# Volume 50, № 4, 1998

### Classification of maximal subalgebras of rank *n* of the conformal algebra *AC*(1, *n*)

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 459–470

We obtain a complete classification of *I*-maximal subalgebras of rank *n* of the conformal algebra *AC*(1, *n*).

### A limit theorem for mixing processes subject to rarefaction. I

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 471–475

We prove a limit theorem on the approximation of point mixing processes subject to rarefaction by general renewal processes. This theorem contains a weaker condition on the mixing coefficient than the known conditions.

### On the modulus of continuity of solid derivatives of a Cauchy-type integral

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 476–484

We establish sufficient conditions for the existence of solid derivatives of a continuous extension of a Cauchy-type integral onto the closure of a domain and obtain an estimate for the moduli of continuity of these derivatives. We prove that the Newton-Leibniz formula is true for certain classes of Jordan rectifiable curves.

### Stochastic integration and one class of Gaussian random processes

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 485–495

We consider one class of Gaussian random processes that are not semimartingales but their increments can play the role of a random measure. For an extended stochastic integral with respect to the processes considered, we obtain the Itô formula.

### Estimates for the best approximation and integral modulus of continuity of a function in terms of its Fourier coefficients

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 496–503

In the integral metric, lower bounds are obtained for the best approximation and the modulus of continuity of a function in terms of its Fourier coefficients.

### Boundary-Value problems with random initial conditions and functional series from sub_{φ} (Ω). I

Koval’chuk Yu. A., Kozachenko Yu. V.

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 504–515

We study conditions for convergence and the rate of convergence of random functional series from the space sub_{φ} (Ω) in various norms. The results are applied to the investigation of a boundary-value problem for a hyperbolic equation with random initial conditions.

### Coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 516–522

We establish sufficient algebraic coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay in the case of a rational correlation between delays. We use (*n* ^{2} + *m*)-parameter Lyapunov functions (*n* is the dimension of the system of equations and *m* is the number of delays).

### Structure theorems for families of idempotents

Kruhlyak S. A., Samoilenko Yu. S.

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 523–533

For *-algebras generated by idempotents and orthoprojectors, we study the complexity of the problem of description of *-representations to within unitary equivalence. In particular, we prove that the *-algebra generated by two orthogonal idempotents is *-wild as well as the *-algebra generated by three orthoprojectors, two of which are orthogonal.

### Groups with elementary abelian commutant of at most $p^2$th order

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 534-539

We obtain a representation of nilpotent groups with a commutant of the type $(p)$ or $(p, p)$ that has the form of a product of two normal subgroups. One of these subgroups is constructively described as a Chernikov $p$-group of rank 1 or 2, and the other subgroup has a certain standard form. We also obtain a representation of nonnilpotent groups with a commutant of the type $(p)$ or $(p, p)$ in the form of a semidirect product of a normal subgroup of the type $(p)$ or $(p, p)$ and a nilpotent subgroup with a commutant of order $p$ or 1.

### Approximate representation for a natural power of the Riemann ζ-function

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 540–551

We obtain an approximate functional equation for *ζ* ^{m} *(z), m ∈ N*.

### Stochastic dynamics and Boltzmann hierarchy. III

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 552–569

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.

### On the smoothness of the Green function for the problem of bounded invariant manifolds

Burilko O. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 570–584

We investigate the smoothness of the Green function for the problem of bounded invariant manifolds of linear extensions of dynamical systems.

### On multiplicators of power series in Hardy spaces

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 585–587

We investigate the accuracy of certain sufficient conditions for multiplicators established by Trigub.

### Construction of an analytic solution for one class of Langevin-type equations with orthogonal random actions

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 588–589

We find an analytic representation of a solution of the Itô-Langevin equations in *R* ^{3} with orthogonal random actions with respect to the vector of the solution. We construct a stochastic process to which the integral of the solution weakly converges as a small positive parameter with the derivative in the equation tends to zero.

### Filtration of random solutions of a system of linear difference equations with coefficients depending on a Markov chain

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 590–592

We solve the problem of the estimation of a random state for a system with discrete time that is described by a system of linear difference equations with coefficients depending on a finite-valued Markov chain.

### On the best $L_1$-approximation of truncated powers by algebraic polynomials

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 593-598

We determine the asymptotic behavior of the best $L_1$-approximations of the kernels $(x - a) + r - 1,\; 1 < r < 2,$ and the classes $W_1^r$ by algebraic polynomials.

### The Third Bogolyubov Readings. International Scientific Conference “Asymptotic and Qualitative Methods in the Theory of Nonlinear Oscillations”

Kolomiyets V. G., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 599

### Yurii L’Vovich Daletskii

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 600