# Volume 62, № 4, 2010

### Kernels of derivations of polynomial rings and Casimir elements

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 435–452

We propose an algorithm for the evaluation of elements of the kernel of an arbitrary derivation of a polynomial ring. The algorithm is based on an analog of the well-known Casimir element of a finite-dimensional Lie algebra. By using this algorithm, we compute the kernels of Weitzenböck derivation $d(x_i ) = x_{i−1},\; d(x_0) = 0,\;i = 0,…, n$, for the cases where $n ≤ 6$.

### On the modularity of a lattice of $τ$-closed $n$-ultiply $ω$-composite formations

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 453–463

Let $n ≥ 0$, let $ω$ be a nonempty set of prime numbers and let $τ$ be a subgroup functor (in Skiba’s sense) such that all subgroups of any finite group $G$ contained in $τ (G)$ are subnormal in $G$. It is shown that the lattice of all $τ$-closed $n$-multiply $ω$-composite formations is algebraic and modular.

### Cubic rings and their ideals

Drozd Yu. A., Skuratovskii R. V.

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 464–470

We give an explicit description of cubic rings over a discrete valuation ring, as well as the description of all ideals of these rings.

### On the strong matrix Hamburger moment problem

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 471–482

We obtain necessary and sufficient conditions for the solvability of the strong matrix Hamburger moment problem. We describe all solutions of the moment problem by using the fundamental results of A. V. Shtraus on generalized resolvents of symmetric operators.

### Self-improvement of summability factors of functions satisfying the reverse Hölder inequality in limit cases

Fomichev V. V., Korenovskii A. A.

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 483–493

We show that the best summability factors of functions that satisfy the reverse Hölder inequality in limit cases can be obtained from the nonlimit case by passing to the limit.

### On continual interpolation nodes for operators in linear topological spaces

Kashpur O. F., Khlobystov V. V., Makarov V. L.

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 494–503

We establish conditions for the existence of continual nodes for interpolation polynomials of the integral type. This result is generalized to the case of multivariable operators. Some examples of these interpolants are analyzed.

### On the Marcinkiewicz–Zygmund law of large numbers in Banach lattices

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 504–513

We strengthen the well-known Marcinkiewicz–Zygmund law of large numbers in the case of Banach lattices. Examples of applications to empirical distributions are presented.

### On a measure of integral square deviation with generalized weight for the Rosenblatt–Parzen probability density estimator

Babilua P., Nadaraya E., Sokhadze G. A.

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 514–535

The limit distribution of an integral square deviation with weight in the form of “delta”-functions for the Rosenblatt–Parzen probability density estimator is determined. In addition, the limit power of the goodness-of-fit test constructed by using this deviation is investigated.

### Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables

Romanyuk A. S., Romanyuk V. S.

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 536–551

We obtain exact order estimates for the best $M$-term trigonometric approximations of the Besov classes $B_{∞,θ}^r$ in the space $L_q$. We also determine the exact orders of the best bilinear approximations of the classes of functions of $2d$ variables generated by functions of d variables from the classes $B_{∞,θ}^r$ with the use of translation of arguments.

### On some systems of nonlinear integral Hammerstein-type equations on the semiaxis

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 552–566

We prove the existence for a one-parameter family of solutions of a system of nonlinear integral Hammerstein-type equations on the positive semiaxis and study the asymptotic behavior of the obtained solutions at infinity.

### New results in $G$-best approximation in $G$-metric spaces

Dehghan Nezhad A., Mazaheri H.

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 567–571

The purpose of this paper is to introduce and discuss the concepts of G-best approximation and $a_0$ -orthogonality in the theory of $G$-metric spaces. We consider the relationship between these concepts and the dual $X$ and obtain some results on subsets of $G$-metric spaces similar to normed spaces.