# Volume 65, № 12, 2013

### Derivations and Identities for Kravchuk Polynomials

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1587–1603

We introduce the notion of Kravchuk derivations of the polynomial algebra. It is proved that any element of the kernel of a derivation of this kind gives a polynomial identity satisfied by the Kravchuk polynomials. In addition, we determine the explicit form of isomorphisms mapping the kernel of the basicWeitzenb¨ock derivation onto the kernels of Kravchuk derivations.

### On the Best Approximation in the Mean by Algebraic Polynomials with Weight and the Exact Values of Widths for the Classes of Functions

Shvachko A. V., Vakarchuk S. B.

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1604–1621

The exact value of the extremal characteristic

is obtained on the class *L* _{2} ^{ r } (*D* _{ ρ })*,* where *r ∈* ℤ_{+}; \( {D}_{\rho} = \sigma (x)\frac{d^2}{d{ x}^2}+\tau (x)\frac{d}{d x} \) *, σ* and τ are polynomials of at most the second and first degrees, respectively, *ρ* is a weight function, 0 *< p* ≤ 2*,* 0 *< h <* 1*, λ* _{ n }(*ρ*)
are eigenvalues of the operator *D* _{ ρ } *, φ*
is a nonnegative measurable and summable function (in the interval (*a, b*)) which is not equivalent to zero, *Ω* _{ k,ρ } is the generalized modulus of continuity of the *k* th order in the space *L* _{2,ρ } (*a, b*)*,* and *E* _{ n } (*f*)_{2,ρ }
is the best polynomial approximation in the mean with weight *ρ* for a function *f ∈ L* _{2,ρ } (*a, b*)*.* The exact values of widths for the classes of functions specified by the characteristic of smoothness *Ω* _{ k,ρ } and the *K*-functional \( \mathbb{K} \) _{m} are also obtained.

### Asymptotic Rate of Convergence of a Two-Layer Iterative Method of the Variational Type

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1622–1635

We present the definition and study the dependence on the initial approximation of the asymptotic rate of convergence of a two-layer symmetrizable iterative method of the variational type. The explicit expression is obtained for the substantial (with respect to the Lebesgue measure) range of its values. Its domain of continuity is described.

### On the Invariants of Root Subgroups of Finite Classical Groups

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1636–1645

We show that the invariant fields *F* _{ q }(*X* _{1} *, . . . ,X* _{ n })^{ G } are purely transcendental over *F* _{ q } if *G* are root subgroups of finite classical groups. The key step is to find good similar groups of our groups. Moreover, the invariant rings of the root subgroups of special linear groups are shown to be polynomial rings and their corresponding Poincaré series are presented.

### Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions

Bai Yongzhen, Li Yannan, Wang Youyu

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1646–1656

We establish some new Lyapunov-type inequalities for one-dimensional *p*-Laplacian systems with antiperiodic boundary conditions. The lower bounds of eigenvalues are presented.

### Theorem on Closure and the Criterion of Compactness for the Classes of Solutions of the Beltrami Equations

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1657–1666

We study the classes of regular solutions of degenerate Beltrami equations with constraints of the integral type imposed on a complex coefficient, prove the theorem on closure, and establish a criterion of compactness for these classes.

### Common Fixed-Point Theorems and *c*-distance in Ordered Cone Metric Spaces

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1667–1680

We present a generalization of several fixed and common fixed point theorems on *c* -distance in ordered cone metric spaces. In this way, we improve and generalize various results existing in the literature.

### Best Bilinear Approximations for the Classes of Functions of Many Variables

Romanyuk A. S., Romanyuk V. S.

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1681–1699

We obtain upper bounds for the values of the best bilinear approximations in the Lebesgue spaces of periodic functions of many variables from the Besov-type classes. In special cases, it is shown that these bounds are order exact.

### On the Topological Fundamental Groups of Quotient Spaces

Mashayekhy B., Pakdaman A., Torabi H.

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1700–1711

Let *p*: *X* → *X/A* be a quotient map, where *A* is a subspace of *X*.
We study the conditions under which *p* _{∗}(π _{1} ^{qtop} (*X*, *x* _{0})) is dense in π _{1} ^{qtop} (*X*/*A*,∗)), where the fundamental groups have the natural quotient topology inherited from the loop space and *p* _{*} is a continuous homomorphism induced by the quotient map *p*. In addition, we present some applications in order to determine the properties of π _{1} ^{qtop} (*X*/*A*,∗). In particular, we establish conditions under which π _{1} ^{qtop} (*X*/*A*,∗) is an indiscrete topological group.

### On Equivalent Cone Metric Spaces

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1712–1715

We explore the necessary and sufficient conditions for the two cone metrics to be topologically equivalent.

### On the Absolute Summability of Fourier Series of Almost Periodic Functions

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1716–1722

We establish new sufficient conditions for the absolute |C, *α*|-summability of the Fourier series of functions almost periodic in a sense of Besicovitch whose spectrum has limit points at infinity and at the origin for \( \alpha \ge \frac{1}{2} \) .

### Index of volume 65 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1723-1725