mqub,fqSqhiied okhlqi dp ruduSq SeSb okhlSkhq ogERFsBCgNE nRFgBC) - - PDF document

mqub,fqSqhiied okhlqi dp ruduSq SeSb okhlSkhq

• gERFsBCgNE nRFgBC) isBAL nEBBT

cBzNEA FCNyGF BO YBVCUNK AwULFRF wAz uGF CCURywGRBAF kARINEFRGL BO mwFQRAPGBAfi iNwGGUNfi HPHFG 0”]W2fi W/0”- uuu~lwse~rtbra~dct.mahreno.idystpdr

Ω Γ f

iHCCBFN ∆ ; ρΓ RF vBEzwA yHEINfi O 9 C { Γ RF yBAOBEVwU- nJFRy CEBaUNV9 QBJ RF PNBVNGEL BO ∆ ENUwGNz GB CEBCNEGRNF BO O(

crd( ) γ

d i a m ( ) γ

γ

w z

β(γ) diam( ) γ

uO δ RF w CUwAwE vBEzwA wEy JRGQ NAzCBRAGF c> Hfi JN FNG9 | zRwV“δ? ; zRwVNGNE BO δ | yEz“δ? ; c × H ; yQBEz UNAPGQ BO δ | “δ? ; UNAPGQ BO δfi | 1“δ? ; “δ? × yEz“δ? ; NKyNFF UNAPGQ | α“δ? ; FHC·c ≡ δ 9 zRFG“c> b?.zRwV“δ?•fi b ; URAN GQEBHPQ c> H

Γ

γ

∆ RF w DtJFRyREyUN RΩ zRwV“δ? ; d“yEz“δ?? OBE δ ◦ ∆- ∆ RF ygBEM,JEy RΩ “δ? ; d“yEz“δ?? OBE δ ◦ ∆-

| rBE w yBAOBEVwU VwCfi O RF ANINE MNEBfi FB UBP O VwTNF FNAFN- | UFB RAGNENFGNz RA iyQJwEMRwA zNERIwGRIN9 R“O? ; O

O

 × 0

W

O

O

W

• | uO ∆ RF w DHwFRyREyUNfi O QwF w DHwFRyBAOBEVwU NKGNAFRBA GB CUwANfi JRGQ

zRUwGRBA / zN¨ANz BA C→ ; ·BcB < 0•- | ∆ ; O“R? RF wFFByRwGNz GB w yBAOBEVwU JNUzRAP Q ; f×0 ∩ O 9 R { R9

h g f

Ω Γ f

SgNBENV “fBVVNENATN) 0867(9 ∆ RF wFLVCGBGRywUUL yBAOBEVwUfi R-N-fi α“δ? { /> wF zRwV“δ? { /> RΩ UBP O RF RA URGGUN nUByQ yUwFF A/ ;

• f QBUBVBECQRy BA C 9 Bf“c?B

0 × BcB { /

• ,

Ω Γ f

SgNBENV “fBVVNENATN) 0867(9 ∆ RF wFLVCGBGRywUUL FVBBGQfi R-N-fi 1“δ? yEz“δ? ; “δ? × yEz“δ? yEz“δ? { /> wF zRwV“δ? { /> RΩ UBP O ≡ lce-

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ

qwyQ BO OBHE yBUHVAF RF w GQNBENV PRIRAP 4 NDHRIwUNAG yBAzRGRBAF- YBAzRGRBAF xNyBVN VBEN ENFGERyGRIN VBIRAP UNOG GB ERPQG-

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ

fBVVNENATN=F GQNBENV9 UBP O ≡ A/ ∗ ∆ wFLVCGBGRywUUL yBAOBEVwU-

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ

fBVVNENATN=F GQNBENV9 UBP O ≡ lce ∗ ∆ wFLVCGBGRywUUL FVBBGQ-

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ

FGwUw]pRAFVNRFGNE GQNBENV9 UBP O ≡ nce ∗ BR“O?BW“0 × BcBW?2zcKzL RF YwEUNFBA, / RF w YwEUNFBA VNwFHEN RO /“Z“K> D?? ; d“D?- nce wEN QBVBVBECQRy OHAyGRBAF FHyQ GQwG Bf“c?BW“0 × BcBW?zKzL RF YwEUNFBA

