# Volume 65, № 7, 2013

### Hereditary Properties between a Ring and its Maximal Subrings

Azarang A., Karamzadeh O. A. S., Namazi A.

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 883–893

We study the existence of maximal subrings and hereditary properties between a ring and its maximal subrings. Some new techniques for establishing the existence of maximal subrings are presented. It is shown that if *R* is an integral domain and *S* is a maximal subring of *R*, then the relation dim(*R*) = 1 implies that dim(*S*) = 1 and vice versa if and only if (*S* : *R*) = 0. Thus, it is shown that if *S* is a maximal subring of a Dedekind domain *R* integrally closed in *R*; then *S* is a Dedekind domain if and only if *S* is Noetherian and (*S* : *R*) = 0. We also give some properties of maximal subrings of one-dimensional valuation domains and zero-dimensional rings. Some other hereditary properties, such as semiprimarity, semisimplicity, and regularity are also studied.

### New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 894–916

We prove new sharp weighted generalizations of Ostrowski-type and generalized trapezoid-type inequalities for Riemann–Stieltjes integrals. Several related inequalities are deduced and investigated. New Simpson-type inequalities are obtained for the \( \mathcal{R}\mathcal{S} \) -integral. Finally, as an application, we estimate the error of a general quadrature rule for the \( \mathcal{R}\mathcal{S} \) -integral via the Ostrowski–generalized-trapezoid-quadrature formula.

### Inverse Problem for a Two-Dimensional Diffusion Equation in a Domain with Free Boundary

Ivanchov N. I., Pabyrivs’ka N. V.

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 917–927

We establish conditions for the existence and uniqueness of a smooth solution to the inverse problem for a two-dimensional diffusion equation with unknown time-dependent leading coefficient in a domain with free-boundary. The equation of unknown boundary is given in the form of the product of a known function of space variables and an unknown time-dependent function.

### Extended Tauberian Theorem for the weighted mean Method of Summability

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 928–935

We prove a new Tauberian-like theorem. For a real sequence *u* = (*u* _{ n }), on the basis of the weighted mean summability of its generator sequence (*V* ^{(0)} _{ n,p }(∆*u*)) and some other conditions, this theorem establishes the property of slow oscillation of the indicated sequence.

### On the Restricted Projective Dimension of Complexes

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 936–945

We study the restricted projective dimension of complexes and give some new characterizations of the restricted projective dimension. In particular, it is shown that the restricted projective dimension can be computed in terms of the so-called restricted projective resolutions. As applications, we get some results on the behavior of the restricted projective dimension under the change of rings.

### Skitovich–Darmois Theorem for Discrete and Compact Totally Disconnected Abelian Groups

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 946–960

The classic Skitovich–Darmois theorem states that the Gaussian distribution on the real line can be characterized by the independence of two linear forms of *n* independent random variables. We generalize the Skitovich–Darmois theorem to discrete Abelian groups, compact totally disconnected Abelian groups, and some other classes of locally compact Abelian groups. Unlike the previous investigations, we consider *n* linear forms of n independent random variables.

### On Supplement Submodules

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 961–966

We investigate some properties of supplement submodules. Some relations between lying-above and weak supplement submodules are also studied. Let *V* be a supplement of a submodule *U* in *M*. Then it is possible to define a bijective map between the maximal submodules of *V* and the maximal submodules of *M* that contain *U*. Let *M* be an *R*-module, *U* ≤ *M*, let *V* be a weak supplement of *U*, and let *K* ≤ *V*. In this case, *K* is a weak supplement of *U* if and only if *V* lies above *K* in *M*. We prove that an *R*-module *M* is amply supplemented if and only if every submodule of *M* lies above a supplement in *M*. We also prove that *M* is semisimple if and only if every submodule of *M* is a supplement in *M*.

### On the Logarithmic Residues of Monogenic functions in a Three-Dimensional Harmonic Algebra with Two-Dimensional Radical

Plaksa S. A., Shpakovskii V. S.

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 967–973

For monogenic (continuous and Gâteaux-differentiable) functions taking values in a three-dimensional harmonic algebra with two-dimensional radical, we compute the logarithmic residue. It is shown that the logarithmic residue depends not only on the roots and singular points of a function but also on the points at which the function takes values in the radical of a harmonic algebra.

### Trees as Level Sets for Pseudoharmonic Functions in the Plane

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 974–995

Let *T* be a finite or infinite tree and let *V* _{0} be the set of all vertices of *T* of valency 1. We propose a sufficient condition for the image of the imbedding ψ: *T* \*V* _{0} → \( {{\mathbb{R}}^2} \) to be a level set of a pseudoharmonic function.

### On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 996–1004

We study oscillatory and asymptotic properties of the third-order nonlinear dynamic equation $$ {{\left[ {{{{\left( {\frac{1}{{{r_2}(t)}}{{{\left( {{{{\left( {\frac{1}{{{r_1}(t)}}{x^{\varDelta }}(t)} \right)}}^{{{\gamma_1}}}}} \right)}}^{\varDelta }}} \right)}}^{{{\gamma_2}}}}} \right]}^{\varDelta }}+f\left( {t,{x^{\sigma }}(t)} \right)=0,\quad t\in \mathbb{T}. $$ By using the Riccati transformation, we present new criteria for the oscillation or certain asymptotic behavior of solutions of this equation. It is supposed that the time scale T is unbounded above.

### On the Geometry of Holomorphic Developable Vector Fields on Almost Hermitian Manifolds

Kirichenko V. F., Kuzakon’ V. M.

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 1005–1008

The determination of conditions for the invariance of geometric objects under the action of a group of transformations is one of the most important problems of geometric research. We study the invariance conditions for almost Hermitian structures relative to the action of a local one-parameter group of diffeomorphisms generated by a developable vector field on a manifold. Moreover, we investigate the relationship between developable (in particular, concircular) vector fields on Riemannian manifolds and locally concircular transformations of the metric of these manifolds.

### On Typical Compact Submanifolds of the Euclidean Space

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 1009–1014

It is shown that typical compact submanifolds of *R* ^{ n } are nowhere differentiable with integer box dimensions.

### On the Best (α;β)-Approximations of Convex Functions by Constants in Integral Metrics

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 1015–1020

We prove inequalities connecting the constants of the best (α;β) -approximation in the space *L* _{ p } for various values of *p*.