# Volume 63, № 5, 2011

### Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 579-595

In this article a new class of Banach manifolds and a new class of mappings between them are presented and also the theory of degree of such mappings is given.

### On generalized derivations satisfying certain identities

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 596-602

Let $R$ be a prime ring with char $R \neq 2$ and $d$ be a generalized derivation on $R$. The goal of this study is to investigate the generalized derivation $d$ satisfying any one of the following identities: $$(i) \quad d[(x, y)] = [d(x), d(y)] \quad \text{for all} x, y \in R;$$ $$(ii) \quad d[(x, y)] = [d(y), d(x)] \quad \text{for all} x, y \in R;$$ $$(iii)\quad d([x, y]) = [d(x), d(y)] \text{either} d([x, y]) = [d(y), d(x)] \quad \text{for all} x, y \in R$$.

### Bernstein-type inequalities for splines defined on the real axis

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 603-611

We obtain the exact inequalities of the Bernstein type for splines $s \in S_{m, h} \bigcap L_2 (\mathbb{R})$ as well as the exact inequalities that, for splines $s \in S_{m, h}, \quad h > 0$, estimate $L_p$-norms of the Fourier transforms of their $k$-th derivative by $L_p$-norms of the Fourier transforms of splines themselves.

### Doubly nonlinear parabolic equations with variable exponents of nonlinearity

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 612-628

We investigate a mixed problem for a class of parabolic-type equations with double nonlinearity and minor terms that do not degenerate and whose indexes of nonlinearity are functions of spatial variables. These problems are considered in the generalized Lebesgue and Sobolev spaces. We obtain conditions for the existence of the generalized solution of this problem by using the Galerkin method.

### Maxwell distributions in a model of rough spheres

Gordevskii V. D., Gukalov A. A.

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 629-639

The Boltzmann equation is considered for the model of rough spherical molecules which possess both translati-onal and rotational energies. The general form of local Maxwell distributions for this model is obtained. The main possible types of corresponding flows of a gas are selected and analysed.

### Stefan problem for a weakly degenerate parabolic equation

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 640-653

In a domain with free boundary, we consider the inverse problem for the determination of time dependent coefficient of the first derivative of unknown fonction in a parabolic equation with weak power degeneration. As overdetermination conditions, the Stefan condition and the integral condition are given. Conditions for the existence and uniqueness of the classical solution of considered problem are established.

### Approximate stabilization for a nonlinear parabolic boundary-value problem

Kapustyan O. A., Kapustyan O. V., Sukretna A. V.

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 654-661

For a problem of optimal stabilization of solutions of a nonlinear parabolic boundary-value problem with small parameter of a nonlinear summand, we justify the form of approximate regulator on the basis of the formula of optimal synthesis of the corresponding linear quadratic problem.

### On strongly $\oplus$-supplemented modules

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 662-667

In this work, strongly $\oplus$-supplemented and strongly cofinitely $\oplus$-supplemented modules are defined and some properties of strongly $\oplus$-supplemented and strongly cofinitely $\oplus$-supplemented modules are investigated. Let $R$ be a ring. Then every $R$-module is strongly $\oplus$-supplemented if and only if R is perfect. Finite direct sum of $\oplus$-supplemented modules is $\oplus$-supplemented. But this is not true for strongly $\oplus$-supplemented modules. Any direct sum of cofinitely $\oplus$-supplemented modules is cofinitely $\oplus$-supplemented but this is not true for strongly cofinitely $\oplus$-supplemented modules. We also prove that a supplemented module is strongly $\oplus$-supplemented if and only if every supplement submodule lies above a direct summand.

### Asymptotic integration of singularly perturbed systems of hyperbolic-type partial differential equations with degeneration

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 668-685

An asymptotic solution of the first boundary-value problem for a linear singularly perturbed system of hyperbolic-type partial differential equations with degeneration is constructed.

### On the regular growth of Dirichlet series absolutely convergent in a half-plane

Sheremeta M. M., Stets' Yu. V.

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 686-698

For the Dirichlet series $F(s) = \sum^{\infty}_{n=1}a_n \exp \{s \lambda_n\}$ with the abscissa of absolute convergence $\sigma a = 0$, conditions on $(λ_n)$ and $(a_n)$ (λn) are established under which $\ln M(\sigma, F) = T_R(1 + o(1)) \exp\{\varrho R/|\sigma|\}$ as $\sigma \uparrow 0$, where$M(σ, F) = \sup\{|F(\sigma + it)| : t \in R\}$ and $T_R$ and $\varrho_R$ are positive constants.

### Analog of the mean-value theorem for polynomials of special form

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 686-698

A mean value theorem for polynomials of a special form is proved. The case of a sum over vertices of a regular polygon is studied and a criterion for the equation of a special form to be satisfied is obtained.

### Finite-dimensional subalgebras in polynomial Lie algebras of rank one

Arzhantsev I. V., Makedonskii E. A., Petravchuk A. P.

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 708-712

Let $W_n(\mathbb{K})$ be the Lie algebra of derivations of the polynomial algebra $\mathbb{K}[X] := \mathbb{K}[x_1,... ,x_n]$ over an algebraically closed field $K$ of characteristic zero. A subalgebra $L \subseteq W_n(\mathbb{K})$ is called polynomial if it is a submodule of the $\mathbb{K}[X]$-module $W_n(\mathbb{K})$. We prove that the centralizer of every nonzero element in $L$ is abelian provided that $L$ is of rank one. This fact allows to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.

### The bidual of *r*-algebras

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 713-717

We prove that the order continuous bidual of an Archimedean *r*-algebra is a Dedekind complete *r*-algebra with respect to the Arens multiplications.

### Nevanlinna characteristics and defective values of the Weierstrass zeta function

Kharkevych Yu. I., Korenkov M. E., Zaionts Yu.

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 718-720

We establish the Nevanlinna characteristics of the Weierstrass zeta function and show that none of the values $a \in \overline{C}$ is exceptional in the Nevanlinna sense for this function.