Том 71
№ 7

All Issues

Article (Russian)

Über Randautgabe von Dirichlet Typus fur Schwingungsgleichung der Saite

Berezansky Yu. M.

↓ Abstract

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 363 - 372

In diese Arbeit untersuchen wir in beschränkte Gebiet G mit Grenze T die Randaufgabe. Wir beweisen, daß solche Gebiete G existieren, in welche unsere Aufgabe hat schwache Lösung für beliebige / und die Lösbarkeit ist stabil über kleine Änderung der Grenze. Wir untersuchen auch die Glattheit der schwache Lösung.

Article (Russian)

Integro-differential Equations of the Nonlinear Theory of Depressed Thin Shells

Berezovsky A. A.

↓ Abstract

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 373 - 380

A method is presented for obtaining nonlinear integro-differential equations equivalent to the differential equations of the nonlinear theory of depressed shells. It is shown fhat in a number of cases the integral equations obtained lead to the same results äs^be nonlinear differential equations.

Article (Russian)

Die Kurven der unveränderlichen Krümmung von projektiven Standpunkt aus

BorawIjow A. F.

↓ Abstract

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 381 - 390

Im Artikel werden Bedingungen festgestellt, unter welchen das projektive Entsprecher zweier Büschel die Kurve unveränderlichen Krümmung (in hyperbolischen, elliptischen und sphärischen Geometrien) bestimmt. Außer den Verhältnissen, die von Charakter der Zentren (eigentlichen und uneingentlichen) der bestimmenden Büschel abhängen, werden auch Formeln betrachtet, die von den Besonderheiten der Stelle der erwähnten Zentren auf der Kurve nicht abhängen. Nebenbei wird bewiesen, da^ die Inversion auf der hyperbolischen Gerade nicht projective Entsprechen ist, es wird auch die Formel der Krümmung gezogen, die für alle Kurven der unveränderlichen Krümmung der hyperbolischen Ebene gleich ist.

Article (Russian)

On Periodic Solutions of Systems of Nonlinear Equations with a Small Parameter

Lykova O. B., Mitropolskiy Yu. A.

↓ Abstract

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 391 - 401

The authors consider a system of nonlinear differential equations containing a small parameter with undifferentiated right parts of types (1) and (37). Making some assumptions, the existence unique and asymptotic stability of a periodic solution is proved for such systems, and an estimate is found for the difference between the exact solution of the systems under consideration and their first approximation, which can be found without any essential difficulty.

Article (Russian)

On the Solution of Filtration Problems by the Method of Majorant Regions

Lyashko I. I., Pakhareva N. A.

↓ Abstract

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 402 - 411

The theoretical principles of the method of majorant regions are outlined in this paper, and its application to the solution of concrete problems of the plane theory of filtration in homogeneous and heterogeneous media is discussed. In § 1 the problem of a two-grooved floodbed with deep-set apron at various bottom marks of the upstream and downstream water in a permeable layer of infinite depth is solved for the case of a homogeneous medium, and the problem of the determination of the discharge through an earthwork weir with mattress type drainage on a permeable base of finite depth is also solved. Strict upper and lower estimates are found for the filtration characteristics, sufficiently close to each other to secure the accuracy necessary in practice. In § 2 the author considers filtration without backwater from a channel of arbitrary cross section in a porous zonally heterogeneous soil (stratified). Making use of known properties of p-analytical functions [4, 51, estimates of the discharge are found by the majorant region method. Numerical examples confirming the efficacy of the method of majorant regions for this class of problems are discussed in all cases

Article (Russian)

Theorems on the Inner Diameter and Their Application to Some Systems of Differential Equations of Nuclear Physics

Skorobogatko V. Y.

↓ Abstract

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 412 - 428

Theorems on unique solutions of boundary value problems are proved for systems of elliptical differential equations of the second order The results are applied to some systems of differential equations of nuclear physics.

Article (Russian)

An Asymptotic Solution of a System ot Linear Differential Equations Having a Small Diameter with Derivatives

Feshchenko S. F., ShkiI N. I.

↓ Abstract

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 429 - 438

The authors consider the problem of constructing an asymptotic solution of a system of linear differential equations in which a real small parameter e > 0 is a multiplier of some of the derivatives. Differential equations in which the small parameter is a multiplier of the lower order derivatives are reduced to such systems.

Article (Russian)

Hydrodynamic Calculation of Drained Aprons. II. Application of the Method of Successive Conformal Mappings

Fil'chakov F. P.

↓ Abstract

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 439 - 462

In tbe first communication [10] an exact hydrodynamie solution is giveri-for the general case of a flat split apron floodbed with T=∞ and then,the most interesting: special cases are discussed. In the present research the earlier developed method of successive con" formal mapping [5; 7] is employed to obtain a very simple approximate solution fôr a fiat drained apron, which is êxtënded in §§ 2 and 3 to the case of a apron of arbitrary practical profile with n drainage openings w ith a finite or infinite depth of water-permeable soil. The application of an extremely simple graphoanalytical method permits solving these problems, in most of the cases encountered in practice, by graphic means in the course of 20—30 minutes, using a compass, ruler and previously constructed nomograms [6; 11]. Conditions (35), (46), (52) are indicated which should be observed in order to avoid large errors on using the graphoanalytical method. The Accuracy of the approximate formulae are illustrated by six examples, in which a comparison is made either with exact bydrodynamical solutions or with résults of simulation on resistance paper [4; 12]. A means of extending the earlier obtained results for anisotropic soil [7; III] and two-layei soil [9] to the case of drained aprons is indicated.

Brief Communications (Russian)

On a Class of Linear Differential Equations with Periodic Coefficients

Breus K. A.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 463 - 466

Brief Communications (Russian)

On an Application of Legendre's Polynomials

Derkach P. Kh.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 466 - 471

Brief Communications (Russian)

Investigation of a MultilinearSystem of Service with Lines and a Limited Stay in the System

Kovalenko I. N.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 471 - 476

Brief Communications (Russian)

Solution of the Inverse Diffraction Problem

Petrina D. Ya.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 476 - 479

Brief Communications (Russian)

Inequalities for a Laplace Operator in an Unlimited Region

Prokopenko L. N.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 479 - 484

Brief Communications (Russian)

Necessary and Sufficient Conditions for Convergence of Monomerie Processes

Sharkovsky O. M.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 484 - 489

Anniversaries (Russian)

Mikhail Alexeyevich Lavrentyev (on his sixtieth birthday)

Fil'chakov F. P., Mitropolskiy Yu. A., Shtokalo I. Z.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 490 - 491

BookReview (Russian)

Linear Differential Equations with Variable Coefficients (asymptotic methods and criteria of stability and instability of solutions)

Mitropolskiy Yu. A., Shtokalo I. Z.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 492 - 493

BookReview (Russian)

List of Booksand Articles on Mathematics Published in the Ukraine in 1959—1960

Sakovich G. N.

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 494 - 497

Index (Russian)

Alphabetical Index ol Volume XII

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 498 - 499

Index (English)

Alphabetical Index ol Volume XII

Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 500 - 501

Modifications (Russian)


Ukr. Mat. Zh. - 1960νmber=4. - 12, № 4. - pp. 504