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İSABUL EKİK ÜİVESİESİ FE-EEBİYA FAKÜLESİ GAUAI PJE SIPLE SUPEGAVIY I IAY SPAE- IE P.v IEUWEHUIZE 90 Ykup Eül 090970 prtt : prtt of Physs Egrg SPIG 00

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HAKS I oul lk to thk to y fly for thr support pt; to Ör Fruk yı for hs tolr support o out stuyg o ths projt; to s lu for hs vlul vss log susso sssos ot oly out th projt ut lso physs; to İsl Hkkı uru Alkr Alv for th sur shool o osology Grvtto t Fz Gürsy sttut; to ll of y frs splly Çğty for thr frshp hlps.

ES ABSA HAKS ES SYBLS A AIS v. HAPE I.. Wss-Zuo ol. HAPE II.. Explt proof of gug vr EFEEES APPEİES A. G trs rvto of So Itts A. rvto of orso Equto A. rvto of Suprsytry rsfortos 6 A. Frz rrgt 0 A. Vrto of Hlrt Equto

SYBLS A AIS g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g EHEst-Hlrt Srt-Shgr tsor s totlly tsytr tsor (Lv-hvt L Lgrg sty L x L Lgrg I Lx Ato h v

. HAPE I.. Wss-Zuo ol Wss-Zuo s th splst o of glol suprsytry th slr A psuoslr B sp ½ fl []. h to s su of th Kl-Goro tos th r to / L ( ( B A ( hs to s glolly (y glol s ostt suprsytr t s vrt ur th follog st of trsforto ruls A B / ( A B ( Hrt gry. o sho th vr of th to o tks th vrto of th to L ( A ( A ( A L ( ( B ( B ( B ( L ( / / ( hr [ / ( A B ] :hrg ojugto trx ( A / / B [] ( / A ( ( / B ( / A ( ( / B

/ / B / A / B A / ( A B. ( hrfor L / ( A B / / ( A B / / ( A B / // ( A B (_ so th totl vrto of th to s L L L L ( A ( ( B ( / ( A B // ( A B // ( / / / g > / /. ( hr So th totl vrto os L [( A ] ( A [ ( B ] ( B [ / ( A B ] / ( A B ( ( B( A. rt th so l of th quto( s [ / ( B ] A / ( A B [ ( g ( A B ] // A ( / / B [ ( A B ] / ( A B ( A ( B ( [ ] (6

th t s sy to s th ltos q. ( (th th proprts. [A q.(] ; At th th vrto of th Lgrg os s totl rvtv L K hr ( ( K / A B. (7

. HAPE II.. Explt Proof of Gug Ivr I ths hptr thr ll gv th xplt proof of gug vr of th gug to. Fro o o ll put κ. L hr EH ( s th tsor s th tsor. h vrto of th ttrs spors sp oto s [A] (usg th. orr forls 0 ( If x s th orry vrl th x x x ( ( A ut f you hv trx th A A ( A h trt of th... ζ......! trx ζ s of ( ( ( A th vrto of xplt ttrs of Hlrt to s. hrfor (. ( L EH ( ( ( (

[ ] [ ] G G G G G L G G (s lso [A] ( h S to ts vrto th rspt to r s follos L ( L ( frst tr s totl rvtv so o (* ll l ltr th o fs (usg [ ] [ ] [ ] [ ] υ (* (* hr [ ]. [A q.s(9 9] L (*

(* (* L ( hr [ ] [ ] [ ] (6 [A q.(-]. L ( [ ( ] [A q.(0]. hr ( [ ( ] ( ( ( [ ]( (7 hr ( hr [A q. (]. L ( [ ] L ( ( ( ( [( ( ( ] [( ( ( ] 6

7 [ ] [ ] [ ] og so rrrgts th th s ls to [ ] G s [A q.(] L G ( th trs th th Est tsor qutos ( ( lrly l. L L [ ] (9 hr

