Volume 62, № 4, 2010

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 $n ≤ 6$.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.