ABTOMOP*M3MbI A/ITEEPbl C OflHOfl MOHAPHOft OflEPAUHEH B CMrHATYPE A.A.TAHMAHOB

Transkrypt

1 Anre6pa H Jiorasa, 6, NO 5 (1967), ABTOMOP*M3MbI A/ITEEPbl C OflHOfl MOHAPHOft OflEPAUHEH B CMrHATYPE A.A.TAHMAHOB KBaHTOpOB B OrpaHHieHHOJS aph^mbthke, KOTOpblfi H6 M6HHM ana onepauhh 2^= 2.x-3^ B nahhofl craxbe nokaabibaroxca o6uihe reopembi 06 B cxaxbe paccmarpabaexca anre6pahhecicaa CHcxeMa flt = <yv; <s> >, cnrhaxypa & Koxopofi coaep>khx OHHH CBMBOJI f(~z:) MOHapaofl onepaunn, incno CHMBOHOB (-x:) oanomecthbix npeahkaros H He conpyrax chmbonos. OpHMepOM TaKofl onepauhh aennerch orpahhhenhaa aph(j)methka tt = <,*>/, ', P± (x) >, d^z, rae JV=t,O,-f,2, -} - MHO^BCTBO uenbix HeoTpHuarenfaHtix sc'^^-x + s, & (x) - oflhomecthbie npeflhkatfai. /? (_&:,,. ^., sc,^, onpenenehhoe B ^, HaabiBaexca sne - ((jxipmynfahbim), ecnh cymectbyer tfiopmyna oc^or.,...^ zc^) cm-harypbi &, ynobnerboparoiuaa ycnobhro: t...,xn ) = ac (->?. - -,0:^ )). (1) (Bcraay -^Z.f= cc oshanaer: (fiopmyjia oc HCTHHHa B cncrsme 3r?,cx=,g e'en, (oc 23.fi ) -^ (j& =? a ) ). Bee nohbtha, nphbeaenhbie B cxarbe 6ea onpenenehhh, BSHTW HS pa- 6oTw A.H.Majibuesa fl]]. B C2D MaKHOTTOH nokaaan, MTO onepamia ^= \ (/ cno>kehha uenwx B orpahhiehhoii apwjimethke He snemenrapha, H nocxabhn Bonpoc 06 onepamw -2" = ^x-^^. flpn STOM OH nphmemui Meroa IIpH-

2 Baomc'3 BHdexHdH H pwadoax CH xextreuo "JxooHhoxBxoot? 'BHHeraoHxo oaohdexhevnei/e BHHeuen - adiio H I w^fmeif en xextreuo ixoowmjoxgoaj-i ' (I) -C,_ O X O a J"J X a 8 Pi H, 2 - < _ -2 ^- ;^BweH({idowoxaB oaogcai; H /- 2 & H H e i i a x o B d x = t i r / BUD ' H W B H - ouo HJMRHH '/O 2^- HHeiieioedxiifjC XBW - e n c j > d o? M O x a B xeoa ndii c o x e B H B d x o o B H H e m o H X O e H H e m B j o g o - y etraox 'etrjox O M I U O X H B H J O X ^^ a w n H d B i -newai/e xetrxg Z - e w e x o H O a eohhei/ - a u a d ii o ' ( ^js-'- --"x)^ a H H a r n o H X O 'S VWW31T ->KB» a qj,oe ox' C2 ^y ^/eoobi/x a OH -dbiheweire B H H a n i B a o g o - 'y oae BtraoH B I T J O X oxii/ox H Btrjox 2JUL a w n H d B x -Haweire xetrxg 2&& ewexoho a e o H H a u a tr e d ii o ' (^Jx'"~f/3C)(y ahhamohxo 'I VWWHlf osjey 2 > wwexona,_ 2 {faheuexobdxqux GH (2 -).,/ H HanHameaooo^ y waaoebh_<_ 1&& iqwaxono /7^^_. etra (7). t _ 2SLH «JHauaxoBdxqirjC oixgcsu a <?xnhedxooduobd OH, oaa ahhamed H BHOXXOHXBW ewsdoax xasxaxiqa xiqdoxosi en go ' 2JL ewaxono a XBHHamoHxo