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ

Y/“C? ; ·O 9 yBAGRAHBHFfi GNAzRAP GB / •> Yc“C? ; ·O 9 BOBW“0 × BcBW?×0zKzL RF w YwEUNFBA VNwFHEN •> Yc/“C? ; ·O 9 BOBW“0 × BcBW?×0zKzL RF IwARFQRAP YwEUNFBA•> bW“zπ? ; ·O 9

• C

BO“c?BW“0 × BcBW?×WzKzL j ∈•> π ; QLCNExBURy wENw

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ

jBzwL=F GwUT RF wxBHG ZRERyQUNG FCwyN ; ·r 9

• Br BWzKzL j ∈•-

jwTQGwSwA wAz jNB zN¨ANz VNGERy BA HARINEFwU jNRyQV+ HUUNE FCwyN “⇒ FCwyN BO DHwFRyREyUNF? GQwG VwTNF RG w zRFyBAANyGNz tRUxNEG VwAROBUz- eAN yBAANyGNz yBVCBANAG RF yUBFHEN BO GQN FVBBGQ yHEINF- qUNVNAGF wEN ywUUNz mNRU,fNsNEFFBA yHEINF: GQNL wEN ENyGR¨wxUN DHwFRyREyUNF-

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ

| ∆ RF mf ∗ ∆ QwF ¨ARGN bBNJANE NANEPL “hBQzN wAz mwAP? | bwEPN zNIRwGRBAF BO ibq“ε? wF ε / “mwAP?- | YBAANyGRBAF GB nEBJARwA UBBC FBHC BO bwJUNE wAz mNEANE “mwAP?- | YBVCHGNE IRFRBAfi VBECQRAP “iQwEBA wAz cHVOBEzfi rNRFMUR wAz dwEwLwA? | ’mf - - - yBEENyG wAwULGRy yQBRyN OBE GQN OBEVHUwGRBA BO Yrj-5 “hwzANUUfi iyQRCCNEF wAz iGwHxwyQ? | YBAANyGRBAF GB ENABEVwURMNz wENwfi DHwAGHV NAGwAPUNVNAGfi zF.Yrj-

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ ∆ RF mf ∗ UBP O RF RA ZRERyQUNG yUwFF “jwTQGwSwA wAz jNB? ∗

• BcBj0

BR“O?“c?B“0 × BcB?zKzL j ∈ “YHR? ∗

• BcB<0

B/“c?Bzπ j ∈ “YHR? ∗ UBP Q ≡ t0. “iQNA? t0.“R? ; ·f 9

• R
• R

Bf“c? × f“H?B Bc × HB BzHBBzcB j ∈•,

UBP O A2 nce lce ZRERyQUNG “UBP O ?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? R“c?“0 × BcB? Y2“C? Yc“C? Yc2“C? b“zπ? / Y2“C? Yc“C? Yc2“C? b“zπ? Q ; f×0 ∩ O FLVVNGERy FGEBAPUL DHwFRFLVVNGERy UBP Q ≡ lce UBP Q ≡ t0. ∆ ; O“R? wFLVCGBGRywUUL yBAOBEVwU nRFQBC]vBANF yBAzRGRBA wFLVCGBGRywUUL FVBBGQ

mQwG PNBVNGERy yBAzRGRBA PBNF RA GQN NVCGL xBK( nce ywFN [ fNGNE vBANF= GEwINURAP FwUNFVwA GQNBENV “jij? PRINF w QRAG-

