The existence of equilibria in certain games,separation for families of convex functions and a theorem of Borsuk-Ulam type

The existence of a Nash equilibrium in the undiscounted repeated two-person game of incomplete information on one side is established. The proof depends on a new topological result resembling in some respect the Borsuk-Ulam theorem.     

Asymptotics of Dirichlet Spectrum on Some Class of Noncompact Domains

We investigate the asymptotic behaviour of the counting function of Dirichlet eigenvalues on some class of noncompact manifolds. We prove that in cases when the volume or the perimeter (the volume of the boundary) of the manifold is infinite, some additional (nonclassical) terms appear in the precise asymptotics. The coefficients at the classical terms in those are regularized in some special way volume (resp. perimeter) of the manifold.     

On some periodic solutions of singularly perturbed parabolic equations

We present results from the theory of singular perturbations and, in particular, from a new branch of this theory (contrast alternating-type structures). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 359–369, March, 2007.     

New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop.     

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.      

Smooth Frame-Path Termination for Higher Order Sigma-Delta Quantization

We study the performance of finite frames for the encoding of vectors by applying standard higher-order sigma-delta quantization to the frame coefficients. Our results are valid for any quantizer with accuracy ε > 0 operating in the no-overload regime. The frames under consideration are obtained from regular sampling of a path in a Hilbert space. In order to achieve error bounds that are comparable to results on higher-order sigma-delta for the quantization of oversampled bandlimited functions, we construct frame paths that terminate smoothly in the zero vector, that is, with an appropriate number of vanishing derivatives at the endpoint.     

Embedding and wavelet decomposition of spaces of minimal splines

On a sequence of embedded nonuniform grids, chains of embedded spaces of minimal splines (not necessarily polynomial) are constructed. The wavelet decomposition is given. The basis wavelets are compactly supported and admit simple analytic representation. The corresponding decomposition and reconstruction formulas are derived. The variety of spaces under consideration is identified with the variety of complete sequences of points of the direct product of an interval and a projective plane. Bibliography: 20 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 15–31     

The free Malcev superalgebra on one odd generator and related superalgebras

A base of the universal multiplicative envelope of the free Malcev superalgebra ℳ on one odd generator is constructed. Some corollaries for skew-symmetric functions and central elements in free Malcev and free alternative algebras are obtained. Moreover, a base of the Poisson-Malcev superalgebra of ℳ is constructed. As a corollary, a set of elements that spans the free alternative superalgebra on one odd generator is obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 97–106, 2004.