3 ABTOMOp4>H3Mbi ajjre6pbi c oflhofi MOHapnofi onepauheft B chrnaxype 23 OSoomaa iiasecthbih Meron Oanoa, ann noicasaxenbctba oxhgiuehha A? (3.,,...^x^), onpeaejiehhoro B W2".nocxaxoHHO yica- 38Tfa abtomop(j)h3m A, H6KOXOpOH CHCX6MH ZffZ J, H6 meft ycnobhio (1). HoaroMy ana Haiueft uenn tfyxtio HayiHTb abtomop4>h3- MBI CHrreMbi <7Z H ee ynbtpactenenefl '^? ' 2. ABTOMOp<{>H3MbI CHCTeMbI 7ft. B CHCT6MS '^ BBGHeM OTHOUieHHe rue, OTHomeHHe ^ Syner OTHomeHHeM 3KBHBa,ieHTHOcTK. Hepes o6o3haihm MHOK6CTBO BC6X 3neM6HTOB, 3KBHBajl6HTHbIX JC, H 3TO MHOJK6CTBO HaaOBSM 6nOKOM TO4KH (Z~. MHOW8CTBO J\/ erca na cymmy SJIOKOB: A/ = -U {CXJ; X -JV}. flna nrosofi TOHKH 3C us yv SJIOK \^x:~] aamkhyr OTHOCHTenfaHO one H Syaer noncucremofl CHcreMbi "0?. flsa 6jioica n, ecnh nonchcrembi Qc] Ha nonchcxemy /1EMMA 3. E c n H /^.- [^3*"*'&''!] H aomop^soe oxo- 6 p a > K 6 H n e, xo o x o 6 p a > K e H H e CHcxeMbiOT H a gt1 ^ o n p e a e n a e M o e p a B e n c x B O M :, e c n H ^ - e c " «., e c n H 6ynex a B x o M O p c f i H S M O M chcxembi ^?. AoKasarenfacxBO oieahflho. BcflKHit abxomop<}>h3m CHCX6MW 'WZ OOCXaB/ieH H3 3BXOMOp(J)H3MOB BHaa <^, H 8BXOMOp<}>H3MOB 6nOKOB \J^-"2, C^D Ka>KflOMy abxomop( )H3- s~ 0 My // CHcreMbi ^ cxashm B cooxaetctbhe

4 24 A.fl.TaftMaHoa TEOPEMA 1. n y c T fa B c H C T e M jien-a 6 H H a p H a n o n e p a u H n mectbyrot a B T O M O p 4 > H 3 M W f-t HZ, y A O B J i e T B o p a i o m n e ycn H c y A. c H c T e M Torna o n e p a H H H ^ Tapna B chcrewe -32" flokasatehbctbo. Hycxb Z otc^c, ^, 2-;, TO ectb -^ /t= 5-= yfac, onpene.iaerca xorna H xontko rorna, 2 t=o<.(x, 2-) cymectbyrox abtomop<}>n3mbi, yao - BneTBOpawmne HM66M: (2).. Flpw abtomopi})h3me a: = 6 Hs ycnobha (2) cneayer:, 6) HM66M: <3,, =& C, TO ecrb <C e ' 3TO npothbopghht Hpn aaromop JleMMa AOKaaana. TEOPEMA 2. flna nio6oro MH cymectbyrot MHo^ecxso ^7~ H 4) H ji b T p _/? Han ^7" raiche, HTO M H O J K e c T B O A< y n f a T p a c T e n e H H yntrpa - O C H O B H O G P. M H O * e C T B O

