### Closed polynomials and saturated subalgebras of polynomial algebras

Arzhantsev I. V., Petravchuk A. P.

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1587–1593

The behavior of closed polynomials, i.e., polynomials $f ∈ k[x_1,…,x_n]∖k$ such that the subalgebra $k[f]$ is integrally closed in $k[x_1,…,x_n]$, is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial $f ∈ k[x_1,…,x_n]∖k$ can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras $A ⊂ k[x_1,…,x_n]$, i.e., subalgebras such that, for any $f ∈ A∖k$, a generative polynomial of $f$ is contained in $A$.

### Estimates for wavelet coefficients on some classes of functions

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1594–1600

Let $ψ_m^D$ be orthogonal Daubechies wavelets that have $m$ zero moments and let $$W^k_{2, p} = \left\{f \in L_2(\mathbb{R}): ||(i \omega)^k \widehat{f}(\omega)||_p \leq 1\right\}, \;k \in \mathbb{N},$$. We prove that $$\lim_{m\rightarrow\infty}\sup\left\{\frac{|\psi^D_m, f|}{||(\psi^D_m)^{\wedge}||_q}: f \in W^k_{2, p} \right\} = \frac{\frac{(2\pi)^{1/q-1/2}}{\pi^k}\left(\frac{1 - 2^{1-pk}}{pk -1}\right)^{1/p}}{(2\pi)^{1/q-1/2}}.$$

### Sharp estimates for inner radii of systems of nonoverlapping domains and open sets

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1601–1618

We study extremal problems of the geometric theory of functions of a complex variable. Sharp upper estimates are obtained for the product of inner radii of disjoint domains and open sets with respect to equiradial systems of points.

### Output stream of a binding neuron

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1619–1638

For a binding neuron with threshold 2 stimulated by a Poisson stream, we determine the intensity of the output stream and the probability density for the lengths of the output interpulse intervals. For threshold 3, we determine the intensity of the output stream.

### Separately continuous mappings with values in nonlocally convex spaces

Karlova O. O., Maslyuchenko V. K.

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1639–1646

We prove that the collection $(X, Y, Z)$ is the Lebesgue triple if $X$ is a metrizable space, $Y$ is a perfectly normal space, and $Z$ is a strongly $\sigma$-metrizable topological vector space with stratification $(Z_m)^{\infty}_{m=1}$, where, for every $m \in \mathbb{N}$, $Z_m$ is a closed metrizable separable subspace of $Z$ arcwise connected and locally arcwise connected.

### On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. I

Kozachenko Yu. V., Perestyuk M. M.

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1647–1660

We establish conditions under which there exists a function *c*(*t*) > 0 such that $\sup\cfrac{X (t)}{c(t)} < \infty$, where *X*(*t*) is a random process from an Orlicz space of random variables. We obtain estimates for the probabilities $P\left\{ \sup\cfrac{X (t)}{c(t)} > \varepsilon\right\}, \quad \varepsilon > 0$..

### Mixed problem for a semilinear ultraparabolic equation in an unbounded domain

Lavrenyuk S. P., Oliskevych M. O.

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1661–1673

We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation $$u_t + \sum^m_{i=1}a_i(x, y, t) u_{y_i} - \sum^n_{i,j=1} \left(a_{ij}(x, y, t) u_{x_i}\right)_{x_j} + \sum^n_{i,j=1} b_{i}(x, y, t) u_{x_i} + b_0(x, y, t, u) =$$ $$= f_0(x, y, t, ) - \sum^n_{i=1}f_{i, x_i} (x, y, t) $$
in an unbounded domain with respect to the variables *x*.

### Generalized boundary values of solutions of quasilinear elliptic equations with linear principal part

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1674–1688

We establish conditions for the nonlinear part of a quasilinear elliptic equation of order $2m$ with linear principal part under which a solution regular inside a domain and belonging to a certain weighted $L_1$-space takes boundary values in the space of generalized functions.

### On the solution of the basic integral equation of actuarial mathematics by the method of successive approximations

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1689–1698

We study the basic integral equation of actuarial mathematics for the probability of (non)ruin of an insurance company regarded as a function of the initial capital. We establish necessary and sufficient conditions for the existence of a solution of this equation, general sufficient conditions for its existence and uniqueness, and conditions for the uniform convergence of the method of successive approximations for finding the solution.

### Leonіd Andrіyovich Pastur (on his 70th birthday)

Baryakhtar V. G., Berezansky Yu. M., Khruslov E. Ya., Korolyuk V. S., Marchenko V. O., Mitropolskiy Yu. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1699-1700

### Re-extending Chebyshev’s theorem about Bertrand’s conjecture

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1701–1706

In this paper, Chebyshev’s theorem (1850) about Bertrand’s conjecture is re-extended using a theorem about Sierpinski’s conjecture (1958). The theorem had been extended before several times, but this extension is a major extension far beyond the previous ones. At the beginning of the proof, maximal gaps table is used to verify initial states. The extended theorem contains a constant *r*, which can be reduced if more initial states can be checked. Therefore, the theorem can be even more extended when maximal gaps table is extended. The main extension idea is not based on *r*, though.

### Sets of linear expansions of dynamical systems on a torus for a fixed Lyapunov function

Astaf’eva M. M., Stepanenko N. V.

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1707–1713

We consider sets of linear expansions of dynamical systems on a torus with general alternating Lyapunov function.

### Absolute asymptotic stability of solutions of linear parabolic differential equations with delay

↓ Abstract

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1714–1721

We establish necessary and sufficient conditions for the absolute asymptotic stability of solutions of linear parabolic differential equations with delay.

### Alexander Ivanovich Stepanets

Gorbachuk M. L., Lukovsky I. O., Mitropolskiy Yu. A., Romanyuk A. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1722-1724

### Index of volume 59 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 12. - pp. 1725-1729