# Volume 64, № 6, 2012

### A new application of generalized quasi-power increasing sequences

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 731-738

We prove a theorem dealing with $|\overline{N}, p_n, \theta_n|_k$-summability using a new general class of power increasing sequences instead of a quasi-$\eta$-power increasing sequence. This theorem also includes some new and known results.

### Realization of a closed 1-form on closed oriented surfaces

Budnyts'ka N. V., Rybalkina T. V.

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 739-751

We study closed 1-forms with isolated zeros on closed orientable surfaces. Conditions under which given invariants generate a closed 1-form are found.

### A comonotonic theorem for backward stochastic differential equations in $L^p$ and its applications

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 752-765

We study backward stochastic differential equations (BSDEs) under weak assumptions on the data. We obtain a comonotonic theorem for BSDEs in $L^p,\quad 1, 1 < p ≤ 2$. As applications of this theorem, we study the relation between Choquet expectations and minimax expectations and the relation between Choquet expectations and generalized Peng’s $g$-expectations. These results generalize the known results of Chen et al.

### On properties of *n*-totally projective abelian *p *-groups

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 766-771

We prove some properties of $n$-totally projective abelian $p$-groups. Under some additional conditions for the group structure, we obtain an equivalence between the notions of $n$-total projectivity and strong $n$-total projectivity. We also show that $n$-totally projective $A$-groups are isomorphic if they have isometric $p^n$-socles.

### Integral manifolds for semilinear evolution equations and admissibility of function spaces

Hà Phi, Nguyễn Thiếu Huy, Vụ Thì Ngọc Hà

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 772-796

We prove the existence of integral (stable, unstable, center) manifolds for the solutions to the semilinear integral equation $u(t) = U(t,s)u(s) + \int^t_s U(t,\xi)f (\xi,u(\xi))d\xi$ in the case where the evolution family $(U(t, s))_{t leq s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term $f$ satisfies the $\varphi $-Lipschitz conditions, i.e., $||f (t, x) — f (t, y) \leq \varphi p(t)||x — y||$, where $\varphi (t)$ belongs to some classes of admissible function spaces. Our main method invokes the Lyapunov-Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces.

### On a *p* -Laplacian system with critical Hardy - Sobolev exponents and critical Sobolev exponents

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 797-810

We consider a quasilinear elliptic system involving the critical Hardy - Sobolev exponent and Sobolev exponent. Using variational methods and analytic techniques, we establish the existence of positive solutions of the system.

### On local near-rings with Miller?Moreno multiplicative group

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 811-818

A near-ring $R$ with identity is local if the set $L$ of all its noninvertible elements is a subgroup of the additive group $R^{+}$. We study the local near-rings of order $2^n$ whose multiplicative group $R^{*}$ is a Miller-Moreno group, i.e., a non-abelian group all proper subgroups of which are abelian. In particular, it is proved that if $L$ is a subgroup of index $2^m$ in $R^{+}$, then either $m$ is a prime for which $2^m - 1$ is a Mersenna prime or $m = 1$. In the first case $n = 2m$, the subgroup $L$ is elementary abelian, the exponent of $R^{+}$ does not exceed 4, and $R^{*}$ is of order $2^m(2^m - 1)$. In the second case either $n < 7$ or the subgroup $L$ is abelian and $R^{*}$ is a nonmetacyclic group of order $2^{n−1}$ and of exponent at most $2^{n−4}$.

### Scalar operators equal to the product of unitary roots of the identity operator

Samoilenko Yu. S., Yakymenko D. Yu.

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 819-825

We study the set of all $\gamma \in \mathbb{C}$ for which there exist unitary operators $U_i$ such that $U_1U_2 ... U_n = \gamma I$ and $U_i^{m_i} = I$, where $m_i \in \mathbb{N}$ are given.

### On the chain equivalence of projective chain complexes

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 826-835

We obtain a necessary and sufficient condition for $n$-dimensional chain complexes composed of finitely generated projective modules to be stabilized by free modules to the chain equivalence.

### Optimality conditions in problems of control over systems of impulsive differential equations with nonlocal boundary conditions

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 836-847

We consider the problem of optimal control in which the state of the controlled system is described by impulsive differential equations under nonlocal boundary conditions, which is a natural generalization of the Cauchy problem. Using the principle of contracting mappings, we prove the existence and uniqueness of a solution of a nonlocal boundary-value problem with pulse action with fixed admissible controls. Under certain conditions for the initial data of the problem, we calculate the gradient of a functional and obtain necessary optimality conditions.

### On the Gauss sums and generalized Bernoulli numbers

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 848-854

Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas.

### On the boundary behavior of open discrete mappings with unbounded characteristic

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 855-859

We study the problem of extension of mappings $f : D → R^n,\; n ≥ 2$, to the boundary of a domain $D$. Under certain conditions imposed on a measurable function $Q(x),\; Q: D → [0, ∞]$, and the boundaries of the domains $D$ and $D' = f(D)$, we show that an open discrete mapping $f : D → R^n,\; n ≥ 2$, with quasiconformality characteristic $Q(x)$ can be extended to the boundary $\partial D$ by continuity. The obtained statements extend the corresponding Srebro’s result to mappings with bounded distortion.

### On generalized solutions of differential equations with several operator coefficients

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 860-864

The theorem on the smoothness of generalized solutions of differential equations with some operational coefficients is proved.