# Volume 66, № 2, 2014

### Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 147–158

We study the inverse problem of reconstruction of the right-hand side of a parabolic equation with nonlocal conditions. The well-posedness of this problem in Hölder spaces is established.

### Nonlocal Problem Multipoint in Time for the Evolutionary Equations with Pseudo-Bessel Operators with Variable Symbols

Horodets’kyi V. V., Martynyuk O. V.

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 159–175

We study the properties of the fundamental solution of a nonlocal problem multipoint in time for the evolutionary equations with pseudo-Bessel operators constructed on variable symbols. The solvability of this problem is proved in the class of bounded continuous functions even on ℝ. The integral representation of solutions is established.

### Topologically Mixing Maps and the Pseudoarc

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 176–186

It is known that the pseudoarc can be constructed as the inverse limit of the copies of [0*,* 1] with one bonding map *f* which is topologically exact. On the other hand, the shift homeomorphism *σ* _{ f } is topologically mixing in this case. Thus, it is natural to ask whether *f* can be only mixing or must be exact. It has been recently observed that, in the case of some hereditarily indecomposable continua (e.g., pseudocircles) the property of mixing of a bonding map implies its exactness. The main aim of the present article is to show that the indicated kind of forcing of recurrence is not the case for the bonding map defining the pseudoarc.\

### Weakly *SS*-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 187–194

A subgroup *H* is said to be an *s*-permutable subgroup of a finite group *G* provided that the equality *HP* =*PH* holds for every Sylow subgroup *P* of *G.* Moreover, *H* is called *SS*-quasinormal in *G* if there exists a supplement *B* of *H* to *G* such that *H* permutes with every Sylow subgroup of *B.* We show that *H* is weakly *SS*-quasinormal in *G* if there exists a normal subgroup *T* of *G* such that *HT* is *s*-permutable and *H \ T* is *SS*-quasinormal in *G.* We study the influence of some weakly *SS*-quasinormal minimal subgroups on the nilpotency of a finite group *G.* Numerous results known from the literature are unified and generalized.

### Semiretractions of Trioids

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 195–207

We introduce and study the notion of semiretraction of trioid. Examples of left, right, and symmetric semiretractions of trioids are given. We also present new theoretical trioid constructions for which some symmetric semiretractions are characterized.

### Nonlocal Parabolic Problem with Degeneration

Isaryuk I. M., Pukalskyi I. D.

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 208–215

We study the problem for a second-order linear parabolic equation with nonlocal integral condition in the time variable and power singularities in the coefficients of any order with respect to the time and space variables. By using the maximum principle and *a priori* estimates, we establish the existence and uniqueness of the solution of this problem in Hölder spaces with power weights.

### Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 216–225

We solve the following extremal problems: (i) \( {\left\Vert {s}^{(k)}\right\Vert}_{L_q\left[\alpha, \beta \right]}\to \sup \) and (ii) \( {\left\Vert {s}^{(k)}\right\Vert}_{W_q}\to \sup \) over all shifts of splines of order *r* with minimal defect and nodes at the points *lh, l ∈* **Z** *,* such that *L*(*s*)_{ p } *≤M* in the cases: (a) *k* =0*, q ≥ p >*0*,* (b) *k* =1*, . . . , r −*1*, q ≥* 1*,* where [*α, β*] is an arbitrary interval in the real line, $$ L{(x)}_p:= \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:a,b\in \mathbf{R},\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\} $$

and \( {\left\Vert \cdot \right\Vert}_{W_q} \) is the Weyl functional, i.e., $$ {\left\Vert x\right\Vert}_{W_q}:=\underset{\varDelta \to \infty }{ \lim}\underset{a\in \mathbf{R}}{ \sup }{\left(\frac{1}{\varDelta }{\displaystyle \underset{a}{\overset{a+\varDelta }{\int }}{\left|x(t)\right|}^qdt}\right)}^{1/q}. $$

As a special case, we get some generalizations of the Ligun inequality for splines.

### Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 226–243

We construct a class of homogeneous vertex representations of $C_l^{(1)},\; l ≥ 2$, and deduce a series of product-sum identities. These identities have fine interpretation in the number theory.

### Approximation of Functions of Many Variables from the Classes $B_{p,θ}^{Ω} (ℝ^d)$ By Entire Functions of Exponential Type

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 244–258

We obtain the decomposition representation of the norm of functions of many variables from the spaces $B_{p,θ}^{Ω} (ℝ^d)$ and establish the exact order estimates for the approximations of functions from the unit balls of these spaces by entire functions of exponential type in the space $L_q (ℝ^d)$.

### Generalized Lebesgue Constants and the Convergence of Fourier–Jacobi Series in the Spaces $L_{1,A,B}$

Motornaya O. V., Motornyi V. P.

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 259–268

Generalized Lebesgue constants for the Fourier–Jacobi sums and the convergence of Fourier–Jacobi series in the $L_{1,A,B}$ spaces are investigated.

### θ-Centralizers on Semiprime Banach *-algebras

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 269–278

We generalize the celebrated theorem of Johnson and prove that every left θ -centralizer on a semisimple Banach algebra with left approximate identity is continuous. We also investigate the generalized Hyers–Ulam–Rassias stability and the superstability of θ -centralizers on semiprime Banach *-algebras.

### On the Third Boundary-Value Problem for an Improperly Elliptic Equation in a Disk

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 279–283

We study the problem of solvability of the inhomogeneous third boundary-value problem in a bounded domain for a scalar improperly elliptic differential equation with complex coefficients and homogeneous symbol. It is shown that this problem has a unique solution in the Sobolev space over the circle for special classes of boundary data from the spaces of functions with exponentially decreasing Fourier coefficients.

### A Sharp Bézout Domain is an Elementary Divisor Ring

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 284–288

We prove that a sharp Bézout domain is an elementary divisor ring.