SEJINURAP iJUNFVJA SgNBENV “vBANF(9 ∆ RF ENyGR¨wxUN RΩ “∆? zRwV“∆? [

• zLwzRy g

αW

∆“g?zRwV“g? j ∈,

iHV RF BINE zLwzRy FDHwENF RA CUwAN-

diam(Q) β Γ 3Q Q

α∆“g? ; zRwV“g? RAO

b FHC·c ≡ ∆ ≤ 2g 9 zRFG“c> b?•

α VNwFHENF UBywU zNIRwGRBA OEBV w URAN-

fNGNE vBANF wAz u CEBINz “0””3? GQwG w DHwFRyREyUN ∆ RF ENyGR¨wxUN RΩ

• Γ

BO“c?BBR“O?“c?BW“0 × BcBW?2zKzL j ∈ nL GQN jij GQRF RF NDHRIwUNAG GB

• g

αW“g?zRwV“g? j ∈,

fNGNE vBANF wAz u CEBINz “0””3? GQwG w DHwFRyREyUN ∆ RF ENyGR¨wxUN RΩ

• Γ

BO“c?BBR“O?“c?BW“0 × BcBW?2zKzL j ∈ nL GQN jij GQRF RF NDHRIwUNAG GB

• g

αW“g?zRwV“g? j ∈, nL aBNxN=F zRFGBEGRBA GQNBENV “OBE O“c? ’ANwE5 g? BO“c?B“0 × BcBW? zRwV“g?,

fNGNE vBANF wAz u CEBINz “0””3? GQwG w DHwFRyREyUN ∆ RF ENyGR¨wxUN RΩ

• Γ

BO“c?BBR“O?“c?BW“0 × BcBW?2zKzL j ∈ nL GQN jij GQRF RF NDHRIwUNAG GB

• g

αW“g?zRwV“g? j ∈, nL aBNxN=F zRFGBEGRBA GQNBENV “OBE O“c? ’ANwE5 g? BO“c?B“0 × BcBW? zRwV“g?, ’YwAyNURAP5 PRINF

• BR“O?“c?BW“0 × BcBW?WzKzL j ∈

∗

• g

αW“g? j ∈, mN wUENwzL TABJ GQwG bti yQwEwyGNERMNF mNRU]fNGNEFFBA yUwFF-

SgNBENV9 ∆ RF mNRU]fNGNEFFBA RΩ RG RF w DHwFRyREyUN wAz

• g

αW“g? j ∈,

SgNBENV9 ∆ RF mNRU]fNGNEFFBA RΩ RG RF w DHwFRyREyUN wAz

• g

αW“g? j ∈, jQRF RF ’¨ARGN GBGwU yHEIwGHEN5- ZNIRwGRBA OEBV ,wGANFF RF FDHwEN RAGNPEwxUN BINE wUU FywUNF wAz UBywGRBAF- YBAGwRAF wUU YW yHEINF- YERGRywU FVBBGQANFF ANwE α“g?

• zRwV“g?-

UUBJF RA¨ARGN FCREwUF GB ByyHEfi FB ABG Y0- jQNEN wEN VwAL NDHRIwUNAG yQwEwyGNERMwGRBAF- ; xBHAzwERNF BO ¨ARGN GBGwU yHEIwGHEN FHEOwyNF RA QLCNExBURy 2]FCwyN-

uMNJ BO CEBBO9 UBP O RA ZRERyQUNG yUwFF RVCURNF O RF ’VBFGUL5 yBAFGwAG ANwE xBHAzwEL- rBUUBJ fBVVNENATN=F NFGRVwGNF GB FQBJ RVwPNF BO FQBEG xBHAzwEL wEyF wEN wUVBFG URAN FNPVNAGF wAz αW j ∈- YBAINEFNULfi GERwAPHUwGN BAN FRzN BO ∆ wAz EN,NyG INEGRyNF HFRAP URANF wC] CEBKRVwGRAP ∆- fb NKGNAFRBA PRINF w gY EN,NyGRBA JRGQ B/“c?B α-jQNA

