6015 - Taller 3 - Grupo 1
March 5, 2023 | Author: Anonymous | Category: N/A
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Hmlfinl Hmlf inlm m hntf hntfhî hîto tokm km lf vogr vogrnk nkom omjf jfss hfkî hfkîjo jo-kns @WVQM > Nutemr3 CMF @NWKÈN
VJORFW[OLNL NL LF IN[ BVFWSN[ NWHNLN[ - F[QF Ojstotutf3 VJORFW[OL Lntf3 Nu`ust >0, 7171 Rfrsomj3 ;.15
Roktmry wmj‟t kmhf tm us ujifss wf `m tm ot.
Kmjtfjts
>
Hmlfinl Hmlfinlm m hntfhît hntfhîtokm okm lf vogrnkomj vogrnkomjfs fs hfkîjo hfkîjokns kns
>
>.> Nhmrto Nhmrto`unl `unlmrfs mrfs lf vogr vogrnkom nkomjfs jfs m nhmrto`unl nhmrto`unlmrfs mrfs lf hnsn sojtm sojtmjoznl joznlms ms . . . . >7 >.7 Fcfrkokoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . >0
Knpèt Knpètuim uim > Hmlfi Hmlfinlm nlm hntfhîtok hntfhîtokm m lf vogrnkomj vogrnkomjfs fs hfkîjokns
Fj hukens npioknkomjfs lf oj`fjofrèn, kmhm in vogrnkoûj lf ims sostfhns hfkîjokms, ims sostfhns sufifj sfr kmhpifcms y lfgfj hmlfinrsf kmhm sostfhns lf hóitopifs `rnlms lf iogfrtnl, im quf ln kmhm rfsuitnlm sostfhns lf fkunkomjfs lobfrfjkonifs iojfnifs mrlojnrons. Hmlfinr uj sostfhn lf oj`fjofrèn kmhpifcm kmhm uj prmgifhn hntfhîtokm hnjfcngif nprmponlm y fstngifkfr ins fkunkomjfs fkunkomjfs lobf lobfrfjk rfjkonifs onifs quf `mgofrj `mgofrjnj nj smj n hfjulm fi prohf prohfrr pnsm lfsn lfsnnjtf. njtf. Fj fstn sfkkoûj, prohfrm sf kmjsolfrn quf uj prmgifhn lf vogrnkoûj iogrf lf uj sostfhn sohpif lf lms `rnlms lf iogfrtnl oiustrn fi prmkflohofjtm prmkflohofjtm gîsokm. Iuf`m sf fstngifkfj ins fkunkomj fkunkomjfs fs lf hmvohofjtm lf uj prmgifhn hîs kmhpifcm, fs lfkor, in vogrnkoûj lf uj nutmhûvoi, quf sf hmlfin kmhm uj sostfhn lf kuntrm `rnlms lf iogfrtnl. Fcfhpim >.> Fcfhpim
Lms gnrrns ujobmrhfs NG y K L lf lfjsolnl lf hnsn pmr ujolnl lf imj`otul ώ fstîj nrtokuinlns fj N y K . Ims rfsmrtfs rmtnkomjn rmtnkomjnifs ifs lf ro`olfz γ > y γ 7 prmpmrkomjnj prmpmrkomjnj rfsostfjkon n ins rmtnkomjfs lfi fxtrfhm N y fi fxtrfhm K , rfspfktovnhfjtf, kmhm sf hufstrn fj in `rîkn. `rîk n. Ins imj` imj`otul otulfs fs lf NG y K L smj s mj I> y I7 , rfspfktovn rfspfktovnhfjtf hfjtf.. In gnrrn NG iifvn ujn hnsn kmjkfjtrnln H fj fi fxtrfhm G . Ins gnrrns fstîj kmjfktnlns pmr uj rfsmrtf lf ro`olfz d . Kunj Kunjlm lm ins vnroiins kufi` kufi`nj nj iogrfhfjtf, iogrfhfjtf, smj vfrtokn vfrtoknifs ifs y jm eny bufrz bufrznn fj fi rfsmrtf. Fi hmvohofjtm lfi sostfhn sf lfskrogf hflonjtf lms îj`uims ε > y ε 7 . Kmj Kmjsol solfrf frf smi smimm pfqufþns mskoinkomjfs, fs lfkor, |ε > |
> y |ε | >, y lfskuolf fi fbfktm lf in `rnvflnl. 7
>. Kmj`urf ins fkunkomjfs lobfrfjkonifs quf ro`fj ims îj`uims lf rmtnkoûj ε > y ε 7 .
7. Qnrn fi knsm fspfkoni kunjlm I> 8 I , I7 8 7 I , H 8 ώ I , γ > 8 γ 7 8 1, y ins
kmjlokomjfs ojokonifs ε > (1) 8 ε >1 >1 , ε 7 (1) 8 ε 7 71 1 , ε > (1) 8 ε 7 (1) 8 1, lftfrhojnr ins rfspufstns lf in vogrnkoûj iogrf.
;. Qnrn fi knsm fspfkoni kunjlm I> 8 I7 8 I , H 8 1, γ > 8 γ 7 8 d I 7 , loskutn ins rfspufstns lf in vogrnkoûj iogrf. [miukoûj >.
♡
Ku Kunjl njlmm in rmt rmtnko nkoûj ûj lf iinn gn gnrrn rrn NG smgrf fi pujtm N fs ε > , fi lfspinznhofjtm
hnsn lf iinn gnrr gnrrnn NG fs h> 8 ώ I> y su emrozmjtni lfi pujtm G fs x> 8 I> soj ε > ≅ I> ε > . In hnsn
hmhfjtm lf ojfrkon smgrf fi pujtm N fs
C > 8
> ;
>
h> I>7 8 ώ I>; ;
Kunjlm in rmtnkoûj lf in gnrrn K L smgrf fi pujtm K fs fs ε 7 , fi lfspinznhofjtm emrozmjtni lfi pujtm L fs x 7 8 I7 soj ε 7
≅ I ε . In hnsn lf in gnrrn K L fs h 7
7
7 8
ώ I7 y
su hmhfjtm lf ojfrkon smgrf
‛7/>5‛
fi pujtm K fs fs C 7 8
> ;
>
h7 I77 8 ώ I7; ;
Qnrn Qn rn fstng fstngifk ifkfr fr ins fku fkunko nkomjf mjfss lobfrf lobfrfjko jkonif nifss qu quff ro`fj ro`fj ims îj` îj`uim uimss ε > y ε 7 , kmj kmjsol solfrf frf ims imslon lon`rn `rnhns hns lf kufrpm iogrf lf ins gnrrns NG y K L, rfspfktovnhfjtf. Fi Qrojkopom lf L‟Nifhgfrt sf npiokn pnrn kmj`urnr in fkunkoûj lf hmvoh hmvohofjtm. ofjtm.
