Ale Mthets 3 4 7-93 htt://oog/436/34746 Pulshe Ole Jul 3 (htt://wwwsog/joul/) Soe Euvlet Fos of Beoull s Ieult: A Suve * u-chu L Cheh-Chh eh 3 Detet of Ale Mthets Ntol Chug-Hsg Uvest Tw Detet of Mthets Ntol Cetl Uvest Tw 3 Detet of Ifoto Mgeet Lughw Uvest of See Teholog Tw El: l@ethhueutw ghe@thueutw gho85@glo eeve M 3; evse Jue 3; ete Jue 9 3 Coght 3 u-chu L Cheh-Chh eh Ths s oe ess tle stute ue the Cetve Coos Attuto Lese whh ets uestte use stuto eouto eu ove the ogl wo s oel te ABSTACT The uose of ths e s to l soe ow eultes whh e euvlet to Beoull s eult Kewos: Beoull s Ieult; oug s Ieult; Jese s Ieult; Höle s Ieult; Cuh s Ieult; Mows s Ieult; Shlöh s Ieult; AGM Ieult; Josthl s Ieult; Euvlet Itouto Bse o the theo of eultes lssl eultes ot ol oote the eveloet of the eult theo ut lso le to ltos ue thets le thets Beoull s eult s oe of the ost stgushe eultes I ths e ew oof of Beoull s eult v the ese oet s gve Soe stegthee fos of Beoull s eult e estlshe Moeove soe euvlet eltos etwee ths eult othe ow eultes e tettvel le The ogzto of ths e s s follows: I Seto ew oof of Beoull s eult es of the oet of est s se I Seto 3 soe stegthee fos of Beoull s eult e estlse I Seto 4 we l soe ow eultes whh e euvlet to Beoull s eult I Seto 5 we ollet soe vts of oug s eult whh e euvlet to Beoull s eult Fo elte esults we efe to [-35] Peles I oe to olete these tss we ee the efto soe s esults of the ove futo s follows: Defto Let f : I e futo whee I s tevl of * Dete to the esete Pofesso Huo Mu ) Suose tht P Q e two ots o the gh of f f the ho PQ ot elow the PQ of the gh of f the we s tht f s ove futo o I Tht s fo two ot I t f t t tf t f () the f s ove futo o I We s tht f s lle ove o I f f s ove o I If fo two ots I wth t f f t t tf t f the we s tht s sttl ove futo o I ) I s s to e ot ove o J-ove o I f fo two ots I f f f () It s well-ow ft tht eve ove futo o tevl s otuous; f f s -ot ove otuous o tevl I the t s ove o I The followg Jese s eult e show the thetl uto etl Le (Jese s eult [3] ge 3) Let f e ove futo o I The fo wth fo I Coght 3 Ses
-C LI C-C EH 7 f f (3) If f s sttl ove the (3) s sttl uless the e ll etll Le 3 Let f : I e futo The the followg stteets e euvlet: ) f s sttl ove o I ) Fo two stt ots I I stsfg f f f ; 3) Fo two stt ots I I stsfg f f f Poof ) ) Let I e stt let e t If z I the z s etwee z It follows fo the stt ovet of f o I tht f f z f Hee ) hols ) 3) Let I e stt let e t If z I the z oth z e stt B the ssuto of () we hve f f f z It follows fo tht 3) hols 3) ) Let I e stt let e t If z the z It follows fo the ssuto of 3) tht f f f z Ths ove ) hols Thus the oof s olete Net we wll ove Beoull s eult es of the oet of est wthout ffeetto o tegto Le 4 foll wth (4) The eult s ovous fo se = o fo se o Poof Let E : Cl : E s ese It suffes to show tht E stsfes the followg thee oetes ) E ) If E the E 3) If E the E E Let e t wth The Thus So 4 < Ths oves ) hee E s oet If E the Ths oves ) Net f e suh tht E the fo eve wth Ths oves the fst t of 3) O the othe h t follows fo E tht tht Theefoe (5) Thus we olete the oof of 3) Se )-3) l E fo Coght 3 Ses
