1 - Reunion1 - EV
July 3, 2022 | Author: Anonymous | Category: N/A
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Nuihdk ae `uhdk tcri piri ijifk as `afkr hk astir itidk i hidi C`iochah i eks dchksiurcks ah ei ni`i" Nbirey Oirnài
]rasahtincþh Aspincks \antkrcieas. »]kr quã; Qstadas sa bih ahnkhtridk, ah su axparcahnci, `unbis vanas nkh nkhfuhtks da aea`ahtks dkhda dkhda tcaha sahtcdk k as hanasirck trijifir su`ihdk aea`ahtks da asa nkh fuhtk k `uetcpecnihdk aea`ahtks da asa nkhfuhtk pkr uh asnieir, kjtahcahdk sca`pra ktrk
aea`ahtk da asa nkhfuhtk. Ask sunadcþ nuihdk trijifirkh nkh vantkras ah Z: k ah Z> 6 ei su`i da dks vantkras da Z: k Z> , diji pkr rasuetidk ktrk vantkr da Z: k Z> raspantcvi`ahta. Ek `cs`k
pisiji nuihdk nienueéji`ks `öetcpeks da uh vantkr. Ti`jcãh sunadcþ, nuihdk astudcirkh luhnckhas, iprahdcarkh i su`ir luhnckhas nkh
ei dahcncþh6 (l + o o)( )(x x) 0 l ( l (x) + o + o((x), ∍x ∂ D Dk` k`((l l )) ∣ Dk` Dk`((o )
y ti`jcãh i `uetcpec `uetcpecnir nir uhi luhncþh pkr uh asnieir6 asnieir6 (gl gl )( )(x x) 0 g gl l (x), ∍x ∂ D Dk` k`((l l )) y ∍ x ∂
Z.
^c ida`és eis luhnckhas arih nkhtchuis, kjtahàis ktri luhncþh nkhtchui7 sc eis
luhnckhas arih darcvijeas kjtahàis ktri luhncþh darcvijea. Ah tkdks astks nisks, ei su`i da aea`ahtks da uh nkhfuhtk \ y ae prkduntk da uh aea`ahtk da asa nkhfuhtk pkr uh hö`ark, sca`pra diji nk`k rasuetidk ktrk aea`ahtk da asa nkhfuhtk. Eis nksis quadijih ahnarridis ah ae nkhfuhtk
ah ae qua trijiféji`k trijiféji`ks. s. Astudcira`ks a`ks eis prkpcadidas nk`uhas da asta tcpk da scsta`is qua nkhstih • Astudcir da uh nkhfuhtk da aea`ahtks, qua vi`ks i eei`ir \, dkhda sa daha uhi su`i y uh prkduntk pkr asnieir (aea`ahtks da uh nkhfuhtk hu`ãrcnk) da lkr`i tie qua sus rasuetidks huhni sa asnipih dae dae nkhfuhtk nkhfuhtk didk.
• ^c bina`ks uh asluarzk da ijstrinncþh y astudci`ks eis prkpcadidas nk`uhas da astks scsta`is, tahdra`ks rasuetidks qua saréh ch`adciti`ahta véecdis tihtk piri ae nkhfuhtk da luhnckhas, nk`k piri ae nkhfuhtk da vantkra vantkras, s, `itrcnas,
pkechk`cks, ahtra ktrks. \i`ks ahtkhnas i dahcr lkr`ie`ahta quã as uh aspinck vantkrcie.
Dahcncþh da Aspinck \antkrcie
Qh aspinck vantkrcie, nkhsti da6 • Qh nkhfuhtk \ da kjfatks qua eei`ira`ks vantkras7 • Qh nkhfuhtk G da asnieiras (G 0 Z, k G 0 N)7 • Qhi kparincþh, eei`idi su`i, dahcdi ahtra eks aea`ahtks dae nkhfuhtk \, qua nu`pea u + u + v v ∂ \7 • Qhi kparincþh eei`idi prkduntk pkr asnieir, qua isknci i nidi asnieir η ∂ G y nidi vantkr u ∂ \ ae vantkr η.u ∂ \.
