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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 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, 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