7 -C LI C-C EH 3 Theefoe E ust e ese Fll f s t E wth the fo eve s Ths oves Sll f fo s t E wth the fo eve s Ths oves fo Theefoe fo eve we hve fo wth It follows fo (5) g tht (4) hols Ths oletes the oof Cooll 5 The followg stteets e euvlet: ) l s sttl ove o ; ) fo ll > wth fo ll ; 3) oug s eult hols tht s X X whee X > wth X wth ; 4) (4) hols Poof hols tht s foll wth foll fo ll wth fo ll fo ll wth fo ll hols tht s fo ll wth fo ll l s sttl ove o tht s hols X X fo X wth X fo ll wth ; hols The eult of oug s eult s le fo se X wth X Ths oletes the oof Net we ove soe euvlet esults whh e elte to l : Le 6 Fo the followg stteets e euvlet: ) l s sttl ove o ; ) l s sttl ove o ; 3) l s sttl ove o Poof Clel ) ) 3) Now we ove 3) ) ) ) Let t e wth e t Se let l t t l t l we hve l tt l tl t l l l Thus f t s sll suh tht t t we ot tht 3) les ) Sll f t s eough lge so tht t t we ot tht ) les ) Ths oletes the oof Le 7 Let stsfg = e ll ostve o ll egtve If fo ll wth the Poof Ths le s tue fo ssuto Suose tht ths le hols fo Let If the lel the oluso hols Now we ssue Se e ll ostve o ll egtve we see tht Theefoe Coght 3 Ses
-C LI C-C EH 73 Ths oletes ou oof 3 Vts of Beoull s Ieult I ths seto we estlsh soe vts of Beoull s Ieult Se l s sttl ove sttl esg o ts vese futo e s sttl ove sttl esg Usg Le 7 we hve the followg Theoe 3 The followg eultes e euvlet: l s sttl ove o ; whee tht s whee < stsf whee whee ; tht s whee stsf whee tht s whee ; whee 3 ; whee ; ; whee ; tht s whee ; whee 3 3 tht s whee ; whee 3 4 4 5 5 6 6 whee ; whee ; tht s whee ; whee > ; whee ; whee ; whee ; whee ; whee ; whee ; Coght 3 Ses
74 -C LI C-C EH 7 7 8 8 9 whee > ; whee ; whee ; whee < ; whee ; 9 whee Poof Let : whee the ove futo : hols : l l l foll wthfoll fo ll fo ll fo ll fo ll z z fo ll z fo ll hols Let WLOG we ssue > > The : s sttl Now we ssue hols fo Set We hve fo wth Ths oves Let It follows fo ove guet the uto ssuto tht s ovous Moeove t follows fo Le 7 tht 3 3 3 3 : B hols Coght 3 Ses
-C LI C-C EH 75 hols : hols whee whee whee : whee hols : hols whee whee : whee hols : hols 3 3 whee whee > > hols 3 : It follows fo hols whee 3 tht whee hols : hols whee whee 4 4 4 > whee whee Coght 3 Ses
76 -C LI C-C EH 4 hols : hols 4 4 hols : hols whee whee whee > 5 3 5 whee whee whee : hols 3 It follows fo 5 : whee tht hols whee whee hols : hols 6 5 6 whee whee whee hols 5 : hols 6 3 6 whee whee whee : whee : 5 Coght 3 Ses