Ida`és eis kparincþh su`i y prkduntk pkr asnieir dajaréh nu`pecr6 + v v 0 v + v + u, u, ∍u, v ∂ \ (nkh`utitcvcdid)7 ^9. u + 0 u u + v + w, ∍u,v,w ∂ \ (iskncitcvcdid)7 ^:. u + (v + w) 0 (u + v) + w 0 ^>. ∎ K\ ∂ \/u /u + + K\ 0 K\ + + u u 0 u (Axcstahnci dae aea`ahtk hautrk piri ei su`i)7 ^4. ∍u ∂ \, ∎ ( √u) ∂ \ tie qua u + (√u) 0 K\ (Axcstahnci dae chvarsk idctcvk piri tkdk aea`ahtk da \ ).
]9. 9.u 0 .u 0 u, u, ∍u ∂ \7 ]:. (ηκ ).u .u 0 0 η. η.((κ .u .u)), ∍η, κ ∂ G y ∍ u ∂ \7 η.(u + + v v)) 0 η.u η.u + + η.v, η.v, ∍η ∂ G y ∍ u, v ∂ \ ]>. η.( ]4. (η+κ ).u .u 0 0 η.u η.u + + κ.u, κ.u, ∍η, κ ∂ G y ∍ u ∂ \
Afa`peks da Aspincks da \antkrcieas
• A`pana`ks pkr ae `és li`cecir6
\
0 Zh , G 0 Z, nkh ei su`i y ae
prkduntk bijctuieas as uh aspinck vantkrcie. ^a dcna qua Z
h
Z
as uh
aspinck vantkrcie.
Hktincþh 6 Iquà sca`pra asnrcjcra`ks i eks vantkras, nk`k vantkras nkeu`hi.
x x 0 .. , x ∂ . x 9
Z
h
:
c
Z
h
∍c 0 9, . . . h .
• Ae nkhfuhtk Nh , da eis h-upeis lkr`idis pkr hö`arks nk`peafks, k sai Nh 0 { Wx9x: . . . . . . xhPT , x9 , . . . xh ∂ N} fuhtk nkh ae nkhfuhtk da asnieiras nk`peafks G N, nkh ei su`i y ae prkduntk pkr 0 asnieir nk`k eks da ihtas lkr`ih uh aspinck vantkrcie. ^a dcna
qua Nh √ N as uh aspinck vantkrcie. • Ae nkhfuhtk Nh , da eis h-upeis lkr`idis pkr hö`arks nk`peafks, nk`k ihtas nkh ae nkhfuhtk da asnieiras raieas G 0 Z, nkh ei `cs`i dahcncþh da su`i y prkduntk pkr asnieir, ti`jcãh lkr`ih uh aspi aspinc nck k van anttkrci krcie. e. Ah as astta ni nisk sk sa dc dcna na qu qua a Nh - Z as uh aspi aspinc nck k
vantkrcie. • Eks nkhfuhtks da `itrcnas da ` eis y h nkeu`his qua sa hktih6 Z`ßh
y N`ßh saoöh sus nkancahtas saih raieas k nk`peafks,
fuhtk nkhfuhtks asnieiras raieas, nuihdk trijifi`ks h ` h nkh Znkh y, eks ah ae nisk dada N ß tihtk nkh ae nkhfuhtk hu`ãrcnk Z nk`k N, lkr`ih uh aspinck vantkrcie nkh ei dahcncþh da su`i y
prkduntk pkr asnieir yi nkhkncdis. • N (C) dahkti ae nkhfuhtk da tkdis eis luhnckhas nkhtchuis nkh dk`chck ah C ⊊ Z y nkdk`chck Z6 N (C) 0 { l 6 C √↚ Z, l as as nkhtchui}, fuhtk nkh ae nkhfuht nkhfuhtk k da asnieir asnieiras as G 0 Z, dkhda sa daha ei su`i
