2019
Том 71
№ 11

# Volume 53, № 7, 2001

Article (Ukrainian)

### On a Parabolic Variational Inequality That Generalizes the Equation of Polytropic Filtration

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 867-878

We obtain conditions for the existence and uniqueness of a solution of a parabolic variational inequality that is a generalization of the equation of polytropic elastic filtration without initial conditions. The class of uniqueness of a solution of this problem consists of functions that increase not faster than e −μt , μ > 0, as t → −∞.

Article (Ukrainian)

### On Interpolation Sequences of One Class of Functions Analytic in the Unit Disk

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 879-886

We establish a criterion for the existence of a solution of the interpolation problem f n ) = b n in the class of functions f analytic in the unit disk and satisfying the relation $$\left( {\exists {\tau }_{1} \in \left( {0;1} \right)} \right)\;\left( {\exists c_1 >0} \right)\;\left( {\forall z,\left| z \right| < 1} \right):\;\left| {f\left( z \right)} \right| \leqslant \exp \left( {c_1 \gamma ^{{\tau }_{1} } \left( {\frac{{c_1 }}{{1 - \left| z \right|}}} \right)} \right),$$ where γ: [1; +∞) → (0; +∞) is an increasing function such that the function lnγ(t) is convex with respect to lnt on the interval [1; +∞) and lnt = o(lnγ(t)), t → ∞.

Article (Ukrainian)

### Brownian Motion in a Hilbert Space with a Semipermeable Membrane on a Hyperplane

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 887-891

In a separable Hilbert space, we construct a continuous Markov process whose behavior coincides everywhere, except for a hyperplane S orthogonal to a given unit vector ν, with the behavior of a homogeneous Gaussian process with a given correlation operator tB, where B is a nonsingular nuclear operator. As the process hits the hyperplane, it receives an impulse infinite in modulus in the direction A such that |(A, ν)| ≤ (Bν, ν).We obtain a stochastic differential equation whose solutions are trajectories of the process constructed.

Article (Russian)

### On Orthogonal Appell-Like Polynomials in Non-Gaussian Analysis

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 892-907

We study an example of the construction of a non-Gaussian analysis using orthogonal generalized Appell-like polynomials with the generating function $$\frac{1}{{\sqrt {1 - 2a{\lambda + \lambda }^{2}} } }\cos \left( {\sqrt x \frac{1}{2}\int\limits_{0}^{\lambda } {\frac{{du}}{{\sqrt {u - 2au^2 + u^3 } }}} } \right),\quad a >1,$$ in the model one-dimensional case. The main results are a detailed intrinsic description of spaces of test functions, a description of generalized translation operators, and the investigation of integral C- and S-transformations.

Article (Russian)

### On One Nonlocal Problem with Free Boundary

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 908-918

We investigate group-theoretic properties of a nonlocal problem with free boundary for a degenerating quasilinear parabolic equation. We establish conditions for the invariant solvability of this problem, perform its reduction, and obtain an exact self-similar solution.

Article (Ukrainian)

### An Analog of the Jackson Inequality for Coconvex Approximation of Periodic Functions

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 919-928

We prove an analog of the Jackson inequality for a coconvex approximation of continuous periodic functions with the second modulus of continuity and a constant that depends on the location of the points at which a function changes its convexity.

Article (Ukrainian)

### $C*$-Algebras Associated with $F_{2^n }$ Unimodal Dynamical Systems

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 929-938

We consider $C*$-algebras associated with simple unimodal one-dimensional ambiguous dynamical systems $(f,R)$ with certain special restrictions. For these algebras, we present a complete classification of irreducible representations in Hilbert spaces and describe the dual space. As an example, we consider the one-parameter family $f_{μ}(x) = μx(1 − x)$.

Article (Russian)

### Scalar Operators Representable as a Sum of Projectors

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 939-952

We study sets $\Sigma _n = \{ \alpha \in \mathbb{R}^1 |$ there exist n projectors P1,...,Pn such that $\sum\nolimits_{k = 1}^n {P_k = \alpha I} \}$ . We prove that if n ≥ 6, then $\left\{ {0,1,1 + \frac{1}{{n - 1}},\left[ {1 + \frac{1}{{n - 2}},n - 1 - \frac{1}{{n - 2}}} \right],n - 1 - \frac{1}{{n - 1}},n - 1,n} \right\} \supset$ $\Sigma _n \supset \left\{ {0,1,1 + \frac{k}{{k\left( {n - 3} \right) + 2}},k \in \mathbb{N},\left[ {1 + \frac{1}{{n - 3}},n - 1 - \frac{1}{{n - 3}}} \right],n - 1 - \frac{k}{{k\left( {n - 3} \right) + 2}},k \in \mathbb{N},n - 1,n} \right\}$ .

Article (Russian)

### Propagation of Perturbations in Quasilinear Multidimensional Parabolic Equations with Convective Term

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 953-969

We establish estimates for the initial evolution of the supports of solutions of a broad class of quasilinear parabolic equations of arbitrary order that have the structure of the equation of strong nonlinear convective diffusion.

Article (Russian)

### Sum of Divisors in a Ring of Gaussian Integers

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 970-982

We construct an asymptotic formula for a sum function for σ a (α), where σ a (α) is the sum of the ath powers of the norms of divisors of the Gaussian integer α on an arithmetic progression α ≡ α0 (mod γ) and in a narrow sector ϕ1 ≤ arg α < ϕ2. For this purpose, we use a representation of σ a (n) in the form of a series in the Ramanujan sums.

Article (Russian)

### Coleman Principles and Krasnosel'skii Genus in Eigenfunction Problems

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 983-992

We consider a simple semilinear elliptic eigenfunction problem. Using it as an example, we demonstrate functional topological methods that give information on the critical numbers almost as detailed (in the qualitative sense) as in the case of separation of variables in an analogous linear problem.

Brief Communications (Ukrainian)

### Games with Pay-off Function and Pulse Influence

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 993-995

We consider differential games with terminal pay-off function and pulse influence at fixed time. We construct optimal strategies for players.

Brief Communications (Russian)

### К вопросу об оценках колмогоровских поперечников классов $B_{p,q}^r$ в пространстве $L_q$

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 996-1001

We obtain an order estimate for the Kolmogorov width of the Besov classes $B_{p,{\theta }}^r$ of periodic functions of many variables in the space $L_q$ for $2 < p < q < ∞$, which complements the result obtained earlier by the author.

Brief Communications (Russian)

### Moments of Markov Random Evolutions

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 1002-1008

We find moments of a process of Markov random evolutions in a finite-dimensional space.