SCIENTIFIC ABSTRACT GELFOND, A. O. - GELFOND, S.

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• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000514620016-6
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• Document Creation Date: November 2, 2016
• Document Release Date: August 23, 2000
• Sequence Number: 16
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

GELIF0414D. A. 0. -
Sur les nombres transcendqnts. C. R. Acad. Sci., 189 (1929), 1224-1228.
0 sed1mony probleme Gillberta. DAN, 2 (1934)p 1-6
Sur le septieme probleme de Hilbert. IA*, Ser. Fiz. jhtem. (1934)v 623-630.
0 Priblizheniy'o-h Transtsendintnykh Chisel algebralcheskird. DAN, 2 (1935)p 177-182.
Transtsondentnyye chisla. Trudy
Vtorogo Vsesoyuzn. Matem S"Ezda, T. 1 (1936). Va-163.
0b odnom obobshchenii neravenstva Ifinkovskogo. DAN, 17 (1937), 443-446
SO: Mpthemtics in thp USSR, 1917-1947
edited b- Kurosh, A. U.#
Markushevich, A. I.,
Rashevshty, P. I..
Moscovr-Ieningrpd, 1948

GELIF0,117), .11%. 0. ((ont.)-
0 priblizhenii al.gebraichez;kim 'chislami otnosheniya logarifmov d-rukh aL~ebraicheskikh
chisel. TAN, C-r. Maten. (19391, 509-518.
Sur 1,, divisibilite do la difforonce doi p4soances &i deux nomlroq entiers par unc
puissanc,Aun ideal pr-mier. Matem, M, 7'(49), (1940) 7-26.
Sur un th-oreme d#, 114. M. Wiegert--IAau. Hatem SB. v36 (1929)p 99-101.
Raziozheniye :aeromorfnoy funktsii V Ryad ratsionallnykh Drobey I ryad Toylora. Hatem. SB.,
2 (41.), (1937), 935-946.
A
Sur une application clu calcul dos diffr~rences finias I Lletudti des fonctions entieres.
Matem. SD., 36 (1929)9 173-183.
Sur 1,?s proprietes arithmetiques des fonations entiores. Toh. Math. J. (19P9) Robots,
Nakhoditsya V Pechati. Sur 1e developpement des fonctlins entigres d1ordre fini en
seric dlinterpolation do Newton. AM Accad. NAZ. Lincei, 11 (1930)p 377-381.
Problema prodstavleniya I edinstvennosti t,-,eloy analiticheskoy funktsii pervogo
poryadka. Usperkhi Matem. Nauk, 3 (1937)p '144-174.
Int-rpolation et unicite des fonctions '~ntiorer:. Matem. SB., 4 (46), (1938) 1-15-148.
jfir les systemes complete des fonctiors analytiques. "'aten. SB.1, 4 (46)(1930), 149-156.
0 nekotorykh int~rpolyatsionnykh Zadachakh. Uspekhi Matems Naukv 1: 5-6 (15-16)
(1946), 236-239.'
Ischisleniye konachnykh Paznostey. 44.,, Onti (1936), 1-176.
SO: Mathematics in the USSR-9 1917-1947
edited by Kurosh, A. G.,
Markushevich, A. 1.9
Rashevshiyp P. I.j
HDacow-leningrad, 1948.

GIMTONDY A. 0).
livedeniye v analiz LN11ira, Stat'ya po istorii materratiki "'ieklinicheskaya Kniga,"
1938-1939 99-

CELTONDY A. 0.
(f0 ryade Teypora, assotsiirovannom s tseloy f=ktsiey, Dokl. Ak. Ilauk, Ma~,23, nc ~-,
PP-756-759- 1939.

GELTONDI '... 0.
0 koeffitsientakh periodicheskikh funktsiy, Izv. Ak. Ilauk, val- 5, no. 2, 1941.