-C LI C-C EH 77 3 hols : hols 7 7 whee < whee whee : hols : hols 7 7 : > whee < whee whee : hols : hols 8 8 : < < whee whee whee : hols : hols 8 8 : whee < whee whee : hols : hols 9 3 9 : > whee whee whee : : Coght 3 Ses
78 -C LI C-C EH 3 hols : hols 9 3 9 whee whee : whee 3 hols Ths ove ou Theoe B Theoe 3 we hve the followg Cooll 3 Let e ostt If the the followg thee eultes e euvlet: ) ) 3) Poof Clel t follows fo Theoe 3 tht hol; ) hols ) hols 3) hols 3 4 M esults hol; hol Now we stte ove soe eultes whh e euvlet to eh othe the followg t 3 Theoe 4 Let whee s ostve tege The the followg soe stteets e euvlet: whee ; whee ; whee ; whee ; whee whee ; ; whee o ; whee ; whee ; whee ; 3 3 whee ; whee o ; 3 whee ; whee ; 3 Coght 3 Ses
-C LI C-C EH 79 3 whee ; 4 4 5 6 whee ; whee whee 7 l s ove o 7 whee ; 8 9 ; whee ; whee ; Thus 3 whee ; ; ; whee hee s esg futo of whee ; whee < < ; hee whee o ; whee ; 4 whee ; 4 f 4 (Cuh s eult); whee ; M M whee s 5 6 s M wth (Höle s eult); : whee whee > > < ; Coght 3 Ses
8 -C LI C-C EH Hee I tul s 6 s = = s (Shlöh s eult); 6 fo ; = 6 fo ; 7 t M M I geel ; 7 7 (Mows eult) (Mows eult) hee t t t t t ; (AGM eult) Sho s eult: 7 (see [7]) G j A j j j j l l ; j j whee j A ; j see the followg fgue: whee j j G 7 j j j j e 7 G G G AA A j j j j A G A : G whh s euvlet to G H ; whee 8 8 ; 8 whee e ostve teges s tol ue; 8 whee ostve tege ; 8 whee ; whee e 8 9 e e e ; s (sttl) esg o ; j ; tht s Coght 3 Ses
-C LI C-C EH 8 9 9 9 s (sttl) esg o ; e s (sttl) eesg o ; whee e l ; e whee t hs followg soe vts: e whee l whee 3 e whee 4 e o e whee 5 e 6 whee e whee l 7 l 8 9 whee whee l whee l whee o e whee l whee 3 l whee e e e whee 4 5 l e whee ; l whee tht s l whee ; l whee ; l whee o ; tht s l ; e e whee ; tht s e e whee ; whee ; whee e e ; tht s (Josthl s eult); Coght 3 Ses
8 -C LI C-C EH ( ) > whee o Poof Tg Cooll 3 we see tht 7 8 9 e euvlet Sll elg Cooll we get tht 3 e euvlet Hee t follows fo Theoe 3 tht - 3 9 e euvlet If the lel 3 3 3 3 4 4 wth = 3 follows tg = + Hee : We see tht hols hols ff > Sll we ove 3 3 follows fo Theoe 3 3 3 follows fo Theoe 3 too : Let stsf 4 B we see tht Thus t follows fo tht Hee 4 hols elg esetvel Sll we ove 3 : Let : whee the tht j It follows fo j j Hee j j j Ths oletes the oof of 3 6 see H et ([8] Theoe 9 6) A tht : It follows fo 8 A A A A A A A A Thus A A (see Mlgs [8] o oo [8]) Hee A A A G 8 hols 8 : Tg Theefoe we see tht Hee Dvg oth ses we get 8 s le : We show the thetl uto o If s ovous Suose hols fo wth Set = + If t s es to see tht eh the ssuto hee Theefoe hols Assue Se we hve j Coght 3 Ses
