Rational matrix functions associated with the Nevanlinna-Pick problem in the class <Emphasis Type="Italic">S</Emphasis>[<Emphasis Type="Italic">a,b</Emphasis>] and orthogonal on a compact interval

We consider the interpolation Nevanlinna-Pick problem with infinitely many interpolation nodes in the class S[a, b] and rational matrix functions associated with this problem and orthogonal on the segment [a, b]. We obtain a criterion of complete indeterminacy of the Nevanlinna-Pick problem in terms of orthogonal rational matrix functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 764–770, June, 2007.     

Factorisation des polynômes de Poncelet

La condition pour qu’il existe des n-gones de Poncelet, inscrits dans un cercle, et circonscrits à un autre cercle, est polynomiale en les paramètres définissant ces deux cercles. Nous décomposons ces polynômes “de Poncelet” en facteurs irréductibles que nous interprétons géométriquement.     

Nash Equilibria in Noncooperative Predator-Prey Games

A noncooperative game governed by a distributed-parameter predator-prey system is considered, assuming that two players control initial conditions for predator and prey, respectively. Existence of a Nash equilibrium is shown under the condition that the desired population profiles and the environmental carrying capacity for the prey are sufficiently small. A conceptual approximation algorithm is proposed and analyzed. Finally, numerical simulations are performed, too.     

On the properties of solutions to the Goursat-Darboux problem with boundary and distributed controls

We consider a control system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls. The controls are subject to the constraints given as multivalued mappings with closed, possibly nonconvex, values depending on the phase variable. Alongside the initial constraints, we consider the convexified constraints and the constraints whose values are the extreme points of the convexified constraints. We study the questions of existence of solutions and establish connections between the solutions under various constraints.     

Higher discriminants of binary forms

We propose a method for constructing systems of polynomial equations that define submanifolds of degenerate binary forms of an arbitrary degeneracy degree. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 147–157, November, 2007.     

On a certain nonlocal problem for a hyperbolic equation

The author proves the existence and uniqueness of a generalized solution of a nonlocal problem with an integral condition for a hyperbolic equation with n spatial variables. This work is a continuation of the studies started in [3–5], where the solvability problem of nonlocal problems with an integral condition was studied for hyperbolic equations on the plane. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Algorithms of motion planning for a six-legged walking machine

This paper investigates algorithms of motion planning for a six-legged walking machine over complex terrain. Application of the Rodrigues-Hamilton parameters for describing the orientation of the body enables one to develop algorithms of motion planning for the body and legs in an absolute coordinate frame with automatic adaptation to the surface. Experiments with a laboratory scale walking machine have demonstrated the efficiency of proposed algorithms. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 197–206, 2005.