nk`k6 Z
l + o o 6 C √↚ /(l + o o)( )(x x) 0 l l ((x) + o + o((x)
y ae prkduntk pkr uh asnieir η ∂ Z nk`k6 (η.l η.l )) 6 C √↚ Z/(η.l η.l )( )(x x) 0 ηl ηl ((x), ∍η ∂
Z
y ∍ l ∂ N N ((C) as uh aspinck vantkrcie. C
Zankrda`ks qua ox ∂ N ( ( ), sa dcna qua l l 0 o sc sa nu` o (xsc) l , y∍ ∂ N C. Ah asta aspinck vantkrcie KN (C) pea qua l (x) 0 as ei luhncþh huei (ae aea`ahtk hautrk dae aspinck vantkrcie),
k sai, KN (C)(x) 0 =, ∍x ∂ C. • Ae nkhfuhtk Zh WxP as ae nkhfuhtk da tkdks eks pkechk`cks nkh nkancahtas raieas da oridk `ahkr a couie qua h, nkh ae nkhfuhtk G 0 Z y ei su`i y prkduntk pkr asnieiras dahcdks nk`k ah ae cta` ihtarckr lkr`ih uh aspinck vantkrcie. ^c ] ∂ ZhWxP ] ] 0 ih xh + ¿ ¿ ¿ + i9 x + + i i=,
ic ∂ Z7 c 0 9, . . . , h .
Didks ] 0 ih xh + ¿ ¿ ¿ + i9 x + + i i= ∂ Zh WxP + j Y 0 jh xh + ¿ ¿ ¿ + j9x + j= ∂ Zh WxP
sa dcna qua ] ] 0 Y ⇔ i c 0 j c ∍c 0 9, . . . , h . Ae aea`ahtk hautrk KZ WxP , ah asta aspinck vantkrcie as ae pkechk`ck huek, as dancr ae pkechk`ck qua tcaha tkdks sus nkancahtas couieas h
i =6 KZ WxP 0 h
=xh + ¿ ¿ ¿ + =x =x + =. =.
]rkpcadidas aea`ahtieas I pirtcr da eks ixck`is sa puadah da`kstrir eis scoucahtas prkpcadi-
das6 • Ae aea`ahtk hautrk K\ ∂ \ as öhcnk. • Ae sc`ãtrcnk da uh aea`ahtk u ∂ \ as öhcnk. • = u 0 K\ piri tkdk u ∂ \. • ηK\ 0 K\ piri tkdk asnieir η ∂
G.
• ^c ηu ηu 0 0 K\ ahtkhnas η η 0 0 = k u u 0 0 K\. • (√9) 9)u u 0 (√u).
Da`kstra`ks ieouhi da astis prkpcadidas • Ae aea`ahtk hautrk K\ ∂ \ as öhcnk. ˕ \ skh aea`ahtks hautrks ah ae aspinck ^upkhoi`ks qua K\ y K
vantkrcie. As dancr K\ + + u u 0 0 u u + + K\ 0 u,
piri tkdk u ∂ \ y ˕ \ + ˕ \ 0 v K + v v 0 v v + + K v,,
piri tkdk v ∂ \. ]ark nk`k astk as ncartk piri tkdk u, v ∂ \, ah pirtcnueir sa
˕ \. ˕ \ y v v 0 K\ . ]kr ek tihtk6 K\ 0 K u 0 0 K nu`pea sc u • = ¿ u 0 K\ piri tkdk u ∂ \.
^ija`ks qua6 0 u + = ¿ u u 0 (9 + =) ¿ u 0 9 ¿ u + = ¿ u 0 u +
ahtkhnas + u 0 (√u) + u (√u) + u + u += ¿ u 0 = ¿ u ⇚ K\
K\ 0
=¿u
K\
Nidi pisk utceczidk daja nkrraspkhdar i uh ixck`i k uhi prk-
pcadid yi da`kstridi.
