Stability of Lotka–Volterra system

The stability of Lotka–Volterra system has been discussed by many authors for two and three species. In this paper, we will discussed the notion of stability for a Lotka–Volterra system with four species. Some criteria and results are given. Our technique depends on the Lyapunov–Razumikhin method. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Compound transient phenomena in multiphase queuing systems. II

The paper continues the article by the author on functional limit theorems in queuing systems under heavy traffic. Theorems are proved for the virtual process of serving jobs when at phases of the queuing system various conditions of heavy traffic are satisfied (compound transient phenomena). Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 343–356, July–September, 1999. Translated by Z. Kryžius     

An initial boundary-value problem for a system of equations of magnetohydrodynamics

In the present paper we consider one initial boundary-value problem for a system of equations of magnetohydrodynamics in the case where it is necessary to take into account the displacement currents in the Maxwell system of equations. We prove a local (in time) unique solvability of this problem in the Sobolev spaces. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 228–254, April–June, 2000. Translated by R. Lapinskas     

The structure of solutions of a hypergeometric equation system

We consider a system of ordinary differential equations which is a multidimensional analogue of a hypergeometric equation. We study the structure and asymptotics of solutions at the singular points and construct a fundamental system of solutions in a neighborhood of each singular point. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 52–64, January–March, 1999. Translated by V. Mackevičius     

The phenomenon of “breaking down” in singularly perturbed systems

A singularly perturbed system can have points of quasi-intersection, where different stable and unstable branches meet. The behavior of solutions in the vicinity of such points is considered. A criterion of “breaking down” is determined for a model problem. Corresponding problems of bifurcation theory are discussed. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 83–91. Translated by A. Ya. Kazakov.     

On the practical μ-stability of solutions of standard systems with delay

We study the problem of μ-stability of a dynamical system with delay. Conditions of the practical μ-stability are established for the general case and for a quasilinear system. The conditions suggested are illustrated by an example. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 204–213, February, 1999.     

Smooth and convergent ε-approximations of the first boundary-value problem for equations of the Kelvin-Voight and oldroyd fluids

Smooth convergent ε-approximations (11)–(13) for the equations of Oldroyd (8) and Kelvin-Voight (9), (10) fluids are constructed. It is shown that the first initial boundary-value problem for two-dimensional system (11) and three-dimensional systems (12) and (13) for every ε>0 has a unique classical solution, and as ε → 0 these solutions converge to classical solutions of the first initial boundary-value problem for Eqs. (8), (9), and (10) respectively. Bibliography: 10 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 246–255. Translated by O. A. Ivanov.     

Unique solvability of the Cauchy problem for a system of Maxwell equations with initial data on a timelike surface

In this paper, a uniqueness theorem on the solution of the Cauchy problem for a system of Maxwell equations is proved in the case where the coefficients ε and μ are analytic functions of coordinates and the initial data are given on an “immovable” surface Σ=Γ×[0≤t≤2T], where Γ is an analytic surface in R³. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 30–37. Translated by N.S. Zabavnikova.     

Uniformly small Δ* Groups

It is established that in each uniformly small Δ*–group, there exists a closed system of representatives; moreover, it turns out that every narrow epimorphism onto a projective Δ*–group splits. (These properties are violated in a class of all Δ*–group). Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 24–39, January–February, 1999.     

Combinatorial optimization in the area of deductive derivability in expert production systems

We consider the problem of discrete optimization with 0–1 variables under the condition that the admissible solutions are logical production systems in the knowledge base of an expert system. A two-stage algorithm for solution of the problem is proposed. Translated fromDinamicheskie Sistemy, Vol. 12, pp. 108–113, 1993.     

Finite-dimensionality and growth of algebras specified by polylinearly interrelated generators

We investigate the finite-dimensionality and growth of algebras specified by a system of polylinearly interrelated generators. The results obtained are formulated in terms of a function ρ. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1435–1440, October, 2005.     

Standardization of distributions of the chord length and distance within oval domains

The distributions of the chord length and distance within oval domains are expressed in terms of the distributions of these values in a circle and of some parameters of oval domains. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 469–474, October–December, 1999. Translated by E. Gečiauskas     

Creep and Long-Term Strength of Press-Powder Composites

The mechanical state of a press-powder filler for pyroautomatics system units of aerospace equipment during their long-term storage is investigated. With the use of the endochronic approach, the rheological properties of a dispersedly reinforced composite are simulated based on the conceptions of linear and nonlinear creep. Analytical expressions of constitutive relations of creep and strength for the composites are obtained. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 803–818, November–December, 2005.     

Deterministic chaos of a spherical pendulum under limited excitation

We investigate the appearance, development, and vanishing of deterministic chaos in a “spherical pendulum-electric motor of limited power” dynamical system. Chaotic attractors discovered in the system are described in detail. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 534–548, April, 2007.     

Asymptotics for the solution of averaged equations for a system of coupled oscillators

The equations obtained by averaging a system of three weakly nonlinear oscillators are considered. Distinctive properties of asymptotics at infinity for the solutions of averaged equations for both constant and variable proper frequencies are studied. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 99–113, 2006.     

Lamb's inverse axially symmetric problem

The problem of restoring the velocity of transversal waves in an elastic half-space by a known wave field on the boundary of a medium is studied. The problem is reduced to a system of “integral” equations similar to the Volterra one. Theorems on existence in the small and on uniqueness of the solution are proved. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 51–67. Translated by T. N. Surkova.     

Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type

We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 443–452, April, 2008.     

An asymptotic formula of the Hellinger integral for renewal processes

Asymptotic formulas of the Hellinger integral are used in the investigation of properties of optimal estimates and statistical criteria. For a certain class of renewal processes, this formula was obtained by the author in [Lith. Math. J.,38(2), 131–143 (1998)]. In this paper, we obtain such a formula for all renewal processes whose intermediate renewal moments have absolutely continuous distributions. We use the traditional representation of the Hellinger integral and the theory of large deviations. Šiauliai University, Višinskio 25, 5400 Šiauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 493–497, October–December, 1999. Translated by V. Mackevičius     

An initial-boundary value problem on the half-line for the MKdV equation

The initial-boundary value problem on the half-line for the modified Korteweg-de Vries equation with zero boundary conditions and arbitrary rapidly decaying initial conditions is embedded in the scheme of the inverse scattering method. The corresponding inverse scattering problem is reduced to the Riemann problem on a system of rays in the complex plane. Supported by the Russian Foundation for Basic Research under grant 98-01-00576. Mathematical Institute, Russian Academy of Sciences, Ufa; e-mail: ihabib@imat.rb.ru. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 65–75, January–March, 2000. Translated by I. T. Habibullin     

Impression of a semiinfinite stamp in an elastic strip with regard for friction and adhesion

We consider the problem of contact interaction between a semiinfinite stamp with rectilinear base and an elastic strip with one rigid side. Friction forces in the contact region are taken into account. These forces lead to the division of the contact region into slipping and adhesion zones. With the use of the Wiener–Hopf method, a system of integral equations is reduced to an infinite system of algebraic equations. The computational results of stresses and strains at the boundary and at inner points of the elastic strip are presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 138–149, January–March, 2008.