5 AaxoMOp4)H3Mbi anrespbi c onnofi Monapnofi onepaimefi B cnrhaxype 25 MHOw:ecTBy ", H B c a K a a n e p e c x a H O B x a M H O w e c T B a ^ ' p a c n p o c T p a H H e T C H no a B- T O M O p ( j ) H 3 M a C H C X e M bl ~ffz i-*'. floicaaaxenbcxbo sxoft xeopembi ochobano na cnenyioihhx nemmax 4-9. fljin npocxotbi H3no>KeHHa MH paccmoxphm cjiynafi c^exhoft CHrnaxypti H npeunonowhm T ^ % 0. J1EMMA 4. E c n n B c n c x e M e ^? H M e e x c a 6 e c K O H 6 H H o e H H C J I O S n o K O B, TO ana n K>- 6oft M O I U H O C T H ^?*- cymectayer yjibrpa - c T e n e H fa ^Z x a K a a, H T O B M H O J K e c T - B e I TfZ. ' /^Ll S n o K o a c H c x e M b i ^? ' M o JK H o B b l S p a X b M H O W e C X B O ^ " M O I U H O C X H - ^ Trt i cocxosinee HS nonapno H S O M O P ^ H S I X 6 ji o K o B. " flokasate.lbctbo. Cymecxsyex He 6onee 2 nonapno Heaao - MoptJiHbix cnexhbix CHcxeM CHrHaxypw <3. Bbi6epeM MHOJKecxBO ~7~,ynbxpa<J>HJibxp 22 rak, Hxo6bi Momnocxfa ynbxpacxenehh MHowecxaaJ^. /ej7' 6bina 6ojifaine TTZ -^. Toraa Momuocxb 6noKOB ynbxpacxenehh^?' ' 6ynex 6onbiiie TTZ.- 2. nycxfav.-_/~.-^- ^/^ _ t '3K/^'\B CHC-. ^. SjieMeHxy J:=(oc>,j. jrjf ace /JT. cxashm B cooxaexcxbhe anemehx (_(f (3:- ), j' e J~), rne (f(ac-) I ecxb npoh3 snemehx us Snoxa <-P(^c-). OneBiiflHO, Hame oxospawehhe 6yaex / / -oxo6pa>kehnem MHOwecxBa /^7-/"- /, *- -,~B CHCxeMy ^^ '. KpOMe ' * xoro, paahbie anemenxw (<S(zc,-), s' J~}, (Zp(zct), SeJ) nopoknaror pas- Hbie 6noKH B tt * J' >>. nosxomy MHOMOCXBO SJIOKOB B ^? ' HMeex MOIHa.ocTbz-m-2. B KamnoM 6noKe cncxembi 7TZ ' Bw6epeM no onnomy sne- H nojiynehhoe MHOKBCXBO o6o3hannm. MomHOcxi» " Qonbuierrz -2 Flycxb JVZ MHHHManbnas aneviehxaphah noachcxema cncxembi^^ ', nopo*flehhaa MnojKecxBOM. nockojibky ^37" nopo>kneha cxonemobckhmh (JjyHKUHHMK, K3>Kflblft 6jTOK B thf? HM66X CHeXHyXD MOIUHOCXb H MHOK6CTBO 6noKOB B 'ZF7Z HMeex MomHOcxb T-T-TT. PasHtie TOIKH MHO>KecxBa nopowaarox pashwe M cnexhbie 6noKH B "&&, H noaxowty B Iff? MO>KHO BW- 6paxi> MHOwecxBO nonapno H3OMOp<J)Hbix SnoKOB, HMeroinee MOmHocxfa?-r7T. CucxeMa TtlTZ ecxb snemenxapnoe pacmnpenhe cncxemw ^7. Ho xeopeme <II, > Kncnepa 3 J cymecxayex ynbxpacxenenb -72 ', H3OMOpi})HaH yjitxpacxe-

6 26 nehh CHCTGMbl 577?. OH6BHHHO, B 0?" ~'^v "''cy UieCTBy 6T HCKOMOe MHO>KeCT- BO 6nOKOB. /1EMMA 5. FlycTfa B c H c T e M e IT? a. a H o M H o- wectbo^", o6naflaiomee c.n e a y» n H M H C B O f t C T B a M H : 1) e c n H 3:, LJ H 3 ", TO p-^d77 C^/J = C1, 2) e c n H ;ar, / H 3 ", TO Qar] n a o M o p < } ) H o f^/]. Torna B c f l K a a n e p e c T a n o B K a?t: "-*- <^ M H o - >KecTBa " M O «e T 6biTfa p a c n p o c T p a H e H a flo a B T O M O p ( } ) H 3 M a C H C T G M b l 177. HOKASATE/lbCTBO. E c n. ^.x e., TO nepea 7L tc o6o3hanhm H3OMOpij)Hoe OTo6pa>KeHHe SnoKa pr] Ha SJIOK d-^itrl]. OTO(3paweHHe J-S, onpeaenaemoe pabenctbom: HCKOMblM abtomop(j)h3mom CHCT6MbI ^7. /1EMMA 6. E C A H B C H C T 6 M 6 "P? H M 6 6 T C fl a n e M O H T r r Taxofi, H T O ana Bcexr77-c7, f^ ^c^-> "^ > O, -r-l^rrt ~-jc (x)^^(x) ito cymectbyet y n b T p a c T e n e H b ^? <J>&> c «c T e M w ^, c o a e p - waraeft 6 e c K O H e H H o e H H c n o SJIOKOB. floka3atehbctbo. B -/". B ^? ' BO3bM6M, HTO Qi^TJ f) 2/^3 = <^ nph ^ ^ / \LU< J ^ C^d- J ^ o. Torna C/^-t-S H cyniectbyrot ^ TaKHe, H /