Gnrrn NG .
Qn Qnrn rn fj fjkm kmjt jtrn rnrr in bu bufr frzn zn lf lfii rf rfsm smrtf rtf npio npiokn knln ln fj fi fxtrf xtrfhm hm G , mgsf mgsfrv rvff fi rfsm rfsmrtf rtf
fj fi fxtrfhm L3 fi fxtrfhm G sf hufvf enkon in lfrfken x> pnrn quf fi rfsmrtf sf kmhprohn y in bufrzn lfi rfsmrtf fhpucf fi fxtrfhm G enkon in ozquofrln. Kmhm fi fxtrfhm L tnhgoïj sf hufvf enkon in lfrfken fj x7 , in kmhprfsoûj jftn lfi rfsmrtf fs x> ∗ x 7 4 pmr im tnjtm, in bufrzn lfi rfsmrtf fs d ( x> ∗ x 7 ).
Kmhm in nkfifrnkoûj lfi pujtm G fs x > 8 I> ε > enkon in lfrfken, in bufrzn lf ojfrkon lf in hnsn H fs fs H I> ε > enkon in ozquofrln.
Kmhm in nkfifrnkoûj nj`uinr lf in gnrrn NG smgrf fi pujtm N fs ε > fj sfjtolm njtoemrnrom, fi hmhfjtm lf ojfrkon lf in gnrrn NG fs C > ε > fj sfjtolm emrnrom. Wffhpinkf in gosn`rn fj N pmr lms kmhpmjfjtfs lf bufrzn lf rfnkkoûj W Nx y W Ny . Fi rfsmrtf `orntmrom fj N prmpmrkomjn uj hmhfjtm lf rfsostfjkon fj fi sfjtolm lf ins n`ucns lfi rfimc γ > ε > . Npioknjlm fi Qrojkopom lf L‟Nifhgfrt y rfsuhofjlm hmhfjtms smgrf fi rfjlohofjtm lfi pujtm N
Qmr tnjtm
Gnrrn K L.
H N 8 1
> ;
3
C > ε > + H I> ε > I > + d x>
( ∗ x ) I
ώ I>7 + H I> ε > + d I> ε >
(
∗ I ε ) 7
7
+
7
γ >
I>
> +
γ > ε > 8 1
ε > 8 1
Qnrn ffjkmj jkmjtrnr trnr in bufrz bufrznn lfi rfs rfsmrtf mrtf npio npioknln knln ffjj fi fx fxtrfh trfhmm L, mgsfrvf fi rfsmrtf
fj fi fxtrfhm G 3 fi fxtrfhm L sf hufvf enkon in lfrfken x 7 pnrn quf fi rfsmrtf sf fxtofjln y in bufrzn lfi rfsmrtf torf lfi fxtrfhm L enkon in ozquofrln. Kmhm fi fxtrfhm G tnhgoïj sf hufvf enkon in lfrfken pmr x > , in fxtfjsoûj jftn lfi rfsmrtf fs x 7 ∗ x> 4 pmr im tnjtm, in bufrzn lfi rfsmrtf fs d ( x7 ∗ x > ). Kmhm in nkfifrnkoûj nj`uinr lf in gnrrn K L smgrf fi pujtm K fs ε 7 fj sfjtolm njtoemrnrom, fi
‛;/>5‛
hmhfjtm lf ojfrkon lf in gnrrn K L fs C 7 ε 7 fj sfjtolm emrnrom. Wffhpinkf in gosn`rn fj K pmr lms kmhpmjfjtfs lf bufrzn lf rfnkkoûj WK x y WKy . Fi rfsmrtf `orntmrom fj K prmpmrkomjn prmpmrkomjn uj hmhfjtm lf rfsostfjkon fj fi sfjtolm lf ins n`ucns lfi rfimc γ 7 ε 7 . Npioknjlm fi Qrojkopom lf L‟ L‟Nifhgfrt Nifhgfrt y rfsuhofjlm ims hmhfjtms smgrf fi pujtm K ln ln
Qmr tnjtm
H K 8 1
3
C 7 ε 7 + d x7
( ∗ x ) I
>
ώ I77 ε 7 + d I7 ε 7
(
;
∗ I ε ) >
>
7 +
>
+
γ 7
I7
γ 7 ε 7 8 1
ε 7 8 1
7. Ku Kunj njlm lm I> 8 I , I7 8 7 I , H 8 ώ I , γ > 8 γ 7 8 1, ins fku fkunko nkomjf mjfss lf hm hmvoh vohofj ofjtm tm sf kmj kmjvof vofrtfj rtfj fj
>
ώ I
7
( ώ I I ε d T I ε ∗ (7 I)ε X
+
;
)
> ;
>
+
7 8 1
>
ώ 7 I 7 ε 7 + d 7 I ε 7
( )
T( ) ∗ I ε X
> 8 1
pmr tnjtm 7
Ϗ1 ε + (ε > ∗ 7 ε 7 ) 8 1, >
Ϗ17 8
;
Ϗ7
8 1
(7ε ∗ ε ) ;
+ ε 7
? d
1
7
,
Ϗ17 8
d
;
Ϗ17 ε > + L7
7 1
7Ϗ
∗;
ε 7 8 1,
Fi lftfrhojnjtf lf in hntroz lf kmfkofjtfs fs
( )
ς L
∗ L
8
7
7 1
Ϗ
+
∗
;
7 Ϗ 1
;
L7 +
;
7 1
7Ϗ
7 1
7Ϗ ;
8
7Ϗ +
7 1
;
ε 7 8 1.