Z:
^
Z>
^
^ujaspincks ^upkhcahdk qua \ as uh G aspinck vantkrcie vi`ks i astudcir nþ`k datantir iquaeeks sujnkhfuhtks da \ qua rasuetih sar uh aspinck vantkrcie vant krcie nkh eis kparinckhas da su`i y prkduntk pkr asnieir dahcdks dahcdks.. ^ujaspincks ^upkhcahdk qua \ as uh G aspinck vantkrcie vi`ks i astudcir nþ`k datantir iquaeeks sujnkhfuhtks da \ qua rasuetih sar uh aspinck aspin ck van vantkrci tkrciee nkh eis kpar kparinckh inckhas as da su`i y prkduntk prkduntk pkr asnieir asnieir
dahcdks. Dahcncþh6 Qh sujaspinck da uh G aspinck vantkrcie \, as uh sujnkhfuhtk ^ ⊊ \, ^ 0 ∁ qua rasueti sar uh aspinck vantkrcie
nkh ei su`i y ae prkduntk pkr asnieir dahcdks. Nuihdk astudci`ks sc ^ nu`pea ei dahcncþh da aspinck vantkrcie,
daja`ks varcnir, ah prc`ar euoir6 • qua ^ rasueti rasueti sar narridk piri eis kparinckhas su`i. • qua ^ rasueti rasueti sar narridk piri prkduntk pkr asnieir.
Ibkri jcah, pkr astir ^ chneucdk ah uh aspinck vantkrcie biy uhi sarca da prkpcadidas qua nkh saourcdid sa vih i nu`pecr. »Nué-
eas; Ei su`i as nkh`utitcvi piri tkdk pir da aea`ahtks da \, pkr ek • tihtk saré nkh`utitcvi, ah pirtcnueir, piri tkdk pir da aea`ahtks ah ^ . Ek `cs`k sunada nkh ei prkpcadid iskncitcvi piri ei su`i
y nkh eis ktris prkpcadidas raeitcvis ie prkduntk pkr asnieir. Si da`kstri`ks qua ah tkdk aspinck vantkrcie sa nu`pea qua =u 0 K\, isà qua sc ^ 0 ∁ ⇚ piri nuiequcar u ∂ ^ , sa vi i nu`pecr
qua = =.u .u 0 0 K\ y nk`k η.u ∂ ^, ∍η ∂ G ⇚ K\ ∂ ^. • Ida`és, as lénce var qua sc uh sujnkhfuhtk da uh aspinck vantkrcie \, as narridk piri ei su`i y ae prkduntk pkr asnieir y K\ ∂ ^ ⇚ ^ rasueti rasueti
sar uh aspinck vantkrcie.
Ahtkhnas, sc asti`ks ah uh aspinck vantkrcie piri var sc uh sujnkhfuhtk da \ as uh aspinck vantkrcie nkh ei su`i y prkduntk pkr asnieir yi dahcdks, sþek hanasctira`ks nbaquair tras prkpcadi-
das. ^c \ as uh G aspinc aspinck k va vant ntkrc krcie, ie, sa dcn dcna a qua ^ ⊊ \ as uh
sujaspinck da \ sc sa nu`pea6 • 9. K\ ∂ ^ 7 • :. ^c u, v ∂ ^ ⇚ u u + + v v ∂ ^ 7 • >. ^c u ∂ ^ y y η ∂
G
⇚ ηu ∂ ^ .
Tahdra`ks nk`k prc`ar tirai astudcir sc dcstchtks sujnkhfuhtks dahc-
dks ah uh aspinck vantkrcie skh k hk skh sujaspincks.
Z:
v
√v u u
√v
έ
v= u=
Afa`pek da nkhfuhtks qua hk skh sujaspincks
Ieouhks afa`peks
A`pana`ks nkh uh afa`pek li`cecir6 ae nkhfuhtk6
^ 0 (x
T
9
»as uh sujaspinck da
x: x> ) ∂ Z Z> ,
>
/x √ :x + + x x 0 = 9
:
>
nkh ei su`i y prkduntk pkr asnieir bijctui-
eas;. Da`kstra`ks ihieàtcni`ahta qua as uh sujaspinck6
i KZ ∂ ^ »]kr quã; ]krqua KZ 0 (= = =)T nu`pea ei nkhdcncþh >
>
didi pkr ei anuincþh qua daha i asta nkhfuhtk, puas = √ :. =+= 0
=.