• αW j ∈

⇔

• B/BWzπ j ∈

⇔ mf,

T v* v T* Γ

hNywUU GQN jij9 RO ∆ ◦ DW RF w vBEzwA wEyfi GQNA “∆? zRwV“∆? [

• g

αW

∆“g?zRwV“g?,

hNywUU GQN jij9 RO ∆ ◦ DW RF w vBEzwA wEyfi GQNA “∆? zRwV“∆? [

• g

αW

∆“g?zRwV“g?,

tNEN RF w FURPQG RVCEBINVNAG9 “∆? ; yEz“∆? [ d  

g

αW

∆“g?zRwV“g?

•  ,

BE 1“∆? ≈ “∆? × yEz“∆?

• g

αW

∆“g?zRwV“g?,

cBzR¨ywGRBA BO w TABJA CEBBO “N-P-fi YQwCGNE 0/fi nRFQBC]fNENF?-

SgNBENV9 ∆ RF mNRU]fNGNEFFBA RΩ RG RF yQBEz]wEy wAz 1“δS? “δS? j ∈

SgNBENV9 ∆ RF mNRU]fNGNEFFBA RΩ RG RF yQBEz]wEy wAz 1“δS? “δS? j ∈ iHV RF BINE w VHUGR]ENFBUHGRBA OwVRUL BO FHxwEyFfi N-P-fi GwTN w zLwzRy zN] yBVCBFRGRBA wAz GERCUN UNAPGQ BO NwyQ wEy- zLwzRy zNyBVCBFRGRBA BO ∆ yBAFRFGF BO FNGF C0 ◦ CW ◦ ± ± ± ◦ ∆ FB GQwG CA zRIRzNF ∆ RAGB WA wEyF BO UNAPGQ W×A“∆?- rBE w yUBFNz yHEINfi GQRF zNCNAzF BA GQN RARGRwU zRIRFRBA RAGB GJB CRNyNF-

SgNBENV9 ∆ RF mNRU]fNGNEFFBA RΩ RG RF yQBEz]wEy wAz 1“δS? “δS? j ∈

• BAsRAtBtF INEFRBA9 ∆ RF mf RΩ RG RF w DHwFRyREyUN wAz
• ∆}∆

1“c> H? Bc × HB2BzHBBzcB j ∈ 1“c> H? ; 1“δ? OBE FQBEGNE FHxwEy δ yBAANyGRAP c> H-

∆A RF CBULPBA PRINA xL AGQ PNANEwGRBA zLwzRy CBRAGF- jQNA “∆A? ⊂ “∆?-

∆A RF CBULPBA PRINA xL AGQ PNANEwGRBA zLwzRy CBRAGF- jQNA “∆A? ⊂ “∆?- SgNBENV9 ∆ RF mNRU]fNGNEFFBA RΩ RG RF w DHwFRyREyUN wAz

∈

• A;0

WA )“∆? × “∆A?fl j ∈

jQN cNAPNE ytEIJstEN BO GQENN CBRAGF K> L> c ≡ DW RF y“K> L> c? ; 0.h JQNEN h RF GQN EwzRHF BO GQN yREyUN CwFFRAP GQEBHPQ GQNFN CBRAGF- bNG “K> L> K? ; BK×LB[BL ×cB[Bc ×KB ; CNERVNGNE BO GERwAPUN “K> L> c?-

x y z R

SgNBENV9 ∆ RF mf RΩ RG RF yQBEz]wEy wAz FwGRF¨NF

• ∆
• ∆
• ∆

y“K> L> c?W “K> L> c? BzKBBzLBBzcB j ∈,

SgNBENV9 ∆ RF mf RΩ RG RF yQBEz]wEy wAz FwGRF¨NF

• ∆
• ∆
• ∆

y“K> L> c?W “K> L> c? BzKBBzLBBzcB j ∈, uG RF TABJA GQwG

∆

• ∆
• ∆

y“K> L> c?WBzKBBzLBBzcB j ∈, RΩ

• g

αW“g?zRwV“g? j ∈, jB PNG wxBIN ENFHUGfi zRIRzN xBGQ FRzNF xL w FywUN OwyGBE-

uO ∆ RF ENyGR¨wxUNfi RG QwF w GwAPNAG zRENyGRBA ∂“c? ≡ R w-N- BA ∆-

z z τ( ) w w τ( )

