# Volume 66, № 7, 2014

### Regular Elliptic Boundary-Value Problems in the Extended Sobolev Scale

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 867–883

We investigate an arbitrary regular elliptic boundary-value problem given in a bounded Euclidean *C* ^{∞}- domain. It is shown that the operator of the problem is bounded and Fredholm in appropriate pairs of Hörmander inner-product spaces. They are parametrized with the help of an arbitrary radial function RO-varying at ∞ and form the extended Sobolev scale. We establish *a priori* estimates for the solutions of the problem and study their local regularity on this scale. New sufficient conditions for the generalized partial derivatives of the solutions to be continuous are obtained.

### Optimal Recovery of *n*-Linear Functionals According to Linear Information

Babenko V. F., Gun’ko M. S., Rudenko A. A.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 884–890

We determine the optimal linear information and the optimal procedure of its application for the recovery of *n*-linear functionals on the sets of special form from a Hilbert space.

### Interpolation by Splines of Even Degree According to Subbotin and Marsden

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 891–908

We consider the problem of interpolation by splines of even degree according to Subbotin and Marsden. The study is based on the representation of spline derivatives in the bases of normalized and nonnormalized *B*-splines. The systems of equations for the coefficients of these representations are obtained. The estimations of the derivatives of the error function in the approximation of an interpolated function by the complete spline are deduced via the norms of inverse matrices of the investigated systems of equations. The relationship between the splines in a sense of Subbotin and the splines in a sense of Marsden is established.

### Modules with Unique Closure Relative to a Torsion Theory. III

Doğruöz S., Harmanci A., Smith P. F.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 922–929

We continue the study of modules over a general ring *R* whose submodules have a unique closure relative to a hereditary torsion theory on Mod-*R*. It is proved that, for a given ring *R* and a hereditary torsion theory τ on Mod-*R*, every submodule of every right *R*-module has a unique closure with respect to τ if and only if τ is generated by projective simple right *R*-modules. In particular, a ring *R* is a right Kasch ring if and only if every submodule of every right *R*-module has a unique closure with respect to the Lambek torsion theory.

### Representation Type of Nodal Algebras of Type D

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 930–938

We establish the representation types (finite, tame, or wild) of nodal algebras of type D.

### Deficiency Values for the Solutions of Differential Equations with Branching Point

Mokhonko A. A., Mokhonko A. Z.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 939–957

We study the distribution of values of the solutions of an algebraic differential equation *P*(*z, f, f′, . . . , f* ^{(s)}) = 0 with the property that its coefficients and solutions have a branching point at infinity (e.g., a logarithmic singularity). It is proved that if *a* ∈ ℂ is a deficiency value of *f* and *f* grows faster than the coefficients, then the following identity takes place: *P*(*z, a,* 0*, . . . ,* 0) ≡ 0*, z* ∈ {*z* : *r* _{0} ≤ *|z| <* ∞}*.* If *P*(*z, a,* 0*, . . . ,* 0) is not identically equal to zero in the collection of variables *z* and *a,* then only finitely many values of *a* can be deficiency values for the solutions *f* ∈ *M* _{ b } with finite order of growth.

### Decomposition of Directed Graphs and the Turán Problem

Novikov B. V., Polyakova L. Yu., Zholtkevich G. N.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 958–969

We consider vertex decompositions of (di)graphs appearing in the automata theory and establish some properties of these decompositions. These decompositions are applied to the problem of forbidden subgraphs.

### On the Problem of Linear Widths of the Classes $B_{p,θ}^r$ of Periodic Functions of Many Variables

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 970–982

We establish order estimates for the linear widths of the classes $B_{p,θ}^r$ of periodic functions of many variables in the space $L_q$ for some relationships between the parameters $p, q$, and $θ$.

### Anisotropic Differential Operators with Parameters and Applications

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 983–1002

In the paper, we study the boundary-value problems for parameter-dependent anisotropic differential-operator equations with variable coefficients. Several conditions for the uniform separability and Fredholmness in Banach-valued *L* _{ p } -spaces are given. Sharp uniform estimates for the resolvent are established. It follows from these estimates that the indicated operator is uniformly positive. Moreover, it is also the generator of an analytic semigroup. As an application, the maximal regularity properties of the parameter-dependent abstract parabolic problem and infinite systems of parabolic equations are established in mixed *L* _{p} -spaces.

### Theorems on Inclusion for Multivalued Mappings

Klishchuk B. A., Tkachuk M. V., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 1003–1005

The paper is devoted to the investigation of some properties of multivalued mappings in Euclidean spaces. Fixed-point theorems are proved for multivalued mappings whose restrictions to a certain subset in the closure of a domain satisfy a “coacute angle condition” or a “strict coacute angle condition.” Similar results for the restrictions of multivalued mappings satisfying certain metric conditions are also obtained.

### A Note on a Bound of Adan-Bante

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 1006–1008

Let *G* be a finite solvable group and let *χ* be a nonlinear irreducible (complex) character of *G.* Also let \( \eta \) (*χ*) be the number of nonprincipal irreducible constituents of \( \upchi \overline{\upchi} \) *,* where \( \overline{\upchi} \) denotes the complex conjugate of *χ.* Adan-Bante proved that there exist constants *C* and *D* such that dl (*G/* ker *χ*) *≤ C* \( \eta \) (*χ*) +*D.* In the present work, we establish a bound lower than the Adan-Bante bound for \( \eta \) (*χ*) *>* 2