j ^c u, v ∂ ^ ⇚ u 0 (x9 x : x >)T , y v 0 (y9 y: y > )T skh dks vantkras ah Z> qua nu`peah ei anuincþh qua daha i ^ . Taha`ks qua varcnir sc ei su`i da astks dks puhtks oahãrcnks da ^ partahana
u + + v v 0 (x9 + + y y9 x: + + y y: x> + + y y> ). i ^ , u
^c raa`peizi`ks ah ei anuincþh da ^ , eis nkrraspkhdcahtas nkkr + y y>) 0 dahidis6 (x9 +y9 )√:( :(x x:+y:)+( )+(x x>+y> ) 0 ( (x x9 √ :x: + + x x>) + (y9 √ :y: +
=+=0 =
0= puas u ∂^
0= puas v ∂^
n ^c η ∂ Z y u ∂ ^, ηu ηu 0 (ηx9 ηx: ηx> )T ⇚∂ ^ , puas ηx 9 √ :ηx: + η = 0 = ηx> 0 η (x9 √ :x: + + x x>) 0 η =
0= puas u ∂^
Nk`k ae Nkhfuhtk ^ nu`pea nkh eis tras nkhdcnckhas vcstis pkda-
`ks ir`ir qua as uh sujaspinck da Z> . Ibkri jcah, ae sujaspinck ^ , ti`jcãh puada dasnrcjcrsa axpecnctihdk
ei lkr`i da eis skeunckhas da ei anuincþh6 x9 √ :x: + + x x> 0 = ⇝ ⇝⇚ ⇚ x > 0 √ x9 + :x : x:
x u ∂ ^ ^ 0⇚ u u 0 0 x 0 9 :
x>
x9 x: √x9 + :x : x:
0
9 = = + x 9 , x , x ∂
0 x u 0 x 9
:
√9
9
:
Z.
:
T
T
Hktir qua W W99 = √ 9P y W= 9 :P partahanah ie sujaspinck vantkrcie ^ , y asti`ks bincahdk su`is y `uetcpecnihdk astks
0 9, :). aea`ahtks, qua astéh ah ^ , pkr asnieiras x c (nkh c c 0 Ihtas da dir huastri prþxc` Ihtas prþxc`i i dahcncþh dahcncþh vai`ks ah ktrk afa`pek, afa`pek, nk`k biy axprasckhas qua qua sa rapctah `és ieeé da ei hiturieazi
dcstchti da eks aea`ahtks dae aspinck vantkrcie. Astudca`ks ae nkhfuhtk6 T T 0 { p ∂
Z: WxP/p /p(9) (9)
vai`ks qua as uh sujaspinck da
Z: WxP,
0 =} ,
nkh ei su`i y ae prkduntk pkr
asnieir yi dahcdks. Taha`ks qua var, ahtkhnas, qua sa nu`peah eis tras nkhdcnckhas da
sujaspinck6 i. Ae pkechk`ck huek KZ WxP nu`pea kjvci`ahta ei nkhdcncþh, puas KZ WxP (x) 0 = ∍x ∂ Z, ah pirtcnuei pirtcnueirr KZ WxP (9) 0 =. ] + Y, nu`pea qua6 j. ^c ] y Y ∂ T ahtkhnas ae pkechk`ck ] :
:
:
(] + Y Y)(9) )(9) 0 ] 0 ] (9) (9) + Y + Y(9) (9) 0 =
n. ]kr öetc`k sc tk`i`ks η ∂ Z y ] ∂ T , ae pkechk`ck η ] , nu`pea qua6 (η] η] )(9) )(9) 0 η] 0 η] (9) (9) 0 η 0 η = = 0 = Nk`k ae Nkhfuhtk T nu`pea nkh eis tras nkhdcnckhas da ei dahc-
ncþh, pkda`ks ir`ir qua as uh sujaspinck da Z: WxP. »Nþ`k skh eks aea`ahtks da asta sujaspinck;
^c ] ∂ T , ahtkhnas sa nu`peah6 • ] ] 0 i: x: + i9 x + + i i= pkr sar uh aea`ahtk da Z:WxP • ] (9) ] (9) 0 i 0 i : + + i i9 + + i i= 0 =, ⇔ i = 0 √ i: √ i9 .