L< + Ϗ17 L7
In fkunkoûj knrnktfrèstokn vofjf lnln pmr ς (δ) 8 1, quf ln δ
<
+
7 1
Ϗ δ
7
8 1
⇗
8
δ
7
δ7
+
Ϗ17
8 1
⇗
8
δ 8 1, 1, o Ϗ1,
∗oϏ
1
Qmr im tnjtm, in smiukoûj kmhpifhfjtnron pnrn ε 7 vofjf lnln pmr
()
ε 7 t
8
K 1 + K > t + N kms Ϗ1 t + G soj Ϗ1 t
In smiukoûj kmhpifhfjtnron lf ε > sf puflf mgtfjfr kmhm
()
ε > t
(K
8 7
1 +
) ∗ N kms Ϗ t ∗∗ G soj Ϗ t
K > t
1
1
Qmr im tnjtm, ins smiukomjfs kmhpifhfjtnrons m ins rfspufstns lf vogrnkoûj iogrf lfi sostfhn fstîj lnlns pmr
( ) 7 (K ε (t ) (K
) ∗ ( N kms Ϗ t G soj Ϗ t ) K t ) ( N kms Ϗ t G soj Ϗ t ) sofjlm (K K t ) 3 Hmv Hmvohofjtm ohofjtm jm mskointm mskointmrom rom y ( N kms Ϗ t G soj Ϗ t ) 3 Mskoinkoûj nrhûjokn kmj Ϗ , ε > t 7
1+
>
8
8
1 +
1 +
K > t >
+
1 +
1
1 +
1
1 +
1
fj fi quf ims lms prohfrms tïrhojms kmrrfspmjlfj ni hmvohofjtm jm mskointmrom kmj ε > 8 7ε 7 (in imj`otul lfi rfsmrtf jm knhgon)4 hofjtrns quf ims lms óitohms tïrhojms kmrrfspmjlfj n in ms-
1
‛5‛
koinkoûj nrhûjokn kmj ε > 8 ∗ε 7 n in brfkufjkon korkuinr Ϗ 1 . Fstms lms hmvohofjtms sf hufstrnj
fsqufhîtoknhfjtf fsqufhîtoknhfj tf fj in so`uofjtf `urn.
Bo`urf >.>3 Hmvohofjtm jm mskointmrom mskointmrom y mskoinkoûj mskoinkoûj nrhûjokn nrhûjokn kmj brfkufjkon Ϗ1
Qnrn ins kmjlokomjfs ojokonifs ε > (1) 8 ε >1 71 1 , ε > (1) 8 ε 7 (1) 8 1, y mgsfrvnjlm >1 , ε 7 (1) 8 ε 7 ε > t
() ε (t )
8 7
∗∗
K > + N Ϗ1 soj Ϗ1 t G Ϗ1 kms Ϗ1 t ,
∗ NϏ soj Ϗ t
7
8
K >
ε 7 1
8
K 1 + N 8 ε 71 71 ε >
1 +
1
G Ϗ1 kms Ϗ1 t .
sf tofjf ε > 1
8 7
K 1 N 8 ε >1 >1
) ln im (quf
∗ K 1 8
()
>
(ε ;
>1 >1 +
1
8 7
()
K > GϏ1 8 1
∗
>
)
7 71 1 +
;
ε 7 1
8
K > + GϏ1 8 1
()
(7ε
ε 71 71 , K > 8 1, N 8
)
ε >>11 , G 8 1
Ins rfspufstns lf in vogrnkoûj iogrf smj 7
() ε (t ) ε > t
8
8
7
(ε ; > (ε ;
>1 >1 +
ε 71 71
>1 >1 +
ε 71 71
) ∗ ;> (7ε ∗ ε ) kms Ϗ t , ) >; (7ε ∗ ε ) kms Ϗ t . +
7 71 1
> >1 1
1
7 71 1
> >1 1
1
;. Ku Kunj njlm lm I> 8 I , I7 8 I , H 8 1, γ > 8 γ 7 8 d I 7 , ins fku fkunko nkomj mjfs fs lf hm hmvoh vohofj ofjtm tm sf kmj kmjvof vofrtfj rtfj fj >
ώ I
;
7
ε + d ( I ε > ∗ I ε 7 ) + >
>
ώ I 7 ε 7 + d I ε 7
( ∗ I ε )
;
>
+
d I7 I
ε > 8 1,
d I7 ε 7 8 1, I
quf rfsuitn ε > + Ϗ7 7ε >
7 8 1
ε 7 + Ϗ7 7ε 7
> 8 1
( ∗ ε )
,
( ∗ ε )
,
Ϗ7
8
Ϗ7
8
; d
ώ I ; d ώ I
Vsnjlm fi mpfrnlmr L , ins fkunkomjfs lf hmvohofjtm sf puflfj fskrogor kmhm 7
7
7
( L 7Ϗ )ε ∗ Ϗ ε ∗Ϗ ε ( L 7Ϗ ) +
7
7 8 1
>
7
> +
7
+
,
8 1
.
Fi lftfrhojnjtf lf in hntroz lf kmfkofjtfs fs
( )
ς L
8
∗
L7 + 7Ϗ7 Ϗ
7
L
∗Ϗ
7
+
7
7Ϗ
7
8
L<
+
5‛
In fkunkoûj knrnktfrèstokn vofjf lnln pmr ς (δ) 8 1, quf ln δ< + vofjf lnln pmr
()
ε > t
8
N> kms Ϗt + G> soj Ϗt + N 7 kms
∓
;Ϗt + G7 soj
∓
;Ϗt
In smiukoûj kmhpifhfjtnron lf ε 7 sf puflf mgtfjfr kmhm
()
ε 7 t
8
∗ ∗
N> kms Ϗt + G> soj Ϗt N 7 kms
∓ ;Ϗt ∗∗ G soj ∓ ;Ϗt 7
Ins smiukomjfs kmhpifhfjtnrons m ins rfspufstns lf vogrnkoûj iogrf lfi sostfhn smj
( ) ( N kms Ϗt ε (t ) ( N kms Ϗt
ε > t 7
8
>
+
8
>
+
∓ ) ( N kms ;Ϗt ∓ G soj Ϗt ) ∗ ( N kms ;Ϗt
G> soj Ϗt
+
>
7
+
7
+
∓ ;Ϗt ), ∓ G soj ;Ϗt ),
G7 soj 7
sofjlm
( N kms Ϗt G soj Ϗt ) 3 Mskoinkoûj lfi prohfr hmlm fj Ϗ y ∓ ∓ ∓ ( N kms ;Ϗt G soj ;Ϗt ) 3 Mskoinkoûj lfi prohfr hmlm fj ;Ϗ. +
>
>
+
7
7
Ins kuntrm kmjstnjtfs N> , G > , N7 y G7 sf lftfrhojnj n pnrtor lf ins kmjlokomjfs ojokonifs ε > (1),
smiukomjfs fs kmrrfspmjlfj ni prohfr hmlm ( ) ( ) y ε (1). Ims lms prohfrms tïrhojms fj ins smiukomj lf mskoinkoûj nrhûjokn n brfkufjkon korkuinr Ϗ Ϗ kmj ε ε 4 hofjtrns quf ims lms∓ óitohms ε 7 1 , ε > 1
7
> 8
> 8
7
tïrhojms kmrrfspmjlfj ni sf`ujlm hmlm lf mskoinkoûj n brfkufjkon korkuinr Ϗ7 8
;Ϗ kmj
∗ε . \fj`n fj kufjtn quf ins brfkufjkons jnturnifs sofhprf sf ugoknj fj mrlfj nskfjlfjtf,
ε > 8
7
fs lfkor, Ϗ > 9 Ϗ7 . Fstms lms hmvohofj hmvohofjtms tms sf hufstrnj fsqufhîtoknhfjtf fj in so`uofjtf `urn.
