# Volume 66, № 5, 2014

### On the Behavior of Algebraic Polynomial in Unbounded Regions with Piecewise Dini-Smooth Boundary

Abdullayev F. G., Özkartepe P.

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 579–597

Let *G* ⊂ *ℂ* be a finite region bounded by a Jordan curve *L* := ∂*G*, let \( \Omega :=\mathrm{e}\mathrm{x}\mathrm{t}\overline{G} \) (with respect to \( \overline{\mathbb{C}} \) ), let Δ := {*w* : |*w*| > 1}, and let *w* = Φ(*z*) be the univalent conformal mapping of Ω onto Δ normalized by Φ (∞) = ∞, Φ′(∞) > 0. Also let *h*(*z*) be a weight function and let *A* _{ p }(*h,G*), *p* > 0 denote a class of functions *f* analytic in *G* and satisfying the condition $$ {\left\Vert f\right\Vert}_{A_p\left(h,G\right)}^p:={\displaystyle \int {\displaystyle \underset{G}{\int }h(z){\left|f(z)\right|}^pd{\sigma}_z<\infty, }} $$ where *σ* is a two-dimensional Lebesgue measure.

Moreover, let *P* _{ n } (*z*) be an arbitrary algebraic polynomial of degree at most *n* ∈ ℕ. The well-known Bernstein–Walsh lemma states that

In this present work we continue the investigation of estimation (*) in which the norm \( {\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)} \) is replaced by \( {\left\Vert {P}_n\right\Vert}_{A_p\left(h,G\right)},p>0 \) , for Jacobi-type weight function in regions with piecewise Dini-smooth boundary.

### Groups with the Same Prime Graph as the Simple Group $D_n (5)$

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 598–608

Let $G$ be a finite group. The prime graph of $G$ is denoted by $Γ(G)$. Let G be a finite group such that $Γ(G) = Γ(D_n (5))$, where $n ≥ 6$. In the paper, as the main result, we show that if $n$ is odd, then $G$ is recognizable by the prime graph and if $n$ is even, then $G$ is quasirecognizable by the prime graph.

### Generalized Twisted Kloosterman Sum Over *ℤ*[*i*]

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 609–618

The twisted Kloosterman sums over Z were studied by V. Bykovsky, A.Vinogradov, N. Kuznetsov, R. W. Bruggeman, R. J. Miatello, I. Pacharoni, A. Knightly, and C. Li. In our paper, we obtain similar estimates for *K* _{ χ }(α, *β*; γ; *q*) over *ℤ*[*i*] and improve the estimates obtained for the sums of this kind with Dirichlet character χ (mod *q* _{1}), where *q* _{1} | *q*.

### Multipoint (in Time) Problem for One Class of Evolutionary Pseudodifferential Equations

Drin Ya. M., Horodets’kyi V. V.

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 619–633

We establish the well-posed solvability of a nonlocal multipoint (in time) problem for the evolution equations with pseudodifferential operators of infinite order.

### Approximation of the Classes $H_p^{Ω}$ of Periodic Functions of Many Variables in the Space $L_p$

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 634–644

We establish upper estimates for the approximation of the classes $H_p^{Ω}$ of periodic functions of many variables by polynomials constructed by using the system obtained as the tensor product of the systems of functions of one variable. These results are then used to establish the exact-order estimates of the orthoprojective widths for the classes $H_p^{Ω}$ in the space $L_p$ with $p ∈ \{1, ∞\}$.

### Multiperiodic Solution of a Boundary-Value Problem for one Class of Parabolic Equations with Multidimensional Time

Abdikalikova G. A., Berzhanov A. B., Kenzhebaev K. K.

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 645–655

We study the existence and uniqueness of the multiperiodic solution of the first boundary-value problem for a system of parabolic equations with multidimensional time.

### On the Diameters of Commuting Graphs of Permutational Wreath Products

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 656–665

Let *G* be a group and let *Z*(*G*) be the center of *G.* The commuting graph of the group *G* is an undirected graph *Γ*(*G*) with the vertex set *G \ Z*(*G*) such that two vertices *x, y* are adjacent if and only if *xy* = *yx.* We study the commuting graphs of permutational wreath products *H* *G,* where *G* is a transitive permutation group acting on *X* (the top group of the wreath product) and (*H, Y*) is an Abelian permutation group acting on *Y.*

### One Inverse Problem for the Diffusion-Wave Equation in Bounded Domain

Lopushanskaya G. P., Lopushanskyi A. O.

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 666–678

We prove the theorems on the existence and unique determination of a pair of functions: *a*(*t*) *>*0*, t* ∈ [0*,T*]*,* and the solution *u*(*x, t*) of the first boundary-value problem for the equation $$ \begin{array}{ll}{D}_t^{\beta }u-a(t){u}_{xx}={F}_0\left(x,t\right),\hfill & \left(x,t\right)\in \left(0,l\right)\times \left(0,T\right],\hfill \end{array} $$

with regularized derivative *D* _{ t } ^{ β } *u* of the fractional order β ∈ (0*,* 2) under the additional condition *a*(*t*)*u* _{ x }(0*, t*) = *F*(*t*)*, t* ∈ [0*,T*]*.*

### Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 679–693

Periodic and bounded solutions of the Coulomb equation of motion in the line are obtained for two and three identical negative point charges in the fields of two and three symmetrically located fixed point charges. The systems possess equilibrium configurations. The Lyapunov, Siegel, Moser, and Weinstein theorems are applied.

### Second Maximal Subgroups of a Sylow *p*-Subgroup and the *p*-Nilpotency of Finite Groups

↓ Abstract

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 694–698

A subgroup *H* of a group *G* is said to be weakly *s*-semipermutable in *G* if *G* has a subnormal subgroup *T* such that *HT* = *G* and *H* ∩ *T ≤* \( {H}_{\overline{s}G} \) , where \( {H}_{\overline{s}G} \) is the subgroup of *H* generated by all subgroups of *H* that are *s*-semipermutable in *G*. The main aim of the paper is to study the *p*-nilpotency of a group for which every second maximal subgroup of its Sylow *p*-subgroups is weakly *s*-semipermutable. Some new results are obtained.

### On the Solvability of a Fourth-Order Operator-Differential Equation with Multiple Characteristic

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 699–707

In the Sobolev-type space with exponential weight, we obtain sufficient conditions for the well-posed and unique solvability on the entire axis of a fourth-order operator-differential equation whose main part has a multiple characteristic. We establish estimates for the norms of the operators of intermediate derivatives related to the conditions of solvability. In addition, we deduce the relationship between the exponent of the weight and the lower bound of the spectrum of the main operator appearing in the principal part of the equation. The obtained results are illustrated by an example of a problem for partial differential equations.

### On the Statistical Convergence of Metric-Valued Sequences

Değer U., Dovgoshei A. A., Küçükaslan M.

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 712–720

We study the conditions for the density of a subsequence of a statistically convergent sequence under which this subsequence is also statistically convergent. Some sufficient conditions of this type and almost converse necessary conditions are obtained in the setting of general metric spaces.