Gel*mf, A. ~1~
. - , __1--_ !~, - F
res ;zi; 10 44, two poin ts. f
a 1,~ ~4 , ..
WSIC Ser. Alat. 11, 541-5(.)U (1917).
Ile authors d6cus3&_-%vrd cas" hi WhicIt. f4,-'(1) 0 f)r
aacquence (,,I and JW(O) -u 14 kor, imply that
Let
a
("1 .0!
1-Then f(a)-G in the folliminj- caxn- (All f(z, i-, mgufnr ir, -
Irl ::;R, where R>v -lim irif or,. (II)f(;) - tin cmire
function, lim a. - ~c and A. - min, z:.tit, I a I i < (6,/vj)Xj- 4
and F-64 < -; (C) 1(t) is an entire fiviction of futitt ordc-r 4
p > I and lim log cr./log
If Isj ls an arithinctic progression, Y,-pi-l, f(s)uiO
if it b entire and of exponential typc a fro.
1 On sp -;.q
. `C, liTing dh-Z .1. Z-1111 "k, (Ij and (2) give,
e.g., the followinp- corollariits. (4) If a and a 96 0, ?F 11 are
il eb-aic and a is of higher rhan the third degree, then at
1119
easL Lwo of the numbers c.-, a-% a-', a*' are alqebraically
independent. If et *,s of the th rd degree, then alrearly a- and
are algebraically indel.)endent. ESeL the paper rited at
t~ beginning of this review.] (5) If a *0. Pe I is algebraic,
3 nd tj-.P&'0 is rationd, then at least two of the numbers
a-P a,) I k -_ 1, 2. S. 4. are algebraically. independent. (6) If
~15-'O is a rational number, then at least one of the numbers
e~p (ell), k - 1. 2, 3, is algebraically independent of e, thus
4Wiscendental;and at least one of the numbers exp
I Theorem 3 is aiso of interest be-
j= 1, 2, 3, is t-raweudentd.
k
I
c4use, e.g., the Thue-Sic-i
4 gel theorem on algebraic nurnbcrs im-
~A
oK ~ 3)1 1 > e-M.
~Ijlies onply the weaker inequality I P I (I a)/(log
rhe roofs of (1)--(3) are rather similar-, they are based
to
on a number of lernmas -which are of int=mrt in them_~elves.
Vor instance, (1) is proved as Wows. A."time that ever)-
two of the I I numIx:rs (a) are algebraically dependent. Then
there exist two transcendental numbers w and .,. where
+c,(cj) =0 and'el(o). c2(w). - - -, r.(O are
ials in w with irm-gral coefficients, such that -he
numbcrs (a) are of the form Sj/7*j (i-~ 1, 11). the S.and
i-.o T~i being polynomials in w of arbitrary degree and in ~;Jj of de-
O-ee at inost P -I with intt-~~--l coefficients. Denote I)%- A' a
zt~fficie.*Itly Ur7ge ixt;i6vt: inte-ger, and put 'I,- T,T': 7,1t.
=4i P - C.V' log-l V3- 1, P, - CNJ/lc)gi N]. X 3 +
'I IX.N7: log~-i ~V~_ In the functioii
F~- L
of thf- ;or-n
L6
-~l 1, C1.
v;,I.. re,tlizt C's aric not all zero. Ilen the nunh-e-5
lo~, N. i 0, 1, 2, art- polynomials in w
a~d wt wifli integral' ctxfficients, and at least 'one of them is
tilt zem. Denote by X,,xj, - - - potsifive constants independ-
CUI Of N. By means, of Dirichlet's principle, integral values

Ail
I-VTEI
'Cv

-nil) Co~
1-1
~
o
Tr,
zo po uum~;
Wit j,- ttj I Ju tu I I u 4:
Mu"~o'LIT till 'a I.-
pur r, LIE t~i

ikr !-"C>1 -
CV, x >

%72 'cv I-Bol Lvx-) &%---w>
(j,- pur

4,4

d%Z)>

;),I XI:U1 S,3

k Gell(ond, A. U. Un the algebraic independence of alge-
--ffirromis of algebr9le numbels.. 0411ally IWIlL
N',kuk SMIt (N.S 64, 2711 ''M
1- .11)), Mns-,ian)
Ixt a be a rotit of :lit irr(A ill M& all"elif"tit: (.11wition of
(legree 3 and a/.0, I ;my algvbraic ntindivi-, Tfic atillitir
allows that, (or any fixcd intcriactalion tif lfrga, ilit! yinin.
bernal-c"'I'land (04-C,8 1"" indt-111-lillent
in the field of the rationA ninnix-rs. 'I'll,! prix-f, which i~' not
given in full detail' IlliAns U'A- of re-jills %aill if) have lit-ell
proved by (lie author ill atiodwr palit-r inaict-s-sibb. (4) Oic
reviewer. It is stated that thc inedioll caii Im.- appliud uitlj-
out appreciable cliange in prove thal if w,, w,, - - -, w,
(w, rational) form it ljwsit; foir the ring of algebraic iiiEt-gers
at -tit algebraic field K. then no rell-stion 11(a,i, ii"i) =0
(iojyd 1) is possible where P(j-, y) is a lxAynoinial in x, y
with rational coefficicllt4; an improvement of this re-,ult,
which is to apix-ar elro.-where, is ionnounmd. [Now. The
reviewer suspects that ill leibina 3 the cas;e, k,-kj-k,-0
is nicant to be exclu&.~d. otherwi,
.0_. tjj(~ lV111111,t Ixt:uFlje.,4
trivial since B, is thell a milltilile of f(,)(fj) which is zero fly
hypothevis'] X A. Rattkin (01mbridge, England).
SOUMe t Eathemucal Reviews, Vol lo, 140. 10