Bo`urf >.73 Mskoinkoûj kmj brfkufjkon Ϗ , Ϗ > 8 Ϗ y Mskoinkoûj kmj brfkufjkon
∓ ;
Ϗ, ε > 8
∗ε
7
Fcfh Fcfhpim pim >.7
Fj uj pinjm emrozmjtni ujobmrhf GK (pmr (pmr fcfhpim, ujn supfrkof iosn), uj mgcftm N> fstî kmjfktnlm n uj pujtm cm Q pmr uj rfsmrtf soj hnsn [ > lf imj`otul jnturni I > . Vj mgcftm N7 fstî kmjfktnlm n N> pmr uj rfsmrtf soj hnsn [ 7 lf imj`otul
jnturni I7 lf tni hnjfrn
quf fi pujtm cm Q y ims kfjtrms lf `rnvflnl N> y N 7 fstîj tmlms fj iojfn rfktn. Fi mgcftm N > sf lfspinzn ujn lostnjkon n> enkon in lfrfken m in ozquofrln lf su pmsokoûj lf fquoiogrom M> , fi mgcftm N 7 sf lfspinzn ujn lostnjkon n 7 enkon in lfrfken m in ozquofrln lf su pmsokoûj lf fquoiogrom M 7 , y fj fi hmhfjtm t 8 1 sf sufitnj ims lms mgcftms. ½Kuîi fs in pmsokoûj lf ims lms mgcftms fj kuniquofr hmhfjtm t : 1=
♡
[upmjf mjfhms hms proh prohfrm frm qu quff fi pin pinjm jm GK fs tn tnjj sunv sunvff quf quf ins ins bufr bufrzn znss lf bro brokk kkoû oûjj pufl puflfj fj sfr sfr [miukoûj [up
‛2/>5‛
lfskuolnln lfsku olnlns. s. \ \nhgo nhgoïj ïj nsuho nsuhohms hms quf qufjoj` joj`ujn ujn bufr bufrzn zn fx fxtfrjn tfrjn nktó nktónn smgrf smgrffi fi sostf sostfhn. hn. [u [upmj` pmj`nhms nhms quf fi mgcftm N > tofjf hnsn h> y fi mgcftm N7 tofjf hnsn h7 . [upmj`n [upmj`nhms hms nlfhîs quf fi rf rfsmrtf smrtf [ > tofjf in kmjstnjtf d > y fi rfsmrtf [ 7 tofjf in kmjstnjtf d 7 .
[fn x> fi lfspinznhofjtm lf N> lfslf su pmsokoûj lf fquoiogrom M > fj fi tofhpm t ≦ 1 y
nsuhor quf x> fs pmsotovm kunjlm N> fstî n in lfrfken lf M > . Lf in hos hoshn hn hnj hnjfrn, frn, sfn x7 fi
≦ 1 y supmj`n quf x 7 fs lfspinznhofjtm lf N 7 lfslf su pmsokoûj lf fquoiogrom M7 fj fi tofhpm t ≦ pmsotovm kunjlm N7 fstî n in lfrfken lf M7 .
Kmjsolfrf ins bufrzns quf nktónj fj N> fj fi hmhfjtm t : 1. Eny lms bu bufrzns frzns,, B > y B 7 , lmjlf B > sf fcfrkf fj fi rfsmrtf [ > y B 7 fj fi rfsmrtf [ 7 . [f`ój in ify lf Emmdf, in bufrzn B > fs lf hn`jotul d > | xx> | . Kmhm fstn bufrzn sf fcfrkf enkon in ozquofrln kunjlm N> fstî n in lfrfken lf M> y enkon in lfrfken kunjlm N > fstî n in ozquofrln lf M> , tfjfhms B > 8 ∗d > x> . Jufv Jufvnhfjtf nhfjtf usnjlm in ify lf Emmdf, in bufrzn B 7 fs lf hn`jotul d 7 s, lmjlf s fs in fimj`nkoûj lf [ 7 fj fi tofhpm t . Kmhm s 8 | xx7 ∗ x > | , vfhms quf in hn`jotul lf B 7 fs d 7 | xx7 ∗ x > | . Nlfhîs, lnlm quf
fstn bufrzn sf fcfrkf enkon in ozquofrln kunjlm x 7 ∗ x> 9 1 y enkon in lfrfken kunjlm x 7 ∗ x> : 1,
vfhms quf B 7 8 d 7 ( x7 ∗ x > ).
Nemrn npioknjlm in sf`ujln ify lf Jfwtmj ni mgcftm N > , mgtfjfhms in fkunkoûj lobfrfj lobfrfjkoni koni h> x> 8
∗d x >
> +
( ∗ x ).
d 7 x7
>
(>.>)
Nemrn vmivfhms ni mgcftm N7 y kmjsolfrnhms ins bufrzns quf nktónj smgrf ïi fj fi hmhfjtm t : 1.
Eny ujn lf fsns bufrzns, B ; , y fstn fs fcfrkoln pmr fi rfsmrtf [ 7 . Npioknjlm in if ifyy lf Emmdf
ujn vfz hîs, mgsfrvnhms quf fstn bufrzn tnhgoïj fs lf hn`jotul d 7 s 8 d 7 | xx7 ∗ x > | . Kmhm B ; sf fcfrkf enkon in ozquofrln kunjlm x7 ∗ x > : 1 y enkon in lfrfken kunjlm x7 ∗ x > 9 1, vfhms
‛0/>5‛
quf B ; 8 ∗d ( x7 ∗ x> ). Npioknjlm in sf`ujln ify lf Jf Jfwtmj wtmj ni mgcftm N 7 , mgtfjfhms in fkunkoûj
lobfrfjkoni
h7 x7 8
∗d ( x ∗ x ). 7
7
>
(>.7)
Nlfhîs lf ins fkunkomjfs lobfrfjkonif lobfrfjkonifss ( (>.> >.>)) y (>.7 ( >.7), ), vfhms fj in lfkinrnkoûj lfi prmgifhn quf ins kmjlokomjfs ojokonifs fstîj lnlns pmr
(1)
(1)
8 1
, x7 1
(1)
8 1
(>.;) In bmrhuinkoûj hntfhîtokn lfi prmgifhn kmjsostf, pufs, fj ins fkunkomjfs lobfrfjkonifs (>.> (>.>)) y x>
8
n >, x>
()
8
n 7, x7
.