SgNBENV9 ∆ RF mNRU]fNGNEFFBA RΩ RG RF yQBEz]wEy wAz

• ∆
• ∆
• ∂“c? × ∂“H?

c × H

• W

BzcBBzHB j ∈, uA BGQNE JBEzFfi ∂ ≡ t0.W“∆? “iBxBUNI FCwyN?-

fEBBO HFNF zRFyENGN wAwUBP FHVVNz BINE VHUGR]ENFBUHGRBA OwVRUL9

• S

“∆S?

• ∆S

B∂“c? × ∂“∆S?BWBzcB j ∈ rwREUL NwFL9 ZRFyENGN INEFRBA ⇔ yBAGRAHBHF INEFRBA ⇔ αW j ∈ ’ αW j ∈ ⇔ zRFyENGN INEFRBA5 HFNF VwEGRAPwUNF-

θ θ

β

itCCBFN GQwG VRzCBRAG BO w zLwzRy wEy RF BA xRFNyGBE BO RGF NAzCBRAGF- jQNA ANJ GwAPNAG zRENyGRBAF zRΩNE xL \τ ; d“α? OEBV BUz BAN- ⇔ CBULPBAwU GwAPNAGF zN¨AN w VwEGRAPwUN BA ∆ - αW j ∈ ⇔ bW xBHAzF BA VwEGRAPwUN- cwKRVwU OHAyGRBAF wAz VwEGRAPwUN yBAINEPNAyN RVCUL ENFHUG-

1

θ

2

θ

∆ β

uA ENJURsLfi VRzCBRAG RF ABG BA xRFNyGBEfi xHG RF JRGQRA 1“∆S? BO xRFNyGBE- YQwAPN RA wAPUNF wEN ABG NDHwUfi xHG FwGRFOL Bτ0 × τWB ∞ α“∆S? ± 1“∆S?.“∆S?, iB CBULPBA wAPUNF wEN w FVwUU CNEGHExwGRBA BO w VwEGRAPwUN- jQRF FHffyNF-

1

θ

2

θ

∆ β

mQNA JN xRFNyG wEy xL UNAPGQfi QwEVBARy VNwFHEN “OBE NRGQNE FRzN? RF wUFB xRFNyGNz HC GB w OwyGBE yBAGEBUUNz xL FHVF BO α=F wAz 1=F- jQHF xRFNyGRAP QwEVBARy VNwFHEN BA BAN FRzN wUVBFG xRFNyGF OBE BGQNE FRzN- jQRF CEBINF iQNA=F GQNBENV9 mf ∗ UBP Q ≡ t0.W, uF GQNEN w QRPQNE zRVNAFRBAwU wAwUBP(

:hBtAM3 αF9 bNG γ∆“g? xN GQN FVwUUNFG γ < / FB GQwG 2g RAGNEFNyGF zRFTF Z0> ZW FNCw] EwGNz xL ∆fi xBGQ JRGQ EwzRHF zRwV“g?.γ wAz zRFG“Z0> ZW? ∞ γ ± zRwV“g?-

