2019
Том 71
№ 8

Volume 50, № 9, 1998

Article (Ukrainian)

A multipoint problem for partial integro-differential equations

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1155–1168

We investigate a multipoint problem for a linear typeless partial differential operator with variable coefficients that is perturbed by a nonlinear integro-differential term. We establish conditions for the unique existence of a solution. We prove metric theorems on lower bounds of small denominators that arise in the course of investigation of the problem of solvability.

Article (Ukrainian)

Description of sequences of zeros of one class of functions analytic in a half-plane

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1169–1176

We describe sequences of zeros of functions ƒ ≠ 0 that are analytic in the right half-plane and satisfy the condition ¦ƒ(z)¦ ≤ 0(1) exp (σ¦ z ¦η(¦ z ¦)), 0 ≤ <+ ∞, Re z > 0, where η: [0; + ∞) → (- ∞; + ∞) is a function of bounded variation.

Article (Ukrainian)

A generalization of the Lindelöf theorem

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1177–1192

We present a generalization of the Lindelöf theorem on conditions that should be imposed on the coefficients of the Taylor series of an entire transcendental function ƒ in order that the relation $ln M_f (r) - \tau r^\rho , r \to \infty , M_f (r) = \max \left\{ {\left| {f(r)} \right|:|z| = r} \right\}$ , be satisfied.

Article (Russian)

On the approximation by Chebyshev splines in the metric of $L_p, p > 0$

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1193-1201

We prove a direct Jackson estimate for the approximation by Chebyshev splines in the classes $L_p, p > 0$.

Article (Russian)

Direct methods for the approximate solution of operator equations with nonzero kernel

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1202–1213

We justify direct methods for the approximate solution of linear operator equations with nonzero kernels and apply these methods to the justification of projective methods for the approximate solution of standard singular integral equations with Cauchy kernels and positive index on the unit disk.

Article (Ukrainian)

Limit distribution of the number of solutions of a system of random Boolean equations with a linear part

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1214–1226

We prove two theorems on the Poisson limit distribution of the number of solutions of an a priori consistent system of nonlinear random Boolean equations with stochastically independent coefficients. In particular, we assume that this system contains a linear part.

Article (Ukrainian)

Asymptotic properties of the norm of extremum values of normal random elements in the space C[0, 1]

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1227–1235

We prove that $$\mathop {\lim }\limits_{n \to \infty } \left( {\left\| {Z_n } \right\| - (2 ln (n))^{1/2} \left\| \sigma \right\|} \right) = 0 a.s.,$$ where X is a normal random element in the space C [0,1], MX = 0, σ = {(M¦X(t2)1/2 t∈[0,1}, (X n ) are independent copies of X, and $Z_n = \mathop {\max }\limits_{l \leqslant k \leqslant n} X_k$ . Under additional restrictions on the random element X, this equality can be strengthened.

Article (Ukrainian)

Topological classification of (n−1)-convex sets

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1236–1243

We investigate the properties of (n−1)-convex sets associated with the properties of conjugate sets. We give a complete topological classification of (n−1)-convex sets.

Article (Ukrainian)

Approximation of continuous functions with random errors in observed values

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1244–1249

We consider the problem of approximation of continuous functions by generalized polynomials in the case where the values of the function at the observation points are known with random errors. We construct confidence limits with a given significance level for the true values of the function at any point of its domain of definition.

Article (Ukrainian)

On the structure of CDN[]-groups

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1250–1261

We describe nilpotent non-Dedekind CDN[]-groups.

Brief Communications (Russian)

On the Poincaré theorem with asymptotically periodic coefficients

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1262–1267

We consider the case where the Poincaré theorem for difference equations with asymptotically constant coefficients is generalized to systems of difference equations with asymptotically periodic coefficients.

Brief Communications (Ukrainian)

On the approximation of a bounded solution of a linear differential equation in a Banach space

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1268–1271

We investigate the problem of approximation of a bounded solution of a linear differential equation by solutions of the corresponding difference equations in a Banach space.

Brief Communications (Ukrainian)

On the problem of renewal of a Poisson field on a plane

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1272–1277

We perform the renewal of a Poisson field at a point on the basis of its values generated by a monotonically increasing curve. We obtain the best mean-square estimate and its error.

Brief Communications (Ukrainian)

On unitarizable modules over generalized Virasoro algebras

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1278–1280

We classify unitarizable modules with highest weight and unitarizable modules of an intermediate series over generalized Virasoro algebras.

Brief Communications (Russian)

On the invariance of a boundary-value problem for a nonlinear evolution equation

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1281–1283

We obtain an invariance group for one boundary-value problem in the physics of the sea.

Brief Communications (Russian)

On the best quadrature formulas for some classes of continuous functions

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1284–1288

We obtain the best quadrature formulas for classes of continuous functions defined by various restrictions on the moduli of continuity with respect to increase and decrease.

Brief Communications (Russian)

On some inequalities for polynomials

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1289–1292

We prove the equivalence between analogs of the Paley and Nikol’skii inequalities for any orthonormal system of functions and for almost periodic polynomials with arbitrary spectrum.

Article (Ukrainian)

On a smooth solution of a boundary-value problem

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1293–1296

We study a periodic boundary-value problem for the quasilinear equation u tt u xx =F[u, u t , u x ], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability.