# Volume 67, № 10, 2015

### Groups of Periodically Defined Linear Transformations of an Infinite-Dimensional Vector Space

Bezushchak O. E., Sushchanskii V. I.

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1299-1308

The notions of periodically defined and residual periodically defined linear transformations of an infinitedimensional vector space $V$ over the field $K$ are introduced. A group of all strictly residual periodically defined transformations and its subgroups of $u$-periodically defined transformations (where $u$ is a supernatural number) are investigated. A continual family of simple groups obtained as infinite-dimensional analogs of $PSL_n (K)$ is constructed.

### Invariant Submanifolds of Trans-Sasakian Manifolds

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1309-1320

We prove the equivalence of total geodesicity, recurrence, birecurrence, generalized birecurrence, Riccigeneralized birecurrence, parallelism, biparallelism, pseudoparallelism, bipseudoparallelism of σσ for the invariant submanifold $M$ of the trans-Sasakian manifold $\tilde{M}$.

### Branching Law for the Finite Subgroups of $SL_4ℂ$ and the Related Generalized Poincaré Polynomials

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1321-1332

Within the framework of McKay correspondence, we determine, for every finite subgroup $Γ$ of $SL_4ℂ$, how the finite-dimensional irreducible representations of $SL_4ℂ$ decompose under the action of $Γ$. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{sl}_4ℂ$ and let $ϖ_1, ϖ_2$, and $ϖ_3$ be the corresponding fundamental weights. For $(p, q, r) ∈ ℕ^3$, the restriction $π_{p,q,r} | Γ$ of the irreducible representation $π_{p,q,r}$ of the highest weight $pϖ_1 + qϖ_2 + rϖ_3$ of $SL_4ℂ$ decomposes as $π_{p, q, r} | Γ = ⊕_{i = 0}^l m_i (p, q, r)γ_i$, where $\{γ_0,…, γ_l\}$ is the set of equivalence classes of irreducible finite-dimensional complex representations of $Γ$. We determine the multiplicities $m_i (p, q, r)$ and prove that the series $${P}_{\varGamma }{\left(t,u,w\right)}_i={\displaystyle \sum_{p=0}^{\infty }{\displaystyle \sum_{q=0}^{\infty }{\displaystyle \sum_{r=0}^{\infty }{m}_i\left(p,q,r\right){t}^p{u}^q{w}^r}}}$$ are rational functions. This generalizes the results of Kostant for $SL_2ℂ$ and the results of our preceding works for $SL_3ℂ$.

### On the Relationship Between the Rate of Polynomial Approximation of an Entire Function and its Order and Type

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1333-1347

We study the relationship between the order and type of an entire function and the rate of its best polynomial approximation for various Banach spaces of functions analytic in the unit disk. The relations specifying the order and type of the entire function via the sequence of its best approximations are deduced. The results are obtained by generalizing the results obtained earlier by Reddy, Ibragimov, Shyhaliev, Vakarchyk, and Mamadov.

### Boundary-Value Problem with Impulsive Conditions and Degeneration for Parabolic Equations

Isaryuk I. M., Pukal'skii I. D.

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1348-1357

We consider the second boundary-value problem for a parabolic equation with power singularities in the coefficients of space variables and impulsive conditions in the time variable. By using the maximum principle and *a priori* estimates, we establish the existence and uniqueness of the solution of posed problem in Hölder spaces with power weights.

### Construction of Lyapunov Functions in the Theory of Regular Linear Extensions of Dynamical Systems on a Torus

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1358-1365

Lyapunov functions are considered in the form of linear combinations of quadratic forms. We study the conditions under which the linear extensions of dynamic systems on a torus are regular.

### On the Best Linear Methods of Approximation and the Exact Values of Widths for Some Classes of Analytic Functions in the Weighted Bergman Space

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1366-1379

We find the exact values of the series of *n*-widths for the classes of functions from the Hardy and Bergman spaces whose averaged moduli of continuity are majorized by a given function obeying certain restrictions.

### On Some New Inequalities of Hermite–Hadamard Type for Functions Whose Derivatives are $s$-Convex in the Second Sense in the Absolute Value

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1380-1397

Several new inequalities of the Hermite–Hadamard type are established for functions whose derivatives are *s*-convex in the second sense in the absolute value. Some applications to special means of positive real numbers are also presented.

### Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. II

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1398-1408

We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema and if a point is not a local extremum, then there exist its neighborhood and a number $n ∈ ℕ$ such that the function restricted to this neighborhood is topologically conjugate to Re $z^n$ in a certain neighborhood of zero. Given $f : M^2 → ℝ$, let $Γ_{K−R} (f)$ be a quotient space of $M^2$ with respect to its partition formed by the components of level sets of the function $f$. It is known that the space $Γ_{K−R} (f)$ is a topological graph if $M^2$ is compact. In the first part of the paper, we introduced the notion of graph with stalks that generalizes the notion of topological graph. For noncompact $M^2$ , we present three conditions sufficient for $Γ_{K−R} (f)$ to be a graph with stalks. In the second part, we prove that these conditions are also necessary in the case $M^2 = ℝ^2$. In the general case, one of our conditions is not necessary. We provide an appropriate example.

### Weakly Periodic Gibbs Measures in the HC-Model for a Normal Divisor of Index Four

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1409-1422

We study an HC-model on a Cayley tree. Under certain restrictions imposed on the parameters of the HC-model, we prove the existence of weakly periodic (nonperiodic) Gibbs measures for a normal divisor of index four.

### Approximation of Functions from the Isotropic Nikol’skii–Besov Classes in the Uniform and Integral Metrics

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1423-1433

We obtain the exact-order estimations for the approximation of the isotropic Nikol’skii–Besov classes of functions of several variables by the de la Vallée-Poussin-type sums in metrics of the spaces $L_{∞}(ℝ^d)$ and $L_1(ℝ^d)$.

### Some Remarks on Spectral Synthesis Sets

Joseph J., Muraleedharan T. K.

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1434-1438

Relations between the difference spectra of unions and intersections are studied and their implications on some problems in spectral synthesis are observed.

### Jacobian and the Darboux Property

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1439-1440

We present a simple proof of the Darboux property for the Jacobian of a differentiable mapping.