Q diam(Q)/ ε ε diam(Q) Γ

rBE FRAPUN gfi JN QwIN α“g? γ“g?fi xHG ABG yBAINEFNUL-

SgNBENV9 ∆ RF mf RΩ RG RF yQBEz]wEy wAz FwGRF¨NF

• g

γW

∆“g? j ∈

γ“g? CEBIRzNF w URAT GB QLCNExBURy PNBVNGEL-

tLCNExBURy VNGERy GQN HARG xwUU RF zπ ;

zF 0×BcBW-

uA GQN HCCNE QwUO]FCwyN D2

[ ; ·“K> L> G? 9 G < /•fi VNGERy RF zπ ; zF.WG,

sNBzNFRyF RA wEN zRwVNGNEF BE yREyHUwE wEyF CNECNAzRyHUwE GB ρ sNBzNFRyF RA D2

[ wEN INEGRywU EwLF BE FNVR]yREyUNF CNECNAzRyHUwE GB DW-

Yt“∆? zNABGNF GQN QLCNExBURy yBAINK QHUU BO ∆ rBE w yHEINfi Yt“∆? RF GQN HARBA BO wUU PNBzNFRyF JRGQ NAzCBRAGF RA ∆-

rBE w yREyUNfi GQRF RF w QNVRFCQNEN- uA PNANEwUfi Yt“∆? QwF ABA]NVCGL RAGNERBE- jQNEN wEN GJB xBHAzwEL FHEOwyNFfi NwyQ wFLVCGBGRy GB ∆-

iHCCBFN Γ RF vBEzwA zBVwRA JRGQ xBHAzwEL ∆- jQN zBVN BO Γ RF HCCNE NAINUBCN BO wUU QNVRFCQNENF JRGQ xwFN zRFT RA Γ- hNPRBA wxBIN zBVN RF RAGNEFNyGRBA BO QwUO]FCwyNFfi QNAyN yBAINK- Yt“∆? RF ENPRBA xNGJNNA zBVNF BO ’RAFRzN5 wAz ’BHGFRzN5 BO ∆-

kCCNE wAz UBJNE xBHAzwERNF BO QLCNExBURy yBAINK QHUU BO w FDHwEN-

bNG β“c? xN GQN QLCNExBURy zRFGwAyN GB OwEGQNE xBHAzwEL yBVCBANAG-

δ( ) z CH( ) Γ z

rBE c ≡ Yt“∆?fi β“c? VNwFHENF ’JRzGQ5 BO yBAINK QHUU ANwE c- β“c? ; / RΩ ∆ RF w yREyUN “QHUU QwF AB RAGNERBE?- rBE DHwFRyREyUNFfi β“c? ≡ b∈ “ABG yBAINEFNUL?-

SgNBENV9 DHwFRyREyUN ∆ RF mf RΩ β“c? ≡ bWfi R-N-fi

• ρYt“∆?

βW“c?zπ“c? j ∈,

SgNBENV9 DHwFRyREyUN ∆ RF mf RΩ β“c? ≡ bWfi R-N-fi

• ρYt“∆?

βW“c?zπ“c? j ∈, jQHF ∆ RF mf RΩ RGF QLCNExBURy yBAINK QHUU RF wFLVCGBGRywUUL ’GQRA5- rREFG PNBVNGERy yBAzRGRBA GQwG RF BxIRBHFUL yBAOBEVwUUL RAIwERwAG- ’jQRyTANFF5 BO yBAINK QHUU wUFB RVCBEGwAG RA jNRyQV+ HUUNE GQNBEL wAz aUNRARwA PEBHCF- cNwFHENF ’yBVCUNKRGL5 BO GQN DHBGRNAG 2]VwAROBUz-

uMNJ BO CEBBO9 kFN γW j ∈ wAz bNVVJ9 uO GJB zRFTF BO zRwVNGNE 0.θ wEN zRFGwAyN θ wCwEG RA CUwANfi GQNA GQNRE zBVNF wEN QLCNExBURy zRFGwAyN θ wCwEG-