]kr ek tihtk6 ] ] 0 i :x: + i9x + (√i: √ i9 ) 0 i : (x: √ 9) + i + i9(x √ 9)
Ahtkhnas, ] ∂ T ⇔ ] ] 0 i: (x: √ 9) + i + i9 7 (x √ 9) 9),, nkh i 9 , i: ∂ Z. Hktir qua (x ( x: √ 9) y ( (x x √ 9) skh aea`ahtks dae sujaspinck T , y asti`ks su`ihdk
`öetcpeks `öetcpe ks da astks aea`aht aea`ahtks, ks, qua astéh ah T . Kjsarvincþh6 Trijifi`ks nkh dks sujaspincks, uhk da aeeks ^ ah Z> y ae ktrk, T ah Z: WxP, piri eks dks ahnkhtri`ks uhi lkr`i (`uy pirancdi) da dasnrcjcr eks aea`ahtks qua partahanàih
i nidi uhk6
u ∂ ^ ⇔ u u 0 0 x x 9 (9 = √ 9)T +x: (= 9 :)T , x9 , x: ∂
Z.
vantkr da ^
vantkr da ^
] ∂ T ⇔ ] ] 0 i : (x: √ 9) +i9 (x √ 9) , nkh i 9 , i: ∂
vantkr da T
vantkr da T
Z.
Nk`jchincþh echaie Dahcncþh6 Qh vantkr w ∂ \ √ G aspinck vantkrcie, as nk` nk`jchi jchincþh ncþh ech echaie aie da uh nkhaxcst cstah ah asn asniei ieira ras, s, η9 , η: , . . . , ηh ∂ G tie tieas as qua qua66 fuhtk da vantk vantkras ras v9 , v:, . . . , vh ah \, sc ax h
w 0 η 0 η 9 v9 + + η η: v: + ¿ ¿ ¿ + ηh vh 0
ηv . c c
c09
hktincþh
HKTINCÞH6 oah{v9, v: , . . . , vh } raprasa raprasahti hti ae nkhf nkhfuhtk uhtk da tkd tkdis is eis nk`j nk`jchinc chinckhas khas echa echaieas ieas pkscjeas ahtra eks vantkras v9 , v: , . . . , vh .
Ieouhks rasuetidks c`pkrtihtas Kjsarvincþh6 ^c { v9 , v: , . . . , vh } ⊊ \ √ G aspinck vantkrcie ⇚ oah {v9 , v:, . . . , vh } sujaspinck ck oaha oaharidk ridk pkr v9 , . . . vh . as sc sca` a`pr pra a uh suja sujasp spin inck ck da \. S sa hk` k`j jri nk nk` `k ae sujaspin
]iri da`kstrir qua asta nkhfuhtk as uh sujaspinck, tahdra`ks qua prkjir6 = v: + ¿ ¿ ¿ + =v =vh . (]uas = =u u 0 K\ ah tkdk i. K\ ∂ oah{v9 , v: , . . . , vh }. Da banbk6 K\ 0 =v9 + =v aspinck vantkrcie).
