# Volume 57, № 4, 2005

### Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 435–457

We study the procedure of averaging in the Cauchy problem for an ordinary differential equation perturbed by a certain Markov ergodic process. We establish several estimates for the rate of convergence of solutions of the original problem to solutions of the averaged one.

### On the Solvability of Impulsive Differential-Algebraic Equations

Perestyuk N. A., Vlasenko L. A.

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 458–468

We establish theorems on the existence and uniqueness of a solution of the impulsive differential-algebraic equation $$\frac{d}{{dt}}[Au(t)] + Bu(t) = f(t,u(t)),$$ where the matrix A may be singular. The results are applied to the theory of electric circuits.

### Congruences of a Permutable Inverse Semigroup of Finite Rank

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 469–473

We describe the structure of any congruence of a permutable inverse semigroup of finite rank.

### Properties of a Solution of an Inhomogeneous Hyperbolic Equation with Random Right-Hand Side

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 474–482

We consider an inhomogeneous hyperbolic equation with zero initial and boundary conditions and a random centered sample-continuous Gaussian right-hand side. We establish conditions for the existence of a solution of the first boundary-value problem of mathematical physics in the form of a series uniformly convergent in probability in terms of a covariance function. An estimate for the distribution of the supremum of a solution of this problem is obtained.

### On Periodic Solutions of One Class of Systems of Differential Equations

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 483–495

We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders.

### Summation of Fourier-Laplace Series in the Space $L(S^m)$

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 496–504

We establish estimates of the rate of convergence of a group of deviations on a sphere in the space $L(S^m),\quad m > 3$.

### On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 505–513

The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation \(f' = \frac{{P(z,f)}}{{Q(z,f)}}\), where *P*(*z, f*) and *Q*(*z, f*) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity.

### Groups with Almost Normal Subgroups of Infinite Rank

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 514–532

We study classes of groups whose subgroups of some infinite ranks are almost normal.

### Best $n$-Term Approximations with Restrictions

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 533–553

We determine exact values of the best $n$-term approximations with restrictions on polynomials used for the approximation of $\lambda, q$-ellipsoids in the spaces $S^{p,\, \mu}_{\varphi}$.

### Multiplicity of Continuous Mappings of Domains

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 554–558

We prove that either the proper mapping of a domain of an *n*-dimensional manifold onto a domain of another *n*-dimensional manifold of degree *k* is an interior mapping or there exists a point in the image that has at least |*k*|+2 preimages. If the restriction of *f* to the interior of the domain is a zero-dimensional mapping, then, in the second case, the set of points of the image that have at least |*k*|+2 preimages contains a subset of total dimension *n*. In addition, we construct an example of a mapping of a two-dimensional domain that is homeomorphic at the boundary and zero-dimensional, has infinite multiplicity, and is such that its restriction to a sufficiently large part of the branch set is a homeomorphism.

### On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from *l *²

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 559–563

We describe the lineal of initial data of a simple conservative scattering system which can be transferred to zero by a sequence from *l *².
The proof is based on the known connection between the Lax - Phillips scattering theory and the theory of unitary operator nodes developed by B. Szokefalvi - Nagy, C. Foias, and M. S. Brodskii.

### Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series

Bodraya V. I., Novikov O. A., Rukasov V. I.

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 564–570

We obtain asymptotic equalities for deviations of rectangular linear means of Fourier series on classes of ψ-integrals of multivariable functions

### On the Stability of the Maximum Term of the Entire Dirichlet Series

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 571–576

We establish necessary and sufficient conditions for logarithms of the maximal terms of the entire Dirichlet series $F(z) = \sum^{+\infty}_{n=0}a_n e^{z\lambda_n}$ and $A(z) = \sum^{+\infty}_{n=0}a_n b_n e^{z\lambda_n}$ to be asymptotically equivalent as ${\rm Re}\;z \rightarrow +\infty$ outside some set of finite measure.