ρ ∼ ε 1/ε 1/ε ε

D1 D2

• BEBUUJEL9 β“c? γ“g? RO c RF QNRPQG zRwV“g? wxBIN g-

SgNBENV “cRTN AMNEFBA) 0872(9 qINEL vBEzwA ∆ ◦ DW RF xBHAz] wEL BO wG UNwFG BAN VRARVwU FHEOwyN R RA D2

[-

jQN VRARVwU FHEOwyN R QwF RA¨ARGN QLCNExBURy wENw9 VRARVwU VNwAF VRAR] VwU HAzNE yBVCwyG IwERwGRBAF- uVCBEsJAs rJys9 AL FHyQ VRARVwU FHEOwyN RF yBAGwRANz RA Yt“∆?-

WAz OHAzwVNAGwU OBEV a“c? VNwFHENF yHEIwGHEN BO wA NVxNzzNz FHEOwyN- hBHPQUL aW ; wW [ xW RO FHEOwyN RF PEwCQ BO wKW [ xLW UBywUUL- SgNBENV “AMENJ iNCCR) W-0”(9 rBE w VRARVwU FHEOwyN R GEwCCNz xNGJNNA ABA]RAGNEFNyGRAP CUwANF zRFGwAyN β wCwEG ANwE cfi a“c? ; d“β?-

WAz OHAzwVNAGwU OBEV a“c? VNwFHENF yHEIwGHEN BO wA NVxNzzNz FHEOwyN- hBHPQUL aW ; wW [ xW RO FHEOwyN RF PEwCQ BO wKW [ xLW UBywUUL- SgNBENV “AMENJ iNCCR) W-0”(9 rBE w VRARVwU FHEOwyN R GEwCCNz xNGJNNA ABA]RAGNEFNyGRAP CUwANF zRFGwAyN β wCwEG ANwE cfi a“c? ; d“β?-

• BE9 uO ∆ RF mf GQNA RG xBHAzF w FHEOwyN JRGQ ¨ARGN GBGwU yHEIwGHENfi R-N-fi
• R

aW“c?zπ“c? j ∈,

SgNBENV “ogJEUNF qCFsNRA) 087”(9 uO R RF w FHEOwyN JRGQ BaB j 0fi GQNA GQN swHFF VwCF sSfi S ; 0> W zN¨AN w DHwFRyBAOBEVwU EN,NyGRBA wyEBFF ∆ JRGQ zRUwGwGRBA B/“s“c??B ; d“a“c??- swHFF VwC ; OBUUBJ ABEVwU PNBzNFRyF OEBV FHEOwyN GB xBHAzwEL-

SgNBENV “ogJEUNF qCFsNRA) 087”(9 uO R RF w FHEOwyN JRGQ BaB j 0fi GQNA GQN swHFF VwCF sSfi S ; 0> W zN¨AN w DHwFRyBAOBEVwU EN,NyGRBA wyEBFF ∆ JRGQ zRUwGwGRBA B/“s“c??B ; d“a“c??- uO R RF QwF ¨ARGN GBGwU yHEIwGHENfi GQNA

• B−∆ B/BWzπ j ∈- jQHF9

SgV9 ∆ RF mf RΩ RG xBHAzF w VRARVwU FHEOwyN BO ¨ARGN GBGwU yHEIwGHEN-

t

R QwF RA¨ARGN QLCNExBURy wENwfi xHG w ’ENABEVwURMNz wENw5 JwF RAGEBzHyNz xL hBxRA sEwQwV wAz qz mRGGNA9 XG ; X ≤ ·“K> L> c? 9 c < G•> h“X ? ; URV

G/

• π“XG? × π“ρXG?
• hNUwGNz GB GQN zi.Yrj yBEENFCBAzNAyN RA FGERAP GQNBEL wAz DHwAGHV

