# Volume 64, № 11, 2012

### Karamata theorem for regularly log-periodic functions

Buldygin V. V., Pavlenkov V. V.

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1443-1463

We generalize the Karamata theorem on the asymptotic behavior of integrals with variable limits to the class of regularly log-periodic functions.

### Homological stabilization for dedekind rings of the arithmetic type

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1464-1476

We study the problem of stabilization in the higher $K$ -theory of rings and improved stabilization. The triviality of the group of standard cycles is established in the case of arithmetic-type rings. Some applications of the obtained results to the problem of homological stabilization are presented.

### Douglis-Nirenberg elliptic systems in Hörmander spaces

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1477-1476

We investigate Douglis-Nirenberg uniformly elliptic systems in $\mathbb{R}^n$ on the class of Hormander Hilbert spaces $H^{\varphi}$, where $\varphi$ is an $RO$-varying function of scalar argument. An a priori estimate for solutions is proved, and their interior regularity is studied. A sufficient condition for these systems to have the Fredholm property is given.

### Boundary-value problems for a nonlinear hyperbolic equation with Levy Laplaciana

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1492-1499

We present solutions of the boundary-value problem $U(0, x) = u_0, \;U(t, 0) = u_1$, and the external boundary-value problem $U(0, x) = v_0,\; U(t, x)|_{Γ} = v_1,\; \lim_{||x||_H→∞} U(t, x) = v_2$ for the nonlinear hyperbolic equation $$\frac{∂^2U(t, x)}{∂t^2} + α(U(t, x)) \left[\frac{∂U(t, x)}{∂t}\right]^2 = ∆_LU(t, x)$$ with infinite-dimensional Levy Laplacian $∆_L$.

### On the behavior of solutions of the Cauchy problem for a degenerate parabolic equation with source in the case where the initial function slowly vanishes

Martynenko A. V., Shramenko V. N., Tedeev A. F.

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1500-1515

We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form $$u_t = \text{div}(\rho(x)u^{m-1}|Du|^{\lambda-1}Du) + u ^p $$ in the case where initial function decreases slowly to zero as $|x| \rightarrow \infty$. We establish conditions for the existence and nonexistence of a global-in-time solution, which substantially depend on the behavior of the initial data as $|x| \rightarrow \infty$. In the case of global solvability, we obtain an exact estimate of a solution for large times.

### Inverse problem for interior spectral data of the hydrogen atom equation

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1516-1525

We consider the inverse problem for second-order differential operators with regular singularity and show that the potential function can be uniquely determined by the set of values of eigenfunctions at some interior point and parts of two spectra.

### Large deviations for impulsive processes in the scheme of Poisson approximation

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1526-1535

Asymptotic analysis of the large deviation problem for impulsive processes in the scheme of Poisson approximation is performed. Large deviations for impulsive processes in the scheme of Poisson approximation are defined by an exponential generator for a jump process with independent increments.

### Consistency of an adjusted least-squares estimator in a vector linear model with measurement errors

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1536-1546

We consider the vector linear errors-in-variables model. For this model, we construct an adjusted least-squares estimator and prove its weak and strong consistency under various assumptions about measurement errors.

### On convolutions on configuration spaces. I. Spaces of finite configurations

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1547-1567

We consider two types of convolutions ($\ast$ and $\star$) of functions on spaces of finite configurations (finite subsets of a phase space) and study some of their properties. A relationship between the $\ast$-convolution and the convolution of measures on spaces of finite configurations is described. Properties of the operators of multiplication and differentiation with respect to the $\ast$-convolution are investigated. We also present conditions under which the $\ast$-convolution is positive definite with respect to the $\star$-convolution.

### Ivan Ivanovych Lyashko (on his 90 th birthday)

Hryshchenko O. Yu., Klyushin D. A., Lyashko S. I., Sergienko I. V.

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1568 - 1571

### A note on noncosingular lifting modules

Amouzegar Kalati T., Keskin Tütüncü D.

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1572-1574

Let $R$ be a right perfect ring. Let $M$ be a noncosingular lifting module which does not have any relatively projective component. Then $M$ has finite hollow dimension.

### On rational functions of the best nonsymmetric approximations in integral metrics

Polyakov O. V., Ruchaevskaya N. O.

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1575-1577

We obtain theorems that characterize the degree of the rational function of the best $(\alpha, \beta)$ -approximation in the space $L_p$ and conditions under which the value of the best rational $(\alpha, \beta)$ -approximation is less than the best $(\alpha, \beta)$ -approximation by algebraic polynomials.

### A matrix approach to the binomial theorem

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1578-1584

Motivated by the formula $x^n = \sum_{k=0}^n\left(n \atop k\right) (x - 1)^k$ we investigate factorizations of the lower triangular Toeplitz matrix with $(i, j)$th entry equal to $x^{i-j}$ via the Pascal matrix. In this way, a new computational approach to a generalization of the binomial theorem is introduced. Numerous combinatorial identities are obtained from these matrix relations.