j. ^c u, w ∂ oah{v9 , v: , . . . , vh }, qucark var quã pisi nkh u u + + v v 6 w 0 0 κ η 9 v9 + + η η: v: + ¿ ¿ ¿ + ηh vh y w κ 9 v9 + + κ κ :v: + ¿ ¿ ¿ + κ h vh u 0 0 u u 0 0 η
nkh η c , κ c ∂ G, ∍c 0 9, . . . h . ]kr ek tihtk6 u + w + w 0 0 (η9 v9 + η + η: v: + ¿ ¿ ¿ + ηh vh ) + (κ ( κ 9 v9 + + κ κ : v: + ¿ ¿ ¿ + κ h vh )
Ipecnihdk Ipecnih dk ei prkpcad prkpcadid id iskncitcvi da ei su`i y iorupihdk iorupihdk tãr`chks6 + κ κ 9)v9 + + ¿ ¿ ¿ + (η (ηh + + κ κ h )vh ∂ oah{v9 , v:, . . . , vh }. u + w + w 0 0 (η9 +
n. ^c u ∂ oah{v9 , v: , . . . , vh } y ι ∂ G, qucark var quã pisi nkh ι u ι u . ]ark sc u ∂ oah{v9 , v: , . . . , vh } ⇚ u 0 u 0 η η 9 v9 + η + η: v: + ¿ ¿ ¿ + ηh vh + η:v: + ¿ ¿ ¿ + ηh vh ) 0 (ιη9 )v9 + (ιη ( ιη: )v: + ¿ ¿ ¿ + (ιη (ιηh )vh ι u 0 ι 0 ι((η9 v9 + η
nk`jchincþh echaie dae nkhfuhtk{v9 ,...,v } h
Oaharidkras Oaharidk ras y sujaspin sujaspincks cks hcti`a hcti`ahta hta oaharidks Da ibkri ah `és eks sujaspincks sa prasahtiréh pkr eis nkhdcnckhas qua nu`peah
sus aea`ahtks k pkr uh nkhfuht nkhfuhtkk da vant vantkras kras qua ek oahari. Nuastckhas Nuastck has da eahouifa y asnrcturi6 Ae sujaspinck ^ da uh
G-aspinck vantkrcie \
sa dcna qua asté oaharidk pkr
eks vantkras {v9, v: , . . . , vh } nuihdk ^ 0 o oah ah{v9 , v: , . . . , vh }.
Ah asta nisk, {v9 , v:,...,vh } saré uh nkhfuht nkhfuhtkk oaharidkr da ^ . ^c ^ tcaha tcaha uh nkhfuhtk oaharidkr nkh hctks aea`ahtks ahtkhnas sa dcré qua ^
as hcti`ahta oaharidk. Ktri kjsarvincþh6 kjsarvincþh6 ^c ^ as uh sujaspinck da uh G- aspinck vantkrcie \ y {u9 , u: , . . . , ug } ⊊ ^ ⇚
oah {u9 , u: , . . . , ug } ⊌ ^ . As uhi nkhsanuahnci dcranti da ei dahcncþh da sujaspinck puas sc ^ as uh sujaspinck ei nk`jchincþh echaie da nuiequcar pir da vantkras da ^ asté asté ah ^ . ]kr ek tihtk,
sc6 v ∂ oah {u9 , u: , . . . , ug } ⇚ v v 0 η 9 u9 +η: u: + ¿ ¿ ¿ + ηg ug
v 0 η 0 η u + η + η u + ¿ ¿ ¿ + η u ∂^
9
9
:
:
g
g
∂^
Ahtkhnas6 Ahtkh nas6 ^c v as nk`jchincþh echaie dae nkhfuhtk { u9 , u: , . . . , ug }, sa nu`pea qua
v ∂ ^ 6
oah {u9 , u: , . . . , ug } ⊌ ^
.
Ae nkhfuhtk oaharidkr da uh sujaspinck hk as öhcnk . As lénce varek ah eks
afa`peks ihtarckras. »Nþ`k pruajk qua dks nkhfuhtks dcstcht dcstchtks ks oaharih ae `cs`k sujaspinck; sujaspinck; Tk`a`ks uh afa`pek, (dcstchtk dae qua ipirana ah ae Apcskdck :). ^upkhoi`ks ^upkhoi `ks qua hks praouhti`ks sc eks sujaspincks6 sujaspincks6 T
T
T
T
^ 0 oah{W9 9 = 9P , W9 9 9 :P } y T T 0 oah {W9 9 √ 9 =P , W9 9 √ : √ 9P }
skh couieas. ]iri da`kstrir qua dks sujaspincks skh couieas, ah asta atipi, tahdra`ks qua da`kstrir da`kstr ir qua sa nu`pea ei dkjea chneuscþh.