-C LI C-C EH 83 Thus hols : Tg 3 8 3 we see tht 8 hols 3 3 : Clel 3 3 Let suh tht : If : the t follows fo tht (5) B Clel B (5) 3 Thus 3 3 3 3 s ove : Let If the : B 3 Ths oletes the oof of 3 7 7 : Wthout loss of geelt we ssue tht > t Se l s sttl esg t t tt tl t l l t t B the efto of the ove futo 7 7 7 7 9 : Tg see tht hols : Let The 7 7 9 : Let The 7 9 : Let The Hee so 3 4 elg 7 7 : If the 3 Thus s ove 4 Sll we ove : Let s 3 6 elg 4 3 the s s s 3 7 we Coght 3 Ses
84 -C LI C-C EH Hee s s M M fo s s M M fo s s f M Ms s If the M Ms s Hee the M M fo s s tht It follows tg 6 hols Sll we ove 4 6 5 6 4 4 : Let A elg A 4 esetvel fo we ot 4 thus we olete the oof 4 4 follows tg = = 4 4 4 : Let f fo 4 tht fo F The t follows F = = F Thus l F s ove o hee l F s ove o Hee fo l F l F l F whh les F F F Lettg the oth ses of the ove eult F F Ths shows tht 4 hols see L Shw [5] 4 5 : Let > : : whee B 4 tht It follows fo (6) jj j Hee 5 hols : Let 5 4 The = = j Thus 5 whee e efe s ove Hee Thus = = j j j= j= j j j j j j Ths oletes the oof of 4 j j (6) 6 4 : Tg s elg 6 6 6 6 follows tg : Tg s 6 fo elg 6 fo we ot 4 6 fo thus Coght 3 Ses
-C LI C-C EH 85 we olete the oof 4 6 : see 55 of Mtov [9] Sll we ove 4 6 6 3 follows tg s 6 6 4 follows tg s 6 6 5 follows tg s 6 6 8 follows tg 6 6 8 follows tg 6 6 7 7 7 (wth 7 7 : elg 7 3 tht oth ) (wth 7 8 7 we ove 7 ) e le Sll we ove 7 7 : Tg t t t we see esl 7 7 Hee 7 3 e euvlet : Wthout loss of geelt we ssue tht A j 3 Ths oletes the oof of 7 A A A A A A A A A A A A A A A A A A j j A A A j Thus 7 7 : Tg the tul logth the oth ses of 7 we get 7 7 we get 7 7 8 : Tg 7 we get G H 7 7 : Tg 7 we see tht 7 hols tul logth of the oth ses of e : Tg : Tg 7 7 e 7 8 7 we get e 7 e 7 usg the followg fgue we get 7 Covesel eletg the 8 3 Coght 3 Ses
86 -C LI C-C EH 8 8 8 8 s le : Tg 8 we get 8 8 follows tg (see [33]): It follows fo tht 8 8 8 8 : Let The 8 Thus 8 8 8 Clel f the the ove eult hols too 8 : Clel f the hols Suose tht 8 hols fo Thus fo f the Hee Ths 8 e 8 7 Thus : If the 7 Thus hols 9 : 9 e olete the oof of e Theefoe e 8 el Let The Thus hols tht s whee hols Theoe 9 : 9 hols tht s e l e s sttl esg futo o whee hols Theoe 3 9 9 : 9 hols tht s : s sttl esg futo o s sttl esg futo o 8 Coght 3 Ses
-C LI C-C EH 87 9 hols Thus 9 s sttl esg futo o s sttl esg futo o 9 9 e euvlet 9 9 : Fo ll se 9 9 I tul fo ll > e euvlet 4 4 ohes to s Thus e l 9 9 we get 9 see [] 9 : B the fst eult of 9 Hee e e B the seo eult of 9 Hee e e It follows fo eh fo tht fo e If o the lel hols Ths oletes the oof of 9 Hee : B e e tht s the fst eul- t of 9 hols Net Hee f the e e whee : Thus the seo eult of 9 hols s le 8 : Tg of A l A A whee Sug ths eultes we get 8 hols 8 : e e A A G A A see Bulle ([3] 7) o Kug ([4] 33) : B follows tg follows tg l l e Thus e e elg we oletes the oof see Clou Dh ([6] 3) e s le 8 : B G e e G Coght 3 Ses