(>.7) >.7) y ins kmjlokomjfs ojokonifs (>.; (>.;)). fskrogofjlm ins fkunkomjfs lobfrfjkonifs fj in bmrhn
h> x> + d > + d 7 x>
(
) ∗ d x 7
7 8 1
,
(>. + d 7 x7 8 1 . 7
vfhms quf bmrhnj uj sostfhn lf fkunkomjfs lobfrfjkonifs iojfnifs emhm`ïjfns kmj kmfkofjtfs kmjstnjtfs.
Fj iu`nr lf rfsmivfr ims prmgifhns `fjfrnifs quf kmjsostfj fj fi sostfhn (>.< (>..;), ), iifvnrfhms n kngm in smiukoûj fj uj knsm pnrtokuinr quf buf fif`olm pnrn bnkoiotnr fi trngncm. (>.; [upmj [up mj`n `nhm hmss quf quf ims imslms lmsmg mgcft cftms ms N> y N7 smjknlnujmlfujolnllfhnsn,pmrimquf h> 8 h 7 8 > . Nlfhîs, supmj`nhms quf ims rfsmrtfs [ > y [ 7 tofjfj kmjstnjtfs lf rfsmrtf d > 8 ; y d 7 8 7, rfspfktovnhfjtf. rfspfktovnh fjtf. Nlfhîs, tmhnrfhm tmhnrfhmss n > 8 ∗> y n 7 8 7 . Fjtmjkfs fi sostfhn (>.< (>. + 6 x>
x 7
∗ 7 x
7 8 1
,
(>.6)
7 x> + 7 x7 8 1,
y ins kmjlokomjfs ojokonifs ( ojokonifs (>.; >.;)) sf vufivfj x>
(1) ∗>, 8
x>
(1)
8 1
()
, x7 1
, x7 1
()
8 7
8 1
.
(>.2)
Fskrogofjlm fi sostfhn ( sostfhn (>.6 >.6)) fj jmtnkoûj lf mpfrnlmr, tfjfhms L7 + 6 x>
∗(
7 x> +
7 x7 8 1,
)( L ∗ 7) x 7
(>.0)
7 8 1
+
,
>.0)), hui huitop topiok ioknhm nhmss in sf` sf`ujl ujlnn fku fkunko nkoûj ûj Npioknhms Npiok nhmsfi fi mpfr mpfrnlmr nlmr ( L7 + 7) ninprohfrnfkunkoûjlf( ninprohfrnfkunkoûjlf(>.0 >.0)) pmr 7, y n`rf`nhms ins lms fkunkomjfs pnrn mgtfjfr lf ( lf (>.0
T( L
7
+
)( L
7
7
+
) ∗ 8 1
⇗
( L
8
<
+
0 L
7
+
)
2 x> 8 1 .
(>.5)
In fkunkoûj nuxoionr kmrrfspmjlofjtf n in fkunkoûj lobfrfjkoni lf kunrtm mrlfj ( mrlfj (>.5 >.5)) fs h<
+
7
0h
+
2 8 1
⇗
8
(h
7
+
)(h
2
7
+
)
>
8 1
.
Qmr im tnjtm, in smiukoûj `fjfrni lf in fkunkoûj lobfrfjkoni (>.5 (>.5)) fs x> 8 k > soj t + k7 kms t + k; soj
∓
2 t +
k< kms
∓
2 t .
(>.?) 7
Nemrn huitopioknhms in prohfrn fkunkoûj lf (>.0 (>.0)) pmr 7, npioknhms fi mpfrnlmr ( L
+ 6
) n in
‛5/>5‛
sf`ujln fkunkoûj lf ( lf (>.0 >.0)) y im n`rf`nhms pnrn mgtfjfr in fkunkoûj lobfrfjkoni
( L
<
+
0 L
7
+
)
2 x7 8 1
(>.>1)
pnrn x 7 . In smiukoûj `fjfrni lf (>.? (>.?)) fs kinrnhfjtf x7 8 d > soj t + d 7 kms t + d ; soj
∓
2 t + d < kms
∓
2 t .
(>.>>)
Fi lftfrhojnjtf lf ims kmfkofjtfs lfi mpfrnlmr fj fi sostfhn (>.0 (>.0)) fs
∗ L7
+
7
6
∗7
L7 + 7
8
L <
+
0 L
7
+
2.
Kmhm sf trntn lf uj mpfrnlmr lf kunrtm mrlfj, in smiukoûj `fjfrni lf (>.6 ( >.6)) lfgf kmjtfjfr kuntrm kmjstnjtfs ojlfpfjlofjtf ojlfpfjlofjtfs. s. Lfgfhms su sustotuor stotuor x> lnlms pmr ( pmr (>.? >.?)) y x7 lnlms pmr (>.>> (>.>>)) fj ins fkunkomjfs lfi sostfhn (>.6 (>.6)) pnrn lftfrhojnr ins rfinkomjfs quf lfgfj fxostor fjtrf ins kmjstnjtfs k> , k7 , k; , k< , d > , d 7 , d ; y d .? (>.?)) y (>.>> (>.>>)) rfprfsfjtf in smiukoûj s miukoûj `fjfrni lf lf ( (>.6 >.6). ).
sustotuyfjlm, sustotuy fjlm, fjkmjtrnhms quf
∗>7 k ,
d > 8 7 k>, d 7 8 7 k7, d ; 8
;
∗ >7 k .
d < 8
<
Qmr im tnjtm, in smiukoûj `fjfrni lfi sostfhn (>.6 ( >.6)) fstî lnln pmr
x> 8 k> soj t + k7 kms t + k; soj
∓ ∓ 2 t
+
x7 8 d > soj t + d 7 kms t + d ; soj
k< kms
∓ ∓ 2 t ,
2 t + d < kms
(>.>7)
2 t .
Nemrn npioknhms ins kmjlokomjfs ojokonifs ( ojokonifs (>.2 >.2). ). Npioknjlm ins kmjlokomjfs x > 8 ∗>, x > 8 1 fj >.>7), ), fjkmjtrnhms t 8 1 ni prohfrm lf in pnrfcn ( pnrfcn (>.>7
∗
> 8 k 7 +
1 8 k > +
k.>;)
∓
2 k ; .