NAGwAPUNVNAG “N-P-fi wAwUBPBHF GB FHEOwyN wENw BO NINAG QBERMBA VNwFHERAP NAGEBCL BO w xUwyT QBUN?-

rRARGN ENABEVwURMNz wENw RF yUBFNUL ENUwGNz GB ¨ARGN GBGwU yHEIwGHEN- UNKwTRF wAz cwMMNB CEBINz GQwG RO ∆ ≡ Y2>χfi GQNA h“X ? j ∈ wAz h“X ? ; ×W굓X ? × 0 W

• X

Ba/BWzπ> JQNEN a/ RF GEwyN OENN WAz OHAzwVNAGwU OBEV “; a OBE VRARVwU FHEOwyNF?- jQHF JN NKCNyG9 ρR ≡ mf ⇔

• R

aWzπ j ∈ ⇔ h“R? j ∈> ats mf yHEINF wEN GBB EBHPQ GB wCCUL GQNRE ENFHUG “ABG NINA Y0?-

rRARGN ENABEVwURMNz wENw RF yUBFNUL ENUwGNz GB ¨ARGN GBGwU yHEIwGHEN- UNKwTRF wAz cwMMNB CEBINz GQwG RO ∆ ≡ Y2>χfi GQNA h“X ? j ∈ wAz h“X ? ; ×W굓X ? × 0 W

• X

Ba/BWzπ> JQNEN a/ RF GEwyN OENN WAz OHAzwVNAGwU OBEV “; a OBE VRARVwU FHEOwyNF?- jQHF JN NKCNyG9 ρR ≡ mf ⇔

• R

aWzπ j ∈ ⇔ h“R? j ∈> ats mf yHEINF wEN GBB EBHPQ GB wCCUL GQNRE ENFHUG “ABG NINA Y0?- tBJNINEfi GQN ENFHUG RG RF FGRUU GEHN wALJwL9 SgV9 R VRARVwU [ ρR ≡ mf ⇔ R QwF ¨ARGN ENABEVwURMNz wENw-

uzNw RF GQwG yLURAzNEF n ; ∆ } “/> 0fl QwIN ¨ARGN ENABEVwURMNz wENw9 π“nG? ;

• ∆

G

DWzDzF ; “∆?“0 G × 0?> π“ρnG? ; 0 G“∆? ⇔ h“n? ; π“nG? × π“nG? ; ד∆?

ZN¨AN w zLwzRy yLURAzNE RA GQN HCCNE QwUO]FCwyN9 n ;

∈

• A;0

∆A } )W×A> W×A[0?> JQNEN ·∆A• wEN RAFyERxNz zLwzRy CBULPBAF RA ∆- ZRFyENGN wAwUBP BO VRARVwU FHEOwyN JRGQ xBHAzwEL ∆-

eHE NwEURNE NFGRVwGN

• A

WA““∆? × “∆A?? j ∈ RVCURNF ∆ RF mf RΩ GQN zLwzRy yLURAzNE QwF ¨ARGN ENABEVwURMNz wENw-

bNVVJ9 uO ∆ RF mNRU]fNGNEFFBAfi GQNA GQN URVRGF URV

G{/)π“RG? × π“nG?fl>

URV

G{/)π“RG? × π“nG?fl>

xBGQ NKRFG wAz wEN ¨ARGN- fEBBO9 BEGQBPBAwUUL CEBSNyG n { R: HFN α> β GB NFGRVwGN wENw.UNAPGQ-

SgV9 R VRARVwU [ ρR ≡ mf ⇔ R QwF ¨ARGN ENABEVwURMNz wENw-

SgV9 R VRARVwU [ ρR ≡ mf ⇔ R QwF ¨ARGN ENABEVwURMNz wENw- ZBNF yBAINEFN QBUz( OBEVHUw BO UNKwTRF]cwMMNB VwL QNUC9 π“RG? × 0 G“ρRG? ; ×W굓RG? × 0 W

• RG

BaBWzπ ×

• ρRG

“εG × 0?zπ

jtdai je qlqhoedq reh buijqduds nqij muitqi je vetd dZ Zed