# Volume 62, № 10, 2010

### On the convergence of positive increasing functions to infinity

Buldygin V. V., Klesov O. I., Steinebach J. G.

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1299–1308

We study the conditions of convergence to infinity for some classes of functions extending the well-known class of regularly varying (RV) functions, such as, e.g., $O$-regularly varying (ORV) functions or positive increasing (PI) functions.

### Averaged model of vibration of a damped elastic medium

Goncharenko M. V., Khruslov E. Ya.

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1309–1329

We consider an initial boundary-value problem used to describe the nonstationary vibration of an elastic medium with large number of small cavities filled with a viscous incompressible fluid. We study the asymptotic behavior of the solution in the case where the diameters of the cavities tend to zero, their number tends to infinity, and the cavities occupy a three-dimensional region. We construct an averaged equation to describe the leading term of the asymptotics. This equation serves as a model of propagation of waves in various media, such as damped soil, rocks, and some biological tissues.

### Cauchy problem for a class of degenerate kolmogorov-type parabolic equations with nonpositive genus

Ivasyshen S. D., Litovchenko V. A.

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1330–1350

We study the properties of the fundamental solution and establish the correct solvability of the Cauchy problem for a class of degenerate Kolmogorov-type equations with $\{\overrightarrow{p},\overrightarrow{h}\}$-parabolic part with respect to the main group of variables and nonpositive vector genus in the case where the solutions are infinitely differentiable functions and their initial values are generalized functions in the form of Gevrey ultradistributions.

### On geometric properties of functors of positive-homogenous and semiadditive functionals

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1351–1359

We investigate the functor $OH$ of positive-homogenous functionals and the functor $OS$ of semiadditive functionals. We prove that $OH(X)$ is an absolute retract if and only if $X$ is an open-generated compactum, and $OS(X)$ is an absolute retract if and only if $X$ is an opengenerated compactum of weight $≤ ω_1$. We investigate the softness of mappings of multiplication of monads generated by these functors.

### Deformations of circle-valued Morse functions on surfaces

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1360–1366

Let $M$ be a smooth connected orientable compact surface and let $\mathcal{F}_{\text{cov}}(M,S^1)$ be a space of all Morse functions $f: M → S^1$ without critical points on $∂M$ such that, for any connected component $V$ of $∂M$, the restriction $f : V → S^1$ is either a constant map or a covering map. The space $\mathcal{F}_{\text{cov}}(M,S^1)$ is endowed with the $C^{∞}$-topology. We present the classification of connected components of the space $\mathcal{F}_{\text{cov}}(M,S^1)$. This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on $∂M$.

### Common fixed points and invariant approximation of $R$-subweakly commuting maps in convex metric spaces

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1367–1376

Sufficient conditions for the existence of a common fixed point of $R$-subweakly commuting mappings are established within the framework of a convex metric space. As applications, we obtain various results on the best approximation for this class of mappings generalizing the results known from the literature.

### On singularities of the Galilean spherical darboux ruled surface of a space curve in $G_3$

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1377–1387

We study the singularities of Galilean height functions intrinsically related to the Frenet frame along a curve embedded into the Galilean space. We establish the relationships between the singularities of the discriminant and the sets of bifurcations of the function and geometric invariants of curves in the Galilean space.

### On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1388–1399

The polyconvolution $∗_1(f,g,h)(x)$ of three functions $f, g$ and $h$ is constructed for the Fourier cosine $(F_c)$, Fourier sine $(F_s)$, and Kontorovich–Lebedev $(K_{iy})$ integral transforms whose factorization equality has the form $$F_c(∗_1(f,g,h))(y)=(F_sf)(y).(F_sg)(y).(K_{iy}h)\;\;∀y>0.$$ The relationships between this polyconvolution, the Fourier convolution, and the Fourier cosine convolution are established. In addition, we also establish the relationships between the product of the new polyconvolution and the products of the other known types of convolutions. As an application, we consider a class of integral equations with Toeplitz and Hankel kernels whose solutions can be obtained with the help of the new polyconvolution in the closed form. We also present the applications to the solution of systems of integral equations.

### Boundary-value problems for a nonlinear parabolic equation with Lévy Laplacian resolved with respect to the derivative

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1400–1407

We present the solutions of boundary-value and initial boundary-value problems for a nonlinear parabolic equation with Lévy Laplacian $∆_L$ resolved with respect to the derivative $$\frac{∂U(t,x)}{∂t}=f(U(t,x),Δ_LU(t,x))$$ in fundamental domains of a Hilbert space.

### Functions of shift operator and their applications to difference equations

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1408–1419

We study the representation for functions of shift operator acting upon bounded sequences of elements of a Banach space. An estimate is obtained for the bounded solution of a linear difference equation in the Banach space. For two types of differential equations in Banach spaces, we present sufficient conditions for their bounded solutions to be limits of bounded solutions of the corresponding difference equations and establish estimates for the rate of convergence.

### On groups with a small number of classes of conjugate noncomplemented subgroups

Baryshovets P. P., Bilotskii N. N.

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1420–1427

We give a description of finite nonprimary groups that contain at most two classes of conjugate noncomplemented subgroups.

### An estimate for the modulus of continuity of a quaternion singular Cauchy integral

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1428–1435

We establish an upper bound for the modulus of continuity of a quaternion singular Cauchy integral in terms of the modulus of continuity of the integrand and a metric characteristic of a curve.

### 2-Simple ore domains of stable rank 1

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1436–1440

It is known that a simple Bézout domain is a domain of elementary divisors if and only if it is 2-simple. We prove that, over a 2-simple Ore domain of stable rank 1, an arbitrary matrix that is not a divisor of zero is equivalent to a canonical diagonal matrix.