(>.>7), Npioknjlm ins kmjlokomjfs x 7 8 7, x 7 8 1 fj t 8 1 ni sf`ujlm lfi pnr (>.>7 ), mgtfjfhms
1 8
> 7
∗ k , ∓ k . 7k ∗
7 8 7k7 >
<
2 7
;
( >.>.>; (>.>;)) y (>.>< k> 8 1, k7 8
; 6
(>.>.6)) y ins Qmr im tnjtm, in smiukoûj pnrtokuinr lfi prmgifhn fspfkèkm quf kmjsostf fj fi sostfhn ( >.6 kmjlokomjfs (>.2 (>.2)) fs
x> 8 ;6 kms t x
Fcfhpim Fcfh pim >.;
∗ ∗
5 kms 6
< 2 7 8 6 kms + 6 kms
t
∓ 2 t , ∓
.
2 t .
Qnrn fstulonr in rfspufstn lojîhokn lf uj nutmhûvoi fj ujn knrrftfrn mjluinln, sf utoiozn y >.< >..; `urns >.; y >. Fi kufrpm lfi vfeèkuim sf hmlfin kmhm uj kufrpm rè`olm lf hnsn h y hmhfjtm lf
‛?/>5‛
ojfrkon C smgrf su kfjtrm lf `rnvflnl K . 7. Ims nhmrto`unlmrfs lfinjtfrms sf hmlfinj kmhm uj rfsmrtf lf ro`olfz d b b y uj nhmrto`unlmr lf kmfkofjtf lf nhmrto`unkoûj k b . Ims nnhmrto` hmrto`unlm unlmrfs rfs tr trnsfrm nsfrmss sf hmlfinj kmhm uj rfsmrtf lf ro`olfz d r r y uj nhmrto`unlmr lf kmfkofjtf lf nhmrto`unkoûj kr . hmlfinlm mlfinlm ;. Fi fcf lfinjtfrm sf h hmlfin mlfin kmhm ujn hnsn pujtuni h b . Fi fcf trnsfrm fstî h kmhm uj pujtm lf hnsn hr . .;3 Vj nutmhûvoi hmvoïjlmsf hmvoïjlmsf fj ujn knrrftfrn mjluinln
t ,
Fi lfspinznhofjtm vfrtokni lf ims jfuhîtokms lfinjtfrms fs y b 1 (t ) 8 y1 (Vt ). Fj fi tofhpm
ims jfuhîtokms trnsfrms fstîj fj x 8 Vt ∗ ∗ NG N G 8 V t ∗ ∗ I , I 8 I b + Ir , y fi lfspinznhofjtm
vfrtokni lf ims jfuhîtokms trnsfrms fs yr 1 t 8 y1 Vt I .
( ) ohpmjfr ( ∗ lfspinznhofjtm ) Fj rfinkoûj kmj ins pmsokomjfs lf rfbfrfjkon, vfrtokni y (t ) ni kfjtrm lf `rnvflnl K y ujn rmtnkoûj ε (t ) nirflflmr lf K n in knrrmkfrèn lfi vfeèkuim, lfspinznhofjtm
‛>1/>5‛
vfrtokni y b (t ) ni fcf lfi lfinjt njtfrm frm (hn (hnsn sn h b ) y vfrto frtokni kni lfs lfspin pinznh znhofj ofjtm tm y r (t ) enko enkonn fi fcf fcf tr trns nsfr frmm (hns (hnsnn hr ) kmhm sf hufstrn fj in `urn >.< `urn >...6 `urn >.6 fstî fstî fj fquoiogrom lojîhokm. Qmr im tnjtm
↛
∗ ∗ ∗ ( ∗ ) ∗ ∗ ∗ ∗ ( ∗ )
h y + k b
B y 8 1
3
H K 8 1
3
C ε ε + k b
Fi fcf lfinjtfrm.
y G
y G
y b
y b
+
d b b
+
d b b
y G
y G
y b
y b
+
kr
I b
y N
kr
yr
y N
+
d rr
yr
y N
+
d rr
yr
y N
8 1
yr
I r 8 1
Qnrn lftfrhojnr in bu bufrzn frzn llfi fi rfsmrtf rfsmrtf npioknln fj fi fcf lfinjtfrm h b pmr
ims nhmrto`unlmrfs lfinjtfrms, mgsfrvf fi rfsmrtf d b b fj fi pujtm G 3 fi fcf lfinjtfrm h b sf hufvf enkon nrrogn pmr y b pnrn quf fi rfsmrtf d b b sf kmhprohn y in bufrzn lfi rfsmrtf fhpucf fi fcf lfinjtfrm h b enkon ngn ngncm. cm. Kmhm fi pujtm G tnhgoïj sf hufvf enkon nrrogn pmr y G , in kmhprfsoûj
‛>>/>5‛
Bo`urf >.63 Vj nutmhûvoi hmvoïjlmsf hmvoïjlmsf fj ujn knrrftfrn mjluinln
jftn lfi rfsmrtf fs y b ∗ y G 4 pmr im tnjtm, in bufrzn lfi rfsmrtf fs d b b ( y b ∗ y G ) en enkon kon ngn ngncm. cm. Lf hnjfrn sohoinr, in bufrzn lf nhmrto`unkoûj npioknln fj fi fcf lfinjtfrm h b pmr fi nhmrto`unlmr
∗ y G ) enkon ngncm. lfinjtfrm fs k b ( y b b ∗ Qnrnlftfrhojnrinbufrznlfirfsmrtfnpioknlnfjfifcflfinjtfrm h b pmr pmr ims jfuh jfuhîtokm îtokmss lfin lfinjtfrm jtfrms, s, mgsfrvf fi rfsmrtf d b b t fj in pnrtf ojbfromr lf ims jfuhîtokms lfinjtfrms3 fi fcf lfinjtfrm h b sf hufvf enkon nrrogn pmr y b pnrn quf fi rfsmrtf d b b t sf fxtofjln y in bufrzn lfi rfsmrtf torf lfi fcf lfinjtfrm h b enkon ngncm. Kmhm in pnrtf ojbfromr lf ims jfuhîtokms lfinjtfrms tnhgoïj sf hufvf hufvf enkon nrrogn fj y b 1 , in fxtfjsoûj jftn lfi rfsmrtf fs y b ∗ y b 1 4 pmr im tnjtm, in bufrzn lfi rfsmrtf fs
( ∗ ) enkon ngncm.
d bb t y b
y b 1
Kmhm in nkfifrnkoûj lfi fcf lfinjtfrm h b fs y b enkon nrrogn, in bufrzn lf ojfrkon fs h b y b enkon ngncm. Npioknjlm fi Qrojkopom lf L‟Nifhgfrt, fi kufrpm iogrf lfi fcf lfinjtfrm kmhm sf hufstrn fj in `urn >.6 `urn >.6 fstî fstî fj fquoiogrom lojîhokm. Qmr im tnjtm
↛ B
y 8 1 3
Fi fcf trnsfrm.
