SCIENTIFIC ABSTRACT KUDRYAVTSEV, L.D. - KUDRYAVTSEV, M.K.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000827210018-2
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: July 12, 2001
• Sequence Number: 18
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

IORYOTSEV, L.D.
Differentiable mappings, Dokl.AN SBSR 95 no.3:921-923 Ap 154.
(NLRA 7:4)
1. Moskovskiy fiziko-tokluilchaskiy Institut.
Predstavleno akademikon M,A.Lavrentlyevym, (Surfaces, Representation of)

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SUBJECV USSR/MATHEMATICS/Integral equations CARD 1/1 PG - 66
AUTHOR KUDRJAVCEV L.D.
TITLE On the properties of the harmonic mappings of plane domains.
PERIODICAL Mat.Sborni~, n. Ser. 26-A.0955) 201-208.
reviewed 6/1956 .9
The contents of this note was 'givvn without proof in Doklady Akad. Nauk
92. 469-471 (1953). The mapping u - u(x,y), v - v(x,y) of the plane domain G
is called harmonic, if the functions u and v are harmonic in 0. Let f(G) -
u(s) + iv(s) be the one-to-one traneformation of the boundary k- of the
unite circle K onto a Jordan curve, the interior of which is denoted by r
It is supposed that u(s) and v(B) have the derivatives of the first order
which satisfy the H81der condition with 0( ~< I, and that there exists a
function qtz) such that the Fourier series of luz) is conjugate to the
Fourier series of u. The author proves a neoessary and sufficient condition
for the existence of the harmonic transformation of If onto F which on 4'
is equal to f(s). Several properties of the harmonic transformation are giveno
INSTITUTION: Moscow

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SUBJECT USSR/ffA REMATICS/The,.iry of flinctions CARD -112 PG - 130
AUTHOR KUDRJAV9V L. D.
TITLE On differentiable mappings.
PERIODICAL Doklady Akad. Nauk 104, 12-14 (1-955)
reviewed 7/1956
The author considers differentiable mappings of a region of the euclidean
n-dimensional space En and generalizes some results of the classical theory
of functions. Theorems If f Is a noDtinuously differentiable mapping of the
region G and if the set of zeros o'~ ~he functional determinant of f is isolated,
then there exists no point sequence x 6 G having a limit point in G and for
which f(X n) - f(xm ) for all n,.2-1,2 ..... .f is called monotone (rasp. compact)
if the origin of every point is a oontinuum (rasp. a compactum). I compact
mapping f of the region G:SEn In En is called local monotone in the point
X yj.-
YO C f(Q) if for every component r- kyc X there exists a neighbor)lood F1 of
this -.omponent x c r S G in wh:'Lch f is mor.o+one. If f is local monotone in
all points yf-f(O), then it Is calle-1 simple leoal monotone. The following
generalization of Hadamard's 4heclrom is va'lj.di
Let f be a local monotonely differentiable orientGd mapping of a bounded
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simply connected region 0 S; E n onto ar. also silaply connected region, where
the boundary of the Image is t"Ite- lae;.ga -)f the b.-rindary. The set of the zer6s

Doklady Akad. Naluk 10 12-14 (1955) CARD 2/2 PC - 130
M.L
of the functional detaxialwLat of thij waf,,plng oball have no inner points.
Then f is a simple monott~ne mappir&. ThE- proef wir-1--b. J0 not carried o-at
bases or, the lemmai Let G be a elmp-iy conuccted, bomd3d region, f a nompact,
orienti-cl, i1fferentlable mapping: '.9 haxe tt.n bwintl~iTy of the image Is the
Image of the boundary, and the set tf of the functional determinant
POSSaBSeS no in.Uel' P,,)I.nt3. Lpt I,.,-V Let
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