# Volume 67, № 12, 2015

### Regularized integrals of motion for the Korteweg – de-Vries equation in the class of nondecreasing functions

Andreev K. N., Khruslov E. Ya.

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1587-1601

We study the Cauchy problem for the Korteweg–de-Vries equation in the class of functions approaching a finite- zone periodic solution of the KdV equation as $x → −∞$ and 0 as $x → +∞$. We prove the existence of infinitely many “regularized” integrals of motion for the solutions $u(x, t)$ of the Cauchy problem, with explicit dependence on time.

### Approximation of analytic functions by the partial sums of Taylor series

Gaevs'kyi M. V., Zaderei P. V.

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1602-1619

We establish the estimates for the deviations of Taylor’s sums on the classes of analytic functions $H_\psi^\infty$, expressed via the best approximations of $\psi$-derivative functions by using the asymptotic equalities for the exact upper bounds of deviations of Taylor’s sums from functions of the same class.

### On the sum of narrow and finite-rank orthogonally additive operators

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1620-1625

It is well known that the sum of two linear continuous narrow operators in the spaces Lp with 1 < p < ∞ need not be narrow. However, the sum of narrow and compact linear continuous operators is narrow. In a recent paper, M. Pliev and M. Popov started the investigation of nonlinear narrow operators and, in particular, of orthogonally additive operators. As our main result, we prove that the sum of a narrow orthogonally additive operator and a finite-rank laterally-to-norm continuous orthogonally additive operator acting from an atomless Dedekind complete vector lattice into a Banach space is narrow.

### Systems of φ-Laplacian three-point boundary-value problems on the positive half-line

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1626-1648

We study the existence of positive solutions to boundary-value problems for two systems of two second-order nonlinear three-point φ-Laplacian equations defined on the positive half line. The nonlinearities may change sign, exhibit time singularities at the origin, and depend both on the solutions and on their first derivatives. Using the fixed-point theory, we prove some results on the existence of nontrivial positive solutions on appropriate cones in some weighted Banach spaces.

### Determination of jumps in terms of linear operators

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1649-1657

A theorem of Luk´acs [4] states that the partial sums of conjugate Fourier series of a periodic Lebesgue integrable function $f$ diverge with a logarithmic rate at the points of discontinuity of $f$ of the first kind. M´oricz [5] proved a similar theorem for the rectangular partial sums of double variable functions.

We consider analogs of the M´oricz theorem for generalized Ces´aro means and for positive linear means.

We consider a similar theorem in terms of linear operators satisfying certain conditions.

### Generalized convex sets and the problem of shadow

Stefanchuk M. V., Vyhovs'ka I. Yu., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1658-1666

The problem of shadow is solved. It is equivalent to the problem of finding conditions for a point to belong to a generalized convex hull of a family of compact sets.

### Singularity and fine fractal properties of one class of generalized infinite Bernoulli convolutions with essential overlaps. II

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1667-1678

We discuss the Lebesgue structure and fine fractal properties of infinite Bernoulli convolutions, i.e., the distributions of random variables $\xi=\sum_{k=1}^{\infty}\xi_ka_k$, where $\sum_{k=1}^{\infty}a_k$ is a convergent positive series and $\xi_k$ are independent (generally
speaking, nonidentically distributed) Bernoulli random variables. Our main aim is to investigate the class of Bernoulli convolutions with essential overlaps generated by a series $\sum_{k=1}^{\infty}a_k$, such that, for any $k\in \mathbb{N}$, there exists $s_k\in \mathbb{N}\cup\{0\}$ for which $a_k = a_{k+1} = . . . = a_{k+s_k} ≥ r_{k+s_k}$ and, in addition, $s_k > 0$ for infinitely many indices $k$. In this case, almost
all (both in a sense of Lebesgue measure and in a sense of fractal dimension) points from the spectrum have continuum many representations of the form $\xi=\sum_{k=1}^{\infty}\varepsilon_ka_k$, with $\varepsilon_k\in\{0, 1\}$. It is proved that μξ has either a pure discrete distribution
or a pure singulary continuous distribution.

We also establish sufficient conditions for the faithfulness of the family of cylindrical intervals on the spectrum $\mu_\xi$
generated by the distributions of the random variables $\xi$. In the case of singularity, we also deduce the explicit formula
for the Hausdorff dimension of the corresponding probability measure [i.e., the Hausdorff–Besicovitch dimension of the
minimal supports of the measure $\mu_\xi$ (in a sense of dimension)].

### On the norm of decomposable subgroups in nonperiodic groups

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1679-1689

We study the relations between the properties of nonperiodic groups and the norms of their decomposable subgroups. In particular, we analyze the influence of restrictions imposed on the norm of decomposable subgroups on the properties of the group provided that this norm is non-Dedekind. We also describe the structure of nonperiodic locally nilpotent groups for which the indicated norm is non-Dedekind . Furthermore, some relations between the norm of noncyclic Abelian subgroups and the norm of decomposable subgroups are established.

### Exact constants in inequalities for the Taylor coefficients of bounded holomorphic functions in a polydisc

Meremelya I. Yu., Savchuk V. V.

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1690-1697

We determine the exact constants $L_{m,n}(X)$ in the inequalities of the form $|\hat f(m)|\leq L_{m,n}(X)(1 − |\hat f(n)|)$ for the pairs of Taylor coefficients $\hat f(m)$ and $\hat f(n)$ on some classes $X$ of bounded holomorphic functions in a polydisc.

### Ultrafilters on balleans

Protasov I. V., Slobodianiuk S. V.

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1698-1706

A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter satellites of a ballean (namely, corona, companion, and corona companion), evaluate the basic cardinal invariants of the corona and characterize the subsets of balleans in terms of companions.

### Almost periodic and Poisson stable solutions of difference equations in metric spaces

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1707-1714

We introduce a new class of almost periodic operators and establish the conditions of existence of almost periodic and Poisson stable solutions of difference equations in metric spaces that can be not almost periodic in Bochner’s sense.

### Finite groups with X-quasipermutable Sylow subgroups

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1715-1722

Let H ≤ E and X be subgroups of a finite group G. Then we say that H is X-quasipermutable (XS-quasipermutable,
respectively) in E provided that G has a subgroup B such that E = N_{E}(H)B and H X-permutes with B and with all subgroups (with all Sylow subgroups, respectively) V of B such that (|H|, |V |) = 1. We analyze the influence of X-quasipermutable and XS-quasipermutable subgroups on the structure of G. In particular, it is proved that if every Sylow subgroup P of G is F(G)-quasipermutable in its normal closure PG in G, then G is supersoluble.

### Index of volume 67 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1723-1729