↛
∗ ∗ ∗ ∗ ∗ ( ∗ ) ( ∗ )
y b + k b h b y
y G
y y b b
+
d b b
y y b
+
y G
d bb t
y y b
y b 1
8 1
Lfi hoshm hmlm, in fkunkoû fkunkoûjj lf hmvohofjtm hmvohofjtm llfi fi fcf trnsfrm fs B y 8 1
3
hr yr + kr
yr
y N
+
d rr
yr
+
y N
d rrt t
yr
y r 1
8 1
Mgsfrvnjlm y G 8 y + I b ε , y N 8 y ∗ I r ε , y b 1 (t ) 8 y1 (Vt ), yr 1 (t ) 8 y1 (Vt ∗ ∗ I ), ins fkunkomjfs lf hmvohofjtm lfi nutmhûvoi smj kuntrm fkunkomjfs lobfrfjkonifs lf sf`ujlm mrlfj
∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ( ) ∗ ∗ ∗
h y + k b + kr
C ε ε + k b I b
y
+
∗ k I
∗ k I
k b I b
r r y
r r
ε
k b y b
kr yr + d b b + d rr +
+
k b I b 7 + kr Ir 7 ε
h b y b
∗ k
b y
k b I b ε + k b y b
d b b y
d rr I r ε d b b y b
k b I b y b + kr Ir yr + d b b I b 7
+
d b b I b
y+
d rr I r
d rr yr 8 1, y +
7
d b I b + d r Ir
y b + d r Ir y yr 8 1, ε d b I b y
d b b I b ε + d b b + d bb t
y b 8 d bb t y1
Vt ,
>.> Nhmrto`un `unlmr lmrfs fs lf vo vogr grnk nkom omjf jfss m nh nhmrt mrto`u o`unlm nlmrf rfss lf hn hnsn sn so sojt jtmj mjoz oznlm nlmss >.> Nhmrto hr yr
∗ k
r y
+ kr Ir ε + kr y ∗ d r r y + d r r I ∗∗ I ). r ε + ( d rr + d rrt t ) y r 8 d r t y1 (Vt r
‛>7/> ‛>7 />5‛ 5‛
>.> Nhmrto Nhmrto`unlmr `unlmrfs fs lf vogrnkomjfs vogrnkomjfs m nhmrto`unlmrf nhmrto`unlmrfss lf hnsn sojtmjoznlms Fj npioknkomjfs lf oj`fjofrèn, hukems sostfhns sf puflfj hmlfinr kmhm sostfhns lf uj smim `rnlm lf iogfrtnl. Qmr fcfhpim, ujn hîquojn hmjtnln fj ujn fstrukturn puflf hmlfinrsf utoioznjlm uj sostfhn lf nhmrto`unlmr lf rfsmrtf lf hnsn, fj fi quf in hîquojn sf kmjsolfrn rè`oln kmj in hnsn h y in fstrukturn lf smpmrtf fs fquovnifjtf n uj rfsmrtf d y uj nhmrto`unlmr k, kmhm sf hufstrn fj in `urn >.2 `urn >.2.. In hîquojn fstî sucftn n ujn bufrzn sojusmolni B 1 soj t , quf
puflf sfr ujn knr`n npioknln fxtfrjnhfjtf m lfgolm n uj lfsfquoiogrom fj in hîquojn.
Bo`urf >.23 Vjn hîquojn hmjtnln smgrf ujn fstrukturn fstrukturn
Qnrn uj ójokm sostfhn lf `rnlm lf iogfrtnl, fs gofj sngolm quf kunjlm in brfkufjkon lf fxkotnkoûj ™ fstî kfrkn lf in brfkufjkon jnturni lfi sostfhn Ϗ1 8
d h
, mkurrf ujn vogrnkoûj
lf `rnj nhpiotul. nhpiotul. Fj pnrtokuinr pnrtokuinr,, kunj kunjlm lm fi sostf sostfhn hn jm fstî nhmrto nhmrto`unl `unlm, m, fs lfko lfkor, r, k 8 1, in rfsmjnjkon mkurrf kunjlm ™ 8 Ϗ 1 , fj fi quf in nhpiotul lf in rfspufstn krfkf iojfnihfjtf kmj fi tofhpm. Qnrn Qn rn rfl rfluko ukorr in vog vogrnk rnkoûj oûj lfi sos sostfh tfhn, n, sf hm hmjtn jtn uj nhm nhmrto` rto`unl unlmr mr lf vog vogrnk rnkomj omjfs fs m uj nhm nhmrto` rto`unl unlmr mr lf hnsn sojtmjoznlm (\HL), quf fs uj sostfhn nuxoionr lf nhmrto`unlmr lf hnsn-rfsmrtf, fj fi sostfhn projkopni kmhm sf hufstrn fj in `urn >.0 `urn >.0(n). (n). In hnsn hnsn,, in ro`olfz llfi fi rfsm rfsmrtf rtf y fi kmfkofjtf lf nhmrto`unkoûj lfi nhmrto`unlmr voskmsm smj h n , d n y k n , rfspfktovnhfjtf, lmjlf fi sugèjlokf "n" so`jokn "nuxoionr". Qnrn lfrovnr in fkunkoûj lf hmvohofjtm lf in hnsn projkopni h, kmjsolfrf su lon`rnhn lf kufrpm iogrf kmhm sf hufstrn fj in `urn >.0 `urn >.0(g). (g). Kmhm in hnsn h sf hufvf enkon nrrogn, fi rfsmrtf d sf sf fxtofjlf y fi rfsmrtf d n sf kmhprohf. >. Lfgolm ni lfspinznhofjtm x lf in hnsn h, in fxtfjsoûj lfi rfsmrtf d fs x . Qmr im tnj tnjtm, tm, fi rfsmrtf d fcfrkf ujn bufrzn lfskfjlfjtf d x y fi nhmrto`unlmr k fcfrkf ujn bufrzn lfskfjlfjtf k x smgrf in hnsn h . 7. Lfgolm n quf in hnsn h n tnhgoïj sf hufvf enkon nrrogn ujn lostnjkon xn , in kmhprfsoûj tnjtm,, fi rfsm rfsmrtf rtf d n y fi nhmrto`unlmr kn fcfrkfj jftn fj fi rfsmrtf d n fs x ∗ x n . Qmr im tnjtm bufrzns lfskfjlfjtfs d n ( x ∗ x n ) y k n ( x ∗ x n ), rfspfktovnhfjtf, smgrf in hnsn h .
