1 - Reunion4 - EV
July 3, 2022 | Author: Anonymous | Category: N/A
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’Gi k` mnbdkn gk fnrrd`s sufurfnk`s gi gs uk l`bfrg hgmgktg äi kukmn t`mn in ogktg m`k ins bnk`s gi gs tnk dkhgpgkhdgktg.....‒ Mlnriy Onrmân Munrtn Vgukdýk Gspnmd`s Sgmt`rdnigs. Murs` >.
]ubn g Dktgrsgmmdýk hg sufgspnmd`s Snb`s n gbpgznr p`r vgr quä pnsn sd rgnidznb`s m`k sufgspnmd`s ins `pgrnmd`kgs bîs gigbgktnigs gktrg m`keukt`s qug s`k in dktgrsgmmdýk y in ukdýk.
Vgm`rhgb`s, p`r ins huhns5 ]d N ]d N y y F s`k h`s m`keukt`s munigsqudgrn5 F s`k
2 {x /x ∌ N y x y x ∌ F } N ∤ F 2
2 {x /x ∌ N ` N ∥ F 2 ` x x ∌ F }
°^uä pnsn munkh` i`s m`keukt`s s`k sufgspnmd`s0 °@ftgkgb`s `tr` sufgspnmd` ni mnimuinr gi m`keukt` qug rgsuitn hg dktgrsgmnr h`s sufgspnmd`s ` hg ukdr h`s sufgspnmd`s0 Qnrn gegbpidmnr, bdrgb`s gi mns` pnrtdmuinr hg fusmnr gk V= in dktgrsgmmdýk hg h`s pink`s qug pnsnk p`r gi `rdogk.
Gk V= , in dktgrsgmmdýk hg h`s pink`s, qug m`ktdgkgk ni `rdogk, gs ukn rgmtn qug pnsn p`r gi `rdogk.
@fsgrvnmd`kgs5 n. ]d ]d ] ] > , ] 6 ⊆ S s`k sufgspnmd`s hg S ⇖ ] > ∤ ] 6 gs uk sufgspnmd` hg S.( ]ýi` tgkgb`s qug hgb`strnr qug sg mubpigk ins trgs m`khdmd`kgs.) M`b` ] M`b` ] > y ] 6 s`k s`k sufgspnm sufgspnmd`s d`s 1S ∌ ] > y 1S ∌ ] 6 ⇖ 1S ∌ ] > ∤ ] 6 . ]d ]d u u > y u 6 ∌ ] > ∤ ] 6 gst` qudgrg hgmdr qug5 p`rqug ] > gs sufgspnmd`. u > , u 6 ∌ ] > ⇖ u > + u 6 ∌ ] > , p`rqug p`rqug ] 6 gs sufgspnmd`. u > , u 6 ∌ ] 6 ⇖ u > + u 6 ∌ ] 6 , p`rqug Q`r i` tnkt` u tnkt` u > + u 6 ∌ ] > ∤ ] 6 . Q`r õitdb`, sd u sd u ∌ ∌ ] > ∤ ] 6 y κ ∌ J ⇖ κu ∌ ∌ ] > y κu ∌ ∌ ] 6 (°p`r quä0) Iugo` κ u ∌ ∌ ] > ∤ ] 6 Hgb`strnb`s qug5 ]d ]d ] ] > , ] 6 ⊆
S
s`k sufgspnmd`s hg S ⇖ ] > ∤ ] 6 gs uk sufgspnmd`.
]d X X ⊆ S gs uk sufgspnmd` hg S tni qug X qug X ⊆ ] > y X ⊆ ] 6 , gkt`kmgs X gkt`kmgs X ⊆ ] > ∤ ] 6 . f. ]d N vgmgs sg hdmg qug ] > ∤ ] 6 gs gi sufgspnmd` "bîs ornkhg‒ dkmiudh` n in vgz gk ] > y ] 6 , pugs gk ] muniqudgr `tr` qug m`ktgkon gigbgkt`s qug gstîk gk ] gk ] > y gk ] 6 , gstî dkmiudh` gk in dktgrsgmmdýk.
In hgb`strnmdýk gs dkbghdntn, pugs sd X sd X ⊃ ] > y X ⊃ ] 6 , sd x ∌ ∌ X ⇖ x ∌ ] > y x ∌ ] 6 ⇖ x ∌ ] > ∤ ] 6 , gk t`kmgs X t`kmgs X ⊆ ] > ∤ ] 6 . Sgnb`s nl`rn qug pnsn m`k in ukdýk hg sufgspnmd`s. @trn vgz bdrgb`s uk mns` buy sgkmdii` sgkm dii`55 in ukdýk hg h`s rgmtns qug m`ktdgkgk ni `rdogk gk V6 .
In õkdmn m`khdmdýk kgmgsnrdn pnrn pr`fnr qug uk m`keukt` gs uk sufgspnmd` qug k` sg mubpig sdgbprg, gs in hg sgr mgrrnh` pnrn in subn. Q`r gs` sg hgkg in subn hg sufgspnmd`s5
Hgkdmdýk5 ]d ] > y ] 6 s`k sufgspnmd`s hg uk gspnmd` vgmt`rdni S, sg iinbn subn hg ] > y ] 6 Hgkdmdýk5 ni m`keukt`5 m`k s > ∌ ] > y s 6 ∌ ] 6 } ] > + ] 6 2 {v ∌ S /v 2 s > + s 6 , m`k s
@fsgrvnmd`kgs5 n. ] > + ] 6 gs uk sufgspnmd`. Xgkgb`s qug pr`fnr qug sg mubpigk ins trgs m`khdmd`kgs qug mnrnmtgrdznk n uk sufgspnmd`. ]d u > ∌ ] > + ] 6 y u 6 ∌ ] > + ] 6 tgkgb`s qug Gk dkbghdnt` qug 1S 2 1S + 1S , ∌ ] > + ] 6 .]d u
∌] >
∌] 6
mlgqugnr sd u > + u 6 ∌ ] > + ] 6 . Qgr` sd u > ∌ ] > + ] 6 ⇖ gxdstgk s > ∌ ] > y s 6 ∌ mlgqugnr gxdstgk t > y t 6 , tni qug u qug u 6 2 t > + t 6 . y i` bdsb` sumghg m`k u m`k u 6 ∌ ] > + ] 6 , gxdstgk t Iugo`5
] 6 , u > 2 s > + s 6
u > + u 6 2 (s > + s 6 ) + ( t > + t 6 )
2
+
(s > + t > )
∌] > pugs
gs sufgspnmd`
∌ ] > + ] 6 .
(s 6 + t 6 )
∌] 6 pugs
gs sufgspnmd`
Xnrgn pnrn gi l`onr hgb`strnr in tgrmgrn m`khdmdýk.
f. ]d ]d ] ] > 2 ogk {v > . . . v j b } ⇖ j } y ] 6 2 ogk {w > , . . . , w b ⇖ ] > + ] 6 2 ogk {v > , . . . v j b } j , w > , . . . , w b m`b` ] > 2 ogk {v > . . . v j ⇑⇖ ⇖ v 2 s > + s 6 , pgr` m`b` ] Qugs v Qugs v ∌ ] > + ] 6 ⇑ b }, j } y ] 6 2 ogk {w > , . . . , w b 2 ζ ζ > w > + ¿ ¿ ¿ + ζ b ∌ J. 2 ξ ξ> v > + ¿ ¿ ¿ + ξj v j s > 2 b w b b : m`kξ> , . . . , ξj , ζ > , . . . , ζb j y s 6 2 2 ξ> v > + ¿ ¿ ¿ + ξj v j + ζ ζ > w > + ¿ ¿ ¿ + ζ b Gkt`kmgs v Gkt`kmgs v 2 s > + s 6 2 ξ b w b b j +
m`bf. idkgni hg v hg v > ,...v j b j ,w > ,...,w b
Q`r i` qug ] qug ] > + ] 6 2 ogk {v > , . . . v j b }. j , w > , . . . , w b
m. (] > ∥ ] 6 ) ⊆ (] > + ] 6 ). Gs hdrgmt`, qughn m`b` tnrgn. h. X`h` sufgspnmd` qug m`ktdgkg n ] n ] > ∥ ] 6 dkmiuyg tnbfdäk n ] n ] > + ] 6 . ]gn x ]gn x ∌ ] > + ] 6 y sgn X sgn X uk uk sufgspnmd` qug m`ktdgkg n ] n ] > ∥ ] 6 , m`b` m`k s s > ∌ ] > y s 6 ∌ ] 6 , m`b` X m`b` X m`ktdgkg m`ktdgkg n ] n ] > ∥ ] 6 s > ∌ X X y s 6 ∌ X X ,, x ∌ ∌ ] > + ] 6 ↖ x 2 s > + s 6 m`k m`b` nhgbîs X nhgbîs X gs gs sufgspnmd` ⇖ s > + s 6 2 x ∌ X X .. X , p`r i` tnkt` qughn hgb`str hgb`strnh` nh` qug Hgb`strnb`s qug ∁x ∌ ] > + ] 6 ⇖ x ∌ X ] > + ] 6 ⊆ X . ]g hdmg qug ] qug ] > + ] 6 gs gi "bgk`r‒ sufgspnmd` qug dkmiuyg n ] n ] > ∥ ] 6 , pugs muniqudgr `tr` sufgspnmd` qug m`ktgkon n in ukdýk c`rz`snbgktg m`ktdgkg n ] n ] > + ] 6 .
Gegbpi` sdbpig5 Hnh`s i`s sufgspnmd`s hg V8 , ] 2 {x ∌ V8 /x ∖ x + x 2 1 , x + x 2 1 } y > 6 8 6 = ] 6 2 ogk {Z> 1 > 1RX , Z1 1 > >RX } gkm`ktrnr gkm`ktrnr ] ] > ∤ ] 6 y ] >> + ] 6. Vgs`iumdýk5 Gbpgmgb`s p`r fusmnr i`s pukt`s gk m`bõk hg ] > y ] 6 5 ξZZ> 1 > 1RX + ζ Z1 1 > >RX , m`k ξ, ζ ∌ ]d ]d x x ∌ ] 6 ⇖ x 2 ξ ∌ V. ]d nhgbîs gstî gk ] > , tdgkg qug mubpidr sus gmunmd`kgs. Gkt`kmgs fusmnb`s ξ, ζ ∌ ∌ V tni X ξ + + ζ ζ ζ R ∌ ] > ⇘ ξ ∖ 1 + ζ 2 2 1 y 1 + (ξ (ξ + + ζ ζ ) 2 1 ⇘ ζ 2 2 ∖ξ. qug x qug x 2 Zξ 1 ξ X X ξZZ> 1 1 ∖ >R . Q`r i` tnkt` x tnkt` x ∌ ] > ∤ ] 6 ⇘ x 2 Zξ 1 1 ∖ ξR 2 ξ X Gkt`kmgs ] Gkt`kmgs ] > ∤ ] 6 2 ogk {Z> 1 1 ∖ >R } y hdb(] > ∤ ] 6 ) 2 > . Mnimuigb`s Mnimuigb` s nl`rn ] > + ] 6 , sgoõk i` qug vdb`s gk in `fsgrvnmdýk f. fnstn m`k m`kstrudr uk m`keukt` ogkgrnh`r c`rbnh` p`r i`s ogkgrnh`rgs hg ] hg ] > y ] 6 . Hg in hgkdmdýk hgkdmdýk hg ] hg ] > , hgspgenkh` hg in prdbgrn gmunmdýk `ftgkgb`s5 x > 2 x 6 ∖ x 8 . T hg in sgoukhn gmunmdýk5 x = 2 ∖x 6 x ∌ ] > sd x 2 Zx 6 ∖ x 8 x 6 ∖ x 6 x 8 RX m`k x 6 , x 8 ∌ V. x ∌ ] > ⇘ x 2 x 6 Z> > ∖ > 1RX + x 8 Z∖> 1 1 >RX X
X
Q`r qug ] > 2 qug ] > qug 1R , Z∖snfgb`s > 1 1 >R qug Z> i` } hdb(] ∖ >nhgbîs Gstgi`m`keukt` gs ogk i.d, { p`r hdb(] > )2 6. M`b` yn gkm`ktrnb`s ] gkm`ktrnb`s ] > ∤ ] 6 , p`hgb`s gkm`ktrnr gkm`ktrnr ukn fnsg hg ] hg ] > qug m`ktgkon ukn X X fnsg hg ] hg ] > ∤ ] 6 , p`r gegbpi`5 F gegbpi`5 F ] > 2 {Z> > ∖ > 1R , Z> 1 1 ∖ >R }. Fusqugb`s nl`rn ukn fnsg hg ] hg ] 6 qug m`ktgkon ukn fnsg ] fnsg ] > ∤ ] 6 , p`r gegbpi`5 X X F ] 6 2 {Z> 1 1 ∖ >R , Z1 1 > >R }.
Xgkdgkh` gstns fnsgs hg ] hg ] > y ] 6 , rgsuitn gvdhgktg qug ni c`rbnr gi m`keukt` hg ogkgrnh`rgs hg ] hg ] > + ] 6 , k` vnb`s n rgpgtdr gi ogkgrnh`r hg in dktgrsgmmdýk hg i`s sufgspnmd`s. Gkt`kmgs `ftgkgb`s 5 ] > + ] 6 2 ogk{Z> > ∖ > 1RX , Z> 1 1 ∖ >RX , Z1 1 > >RX }. @fvdnbgktg hdb(] > + ] 6 ) 2 = . ]g mubpig qug 5 hdb ] ] hdb ] hdb ] hdb ] ] 6) > 6) ∖ >) + > + 6 ) 2 ( ( ( Gstn õitdbn dounihnh sg mubpig pnrn t`h`(pnr∤hg sufgspnmd`s kdt`s. Xg`rgbn5 Hnh`s ] Hnh`s ] > y ] 6 sufgspnmd`s hg hdbgksdýk kdtn, gkt`kgs5 hdb(] > + ] 6 ) 2 hdb(] > ) + hdb(] 6 ) ∖ hdb(] > ∤ ] 6 )
Hgb`strnmdýk5 ]d hdb(] > ) 2 k , hdb(] 6 ) 2 b y y hdb(] > ∤ ] 6 ) 2 j , j ≧ 1, vnb`s n hgb`strnr qug + b ∖ j . hdb(] > + ] 6 ) 2 k + In c`rbn hg hgb`strnri` sgrî ukn ogkgrnidznmdýk hg in rgs`iumdýk hgi gegbpi`. ]up`konb`s prdbgr` j prdbgr` j ≧ >, gkt`kmgs gxdstg ukn fnsg F fnsg F ] > ∤] 6 2 {v > , . . . , v j j }, p`hgb`s gxtgkhgr gsn fnsg n ukn fnsg hg ] hg ] > , F ] > 2 {v > , . . . , v j k } y tnbfdäk n ukn fnsg hg j , v j j +> , . . . , v k ] 6 , F ] 6 2 {v > , . . . , v j b } j , w j j +> , . . . , w b Q`r i` vdst` snfgb`s qug ] qug ] + + X 2 ogk {v > , . . . , v j k , w j b } j +> , . . . , v k j , v j j +> , . . . , w b
k gigbgkt`s
b ∖j gigbgkt`s gigbgkt`s
]ýi` tgkgb`s qug vgr qug gstg sdstgbn hg ogkgrnh`rgs gs i.d. Douninb`s ukn m`bfdknmdýk idkgni n 1 S 5
κ> v > + ¿ ¿ ¿ + κj v j + κ κj +> v j + ζ ζ > w j k + b 2 1 S (>) j + j +> + ¿ ¿ ¿ + κk v k j +> + ¿ ¿ ¿ + ζ b b∖ j w b κ> v > + ¿ ¿ ¿ + κj v j + κj +> v j b k 2 ∖ζ > w j b∖ j w b j +> ∖ ¿ ¿ ¿ ∖ ζ b j + κ j +> + ¿ ¿ ¿ + κk v k
∌] >
∌] 6
Gkt`kmgs5 ∖ζ > w j b ∌ ] > ∤ ] 6 ⇖ b∖ j w b j +> ∖ ¿ ¿ ¿ ∖ ζ b ⇖ ∖ζ > w j b 2 ν > v > + ¿ ¿ ¿ + ν j j +> ∖ ¿ ¿ ¿ ∖ ζ b b∖ j w b j v j j 2 ν ν > v > + ¿ ¿ ¿ + ν j + ζ > w j 1S 2 b j v j j + ζ j +> + ¿ ¿ ¿ + ζ b b∖ j w b , . . . , , } gs uk m`keukt` i.d. pugs gs ukn fnsg hg ] hg ] 6 m`kmiudb`s M`b` {v > , . . . , v j w w b b j j +> j 2 ζ 6 2 ¿ ¿ ¿ 2 ζ b qug t`h`s i`s gsmninrgs s`k kui`s, gk pnrtdmuinr5 ζ > 2 ζ b∖ j 2 1. Vggbpinznb`s gk (>). T `ftgkgb`s5
κ> v > + ¿ ¿ ¿ + κj v j + κ κj +> v j k 2 1 S j + j +> + ¿ ¿ ¿ + κk v k `ftgkgb`s qug κ > 2 ¿ ¿ ¿ 2 κ k 2 1 T hg nquâ, m`b` {v > , . . . , v kk } gs uk m`ke. i.d. `ftgkgb`s M`b` i`s gsmninrgs ζ > , . . . , ζb ∖j , κ> , . . . , κk vdgkgk hg in m`bfdknmdýk idkgni (>), m`kmiudb`s qug {v > , . . . , v j k , w j b ∖ j } gs i.d. j , v j j +> , . . . , v k j +> , . . . , w b Gkt`kmgs p`hgb`s nrbnr qug 5 + b ∖ j 2 hdb(] > ) + hdb(] 6 ) ∖ hdb(] > ∤ ] 6 ) hdb(] > + ] 6 ) 2 k + ^ughn m`b` tnrgn pnrn gi l`onr, vgrdmnr in cýrbuin munkh` in dktgrsgmmdýk gs gi sufgspnmd` kui`, gk gsg mns` k` gxdstg fnsg hg in dktgrsgmmdýk y hdrgmtnbgktg trnfnlenb`s m`k ins fnsgs hg mnhn sufgspnmd`.
]ubn hdrgmtn hg sufgspnmd`s. Q`r hgkdmdýk, mnhn gigbgkt` hgi sufgspnmd` ] sufgspnmd` ] > + ] 6 , pughg gxprgsnrsg gk in c`rbn m`k s > ∌ ] > y s 6 ∌ ] 6 , pgr` gsn hgsm`bp`sdmdýk k` sdgbprg gs õkdmn. v 2 s > + s 6 , m`k s Gk gi gegbpi` qug vdb`s, sd t`bnb`s5 v 2 Z6 > 1 1RX 2 Z> > ∖ > 1RX + Z> 1 1 ∖ >RX + Z1 1 > >R
∌] >
∌] 6
v 2 Z6 > 1 1RX 2 Z> > ∖ > 1RX + Z> 1 > 1RX
∌] >
∌] 6
Munkh` mnhn v mnhn v ∌ ] > + ] 6 pughg hgsm`bp`kgrsg gk c`rbn õkdmn m`b` subn hg uk gigbgkt` hg ] hg ] > y uk gigbgkt` hg ] hg ] 6 , sg hdmg qug in subn gs hdrgmtn. Hgkdmdýk5 ]g hdmg qug in subn hg ] > y ] 6 gs hdrgmtn, ` qug ] > y ] 6 gstîk gk subn hdrgmtn sd, pnrn mnhn v mnhn v ∌ ] > + ] 6 gxdstgk õkdm`s s õkdm`s s > ∌ ] > y s 6 ∌ ] 6 tni qug v qug v 2 s > + s 6 . Munkh` Munkh ` in subn gs hdrgmtn, hdrgmtn, sg k`tn5 ] k`tn5 ] > ⊗ ] 6
@fsgrvnmdýk5 ] > y ] 6 gstîk gk subn hdrgmtn sd y sýi` sd ] sd ] > ∤ ] 6 2 {1S } ⇖) ]up`konb`s qug ] qug ] > y ] 6 gstîk gk subn hdrgmtn hdrgmtn y sgn v sgn v ∌ ] > ∤ ] 6 . v ∌ ] > ⇖ v 2 v + 1v , v ∌ ] 6 ⇖ v 2 1v + v .
