Nonlocal problems for the equations of motion of Kelvin-Voight fluids

For the equations of the motion of Kelvin-Voight fluids (0.1) and for the semilinear abstract differential Eqs. (0.11)–(0.17) arising in the theory of Sobolev equations (to which Eqs. (0.1) also belong), we study the four following nonlocal problems: 1) the solvability of the initial boundary problem for Eqs. (0.1) and the Cauchy problem for Eqs. (0.11)–(0.17) on the semiaxis 0<t≤∞; 2) the existence of periodic solutions of Eqs. (0.1) and Eqs. (0.11)–(0.17) with a free term periodic in t; 3) exponential stability theory for solutions of Eqs. (0.1) and Eqs. (0.11)–(0.17) as t→∞ and related problems; 4) attractor theory for Eqs. (0.1). Bibliography: 40 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 120–158, 1992.     

The Lévy–Khinchin representation of a class of stable signed measures

We study properties of symmetric stable measures with index of stability α ∈ (2, 4) ∪ (4, 6). For such signed measures, we construct a natural analog of the Lévy-Khinchin representation. We show that, in some special sense, these measures are limit measures for sums of independent random variables. Bibliography: 6 titles. Translated from Zapiski Nauchnykh. Seminarov POMI, Vol. 361, 2008, pp. 145–166.     

Generalizations in the problem of Lipschitz stability

The problem of Lipschitz stability of nonlinear systems of differential equations is considered. Theorems extending previous results are stated. Translated fromDinamicheskie Sistemy. Vol. 12, pp 101–105, 1993.     

Free boundary surface bifurcation in the phase transition problem of elasticity

The paper is concerns the phase transition problem in an elastic medium lying in a one-parameter force field. The stability and bifurcation problem for the interface between the two phases is studied. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 169–200. Translated by N. A. Karazeeva.     

Fluctuations in one-dimensional dynamic systems

We consider one-dimensional dynamic systems with random fluctuations that are encountered in applications, analyze solutions, and investigate the stability of stationary points. Translated fromDinamicheskie Sistemy. Vol. 12, pp. 96–100, 1993.     

Some properties of autostable models

Established are (1) a nonuniform criterion for the stability of models in terms of enumeration reducibility of constructivizations; (2) a criterion for the autostability of certain particular classes of models close to algebraic number fields; (3) a uniform autostability of each 1-constructive model that is autostable. Supported by RFFR grant No. 096-01-01525 and by ISF grant NQ 6000. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 685–698, November–December, 1996.     

Billiards in rational polygons: Periodic trajectories, symmetries, and d-stability

Periodic trajectories of billiards in rational polygons satisfying the Veech alternative, in particular, in right triangles with an acute angle of the form π/n with integern are considered. The properties under investigation include: symmetry of periodic trajectories, asymptotics of the number of trajectories whose length does not exceed a certain value, stability of periodic billiard trajectories under small deformations of the polygon. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 66–75, July, 1997. Translated by V. N. Dubrovsky     

On the stability of branching continued fractions

We study two aspects of the stability problem for various types of branching fractions using the domains of the elements, the region of convergence, and limit-periodic fractions. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 3–7.     

Stability of one difference equation with positive coefficients

We consider a linear homogeneous difference equation of ordern with positive coefficients, which is presented in the normal form. We obtain necessary and sufficient conditions for the stability and asymptotic stability. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1276–1280, September, 1990.     

Integral metrics and their applications

We define a method of construction of integral metrics and estimate 1) the rate of convergence in the central limit theorem; 2) the stability of one statistical characterization of the normal law in terms of these metrics. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 83–89, Perm, 1990.     

A study of the stability of the solutions of certain nonlinear systems

We use the method of Lyapunov functions to study the stability of the solutions of certain nonlinear systems close to Bazhevskii systems, and also stability in the cone x≥0, x∈Rⁿ, of a system of differential equations with a polynomial right-hand side. We consider an application to the method of Lyapunov vector functions in the case when the system of equations is essentially nonlinear. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 6–12.     

Geometry of distributions of flags on Grassmann manifolds of a projective space. I

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 214–227, April–June, 2000. Translated by V. Mackevičius     

Stability of solutions of an equation arising in hydromechanics

We study the strong stability of the equation describing small oscillations of an elastic pipe conveying an ideal fluid. We prove the existence and uniqueness theorem for the generalized solution and find sufficient conditions for strong stability in the case of a pulsing fluid. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 15–24, January, 2000.     

Geometry of distributions of flags on Grassmann manifolds of a projective space. II

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 335–349, July–September, 2000. Translated by V. Mackevičius     

The problem of applying representative sets in problems of constructing domains of stability and quality of dynamic control systems

We introduce the concept of a representative set as the set of values of the stability and quality characteristics possessing the properties of informativeness and nonredundancy. We obtain relations that determine representative sets for problems of constructing domains of stability and quality for linear stationary systems and retarded systems. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 16–20.     

On partial stability of a dynamic system that is integrally continuous with respect to a small parameter

For a dynamic system described by Carathéodory ordinary differential equations with a small parameter we introduce the definitions of a type of partial stability, attraction, and asymptotic stability. We state theorems giving sufficient conditions for stability in the new definitions. In particular, in terms of perturbed Lyapunov functions we obtain conditions for partial asymptotic stability that generalize known results. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 29–36.     

The method of Lyapunov functionals in stability analysis of functional-differential equations

We develop the method of Lyapunov functionals in the stability analysis of linear nonautonomous functional-differential equations of neutral type. The approach is based on the construction of limit equations and limit Lyapunov functionals. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 323–331, September, 2000.     

Some algorithms for inversion of linear dynamic systems

We study the problem of inversion of linear finite-dimensional dynamic systems in terms of the output. For different types of systems we propose several algorithms for inversion. We study the stability of the algorithms with respect to various perturbations. Translated fromAlgoritmy Upravleniya i Identifikatsii, pp. 156–167, 1997.     

Smooth periodic solutions of the penalized Maxwell fluid dynamics equations

The existence of smooth periodic solutions of the penalized Maxwell equations (2) is proved. The limit passage as ε→0 is investigated. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 85–90. Translated by Ya. Belopol'skaya.     

Time periodic solutions of smooth convergent dissipative ε-approximations of the modified Navier-Stikes equations

In this paper, we prove the global existence of time periodic classical solutions v' of dissipative ε-approximations (4)–(6) for the three-dimensional modified Navier-Stokes Eqs. (1)–(3) that satisfy the first boundary condition. We also study the convergence for ε → 0 of solutions {v'} to time periodic classical solutions v of Eqs. (1)–(3). Bibliography: 21 titles. Dedicated to V. A. Solonnikov on his sixtieth anniversary Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 116–130.