# Volume 53, № 11, 2001

### On differential operators with singularity and conditions of discontinuity inside an interval

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1443-1457

We investigate boundary-value problems for differential equations with singularity and discontinuity conditions inside an interval. We describe properties of the spectrum, prove a theorem on the completeness of eigenfunctions and associated functions, and study the inverse spectral problem.

### On the Existence of a Noninner Automorphism of Order $p$ for $p$-Groups

Bondarchuk L. Yu., Pylyavs'ka O. S.

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1458-1467

We obtain sufficient conditions for the existence of a noninner automorphism of order *p* for finite *p*-groups. We show that groups of order *p* ^{n} (*n* < 7, *p* is a prime number, *p* > 3) possess a noninner automorphism of order *p*.

### Fourier Problem for a Coupled Diffusion System with Functionals

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1468-1481

We investigate the well-posedness of a problem for a system of functional differential equations of different types without initial conditions. Each equation consists of two parts one of which has the same structure as a parabolic equation or an ordinary differential equation with parameters, while the other contains functionals defined on a space of functions continuous in space variables.

### Extremal Versions of the Pompeiu Problem

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1482-1487

We investigate the local Pompeiu problem of functions with zero integrals over balls and cubes and related problems.

### Sylow Structure of Idempotent $n$-Ary Groups

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1488-1494

We study idempotent *n*-ary groups. We describe the Sylow structure of finite idempotent *n*-ary groups.

### Bounded Solutions for Some Classes of Difference Equations with Operator Coefficients

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1495-1500

We obtain necessary and sufficient conditions for the existence and uniqueness of bounded solutions for some classes of linear one- and two-parameter difference equations with operator coefficients in a Banach space.

### A Linear Method for the Recovery of Functions Based on Binary Data Completion

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1501-1512

We construct a linear recovery method based on binary data completion using the Bessel interpolation formula. We find an asymptotic value of the error of this method, determine its norm, and study its properties.

### Group Classification of Generalized Eikonal Equations

Egorchenko I. A., Popovich R. O.

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1513-1520

By using a new approach to a group classification, we perform a symmetry analysis of equations of the form *u* _{a} *u* _{a} = *F*(*t*, *u*, *u* _{t}) that generalize the well-known eikonal and Hamilton–Jacobi equations.

### One-Sided Nonlocal Boundary-Value Problem for Singular Parabolic Equations

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1521-1532

In spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of a one-sided nonlocal boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients. We obtain an estimate for the solution of this problem in the corresponding spaces.

### On Polymer Expansions for Equilibrium Systems of Oscillators with Ternary Interaction

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1532-1544

For Gibbs lattice systems characterized by a measurable space at sites of a *d*-dimensional hypercubic lattice and potential energy with pair complex potential, we formulate conditions that guarantee the convergence of polymer (cluster) expansions. We establish that the Gibbs correlation functions and reduced density matrices of classical and quantum systems of linear oscillators with ternary interaction can be expressed in terms of correlation functions of these systems.

### Investigation of Exponential Dichotomy of Itô Stochastic Systems by Using Quadratic Forms

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1545-1555

For linear stochastic systems, we obtain sufficient conditions for mean-square exponential dichotomy in terms of Lyapunov functions that are quadratic forms.

### Criteria for Invertibility of Elements in Associates

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1556-1563

We continue the investigation of invertible elements in associates, i.e., in (*n* + 1)-ary groupoids that are (*i*, *j*)-associative for all *i* ≡ *j* (mod *s*), where *s* is a divisor of a number *n*. For *s* = 1, an arbitrary associate is a semigroup. We establish two new criteria for the invertibility of elements, which generalize the results obtained earlier, and formulate corollaries for (*n* + 1)-groups and polyagroups, i.e., quasigroup associates.

### Factorial Analog of Distributive Bezout Domains

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1564-1567

We investigate Bezout domains in which an arbitrary maximally-nonprincipal right ideal is two-sided. In the case of *At*(*R*) Bezout domains, we show that an arbitrary maximally-nonprincipal two-sided right ideal is also a maximally-nonprincipal left ideal.

### Asymptotic Behavior of Logarithmic Potential of Zero Kind

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1568-1574

Under a fairly general condition on the behavior of a Borel measure,we obtain unimprovable asymptotic formulas for its logarithmic potential.

### Orders of Trigonometric and Kolmogorov Widths May Differ in Power Scale

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1575 -1579

We present a class of functions for which trigonometric widths decrease to zero slower than the Kolmogorov widths in power scale.

### Relations of Borel Type for Generalizations of Exponential Series

Skaskiv O. B., Trusevich О. M.

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1580-1584

We prove that the condition \(\sum\nolimits_{n = 1}^{ + \infty } {\left( {n{\lambda }_n } \right)^{ - 1} < + \infty }\) is necessary and sufficient for the validity of the relation ln *F*(σ) ∼ ln μ(σ, *F*), σ → +∞, outside a certain set for every function from the class \(H_ + \left( {\lambda } \right)\mathop = \limits^{{df}} \cup _f H\left( {{\lambda,}f} \right)\) . Here, *H*(λ, *f*) is the class of series that converge for all σ ≥ 0 and have a form $$F\left( {\sigma} \right) = \sum\limits_{n = 0}^{ + \infty } {a_n f\left( {{\sigma \lambda}_n } \right),\quad a_n \geqslant 0,\;n \geqslant 0,}$$ and *f*(σ) is a positive differentiable function increasing on [0, +∞) and such that *f*(0) = 1 and ln *f*(σ) is convex on [0, +∞).