# Volume 48, № 4, 1996

### Differential structures in lie superalgebras

Daletskii Yu. L., Kushnerevich V. A.

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 435-442

We develop a formal construction of an U-system as a fundamental concept of noncommutative differential geometry. By using the notion of “conditional differential” (an analog of the Hamiltonian mapping), we construct a series of brackets that generalize the classical Poisson brackets.

### Introduction of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 443-451

We establish conditions of the existence of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold and present the form of this system in these coordinates.

### Abstract Lax-Phillips scattering scheme for second-order operator-differential equations

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 452-463

We construct an analog of the Lax-Phillips scattering scheme for an abstract operator-differential equation *u* _{u}=-*Lu* under certain restrictions imposed on the operator *L*. In particular, we construct the incoming and outcoming subspaces and describe singularities of the scattering matrix in terms of the spaces of boundary values.

### Second bogolyubov theorem for systems of difference equations

Danilov V. Ya., Martynyuk D. I., Pan'kov V. G.

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 464-475

We establish an analog of the second Bogolyubov theorem for a system of difference equations.

### Some boundary-value problems for a class of fourth-order differential equations on a graph

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 476-482

We study the problem of nondegeneracy of some boundary-value problems for a class of fourth-order differential equations on a geometric graph (topological net), present conditions for the existence of the Green function for boundary-value problems under consideration, and analyze its principal properties.

### Asymptotics of eigenvalues of A regular boundary-value problem

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 483-519

We study a boundary-value problem *x* ^{(n)} + *Fx* = λ*x*, *U* _{h}(*x*) = 0, *h* = 1,..., *n*, where functions *x* are given on the interval [0, 1], a linear continuous operator *F* acts from a Hölder space *H* ^{y} into a Sobolev space W _{1} ^{n+s} , *U* _{h} are linear continuous functional defined in the space \(H^{k_h } \) , and *k* _{h} ≤ *n* + *s* - 1 are nonnegative integers. We introduce a concept of *k*-regular-boundary conditions *U* _{h}(*x*)=0, *h* = 1, ..., *n* and deduce the following asymptotic formula for eigenvalues of the boundary-value problem with boundary conditions of the indicated type: \(\lambda _v = \left( {i2\pi v + c_ \pm + O(|v|^\kappa )} \right)^n \) , *v* = ± *N*, ± *N* ± 1,..., which is true for upper and lower sets of signs and the constants κ≥0 and *c* _{±} depend on boundary conditions.

### Refinement of Kneser theorem on zeros of solutions of the equation $y" + p(x)y = 0$

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 520-524

We find conditions for a linear homogeneous second order equation to be nonoscfflatory on the half-axis and such that its solutions have infinitely many zeros.

### Complexity of fredholm equations of the second kind with kernels from anisotropic classes of differentiable functions

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 525-534

We establish the exact order of complexity of the approximate solution of Fredholm equations with periodic kernels with dominant mixed partial derivative.

### Groups with incidence condition for noncyclic subgroups

Chechulin V. L., Chernikov N. S., Polovitskii Ya. D.

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 533-539

We give a description of groups with incidence condition for noncyclic subgroups to within minimal noncyclic subgroups. We present a complete constructive description of locally graded groups (in particular, arbitrary locally finite groups) satisfying this condition.

### On universality of countable powers of absolute retracts

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 540-542

We construct an absolute retract *X* of arbitrarily high Borel complexity such that the countable power *X* ^{ω} is not universal for the Borelian class A _{1} of sigma-compact spaces, and the product *X* ^{ω} x ∑, where ∑ is the radial interior of the Hilbert cube, is not universal for the Borelian class A _{2} of absolute *G* _{δσ}-spaces.

### On exact order estimates of *N*-widths of classes of functions analytic in a simply connected domain

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 543-547

In the spaces *E* _{q}(Ω), 1 < *q* < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral *n*-widths of the classes *W* ^{r} *E* _{p}(Ω) and *W* ^{r} *E* _{p}(Ω)Ф in the case where *p* and *q* are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.

### Averaging of boundary-value problems of control over a standard system with delay

Efendiev V. V., Zheltikov V. P.

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 548-553

For the terminal problem of optimal control over systems of standard form with constant delay, according to the Pontryagin maximum principle, we study a boundary-value problem with deviating arguments with delay and anticipation. We justify an averaging method for an asymptotic solution of the boundary-value problem obtained.

### Generalized adequate rings

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 554-557

We introduce a new class of rings of elementary divisors which generalize adequate rings. We show that the problem of whether every commutative Bezout domain is a domain of elementary divisors reduces to the case where the domain contains only trivial adequate elements (namely, the identities of the domain).

### Estimates for distributions of components of mixtures with varying concentrations

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 558-562

For the data of sampling from a mixture of several components with varying concentrations, we construct nonparametric estimates for the distributions of components and determine the rank correlation coefficient. We prove the consistency of the rank coefficient and the efficiency of the estimates of distributions.

### On the properties of an operator of stochastic differentiation constructed on a group

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 563-568

We construct a differential operator by an admissible group in the space *L* _{2} (ℝ^{m},*P*)and study its properties.

### Remarks on summability of series formed of deviation probabilities of sums of independent identically distributed random variables

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 569-573

We make some remarks leading to a refinement of the recent work of Klesov (1993) on the connection between the convergence of the series \(\Sigma _{n = 1}^\infty \tau _n P(|S_n | \ge \varepsilon n^\alpha )\) for every ε > 0 and the convergence of \(\Sigma _{n = 1}^\infty n\tau _n P(|X_1 | \ge \varepsilon n^\alpha )\) again for every ε > 0.

### Symmetry reduction and some exact solutions of a nonlinear five-dimensional wave equation

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 573-576

By using decomposable subgroups of the generalized Poincaré group *P*(1,4), we perform a symmetry reduction of a nonlinear five-dimensional wave equation to differential equations with a smaller number of independent variables. On the basis of solutions of the reduced equations, we construct some classes of exact solutions of the equation under consideration.

### Erratum

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 577