional indecomposable representation.     

Rate of strong differentiation of integrals

The results of the present paper can be viewed as many-parameter analogs of Oskolkov's estimates for the rate of almost everywhere convergence of Steklov means. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 564–584, April, 1996. In conclusion, the author expresses his gratitude to Yu. V. Kryakin for useful discussions and for his help in preparing the paper for publication.     

K-groups of quadratic extensions of rings

We construct the Levin braid connecting the Tate cohomology ofK-groups for quadratic extensions of rings with antistructures. Explicit formulas for isomorphisms of relative cohomology groups for the induced mapping and for the transfer mapping are obtained; these formulas are necessary in the construction of the Levin braid.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 272–280, August, 1995.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-1402.     

Method of solving equations of motion of viscoelastic media

A new method is proposed for solving dynamic problems for viscoelastic media based on the introduction of potential functions and transformation of equations of motion. The equations obtained for potential functions are used for constructing the general solution in the case of the effect of moving loads on viscoelastic media with plane-parallel interfaces. The problem of the propagation of Rayleigh surface waves is solved independently of the form of the kernels of the linear operators; a formula is obtained for determining the velocity of the Rayleigh surface wave with an arbitrary form of the viscoelastic operators. A method of experimental determination of the kernels determining the linear viscoelastic operators is proposed.V. V. Kuibyshev Moscow Civil Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 429–435, May–June, 1973.     

Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein?s method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein–Uhlenbeck processes is discussed.     

Derivation of equations of seismic and acoustic wave propagation and equations of filtration via homogenization of periodic structures

A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve nonsmooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results for different ratios between physical parameters, we derive Biot’s equations of poroelasticity, a system consisting of nonisotropic Lamé’s equations for the solid component and acoustic equations for the liquid component, nonisotropic Lamé’s equations or equations of viscoelasticity for one-velocity continuum, decoupled system consisting of Darcy’s system of filtration or acoustic equations for the liquid component (first approximation) and nonisotropic Lamé’s equations for the solid component (second approximation), a system consisting of nonisotropic Stokes equations for the liquid component and acoustic equations for the solid component, nonisotropic Stokes equations for one-velocity continuum, or, finally a different type of acoustic equations for one- or two-velocity continuum. The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures.     

Toward determination of the eigenvalues of problems on description of the rotation of deep-water drill columns

This paper presents a method for computing the complex eigenvalues for a boundary-value problem describing the rotation of deep-water drill columns for drilling at depths of thousands of meters. A solution is sought for a singularly perturbed boundary-value problem in the form of an asymptotic expansion in a small parameter ε for two parts of the of the drill column: a lower compressed length, and the upper length. The method can be used to investigate the stability of drill columns used for deep-water drilling. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 29–36, 1993.     

Generalizations of a theorem of Rolewicz

The aim of this article is to give a unified treatment for the theorems of Rolewicz and Neerven type for uniform exponential stability of evolution families. We obtain necessary and sufficient conditions for uniform exponential stability of evolution families, generalizing a stability theorem due to Rolewicz and we present a new proof for the Rolewicz theorem, based on the theory of Banach function spaces. Finally, we apply our results and we deduce a generalization for a classical stability theorem due to Przyluski and Rolewicz.     

Generalization of the method of functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media

A new approach for constructing functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media is proposed. Solutions of the equations of motion in displacements and potentials, which express plane waves and waves from a point source, and also complex solutions of a general type are obtained and investigated. The problem of the reflection of plane waves from the boundary of a half-space is solved for comparison with earlier results [1]. The solutions obtained agree with the physical meaning of the problems and with the solutions for isotropic media.     

Effective moduli of elasticity of a composite composed of anisotropic layers

Relations are obtained for the effective moduli of elasticity and Poisson's ratios of a laminated fiber-reinforced composite, each layer of which has at least orthorhombic symmetry. The elastic properties of the composite in terms of the elastic constants of the layer are expressed exactly, and the elastic constants of the individual layer in terms of the values for the fiber and the matrix are expressed approximately. Two approximations are considered: one corresponds to the Hashin-Shtrikman variational approach, while in the second the comparison material is assigned elastic properties equal to the Voigt or Reuss means of the values for each layer. A numerical example is worked for the combination boron fibers-epoxy resin. The results of the calculation are compared with the exact solution of the problem for a composite composed of alternating layers of boron and epoxy resin.     

The properties of solutions of a class of isoperimetric problems of stability optimization

A class of isoperimetric problems of stability optimization is considered. These arise, for example, when maximizing the Euler force in the destabilization of a column (rod) of varying cross-section and given volume (Lagrange's problem). It is well known that an extremum which depends on the form of the boundary conditions can be achieved for both simple and double eigenvalues. A class of problems is identified for which a global maximum is found at a simple eigenvalue. The possibility of achieving a local extremum for the first (simple) eigenvalue at stationary points is analysed qualitatively in terms of the parameter values and the form of the boundary conditions.     

On equivalence of moduli of smoothness of polynomials in ,

It is well known that for functions , 1p∞. For general functions fL_p, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_p space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with .     

The order of magnitude of the moments of the modulus of continuity of multiparameter poisson and empirical processes

In this note we derive the exact order of magnitude of the moments of the modulus of continuity for multiparameter Poisson and, almost as a corollary, for multivariate empirical processes.     

An estimate of the error of the linearized equations of motion of an axisymmetric projectile

An estimate is obtained, by small-parameter methods, of the error of the solution for the system of equations of motion of a projectile in the atmosphere, linearized for small angles of attack with respect to the variables describing the angular oscillations of the axis of symmetry, compared with the solution of the initial non-linear system for the same initial conditions.     

Solvability of Control Problems for Stationary Equations of Magnetohydrodynamics of a Viscous Fluid

We consider the boundary-value problem for the stationary equations of magnetohydrodynamics of a viscous incompressible fluid with nonhomogeneous boundary conditions for the velocity and electromagnetic field. We study global solvability of this problem and establish some sufficient conditions for uniqueness of its solution. We state control problems for the model of magnetohydrodynamics under consideration, study their solvability, give and examine optimality systems for both arbitrary and particular quality functionals.     

Formation of singularities of solutions of the equations of motion of compressible fluids subjected to external forces in the case of several spatial variables

One considers a class of solutions with finite total energy and moment of inertia for the equations of motion of compressible fluids. It is shown that for a wide class of right-hand sides, including the viscosity term, initially smooth solutions may acquire singularities on a finite time interval. A sufficient condition for the appearance of singularities is found. This condition may be called “the best possible sufficient condition” in the sense that one can explicitly construct a time-global smooth solution for which this condition does not hold to within arbitrary infinitely small quantities. For a nontrivial constant state, perturbations with compact support are considered. A generalization is proved for the known theorem on the initial conditions for which the solution acquires singularities on a finite time interval. The effect of dry friction and rotation on the formation of singularities of smooth solutions is examined. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 274–308, 2007.