7 anre6pbi c oahofi MOHapnoii onepauneft B curharype 27 C npyrofi CTOPOHW, ana acex 3HaieHnit Kpotvie KOHGHHOFO. CnenoBarenbHo, FIOHTH ana Bcex c (x) rionynehhoe npothbopeine nokasbiaaet, HTO ^Ct^\ p-^vl] == <-". /1EMMA 7. ECHH B ciicreme ~2~Z a.,1 a K a >K - noro uenoro n o n o w H T e n f a H o r o H H C - na ri. * O cyiuectbyer s n e M B H T sir_ T a K o ft, H T O nonapno He p a B H bi, TO c y m e c T B y e r y j i b T p a c T e n e n b W ' c H c T e M w, B K O T O p o f t ectt. a n e M B H T ^ r. y f l O B J i e T B o p a i o m H f t "yc- JIOBHKJ: ana scex»tt, -rz H a /T ^ >^T c n e a. y e T CX). AOKASATEHbCTBO. OonoMHM J~= <1 -/, 2,3,... >,./7 - ynbtpa - -,- _^ < JGJ7? B _/ H B #? ' paccmotphm 3neM6HT us /7T, -. (tt ) npvi rz. ^ rrz. UonySnoKOM srj TOHKH HaaoBeM MHHHManbHoe MHowecTBO, aamkhyroe OTHOCHTe.TbHO onepauhh JL H coaepjramee TOKKy &:. (Onepauaa j MOWOT 6biTb MHO- IlonySnoK/icJ BMecre c TOHKOft // coaep>kht ace eg npo- X7 ( _ o6pasbi 2" y TO ectb f(z) t/ BnenftT ^ e -j^-xrj. UBS TOHKH jzr, / aa- 30B6M nonyskbhbanehthwmh, ecna nony6nokh {jx:~\ \t/~^ H3OMop4>Hbi. /1EMMA 8. IlycTb cyraectbyiot uenoe HHcno n. 7- o H s n e M S H T C ^ B "^T" T a K n e, HTO.v^ (jc) = a:) A yv<2^ =^-. Torna Ana nndsofi MOUIHOCTH i^t- c y m e c T- ayex y n b T p a c T e n e a f a TfZ ' =:</V, c5>chcre MH tt H M H O J K e C T B O ^ " M O m H O C T H **i T 3 K O 6, H T o " C. A/ H B c a K a a n e p e c r a H O B K a MHOwecTBa^ pacnpoctpanaetca ao BBTO-

8 28 M O p 4 > H 3 M a C H C T 6 M bl "ZT? flokaaaxejibctbo npoboflhm HHflyKuneft no 77. FIpn T-T.= Y MHOwecx- BO A/ HMeeT ana: A/, = { He cymectfiver snemenxa ^ 6ynex nonysjiokom, n6o, o6paaom Koroporo 6ynex ^zr UneBH/iHo, cyinectbyex ynbtpactenehfa^?" ' xakan, HXO BA/=j 3*? a ^ conepwnxch MHOKSCXBO ~ MOIUHOCXH rr-i.g, ^A/,. MHOXOCTBO, cocxoamee HS nonapno HeH3OMop4)Htix QTIOKOB, HMeex Momnocxfa He 6onee 2.S. HosTOMy B MHowecxse Haftaexca MHOMGCXBO f, MOUIHOCXH Vm. H coctoaraee HS HaoMoptjjHwx Mexay co6ofi nonysnokob. Toraa aca-- Kaa nepecxahobka MHowecxBa Mower 6bixb pacnpocxpahena ao aaxo - MOp(})H3Ma VZ ' flonycthm nemma nokaaana ana T-L< f< H.^-z.=/^. Toraa MHo«ecx - BO A/ HMeex sun A/=y\/, c/ A^ cy.,. c/ A/r_/ c/ \c^\jv T0 o6pasy'et cncxeviy '&?,='(& ^ sif^ /? fir9>, ana Koxopofl nemma sepna no HHf^-i^-^.-^> aykthbhomy aonymehhio, xo ecxb cyraecxsyex ynbxpacxenenb (o J, ^ <5^ > conep«araaa HCKOMOB MHowecxao c=. MOUIHOCXH *-^z_. Torna 7>7 conep>khx { t^j -/' ' H ynbxpacxenehfa ^Z ~7' 7 yflosnexbopaex rpe6o- HH9M nemmbl. <j x o /IEAWA. 9. ECJIH cymecxbyex/-r>c >TaKoe,... v ^; =,, (3) xo nna nw6oii M O U I H O C X H»*^- cymectbyer <J;D> y n b x p a c x e n e Hfa -^Z ', c o a e p > K a m a a M H O - jkectbo^ M O I D H O C T H i^-t-.kaxflaa nepecra- H O B K a K o x o p o f t p a c n p o c x p a H a e r c a ao abtomop(j)h3ma ^2 -O -"- *. flokaaaxenbctbo ahanorhhho nokaiarenscxby nemmbi 8. IIpH npnxoflhxca paccmaxphbaxb cnyiah, Koraa B ^? ecxb UHKnw STOM