∌] >
∌] >
∌] 6
∌] 6
Qgr` sd ] sd ] > y ] 6 gstîk gk subn hdrgmtn in hgsm`bp`sdmdýk gs õkdmn, p`r i` tnkt` v 2 1 S ⇖ ] > ∤ ] 6 2 {1S }. ⇑) Nl`rn sup`konb`s qug ] qug ] > ∤ ] 6 2 {1S } y sgn v sgn v ∌ ] > + ] 6 tni qug v qug v 2 s > + s 6 y m`k s s > , t > ∌ ] > y s 6 , t 6 ∌ ] 6 . v 2 t > + t 6 m`k Gkt`kmgs5 v Gkt`kmgs5 v 2 s > + s 6 2 t > + t 6 ⇖ s > ∖ t > 2 t 6 ∖ s 6 ∌] >
∌] 6
] > ∤ ] 6 2 {1S } ⇖ s > ∖ t > 2 1 S 2 t 6 ∖ s 6 , s > 2 t > y s 6 2 t 6 . Hgb`strnb`s qug sd ] > ∤ ] 6 2 {1S } ⇖ ] > y ] 6 gstîk gk subn hdrgmtn.
Xnbfdäk sg pughg pr`fnr qug sd F sd F > gs fnsg hg ] hg ] > y F 6 gs fnsg hg ] hg ] 6 , in subn ] subn ] > + ] 6 gs 2 F > ∥ F 6 gs i.d. hdrgmtn sd y sýi` sd F sd F 2
]d S gs uk gspnmd` vgmt`rdni hg hdbgksdýk kdtn y ] gs gs uk sufgspnmd` hg S, gxdstg uk sufgspnmd` _ sufgspnmd` _ tni tni qug 5 ] 5 ] ⊗ _ 2 S 2 S ` ] 2 2 {1S } ⇖ _ 2 {1S } y _ 2 S rgspgmtdvnbgktg. ]d ]d ] ] 2 ]d > ≨ hdb hdb] 2 j ≨ k ∖ >, gxdstg ukn fnsg F fnsg F ] 2 {v > , . . . , v j ] 2 j } ]nfgb`s qug gstn fnsg pughg gxtgkhgrsg n ukn fnsg hg S, ` sgn, gxdstgk v j +> , . . . , v k tnigs qug F qug 2 {v > , . . . , v j F 2 k } gs ukn fnsg hg S. j , v j j +> , . . . , v k Gkt`kmgs sd m`ksdhgrnb`s _ m`ksdhgrnb`s _ 2 ogk {v j qug ] ⊗ _ 2 S. k }, sg mubpig qug ] j +> , . . . , v k Hg gstn hgb`strnmdýk gs gvdhgktg qug gi sufgspnmd` _ mubpig ] ⊗ _ 2 S k` gs õkdm`, pnrn muniqudgr sufgspnmd` k` trdvdni hg
S.
Hgkdmdýk5 Hnh` uk sufgspnmd` Hgkdmdýk5 sufgspnmd` ] ⊆ hg hg ] ] sd sd ] ] ⊗ _ 2 S
S,J
gspnmd` vgmt`rdni, sg hdmg qug _ gs uk supigbgkt`
Gegbpi`s5 hg ] . gkmugktrg _ > y _ 6 , sufgspnmd`s supigbgkt`s hg ] >. Gk V6 , ] 2 2 {x ∌ V6 /6x > ∖ x 6 2 1 }, gkmugktrg _ Vgs`iumdýk5 Gs dkbghdnt` qug i`s sufgspnmd`s qug fusmnb`s, s`k sufgspnmd`s hg hdbgksdýk >. Qugs, sd gstîk gk subn hdrgmtn m`k ] , m`b` in hdbgksdýk hg ] gs >, hgfg mubpidrsg qug mnhn uk` hg gii`s tgkon hdbgksdýk >, pugs hdb(] + + _ dd ) 2 hdb(] ) + hdb(_ d d ) 2 > + hdb(_ d d ) 2 6 ⇖ hdb (_ dd ) 2 > , d 2 > , 6. 