88 -C LI C-C EH A G G Thus 8 Hee e e e hols see [3] [9] 8 9 e euvlet we lso efe to [] : Wthout loss of geelt we 8 ssue tht Hee B Thus 8 hols e l l 8 : Fo > t follows fo e e e Tg e e 8 see H et ([8] 4-4) o Wg Su Wg [33] : Tg Hee tht 8 : Tg A we get A A Hee A A A G Ths oletes the oof of 8 see Bulle ([3] 98) o Kug ([4] 33) follows tg = fo = 8 7 : B 8 l l l Hee l s ot ove o Se l s otuous o l s ove futo o Thus 7 hols We lso ove 8 8 usg the thetl uto Thus ou oof s olete 5 Othe Euvlet Fos of Beoull s Ieult I ths seto we shll ollet soe vts of oug eult whh s euvlet to the Beoull s eult Theoe 5 Let s e ostve ues fo If whee the el ues stsf the the followg soe eultes e euvlet: whee ; whee ; whee ; whee ; whee ; whee ; t wheet ; t t wheet ; t t wheet ; t Coght 3 Ses
-C LI C-C EH 89 t 3 t whee t ; t 3 t whee t ; 3 t whee ; t t 4 4 4 t t whee t ; t t whee t ; t t t 5 5 whee ; t wheet ; t t wheet ; t 5 t wheet ; t 6 s whee ; s 6 s whee ; s 6 s whee ; s 7 7 7 t t whee ; t t whee ; t t 8 9 9 whee ; whee ; l e whee ; e whee ; whee ; whee ; whee ; Coght 3 Ses
9 -C LI C-C EH h h h h whee h ; h h h h h h whee ; h h h h h h h whee ; h h whee ; whee thee ests etl oe of s ostve the othe e egtve; 3 whee stsfg ; 3 whee stsfg ; Poof Clel euvlet s le e vt of : elg 8 Hee Thus : Let Ths oletes ou oof 7 9 8 esetvel Hee t t esetvel we get whee The whee : Fo ll stsfg Ths ove the oof of follows tg : Let see lso Su [3] stsf The fo e Coght 3 Ses
-C LI C-C EH 9 Thus hols follows elg s le Sll we ove If the h h h h e ove sll follows usg the thetl uto Sll 4 4 6 6 7 t : elg t whee follows elg : elg t t whee : elg s : elg s : elg whee h : elg h h esetvel 4 esetvel we get t t esetvel Covesel esetvel whee Sll we ove esetvel we get 6 esetvel we get 6 t t esetvel we get 7 Sll we ove 7 h h : It follows fo Theoe 3 tht 3 elg 3 tht we ssue tht It follows fo esetvel we get whee e euvlet Wthout loss of geelt 3 esetvel whee 3 hols 3 3 5 3 whee 5 3 : elg t : elg t : elg t t we get 3 whee t 3 esetvel we get 5 t we get 3 follows elg t 6 follows tg whee t 5 s 6 whee Sll we ove 3 esetvel whee Coght 3 Ses
9 -C LI C-C EH get 6 6 follows elg s s esetvel 8 9 : Se l l l e 9 9 : Tg 9 we get 9 9 9 : Lettg u s s 9 e l Ths oletes the oof of 9 9 s vt of 3 3 3 3 follows tg h e s e se s theefoe 8 we get 9 A the tg sl we see Su [3] whee : B Theoe 3 3 e euvlet It suffes to show 3 elg We ssue the elg 3 tg hols 3 : It follows fo esetvel t follows fo tht 3 tht elg we olete ou oof Sll we ove 3 3 3 e le e 5 Fo eult 6 we efe to Isuo Tog [3] Fo eult 8 we efe to [3] Fo eult 9 9 we efe to Su [3] we efe to Kug Fo eult [4] Fo eult Fo eult 3 we efe to [34] we efe to Su [3] e 53 Thee e vts of Höle s eult Shlöh s eult AGM eult Mows s eult so o we ot the etl EFEENCES [] E F Beeh Bell Ieult 4th