9 ABTOMOp<J>H3Mbi anre6pt.i c OflHOfl MOHapnofl onepauheft B cnrnarype29 = x, HOKA3ATEnbCTBO reopsmw 2. PaccMOtpHM pasnhhhbie CnVHAfi 1. B chctsme &Z ectb snemeht X TaKofi, HTO cnenyet >. (4) Mbi VZ B STOM cnynae no nemme 6 cymecrsyet ynfarpacrenehfa -&Z, cacre -, B Koropofi ectfa 6ecKOH6HHoe HHcno 6jioKOB. Ho nemme 4 cyiue- CTByer ynbtpacxenehb TZg chctemu ^7,, B Koropofl cymecrayet MHOJKS- CTBO MOIUHOCTH? ~m, COCTOHmee H3 nonapho H3OMOp({>HbIX SnOKOB. Ilo nemme 7 B "^^ cyiuecrbyer MHOXCBCTBO E' MOIUHOCTH ^ -m H niodyio nepecxahobky ft. - MOJKHO pacnpocrpahhtb ao abtomop4>h3ma CHC- T6MbI 1fZ CnyHAft 2. B ^ Her anemehra, yaosnetbopaiomero ycnobhio (4), & HO nna Kawnoro -ri y- O cyiuectbyer anemeirr ar_ rakoft, HTO 7-(zCn )t..(zft+o p. f^rt) nonapho HepasHbi. Torna no JISMMB 7 cymectbyer ynbrpacrenehb "97/ cncrembi TTZ, B Koropoft ecrb snemeht, ynosnetbopfliomhft ycnobhio (4). C/iyHAPt 3. (4^ HJIH ycnobhio. nemmw 7. B chcteme Her anemehtob, y aobnetbop9k>iunx ycnobhio B 3TOM cnynae nphmehflem nemmw 8, 9. Ha reopembi 1, 2 nonynaem C/lEflCTBHE 1. ECJIH onepauaa %=-V>(artc/), 3 a - n a H H a a B 6 e c K O H 6 K H o f t cacrem y a o B J i e T B O p n e r ycnobhio TO OHa He a n e M e H T a p n a B "P?. (ty a:)) } (5) C/IEflCTBHE 2. B o r p a H H n e H H o f t apa({)me- T H K e *&?.= <^A^ ^ P^ Csc) > o n e p a u H H He s n e M e H T a p u a. (TeopeMa MaKHOTTOHa). c n o > K e H H S i Z=*K+u flokasatejlbctbo. OnepauHH S' jc+e./ ynosnetbopaet ycnobhio </ (5) cnenctbha 1. CHEflCTBHE 3. B o r p a H H n e H H o f t apnijime-