6 i.d. m`k Gkt`kmgs, hgkdr uk`i`s hgvgmt`rs gst`s sufgspnmd`s kgmgsdtnb`s gigodr vgmt`rgs pnrn mnhn V6 hg ] . Fnstn sýi` gkt`kmgs m`k gkm`ktrnr hguk` hg gii`s ukpnrn vgmt`r gk Vmnhn > qug k` gstäk gk gi sufgspnmd` ] , ` sgn qug k` mubpink su gmunmdýk. Q`r gegbpi`5 v > 2 y > > v 6 2 s`k h`s hg i`s dkkdt`s vgmt`rgs qug mubpigk sgr i.d. m`k muniqudgr vgmt`r hg ] . 1 > > Gkt`kmgs gide` hgkdr hgkdr _ _ > 2 ogk y _ 6 2 ogk
>
1
> 6 > > 6. Gk V8 , hnh`s i`s sufgspnmd`s ] > 2 ogk , 1 > > > 8 8 sufgspnmd` X sufgspnmd` X ⊆ V , tni qug ] qug ] > ⊗ X 2 V 2 ] 6 ⊗ X X 0 0
y ] 6 2 ogk
> > , > >
> 6 6 >
, °gxdstg uk
Vgs`iumdýk5 ]d gxdstg X gxdstg X ,, gbpgmgb`s p`r vgr muîi tgkhrân qug sgr in hdbgksdýk hg gstg sufgspnmd`. M`b` hdb(] > ) 2 hdb (] 6 ) 2 6 ⇖ hdb(X ) 2 6 Kgmgsdt` gkt`kmgs gkm`ktrnr h`s ogkgrnh`rgs i.d. pnrn hgkdr gi sufgspnmd` X sufgspnmd` X .. M`b` qudgr` qug sgn uk supigbgkt` hg ] hg ] > y ] 6 , gk hgkdtdvn kgmgsdt` gkm`ktrnr h`s vgmt`rgs hg V8 qug sgnk n in vgz i.d m`k i`s ogkgrnh`rgs hg ] hg ] > y m`k i`s ogkgrnh`rgs hg ] hg ] 6 Gbpgmgb`s p`r vgr sd gst`s h`s sufgspnmd`s hnh`s, ] > y ] 6 gstîk gk subn hdrgmtn `, p`r gi m`ktrnrd`, hdb(] > ∤ ] 6 ) ; 1 .
> > , 1 >
] > + ] 6 2 ogk
6 > , > >
> > , > >
> 6 6 >
Gst`s ogkgrnh`rgs s`k i.d. p`häs ‒mlgqugnri`‒ trdnkouinkh` i`s muntr` vgmt`rgs pugst`s m`b` ins hg ukn bntrdz ` mnimuinkh` gi hgtgrbdknktg hg in bntrdz hg 8 ß 8 y vdgkh` qug gs k` kui`.
> 6 > >
> > > 6
1 > > 6
> > > >
C 6 ∖6C > /C = ∖C >
∖∖∖∖∖∖∖∖∖↖ C 8 ∖C >
> > 1 ∖> 1 1 1 >
1 > > ∖> > 1 6 1
C +C
8 6 ∖∖ ∖∖↖
> > 1 ∖> 1 1 1 1
1 > > ∖> > 1 = ∖>
C ∖ =C =
8 ∖∖ ∖∖∖↖
> > 1 ∖> 1 1 1 1
I`s vgmt`rgs qug ogkgrnk ] ogkgrnk ] > + ] 6 s`k i.d. y, p`r i` tnkt`, snfgb`s qug hdb(] > + ] 6 ) 2 8 ⇖ ] > ∤ ] 6 2 {1V8 } Gkt`kmgs tgkgb`s qug gkm`ktrnr uk sufgspnmd` ogkgrnh` p`r h`s vgmt`rgs hg V8 qug rgsuitgk i.d. m`k i`s ogkgrnh`rgs hg ] hg ] > p`r uk inh` y i.d. m`k i`s ogkgrnh`rgs hg ] hg ] 6 . Fusmnb`s v Fusmnb`s v > , v 6 hg bnkgrn bnkgrn tni qug sg mubpin5 mubpin5 6.>
6.6
> > , 1 >
> > , > >
6 > , v > , v 6 > >
i.d5
> 6
> >
1 >
> >
> 6 , v > , v 6 i.d5 6 >
> >