Eto Sge-Velg Bel 984 [] E F Beeh W Wle Geel Ieultes III Bhäuse Velg Bsel 983 [3] P S Bulle Hoo of Mes The Ieultes Kluwe Ae Pulshes Doeht 3 o:7/978-94-7-399-4 [4] P S Bulle D S Mtovć P M Vsć Mes The Ieultes D eel Pulshg Co Doeht 95 [5] P S Bulle A Chte o Ieultes Southest As Bullet of Mthets Vol 3 979 8-6 [6] M J Clou B C Dh Ieultes wth Altos to Egeeg Sge Velg New o 998 [7] C Geogs O the Ieult fo the Athet Geoet Mes Mthetl Ieultes Altos Vol 5 5-8 [8] G H J E Lttlewoo G Pól Ieultes Eto Cge Uvest Pess Cge 95 [9] Z Ho Note o the Ieult of the Athet Geoet Mes Pf Joul of Mthets Vol 43 No 99 43-46 [] J How J How Euvlet Ieultes The College Mthets Joul Vol 9 No 4 988 35-354 [] K Hu Soe Poles of Alt Ieultes ( Chese) Wuh Uvest Pess Wuh 3 [] C A Iftozz A Itouto to eltos og Ieultes Notes of the Ae Mthetl Soet Vol 4 97 A98-A8 [3] S Isuo M Tog Estto Höle s Te Ieult Mthetl Ieultes Altos Vol 4 63-87 [4] J Kug Ale Ieultes ( Chese) 3 Eto Shog See Teholog Pess Shog Coght 3 Ses
-C LI C-C EH 93 4 [5] -C L S Shw A Poof of Höle s Ieult Usg the Cuh-Shwz Ieult Joul of Ieultes Pue Ale Mthets Vol 7 No 6 [6] C K L Cove Futos Jese s Ieult Legee Tsfoto ( Chese) Mthe Ae S Vol 9 995 5-57 [7] C K L The Essee Sgfe of Cuh- Shwz s Ieult ( Chese) Mthe Ae S Vol 4 6-4 [8] L Mlg Wh Höle s Ieult Shoul Be Clle oges Ieult Mthetl Ieultes Altos Vol 998 69-83 [9] A W Mshll I Ol Ieultes: Theo of Mjozto Its Altos Ae Pess New o 979 [] D S Mtovć Alt Ieultes Sge-Velg Bel 97 [] D S Mtovć J E Pečć A M F Clssl New Ieultes Alss Klue Ae Pulshe Doeht 993 o:7/978-94-7-43-5 [] D S Mtovć J E Pečć Beoull s Ieult eot el Colo Mteto Pleo Vol 4 No 3 993 37-337 [3] D J New Athet Geoet Ieult The Ae Mthetl Mothl Vol 67 No 9 96 886 o:37/3946 [4] N O Oze M K Ao Ieultes ( Jese) 3 Eto M Shote Too 967 [5] J E Pečć O Beoull s Ieult A Nu Ujet Bos Heegov Oelj Po Mt Nu Vol 983 6-65 [6] J Pečć K B Stols Cle s Ieult: Hsto New Geelztos Aeutoes Mthete Vol 6 No - 49-6 o:7/s56 [7] J Pečć S Vše A New Poof of the Athet Me The Geoet Me Ieult Joul of Mthetl Alss Altos Vol 5 No 997 577-578o:6/j997566 [8] J oo Soe New Poofs fo the AGM Ieult Mthetl Ieultes Altos Vol 7 No 4 4 57-5 [9] N Shuege A Coote Aoh to the - GM Ieult Mthets Mgze Vol 64 No 4 99 73 o:37/69837 [3] S-C Sh Covet Dl Uvest of Teholg Pess Dl [3] X H Su O the Geelze Höle Ieultes Soohow Joul of Mthets Vol 3 997 4-5 [3] C L Wg Ieultes of the o-poovu Te fo Futos The Altos Joul of Mthetl Alss Altos Vol No 984 436-446 o:6/-47x(84)99- [33] X T Wg H M Su F H Wg Ieultes Theo Methos ( Chese) He Euto Pulto Zhegzhou 967 [34] J We W Wg H Zhou Z g A Clss of Cl Ieultes of Jous Te ( Chese) Joul of Chegu Uvest Vol 3 5-9 [35] C X Xue Isolto Eteso of Beoull Ieultes ( Chese) Joul of Gsu Euto College Vol 3 No 3 999 5-7 Coght 3 Ses