# Volume 52, № 3, 2000

### Averaging of a multifrequency boundary-value problem with linearly transformed argument

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 291-299

We establish the existence of a solution and obtain an estimate of the error of the averaging method for a multifrequency system with linearly transformed argument and multipoint boundary conditions.

### On asymptotic properties of the empirical correlation matrix of a homogeneous vector-valued Gaussian field

Buldygin V. V., Demyanenko O. O.

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 300-318

We investigate properties of the empirical correlation matrix of a centered stationary Gaussian vector field in various function spaces. We prove that, under the condition of integrability of the square of the spectral density of the field, the normalization effect takes place for a correlogram and integral functional of it.

### On lower bounds for the widths of classes of functions defined by integral moduli of continuity

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 319-328

We establish lower bounds for the Kolmogorov widths *d* _{2n-1}(*W* ^{ r } *H* _{1} ^{ω} .*L* _{ p }) and Gel’fand widths *d* ^{2n-1}(*W* ^{ r } *H* _{1} ^{ω} .*L* _{ p }) of the classes of functions *W* ^{ r } *H* _{1} ^{ω} with a convex integral modulus of continuity ω(*t*).

### Inverse problem of simultaneous determination of two coefficients in a parabolic equation

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 329-335

We establish conditions for the unique existence of a solution of the inverse problem of simultaneous determination of two unknown coefficients in a parabolic equation. One of these coefficients is the leading coefficient that depends on time, and the other coefficient depends on a space variable.

### Diffusion approximation of the Wright-Fisher model of population genetics: Single-locus two alleles

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 336-345

We investigate an autoregressive diffusion approximation method applied to the Wright-Fisher model in population genetics by considering a Markov chain with Bernoulli distributed independent variables. The use of an autoregressive diffusion method and an averaged allelic frequency process lead to an Orn-stein-Uhlenbeck diffusion process with discrete time. The normalized averaged frequency process possesses independent allele frequency indicators with constant conditional variance at equilibrium. In a monoecious diploid population of size *N* with *r* generations, we consider the time to equilibrium of averaged allele frequency in a single-locus two allele pure sampling model.

### Groups all proper quotient groups of which have Chernikov conjugacy classes

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 346-353

We study groups all proper quotient groups of which are *CC*-groups.

### Moduli of multiply-connected domains on a Riemannian Mobius strip

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 354-358

We investigate the modulus problem for families of curves in multiply-connected nonorientable domains on a Riemannian Mobius strip. We determine the extremal metric and the modulus of a “cross” family of arcs.

### Integro-differential equations with multivalued solutions

Plotnikov A. V., Tumbrukaki A. V.

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 359-367

We prove a theorem on the unique existence of a classical solution of an integro-differential equation with Hukuhara derivative. We also justify an averaging scheme for equations of this type in the standard form.

### Singular and fractal properties of distributions of random variables digits of polybasic representations of which a form homogeneous Markov chain

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 368-374

We study the fractal properties of distributions of random variables digits of polybasic *Q*-representations (*a* generalization of *n*-adic digits) of which form a homogeneous Markov chain in the case where the matrix of transition probabilities contains at least one zero.

### Approximation by fourier sums and best approximations on classes of analytic functions

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 375-395

We establish asymptotic equalities for upper bounds of approximations by Fourier sums and for the best approximations in the metrics of *C* and *L1* on classes of convolutions of periodic functions that can be regularly extended into a fixed strip of the complex plane.

### On groups factorized by two subgroups with Chernikov commutants

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 396-402

We establish results concerning the almost solvability and other properties of groups factorized by two subgroups with finite or Chernikov commutants.

### Linearly convex domains with singularities on the boundary

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 403-408

We present a topological classification of linearly convex domains with almost smooth boundary whose singularities lie in a hyperplane. We investigate sets with linearly convex boundary and the closures of linearly convex domains.

### On a solution of a boundary-value problem for a hyperbolic equation on a disk with random initial conditions

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 409-415

We establish conditions under which a solution of a boundary-value problem for a hyperbolic equation on a disk with random initial conditions can be represented as a series uniformly convergent with probability one.

### On the existence and uniqueness of solutions continuous and bounded on the real axis for nonlinear functional equations

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 416-418

For one class of nonlinear functional equations, we establish conditions for the existence and uniqueness of solutions continuous and bounded on the real axis.

### Averaging of a multipoint problem with parameters for an impulsive oscillation system

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 419-423

By using the averaging method, we prove the solvability of a multipoint problem with parameters for a nonlinear oscillation system with pulse influence at fixed times. We establish estimates for the deviation of solutions of the original and averaged problems.

### Analog of the black-scholes formula for option pricing under conditions of (*B, S, X*)-incomplete market of securities with jumps

Kalemanova A. V., Svishchuk A. V., Zhuravyts'kyi D. G.

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 424-431

We describe a (*B, S,X *)-incomplete market of securities with jumps as a jump random evolution process that is a combination of an ltô process in random Markov medium and a geometric compound Poisson process. For this model, we derive the Black-Scholes equation and formula, which describe the pricing of the European call option under conditions of (*B,S,X*)-mcomplete market.