[ ] so th totl vrto of L os s follos: L rst L L rst hr L (0 th th rst of th totl vrto s (fro q.s 9 0 L rst ( s th so tr os th frst tr os L rst ( s κ th torso quto [A q.7] th th so tr os L rst (-

9 osr frst tr [ ][ ] us. rrrg (Frz rrrgt [A] th prtrs k th th slr orr to th so o so th frst tr os [ ][ ] j j rplg th sr P < so th frst tr [ ] [ ] spors r t-out hr [ ] P P r [ ] [ ] [ ][ ] [ ][ ] [ ][ ] [ ][ ] [ ][ ] [ ][ ] 6 6 * 6 6 6 6 6 (- osr th frst tr of th q.(-.

( ( ( ( ( ( 0. ( hr us (th rvto of ths tty s [ ppx ] so th frst tr s vshs. h f o tks look th th trs fs ( ( A q. ( ( ( ( ( 0. [ ] ( ( ( A q. ( ( ( ( 0. [ ] ( So th th th trs r lso vshs. h thr tr s lso vshs us of th tty 0 [A q. (]. Usg th tty [A q. (] for th tr th o fs th rst of totl vrto s (/ L ( ( ( rst ( ( ( hr so q. ( vshs. hs olus th proof of gug vr. It hs sho tht th to q.( s vrt ur q. ( []. 0

EFEEES [] P. v IEUWEHUIZE Suprgrvty PHYSIS EPS 6 o. (9 9-9.

APPEIES A. G trs rvto of So Itts h r trs us stsfy { } < g th th. All fv trs r Hrt. A ss for trs s gv y th 6 lts ( th < ( hh stsfs tr( j j. W f [ ] jor rprstto:. 0 0 0 0 0 0 0 0 ( hr k r th Pul sp trs. portt thk s g trs th Lt s r ostt ut th Grk s o hs. h trx s lys ostt. So tts thr rvtos: h trspos of g tr s f y [] hr s th hrg ojugto trx ( ot tht hv us st of th Wss- Zuo ol. For or susso out s th Apx B Appx of []. ( ( ( ( ( (

( (spors r to. ( hr so tly qvl or. ( v s ( v s v s (spors r tout (-

( ( ( ( ( ( s v ( ( ( ( ( ( ( ( ( v th q.(- os ( s (- (- [ ] ( ( ( ( ( ( ( ( ftr so lto o gts (- ftr so rrgts th s o gts (- th q.( os ( t s sy to s th ltos [ ] ( (

s f y t ultplyg oth ss y! x y!! x y ( g ( g ( g ( g x x y y * x y th ( g ( g x y x y th (. ( h vrto of tsor th rspt to sp oto s (usg th ftos ( [A] ( ( ( ( ( ( ( ( ( ( ( ( ( ( I th lst rkt hgg th s gvs us sg for h tr (sp oto s tsytr. h shoul lso rrg th uy s to stsfy th otrto (org to orthst southst ovto thrfor t os

6. (6!! 6 6 [ ] [ ] 6 6 6 6 6 (7!!

6 6 [ ( ( ] 6 [( ( ] 6 6 6 ( ( 6 ( ( ( (7 s totlly tsytr tsor ( g g g ultply g g g ( g g ( g g ( g g ( [ ] (9 ( ( ( ( ( ( 7

so of trs l th o fs [ ] [ ] (9 hr [ ] l l k k l k k l [A q.] [ ] l l k lk l k k k k k l k l l l k lk l k k k k k k l l l l k k l l l l k k k k

9 [ ] [ ]. (9 So [ ]. (9 th usg thtrsf. (0 hr. [Aq.(].! z y x z y x! *! g g g z y x g g g z y x!! (

0 [ ] [ ] [ ] [ ] 0 0 ( ( (

A. rvto of orso Equto I ths ppx y solvg th fl quto of th sp oto th torso u y grvtos ll vlut. h Hlrt to s f y [] L hr ( κ ( EH o rrt ths to s th follog oto L ( hr κ ( ( κ κ κ ( ( ( hr ( [A]. Ur th follog rrrgt th Hlrt to os: κ L EH ( κ κ ( (. ( h ftos of tsor th ovrt rvtv of sp oto ttr r []

