2019
Том 71
№ 11

# Volume 16, № 3, 1964

Article (Russian)

### On the application of the Chebyshev-Bernstein method to a class of extremal functions satisfying some relationships, linear with respect to the coefficients

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 283-291

The aim of the article is to establish the form of nonnegative polynomials $y(x)$ of a power which does not exceed the given one, limiting the integral $$\int_{-1}^1p(x)y_n(x)dx$$ where $p(x)$ is the given summed function, if the coefficients of the polynomial $y(x)$ satisfy any number s of linear relationships.

Article (Russian)

### On periodic solutions of an irregularly perturbed weakly nonlinear equation

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 292-299

The conditions of the existence, uniqueness and stability of the periodic solution of a weakly nonlinear equation of the second order, perturbed by terms containing a small parameter with the higher derivatives are established in this article.

Article (Russian)

### On singular lineals in $\Pi_{\chi}$ spaces

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 300-308

The author consider the Hilbert space $\mathfrak{H}$ in which a $J$-metric $[x, y] = (yx, y)$ is introduced, where $yj$ is the difference of two orthoprojections in $\mathfrak{H}$. The lineal $\mathfrak{L} \subset \mathfrak{H}$ is called definite if the form $[x, x] (x \in \mathfrak{L})$ has a constant sign; the lineal is called singular if the norms $(x, x)^{1/2}$ and $|[ x , x]|^{1/2}$ are nonequivalent. The properties of singular lineals are studied. In particular, it is shown that an arbitrary infinite-dimensional lineal with a positive Hermitian-bilinear metric $[x, y]$, complete with respact to the norm $|x| = [x, x]^{1/2}$ may, preserving the form $[x, y]$, be embeded into space $\Pi_{\chi}$ with an arbitary integer $\chi$ so that it proves to be a singular lineal with a given measure of singularity $m \leq \chi$.

Article (Russian)

### Theorem on elementary divisors for a ring of differential operators

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 309-318

This paper presents the proof of the following theorem:
For any matrix with elements of a differential ring $\Delta_{\tau}$ from $n$ differentiations there exist such reversible over $\Delta_{\tau}$ matrices $P$ and $Q$ for which relationship (1) holds.
If the matrix $A$ is of quadratic order and rank $n$ then (1) has the form of (2).

Article (Russian)

### Structure of a set of solutions of equations of the parabolic type

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 319-333

The basic result of this research is formulated as follows: if the mixed boundary value problem for a nonlinear parabolic equation has two solutions, it has a continuum of solutions.

Article (Russian)

### On the investigation of a integral manifold for a system of nonlinear equations, close to equations with variable coefficients, in a Hilbert space

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 334-338

The author considers a system of differential equations $$\frac{d\varphi}{dt} = \omega(t) + P(t, \varphi, h, \varepsilon)$$ $$\frac{dh}{dt} = H(t)h + Q(t, \varphi, h, \varepsilon)$$ where $h$ and $Q$ are vector functions with values in Hilbert space $H$, $\omega(t)$ is a limited operator function in Hilbert space $H$, for which the real parts of all points of the spectrum are negative. The existence and stability of a one-dimensional integral manifold for system (1) is proved with certain assumptions.

Article (Russian)

### Extension of a theorem of N. N. Bogoliubov to the case of a Hilbert space

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 339-350

The author considers an equation in standard form $$\frac{dx}{dt} = \varepsilon X(t,x) \quad(1)$$ where $x(t), X(t,x)$ are vector functions with values in the Hilbert space $H \varepsilon$ is a small parameter. A theorem is proved on the existence and uniqueness of an almost periodic solution of equation (1) in the neighborhood of the equilibrium position of the corresponding averaged equation $$\frac{dx}{dt} = \varepsilon X_0(x) \quad(2)$$ The question oi the stability of this solution is also decided.

Article (Russian)

### On'a multiple stochastic integral

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 351-364

A multiple stochastic integral on Brownian movement is defined, and the formula of its transformation, analogues to K. Ito's formula for a single integral, is proved.

Article (Russian)

### Melirdimensionaler Grenzwertsatz feur Wahrscheinlichtkeitsdichten

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 365-373

Im ersten Teil vorliegender Arbeit werden notwendige und hinreichende Bedingungen dafür angegeben daß die Dichten der normierten Summen unabhängiger identisch verteilter $s$-dimenensionaler Zufallsvektoren gegen eine stabile Verteilunsdichte konvergieren. Der zweite Teil enthält eine asymptotische Zerlegung der Differenz der Dichte der normierten Summe von $n s$-dimensionalen unabhängigen Zufallsvektoren (sowohl nicht identisch als auch identisch verteilter) und der Dichte der $s$-dimensionalen Normalverteilung nach Potenzen von $\cfrac1{\sqrt{n}}$.

Brief Communications (Russian)

### Remarks on stationary streams of homogeneous events

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 374-375

Brief Communications (Russian)

### On a method of solving boundary value problems for linear differential equations of the second order

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 375-382

Brief Communications (Russian)

### On quasismooth functions of two variables

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 383-389

Brief Communications (Russian)

### Representation of coded regular events in abstract finite automata

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 385-389

Brief Communications (Russian)

### On a non-standard iterative method of the approximate solution of linear operator equations

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 389-396

Brief Communications (Russian)

### On the properties of the periodicity of events represented in finite automata

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 396-402

Brief Communications (Russian)

### On the equivalence of two determinations of a finite annular group

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 402-406

Brief Communications (Russian)

### «Trigonométrie» Chebyshev polynomials are also hyperbolic

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 406-408

Brief Communications (Russian)

### On adjoint orders and types of whole functions with many variables

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 408-413

Brief Communications (Russian)

### Similitude theorem for surface $R_n$ in Riemannian space $R_m$

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 413-417

Brief Communications (Russian)

### On the uniqueness of the solution of Cauchy's problem for a differential equation with constant coefficients

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 417-421

Brief Communications (Russian)

### On problems of multistage queues with losses

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 421-428

Chronicles (Russian)

### First Republican Scientific Conference of Young Research Workers in the Field of Mathematics

Ukr. Mat. Zh. - 1964. - 16, № 3. - pp. 428