# Volume 69, № 7, 2017

### Co-coatomically supplemented modules

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 867-876

It is shown that if a submodule $N$ of $M$ is co-coatomically supplemented and $M/N$ has no maximal submodule, then $M$ is a co-coatomically supplemented module. If a module $M$ is co-coatomically supplemented, then every finitely $M$-generated module is a co-coatomically supplemented module. Every left $R$-module is co-coatomically supplemented if and only if the ring $R$ is left perfect. Over a discrete valuation ring, a module $M$ is co-coatomically supplemented if and only if the basic submodule of $M$ is coatomic. Over a nonlocal Dedekind domain, if the torsion part $T(M)$ of a reduced module $M$ has a weak supplement in $M$, then $M$ is co-coatomically supplemented if and only if $M/T (M)$ is divisible and $TP (M)$ is bounded for each maximal ideal $P$. Over a nonlocal Dedekind domain, if a reduced module $M$ is co-coatomically amply supplemented, then $M/T (M)$ is divisible and $TP (M)$ is bounded for each maximal ideal $P$. Conversely, if $M/T (M)$ is divisible and $TP (M)$ is bounded for each maximal ideal $P$, then $M$ is a co-coatomically supplemented module.

### Existence and uniqueness theorem to a model of bimolecular surface reactions

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 877-888

We prove the existence and uniqueness of classical solutions to a coupled system of parabolic and ordinary differential equations in which the latter are determined on the boundary. This system describes the model of bimolecular surface reaction between carbon monoxide and nitrous oxide occurring on supported rhodium in the case of slow desorption of the products.

### Indecomposable and isomorphic objects in the category of monomial matrices over a local ring

Bondarenko V. M., Bortos M. Yu.

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 889-904

We study the indecomposability and isomorphism of objects from the category of monomial matrices $\mathrm{M}\mathrm{m}\mathrm{a}\mathrm{t}(K)$ over a commutative local principal ideal ring $K$ (whose objects are square monomial matrices and the morphisms from $X$ to $Y$ are the matrices $C$ such that $XC = CY$). We also study the subcategory $\mathrm{M}\mathrm{m}\mathrm{a}\mathrm{t}_0(K)$ of the category $\mathrm{M}\mathrm{m}\mathrm{a}\mathrm{t}(K)$ with the same objects and only those morphisms that are monomial matrices.

### Problem of optimal strategy in the models of conflict redistribution of the resource space

Koshmanenko V. D., Verigina I. V.

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 905-911

The theory of conflict dynamical systems is applied to finding of the optimal strategy in the problem of redistribution of the resource space between two opponents. In the case of infinite fractal division of the space, we deduce an explicit formula for finding the Lebesgue measure of the occupied territory in terms of probability distributions. In particular, this formula gives the optimal strategy for the occupation of the whole territory. The necessary and sufficient condition for the parity distribution of the territory are presented.

### Approximative properties of biharmonic Poisson integrals on Hölder classes

Hembars'ka S. B., Zhyhallo K. M.

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 925-932

We establish asymptotic expansions for the values of approximation of functions from the H¨older class by biharmonic Poisson integrals in the uniform and integral metrics.

### Viscous solutions for the Hamilton – Jacobi – Bellman equation on time scales

Danilov V. Ya., Lavrova O. E., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 933-950

We introduce the concept of viscous solution for the Bellman equation on time scales and establish сonditions for the existence and uniqueness of this solution.

### Hyperbolic cross and complexity of various classes of linear ill-posed problems

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 951-963

The present paper is a survey of the latest results obtained in the fields of information and algorithmiс complexity of severely ill-posed problems.

### Characterizations of groups with almost layer-finite periodic part

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 964-973

Construction of the set of finite subgroups of the form $L_g = \langle a, a^g\rangle$ in Shunkov’s groups is studying. As a corollary of this result follows two characterizations of groups with an almost layer-finite periodic part.

### A note on property (gaR) and perturbations

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 974-983

We introduce a new property $(gaR)$ extending the property $(R)$ considered by Aiena. We study the property $(gaR)$ in connection with Weyl type theorems and establish sufficient and necessary conditions under which the property $(gaR)$ holds. In addition, we also study the stability of the property $(gaR)$ under perturbations by finite-dimensional operators, by nilpotent operators, by quasinilpotent operators, and by algebraic operators commuting with $T$. The classes of operators are considered as illustrating examples.

### Exponential estimates for the maximum scheme

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 984-991

Exponential estimates are obtained in the law of iterated logarithm for the extreme values of sequence of independent random variables.

### Well-posedness of mixed problems for multidimensional hyperbolic equations with wave operator

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 992-999

We establish the unique solvability and obtain the explicit expression for the classical solution of the mixed problem for multidimensional hyperbolic equations with wave operator.

### Homeotopy groups for nonsingular foliations of the plane

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 1000-1008

We consider a special class of nonsingular oriented foliations $F$ on noncompact surfaces $\Sigma$ whose spaces of leaves have the structure similar to the structure of rooted trees of finite diameter. Let $H^+(F)$ be the group of all homeomorphisms of $\Sigma$ mapping the leaves onto leaves and preserving their orientations. Also let $K$ be the group of homeomorphisms of the quotient space $\Sigma /F$ induced by $H^+(F)$. By $H^+_0(F)$ and $K_0$ we denote the corresponding subgroups formed by the homeomorphisms isotopic to identity mappings. Our main result establishes the isomorphism between the homeotopy groups $\pi_0 H^+(F) = H^+(F)/H^+ _0 (F)$ and $\pi_ 0K = K/K_0$.

### Decay estimates for a kind of linear wave equations

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 1001-

We consider one kind of dissipative wave equations with exponential speed of propagation. An arbitrary power decay rate for the $L^2$-norm and energy is obtained by using the multiplier method.