7-L
7~ 7,
UfAlond, A. 0. On the elp!braic of tran-
'-7mmdVTT1'r"nGmbers
of cettain clai;sci; D,d:h,ly Akad.
Nank sS4?rTM,"T-)b7, t,3-1.1 (vI19.).
thrOrOlli Ntil.tl 0 WhiCh. it Oilled,
Can be ficillivol by a varktioll of Ow llw:kt~d- Wit~d ill All
tiather 1,uIx-r Dokl,iily (N.S.) 61, 27;-2,14) (19-19);
tLt,-~! kcv, 10, 082]. Thf~t, tl~cotciits art: (ompl ca(~~d
It, st-M, herl" but the followitil, colitwillicl,li(~ Imly bi: 11"Cil-
6imcd (1) Ilij tjZtI (I Iltillik-t!i'm 1,11 ,-0, I'and
If lilt, (It ~Tvo of uwv.!vf,d, 2, 1111,11 it ~"A JlQt
C-Ach If the four itumbf.13 al, fr". 1j"', a,'
tt~mliof une of 'hinli. Whvtl ev is a fulsic it 1-41oW3
Ot'It it' a-ld a` aw id"'t-blait'.1113, illdrpCildclit 1.1 Ih,- 1-:111 of
tit,- rittiowd imi illcrii, a vut;iih Arti-viv lir(ovd i;1 fl"7 jl.~pcr
it-f,.rrid t-,Y. (11) If a k, a,; 1144ow awl v 6 not
th-11 it i.- If)( p.,):,iUv t,i iv~,.prt:!75 (if tho four
ill 1, ; ~ ~1 ~ of ww of
Y
11irm ond, ill p.micular, at o;tv -if th,:111 I., I*
dor.Lil- A imnifm, rcsidt. hold-; fi~ r (lit: fin-t Ihri- -f i'lo b-or
RL A, Rtilikin
SoUrcla: Vathemqtlaal Vol 140,

Gellond,-A, O~-- The-afifiroximntin"t
aA
ad. SL i
A
L Iloopar. 1. 119-260
This p.Ajxr was already priotol undcr lite _&wc, Wit in
Uspehi Matem. Nauk (N.5 4, no. 4 (32), W-49 (1949);
tbc,.,c Rev. 11, 231, K. Ifijh!cr (Manche5tor).
Vol 3 No. 7

%'01,11111111 A. 0. 011 1he reneralized polyimmink of S. N.
Akad. Natil, S,'~Sk. S,-r. M it. I
If
2ri
O"a'44 ... 0- [ak aj 0 -'n* k 7.
where [ak ... a.] are the divided differences of tPe func-
tion x'. then the "polynomial" B.(f.x)=Ek".uf(T,.)qk.(x)
converges uniformly towards J(x) for any f(x) continuous
on [0, 1]. The qa.(x) are linear aggregates of (unctions
V4 log, x, where P is a nonnegative integer less thart the
multiplicity p., of ak, In case juj - 1, k - 1, 2, - - -, thiii has
been given by Hirschman and Widder [Duke Math, J. 16,
433-438 (1949); these R(:v. 11, 29). The author uses a more
Lonvenient technique which enables him to estimate the
degree of approximation. Two inaccuracies on pp. 417-418
are easily corrected. G. Lorenix (Toronto, Cot.).
Mathematioal Reviews,
Vol 12 No,
4.

V
Or a measure (it
_j2jLdA. 0., and Fel'dinspAj. On th
re aniscendent-a-Irt-yof t7r-Wa number - Izvestiya
Akall, Nauk S.SSR. Ser, Mat. 14, 493-SUO (195qf.,
Thc -mthors improve art carlier result due to Crel'fond
rDA-Litly Ak,id. Nauk SSSR (N.S.) 64, 277-280 (1949;,;~
these Rev. 10, 682]. Let a be a root of an irreducible -qu:-,-
finji of dvgree 3nnd (ve% I anatgebraic number. Let P(x,
be.my polynomial iii xand y having integral coefficients each
of mr0uhis not exece(Eng Ir and of degrees nj and no In x
and y. Theii, for any 4>0,
4+-
1)(4-, a-') e--*
providcd that o - in.
ax (nj+n,,IogH)>v*. Theproofmakes
Wx of Weas similar tu tbose introduced In the pipcr referred
to alxwc. R. A. Pankits (Cambridge, England).
Kgth-11ratical Reviews# Vol No.

C,
OM/Natbematles - Ajpvz1=ftmw ZM/IP*b 51
"Quasipol7nmials ftat Deviate the Leant From Zero
on the Interval (0, 1)p" A. 0. Gellfond
*Iz A Nauk SSSR, Ser Matemat" Vol XV, No 1, pp 9-
16
Following S. W. Pernsht"n's mthods, i3ellfond
flntU lower AM uppe. bounds of divergence of
quasipolynostial that deviates least rrm zero.
ftbuitted 12 Oct 50.
170T51

L
Gel'fond. A- 0 and Leont'ev, A. F. On a generalization
Mat. Sbornik N.S. 29,71), 477-500
~1955T~. ~Russian)
n=0,I,---)be!AhentJrefunc- -1Z
tion of'order p and finite, non-zero type _~. with (a)-
1, --be, a arbi-
1 im a. (crep) 110; and let F(:) 6b a
trar%. an aly-zic function, regulax in'z: < R (R Define dhe
operator This)Lf
~erit-s for D*F converges in j.-1