( xt th vrto (th rspt to sp oto of L EH s L ( ( EH hr κ.[a] h otug th lulto of th vrto of Hlrt-Est to gvs L ( κ κ og so rrgts th th s th sot thr of th ov quto ls to th rsult κ κ. κ ( r κ κ κ ( ( ( ( κ ( th frst tr s totl rvtv so th rst L ( ( rst (6 κ κ og th follog rrrgts th so tr ls to

κ κ L κ ( ( ( ( ( κ κ. (7 h rt-shgr to [] s L ( S hr ( so th vrto of sp oto s L ( ( L ( ( (. (9 S rtt s vtor trs xl vtor trs y oposg. hr [A]. ( (. hr [A q(]. So ( (0

( ( L ( ( (. ( ( ( ( ( ( og th rrgt ( ( osrg th so tr of th so tr of th ov quto ls to ( L osr th frst tr q.( ( ( th [A q.(]. ( ( 0 ( ( (. ( L (

L ( plus L shoul l h othr[] so ( 0 κ Hr L ( (q.7 rtt s L ( { ( ( } κ (. (6 κ h (fro q. ( ( κ ( ( ( ( κ κ ( (7

A. rvto of Suprsytry rsfortos Assotg th vry grtor gug fl prtr s follos [] V P κ Q ( Λ P Q g V Λ Λ [ V Λ ] ( th supr-por lgr s [] [ rs ] r s s r r s s r ( [ Ps rs ] r Ps spr [ P ] 0 ( P ( [ Q P ] 0 [ Q Q ( ] { Q Q } ( P. ( Applyg th ruls f q.( to th supr-por lgr gvs V P κ Λ [ V Λ ] Q P k rs P κ Q Pk Q rs P Q k k [ P ] κ [ Q P ] [ P P ] k k k rs κ rs [ P ] [ ] [ Q ] rs rs rs k rs 6

[ P Q ] [ Q ] κ[ Q Q ] Q so of th trs vshs ous of th supr-por lgr th [ P ] k P k rs κ rs [ P ] [ ] [ Q ] rs rs rs [ Q ] κ[ Q Q ] Q ( h frst l of th ov quto s ots th oort trslto prtr th so l ots Lortz prtr th thr l ots suprsytry prtr. h lst thr s ofto us of th supr-por lgr so [ Q Q ] Q Q Q Q rs Q Q Q Q [ Appx ] Q Q Q Q ( Q s tout [] Q Q Q Q fro q.( { Q } Q { Q Q } ( P [ Q Q ( Q Q Q Q ] { } { } ( P P 7

( P ( ( P P k V P k κ rs rs rs Q k ( P P r s rs ( r Q s r s ( ( k P k P k P rs r s s κ κ rs rs rs P r r s r Q s rs rs rs rs s r s r k P ( s s r r s s κ r Q ( r ( s ( r ( P Q k ( k P r s κ rs [ r s ] Q rs r Ps rs (6 κ ( P Q P

h frst l hs th oort trslto prtr so h grl oort trsf.: 0 0 (7 h so l hs th Lortz prtr so Lol lortz trsf. : r s r s ( κ (7 th thr l hs th suprsytry prtr so th suprsytry rsf. : κ κ ( ( 0 (7 9

0 A. Frz rrgt χ χ χ ( osr P ( : trl s : outr s Exp P o oplt ss (of r trs P s r. j j ( j j j j P r r P r ( rplg y o ots th rsult: χ χ χ ( hr ovto P P P r SW E / ( P (... ( : ( (totl # 66 ( o to go y futo us τ τ τ τ τ χ χ χ χ χ Μ Ν Μ Ν Μ Ν ΝΜ Ν Μ ( ( ( f o fs th s th 6 opot s

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A. Vrto of Hlrt Ato L EH ( L EH hr g g g g. ( h L EH G G L G G (