]rc`ark da`kstrira`ks qua ^ ⊌ T y euaok qua T ⊌ ^ . ]kr ei öetc`i kjsarvincþh sija`ks qua sc uh nkhfuhtk da aea`ahtks asti ah uh sujaspinck, tkdi nk`jchincþh echaie da aeeks ek asté. \i`ks i ipecnir asti prkpcadid ah
eks dks nisks. ^ ⊌ ⊌ T \i`ks i da`kst da`kstrir rir qua nidi uhk da eks oahar oaharidkr idkras as da ^ asté ah T . »Yuã tahok qua binar; Tahok qua ‐ nbaquair‐ qua nidi uhk da eks oaharidkras da ^ asté asté as nk`jchincþh echaie da eks vantk vantkras ras dae nkhfuhtk {W9 9 √ 9 =PT , W9 9 √ : √ 9PT }
\ai`ks qua W9 9 = 9PT ∂ T y qua W9 9 9 :PT ∂ T W9 9 = 9PT 0 ιW9 ι W9 9 √ : √ 9PT + κ W9 W9 9 √ 9 =PT ⇔
⇔ W9 9 = 9PT 0 Wι + κ + κ
ι + + κ κ √ :ι √ κ √ ιPT ⇔
9 0 ι ι + + κ κ 9 0 ι ι + + κ κ ⇔ = 0 √:ι √ κ ⇔ ι 0 √9, κ 0 : . 9 0 √ι W9 9 = 9PT ∂ T
Ibkri nk`prkja`ks qua ti`jcãh W9 W9 9 9 :PT ∂ T W9 9 9 :PT 0 ιW9 ι W9 9 √ : √ 9PT + κ W9 W9 9 √ 9 =PT ⇔
⇔ W9 9 9 :PT 0 Wι + κ + κ
ι + + κ κ √ :ι √ κ √ ιPT ⇔
9 0 ι ι + + κ κ 9 0 ι ι + + κ κ ⇔ 9 0 √:ι √ κ ⇔ ι 0 √:, κ 0 > . : 0 √ι W9 9 9 :PT ∂ T
Isà da`kstri`ks qua ^ ⊌ T , puas sc eks dks oaharidkras dar ^ astéh ah T tkdi
nk`jchincþh nk`jchin cþh echaie da aeeks asté, puas T as uh sujaspinck ⇚ ^ ⊌ T . da`kstrir kstrir astk, taha` taha`ks ks qua var qua nidi nidi oaharidkr da T asté ah T ⊌ ^ ]iri da` ae sujaspinck ^ . Ahtkhnas ibkri taha`ks qua ‐ nbaquair‐ qua W9 9 √ 9 =PT ∂ ^ y qua W9 9 √ : √ 9PT ∂ ^ , ahtkhnas taha`ks qua var qua astks dks vantkras skh nk`jchincþh
echaie da eks oaharid oaharidkras kras da .Taha`ks qua var sc axcstah y tie qua 6 ^
ι κ
ιW9 W9 9 = 9PT + κ W9 W 9 9 9 :PT W9 9 √ 9 =PT 0 ι W9 9 √ 9 =PT 0 Wι + κ + κ ι + κ + κ κ ι + :κ : κ PT ⇔
9 0 ι + ι + κ κ 9 0 ι + ι + κ κ ⇔ √9 0 κ ⇔ ι 0 :, κ 0 √9 . = 0 ι + ι + :κ :κ W9 9 √ 9 =PT ∂ ^
Yuadi nk`k tirai piri ae bkoir nbaquair qua W9 9 √ : √ 9PT ∂ ^ Qhi vaz nk`prkjidk
astk, quadi prkjidk qua T ⊌ ^ Da asti `ihari prkji`ks qua ^ ⊌ ⊌ T y T ⊌ ^ ⇚ ⇚ ^ 0 T
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