# Volume 61, № 7, 2009

### Classical solvability of a problem with moving boundaries for a hyperbolic system of quasilinear equations

Andrusyak R. V., Burdeina N. O., Kirilich V. M.

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 867-891

Using the method of characteristics and the method of contracting mappings, we establish the local classical solvability of a problem for a hyperbolic system of quasilinear first-order equations with moving boundaries and nonlinear boundary conditions. Under additional assumptions on the monotonicity and sign constancy of initial data and a restriction on the growth of the right-hand sides of the system, we formulate sufficient conditions for the global classical solvability of the problem.

### Generalized procedure of separation of variables and reduction of nonlinear wave equations

Barannyk A. F., Barannyk T. A., Yuryk I. I.

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 892-905

We propose a generalized procedure of separation of variables for nonlinear wave equations and construct broad classes of exact solutions of these equations that cannot be obtained by the classical Lie method and the method of conditional symmetries.

### Lax-integrable Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems and its finite-dimensional reduction of Neumann type

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 906-921

A compatibly bi-Hamiltonian Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems is obtained by using a relation for the Casimir functionals of the central extension of a Lie algebra of superconformal even vector fields of two anticommuting variables. Its matrix Lax representation is determined by using the property of the gradient of the supertrace of the monodromy supermatrix for the corresponding matrix spectral problem. For a supersymmetric Laberge–Mathieu hierarchy, we develop a method for reduction to a nonlocal finite-dimensional invariant subspace of the Neumann type. We prove the existence of a canonical even supersymplectic structure on this subspace and the Lax–Liouville integrability of the reduced commuting vector fields generated by the hierarchy.

### Convergence of solutions of backward stochastic equations

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 922-938

We establish conditions for the weak convergence of solutions of backward stochastic equations in the case of the weak convergence of coefficients.

### Arithmetic of semigroups of series in multiplicative systems

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 939–947

We study the arithmetic of a semigroup $\mathcal{M}_P$ of functions with operation of multiplication representable in the form $f(x)=∑^{∞}_{n=0} a_nχ_n(x)\left(a_n≥0,\; ∑^{∞}_{n=0}a_n =1 \right)$, where $\{χ_n|\}^{∞}_{n=0}$ is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup $\mathcal{M}_P$ , analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in $R_n$ are true. We describe the class $I_0(\mathcal{M}_P)$ of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in $\mathcal{M}_P$ in the topology of uniform convergence.

### Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

Perestyuk N. A., Slyusarchuk V. Yu.

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 948-957

We establish conditions for the existence of an invariant set of the system of differential equations $$\frac{dφ}{dt} = a(φ),\quad \frac{dx}{dt} = P(φ)x + F(φ,x),$$ where $a: Φ → Φ, P: Φ → L(X, X)$, and $F: Φ × X→X$ are continuous mappings and $Φ$ and $X$ are finite-dimensional Banach spaces.

### On expansions of numbers in alternating s-adic series and Ostrogradskii series of the first and second kind

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 958-968

We present expansions of real numbers in alternating $s$-adic series $(1 < s ∈ N)$, in particular, $s$-adic Ostrogradskii series of the first and second kind. We study the “geometry” of this representation of numbers and solve metric and probability problems, including the problem of structure and metric-topological and fractal properties of the distribution of the random variable $$ξ = \frac1{s^{τ_1−1}} + ∑^{∞}_{k=2}\frac{(−1)^{k−1}}{s^{τ_1+τ_2+...+τ_k−1}},$$ where $τ_k$ are independent random variables that take natural values.

### Generalization of one Poletskii lemma to classes of space mappings

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 969-975

The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings $f ∈ W^{1,n}_{\text{loc}}$ such that their outer dilatation $K_O (x, f)$ belongs to $L^{n−1}_{\text{loc}}$ and the measure of the set $B_f$ of branching points of $f$ is equal to zero have finite length distortion. In other words, the images of almost all curves $γ$ in the domain $D$ under the considered mappings $f : D → ℝ^n,\;n ≥ 2$, are locally rectifiable, $f$ possesses the $(N)$-property with respect to length on $γ$, and, furthermore, the $(N)$-property also holds in the inverse direction for liftings of curves. The results obtained generalize the well-known Poletskii lemma proved for quasiregular mappings.

### Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 976-985

We study mathematical models of the structure of nilpotent subsemigroups of the semigroup $PTD(B_n)$ of partial contracting transformations of a Boolean, the semigroup $TD(B_n)$ of full contracting transformations of a Boolean, and the inverse semigroup $ISD(B_n)$ of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For $PTD(B_n)$ and $TD(B_n)$, the capacity of a maximal nilpotent subsemigroup is calculated. For $ISD(B_n)$, we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small $n$. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree $k$ and determine the number of elements and subsemigroups for some special cases.

### Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 986-995

We study inequalities of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in the quasinorm of $L_0$ and derivatives of any order. We present expressions for constants in these inequalities and obtain double-sided estimates for them.

### Inequalities of the Bernstein type for splines of defect 2

Babenko V. F., Parfinovych N. V.

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 995-999

We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order *r* and defect 2.

### On the action of derivations on nilpotent ideals of associative algebras

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 1000-1004

Let *I* be a nilpotent ideal of an associative algebra *A* over a field *F* and let *D* be a derivation of *A*. We prove that the ideal *I* + *D*(*I*) is nilpotent if char *F* = 0 or the nilpotency index *I* is less than char *F* = *p* in the case of the positive characteristic of the field *F*. In particular, the sum *N*(*A*) of all nilpotent ideals of the algebra *A* is a characteristic ideal if char *F* = 0 or *N*(*A*) is a nilpotent ideal of index < *p* = char *F*.

### Equivalence of two methods for construction of regular continued *C*-fractions

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 1005-1009

A regular continued *C*-fraction is associated with a power series. The coefficients of this fraction are determined via either Hankel determinants or inverse derivatives. We prove the equivalence of these approaches to the construction of regular continued *C*-fractions.