10 30 A.fl.TaftMaHOB T n K e Tn:=<! A/} ' t P. f^c) > o n e p a u H H Z=2^3 ^ H e a n e- MeHxapaa. (PemeHHe npo6nembi MaKHoxxona). C/lEflCTBHE 4. B n K> 6 o & SecKOHeiHOft rpynne o n e p a u H K ) y M H o «e H H f l H e n b 3 a B bip a 3 H T b H e p e 3 onepaunio B S H T H H 06- p a x H o r o a n e M e n x a H ji K> 6 o e H H C J I O cboftctb 3JieM6HTOB rpynnfai. C/IEflCTBHE 5. flna nro6ofl 6ecKOHeHHott CHC- T e M si ^2 CHTHaTypbi H ana Jiro6oii rpynnbi<y HaftfleTca y'n fatpacxenehb. /2T ' T a K a a, HTO rpynna a B T O M O p c } ) H 3 M O B ynbrpac T e n e H H TTZ "^ conepkht rpynny & CneayeT HS reopembi KenH H reopembi 2. CHEflCTBHE 6. Una nro6ofi S e c K O H e n H o i l c n c T e M w "ZT C H r H a T y p w O,ana nio6oro c T\n G^ H a 8 a. e T - ca T a K a a yjibtpactenenfa TFZ ~'' CHcre- M H 717, H T O rpynna a B T O M O p 4 > i i 3 M O B ccu. C^Z- ' ) y n f a r p a c r e n e H H ^? > c o n e p x H T n K)6 y i o r p y n n y r H S M H O J K e c T B a ^. floka3ate/ibctbo. Dyers ^9= { 6^., c T }. BoatMeM CHcreviy nonapno HenepeceKawraHxca MHOJKGCTB < ", e Z~>,, f^: O *~ s f, ff npn t^/ n "^ = G^ nph c T. OycTb " = C/ {< ^-.,<:'< -T } ynbrpacrenehfa 92 ^-^> xak, irosbi GUL/. C'tt*-'*' ) coaep - rpynny nepecrahobok MHOJKecTsa ". T'ora.aC&-sZ(2Z ' 0 coaep - JKHT rpynny nepecxanobok MHOJKecxBa ". nph e7 H no xeopeme Kena coaepjkmx rpynny r^ npn ez riycxfa Sn - rpynna nepecxahobok MHOxecxsa Co, */, 2....,7-1 7) OxHoiaeHHe /? (^2C/ }... _, ^>j_ ), onpenenehhoe B ^? aasoaem (one - ziya 3peH<f>oftxTy C^J) caaaa HHMM, ecnn ZOTH jno6oft nocneaobaxenbhocth paanahhhx anemenxos jrro, OC/,..., jr^ us JV H3 5^ rakoft, HTO VZ f= ff (ZCS> '^scn-sj- naiiaexca anemsht.s* OrHouieHHe (x-/3...,3cn), onpenenehhoe B ^T_, HasoaeM M6Tpn«iHbiM, ecjih ana jik>6ofi nocnenobaxenfahocth paananabix ^ ot^y >'2vr H3 -^ Haftnexca snemeht 5" na 5, xaxofl, ITO

11 ABTOMOp4>H3Mfai ajire6pt>i c onhoft MOHapnoit onepaimeft B CHrnarype 31 *= 7 Cx sco},.. C/IEflCTBHE 7. B c a K o e casaahhoe H a H T H C H M M 6 T p H H H O e O T H O U I G H H e A? C^/y o n p e a e n e H H o e B "ft?, H e B b i p a s H M O M e H T a p H O. B nacthocth, OTHomeHHe nhhefthoro nopflaxa He BbipaaHMO xapno B CHCT6M6 '&Z. OrHoineHHe zc / B apni})methke-^2r He BbipasHMO snemehtapno. YcHneHHe cnencrbhh 1 nojiyiaerch HS onhoft reopemw M.A.Tafiunn na [8j ABTOP 6naronapeH KD.Jl.Epmoay, aamenahna Koroporo nosbonnnn ynyiuihtfa aanowehhe Marepnana. HocTyniuia B penakumo 18. yn r. JlHTepaxypa 1. A.H.ManfaueB. 2. B.Uc Naugton, HeKoxopbie aonpocbi reophh KJIBCCOB Moneneft. - Tpyflbi HeTBepxoro BcecoraaHoro MaxeMaTHiecKoro cieana, /lehhhrpan, T. 1, cxp Undetlnabitity of addition from one unary operator. Trans. JUer.lteth.Soc.v117, A.fl.TailMaHOB. 4-. A.EhrenTeuoht, O TeopeMe Bexa-KoneHa. -Anredpa H jiorhka. Hap, 1, N9 6 (1962), On theories categorical in power,fund.math.,44 (1957), J.Keisler, Ultraproducts und Saturated models.proc.konintt. Hederl.jUcad.Wetenach.1964,A67,N2,