2019
Том 71
№ 11

# Volume 61, № 6, 2009

Article (Ukrainian)

### On semigroups of order-preserving transformations of countable linearly ordered sets

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 723-732

We consider semigroups of endomorphisms of linearly ordered sets ℕ and ℤ and their subsemigroups of cofinite endomorphisms. We study the Green relations, groups of automorphisms, conjugacy, centralizers of elements, growth, and free subsemigroups in these subgroups.

Article (English)

### Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 733-764

We review some recent results related to stochastic integrals of the Hitsuda–Skorokhod type acting on the extended Fock space and its riggings.

Article (Russian)

### On some extremal problems of different metrics for differentiable functions on the axis

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 765-776

For an arbitrary fixed segment $[α, β] ⊂ R$ and given $r ∈ N, A_r, A_0$, and $p > 0$, we solve the extremal problem $$∫^{β}_{α} \left|x^{(k)}(t)\right|^qdt → \sup,\; q⩾p,\; k=0,\; q⩾1,\; 1 ⩽ k ⩽ r−1,$$ on the set of all functions $x ∈ L^r_{∞}$ such that $∥x (r)∥_{∞} ≤ A_r$ and $L(x)_p ≤ A_0$, where $$L(x)p := \left\{\left( ∫^b_a |x(t)|^p dt\right)^{1/ p} : a,b ∈ R,\; |x(t)| > 0,\; t ∈ (a,b)\right\}$$ In the case where $p = ∞$ and $k ≥ 1$, this problem was solved earlier by Bojanov and Naidenov.

Article (Russian)

### Construction of a fundamental system of solutions of a linear finite-order difference equation

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 777-794

We present an efficient algorithm for the construction of a fundamental system of solutions of a linear finite-order difference equation. We obtain expressions in which all elements of this system are expressed via one of its elements and find a particular solution of an inhomogeneous equation.

Article (Ukrainian)

### Properties of solutions of a mixed problem for a nonlinear ultraparabolic equation

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 795-809

Mixed problems for a nonlinear ultraparabolic equation are considered in domains bounded and unbounded with respect to the space variables. Conditions for the existence and uniqueness of solutions of these problems are established and some estimates for these solutions are obtained.

Article (English)

### Tame comodule type, roiter bocses, and a geometry context for coalgebras

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 810-833

We study the class of coalgebras $C$ of $fc$-tame comodule type introduced by the author. With any basic computable $K$-coalgebra $C$ and a bipartite vector $v = (v′|v″) ∈ K_0(C) × K_0(C)$, we associate a bimodule matrix problem $\textbf{Mat}^v_C(ℍ)$, an additive Roiter bocs $\textbf{B}^C_v$, an affine algebraic $K$-variety $\textbf{Comod}^C_v$, and an algebraic group action $\textbf{G}^C_v × \textbf{Comod}^C_v → \textbf{Comod}^C_v$. We study the $fc$-tame comodule type and the fc-wild comodule type of $C$ by means of $\textbf{Mat}^v_C(ℍ)$, the category $\textbf{rep}_K (\textbf{B}^C_v)$ of $K$-linear representations of $\textbf{B}^C_v$, and geometry of $\textbf{G}^C_v$ -orbits of $\textbf{Comod}_v$. For computable coalgebras $C$ over an algebraically closed field $K$, we give an alternative proof of the $fc$-tame-wild dichotomy theorem. A characterization of $fc$-tameness of $C$ is given in terms of geometry of $\textbf{G}^C_v$-orbits of $\textbf{Comod}^C_v$. In particular, we show that $C$ is $fc$-tame of discrete comodule type if and only if the number of $\textbf{G}^C_v$-orbits in $\textbf{Comod}^C_v$ is finite for every $v = (v′|v″) ∈ K_0(C) × K_0(C)$.

Article (English)

### Asymptotic behavior and periodic nature of two difference equations

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 834-838

We discuss the global asymptotic stability of the solutions of the difference equations $$x_{n+1} = \frac{x_{n−2}}{±1 + x_nx_{n−1}x_{n−2}}, \quad n = 0,1,…,$$ where the initial conditions $x_{−2}, x_{−1}, x_0$ are real numbers.

Article (Russian)

### Asymptotic representations of one class of solutions of a second-order difference equation with power nonlinearity

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 839-854

We obtain asymptotic representations for one class of solutions of a second-order difference equation with power nonlinearity.

Article (Russian)

### On exact values of quasiwidths for some classes of differentiable periodic functions of two variables

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 855-864

We determine the exact values of Kolmogorov and linear quasiwidths for some classes of differentiable periodic functions of two variables in the Hilbert space $L_2(Q)$.