> 6
> 6
> >
> > 1 ∖>
↖ ∖
↖ ∖
> 1
> >
> >
1 > (n) > ∖>
> (f) 1
"Bdrnkh` e`"(n) y (f), vgb`s qug sd n ins h`s bntrdmgs igs norgonb`s i`s vgmt`rgs g = y g 8 `ftgkgb` `ftgkgb`s s bntrdmgs hg rnko` 8, gkt` gkt`kmgs, kmgs, p`r gegbpi`, gigodb` gigodb`s s gi sufgspnmd` 1 1 1 1 X 2 ogk , > 1 1 >
=. Gk V 8 gi bdsb` pr`figbn, lniinr X ⊆ V8 , tni qug ] > ⊗ X 2 V8 2 ] 6 ⊗ X pgr` nl`rn ] > y ] 6 s`k sufgspnmd`s qug K@ gstîk gk subn hdrgmtn5
1 > > ∖> > 1 1 ∖>
] > 2 ogk
> > , 1 >
6 6 > >
y ] 6 2 ogk
> 6 , > >
> 1 ∖6 6
Sgnb`s sd gstîk gk subn hdrgmtn ` k`5
> > 1 , >
] > + ] 6 2 ogk
> 6 > >
> 1 6 > 6 > 1 ∖6
> > > 6
C 6 ∖6C > /C = ∖C >
∖∖∖∖∖∖∖∖∖↖ C 8 ∖C >
6 6 > , >
> > 1 > 1 1 > ∖> 1 > > 1 1 ∖> ∖6 >
> 6 > , >
> 1 ∖6 6
> 1 1 1
C 8 +C =
∖∖∖∖↖
> 1 > 1 > ∖> > > 1 1 ∖> >
C 8 +C 6
∖∖∖∖↖
Gkt`kmgs5
] > + ] 6 2 ogk
> > , 1
6 6 , >
> 6 , >
>
>
>
> 1 ∖6
2 ogk
Nvgrdounb`s qug hdb(] > + ] 6 ) 2 = ⇖ hdb(] > ∤ ] 6 ) 2 > . Fusqugb`s in dktgrsgmmdýk hg gst`s sufgspnmd`s5 Hgf` gkm`ktrnr uk vgmt`r qug mubpin5 > > \ 2 ξ + ζ 1 >
> > , 1 >
6
6 6 (=) > >
> > 6 1 + κ (8) \ 2 ν > ∖6 >
Douninkh`5
> > \ 2 ξ + ζ 1 > > > ξ + ζ 1 >
6
6 6 2 ν > >
6 6 ∖ ν > >
> > 6 1 + κ ∖6 > 6 >
> > 1 6 1 1 ∖κ 2 > ∖6 1 > 6 1
Vgs`ivdgkh` gi sdstgbn `ftgkgb`s5
ν 2 2 κ, ζ 2 2 ∖κ, ξ 2 8 κ Vggbpinznkh` gk (=) ` gk (8), `ftgkgb`s5
6 6 \ 2 κ ∖>
1 1 , >
∖>
1 > > 1
> 1 1 1
> 1 > 1
1 > > ∖> > 1 1 1
=
Gkt`kmgs nl`rn p`hgb`s m`kstrudr ukn fnsg pnrn mnhn sufgspnmd` qug m`ktgkon n gstg ogkgrnh`r hg in dktgrsgmmdýk5 > 6 6 > > 6 6 1 , y ] 6 2 ogk , ] > 2 ogk 1 ∖> ∖> ∖6 > = = 6
T nl`rn p`hgb`s nmtunr m`b` gk gi dtgb nktgrd`r5
Fusmnb`s v Fusmnb`s v > , v 6 hg bnkgrn bnkgrn tni qug sg mubpin5 mubpin5 >.
6.
> > , 1 >
6 6 , v > , v 6 ∖> =
> 1 , ∖6 6
i.d5
6 6 , v > , v 6 i.d5 ∖> =
> > 1 > 6 6 ∖> =
↖ ∖
> 1 ∖6 6 1 6 = ∖>
> > 1 > (n) 1 1 ∖> >
↖ ∖
> > > > (f) 1 > > 1
Gkt`kmgs, p`r gegbpi`, i`s vgmt`rgs g vgmt`rgs g 6 y g 8 pughgk sgr norgonh`s gk ins h`s bntrdmgs y `ftgkgr bntrdmgs hg rnko` 8 y gigodb`s5 1 1 > , 1 X 2 ogk 1 1 1 >
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