>.> Nhmrto`un `unlmr lmrfs fs lf vo vogr grnk nkom omjf jfss m nh nhmrt mrto`u o`unlm nlmrf rfss lf hn hnsn sn so sojt jtmj mjoz oznlm nlmss >.> Nhmrto
‛>;/> ‛>; />5‛ 5‛
Bo`urf >.03 Vj ngsmrgflmr ngsmrgflmr lf vogrnkomjfs hmjtnlm hmjtnlm fj fi sostfhn projkopni
In sf`ujln ify lf Jfwtmj Jfwtmj rfquofrf
↞
hx 8 B 3
hx 8
∗d x ∗ k x ∗ d ( x ∗ x ) ∗ k n
n
∗ x
n
m
(
)
+
B 1 soj ™t
) ∗ k x ∗ d x
hx + k + kn x + d + d n x
(
x n
n n 8 B 1 soj ™t
n n
Kmhm hm iinn hn hnsn sn hn sf Lfi hoshm hmlm, kmjsolfrf fi lon`rnhn lf kufrpm iogrf lf hnsn hn . Km hufvff enkon nrrogn ujn lostnjkon x n (t ), fi rfsmrtf d n sf fxtofjlf. In fxtfjsoûj jjftn hufv ftn lfi rfsmrtf d n
fs x n ∗ x . Qmr im tnjtm, fi rfsmrtf d n y fi nhmrto`unlmr k n fcfrkfj bufrzns enkon ngncm d n ( xn ∗ x )
y k n ( xn ∗ x ), rfspfktovnhfjtf. Npioknjlm in [f`ujln Ify lf Jfwtmj ln
↞
hn xn 8
B 3 hn xn 8
∗d ( x ∗ x ) ∗ k n
n
n
pmr im tnjtm
∗ k x ∗ d x
hn xn + kn xn + d n xn
n
n
∗ xn
x
8 1
Ins fkunkomjfs lf hmvohofjtm sf puflfj fskrogor utoioznjlm fi mpfrnlmr L kmhm
hL 7 + x + kn L + d + d n x
( ) ∗ (k L n
+
( d ) x n
) ∗ (k L
+
hn L7
+
)
∗∗
d n xn
8
B 1 soj ™t
kn L + d n x n
8
1
n
+
Lfgolm n in fxostfjkon lf nhmrto`unhofjtm, ins rfspufstns lf vogrnkoûj iogrf (smiukomjfs kmhpifhfjtnrons) lfknfj fxpmjfjkonihfjtf y sf nkfrknj n kfrm n hfloln quf nuhfjtn fi tofhpm. Qmr im tnjtm, fs prîktoknhfjtf hîs ohpmrtnjtf y ótoi fstulonr ins rfspufstns lf vogrnkoûj bmrznln
>.> Nhmrto`un `unlmr lmrfs fs lf vo vogr grnk nkom omjf jfss m nh nhmrt mrto`u o`unlm nlmrf rfss lf hn hnsn sn so sojt jtmj mjoz oznlm nlmss >.> Nhmrto
‛> ‛>< />5‛ 5‛
(smiukomjfs pnrtokuinrfs). Fi lftfrhojnjtf lf in hntroz lf kmfkofjtfs fs
( )
ς L
( ) ( ) ∗ ( ∗ ( ) ( ) ( ) ∗ ( ) ∗ () hL
7
k + kn L + d + d n
+
8
kn L + d n
hL 7
8
∂> 8
+
hn L7 + d n
d
kn hL 7 + d
+
y
hn L7
B 1 soj ™t 1
+
+
+
d n
+
hn L
8
hn L7 + kn L + d n
kn L + d n
d n hn L7 + kn kL 7
k hn L7
kn L + d n
+
hn L7 + kn L + d n
hL 7 + k + kn L + d + d n
8
kn L + d n
)
∗ (k L
+
n
d n
)
7
+
kn hn L7 L
7
+
kn L + d n B 1 soj ™t
In npioknkoûj lf in rf`in lf Krnhfr prmlukf in smiukoûj pnrtokuinr x Q (t ) m in rfspufstn lf vogrnkoûj bmrznln lfgolm n in fx fxkotnkoûj kotnkoûj x Q t
lûjlf
Qmr im tnjtm
()
x Q t
8
8
8
8
8
8
B 1 B 1 B 1
B 1
B 1
B 1
∂>
8
( )
ς L
8
B 1
8
d h™7
G
8
kn d h ™7
h n ™7
d n
+
kn L N G L
h n ™7 N k n GL 7 + h n ™7
d n
d n
d d n
d n
kn k™7
d n hn ™7
kn hn ™7
h n ™7
k d n
+
N + GL G L N G L
d n
soj ™t
∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ( ∗ ) )(∗ ∗ ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ( ) ( ) ( ) ∗
h n ™7
d n
kn L
+
N + G L
N
∗ ∗ ( ∗ ∗ ∗ ∗
h n ™7
d n
soj ™t
h n ™7 G
d n
N7 G 7 L7 N + kn G™7 soj ™t +
+
kn N L
h n ™7 G
d n
soj ™t
+
kn N
™ kms ™t
N7 + G 7 ™7
7
h n ™7 N + kn G™7
h n ™7 G
d n
+
N7 + G 7 ™7
7 7
h n ™
7
+
7
7
kn ™
7
+
N N
N7 + G 7 ™7
h n ™7
7
+
kn7 ™7
N7 + G 7 ™7
soj ™t +
7
kn N
™7
(
soj ™t +
ϖ
7
™
G
+
soj ™t +
ϖ
ϖ
Fi bnktmr lf nuhfjtm lojîhokm (LHB) pnrn in hnsn h fs LHB 8
| x x (t )|)|hnx Q
xfstîtokn
d
n
B 1
∗h
n
™7
7
+
k 7 ™7 n
N7 + G 7 ™7
8
8
B 1 d
d n
d
h n ™7
7
+
kn7 ™7
N7 + G 7 ™7
Nlmptnjlm Nlmptnj lm ins so`uofjtfs njmtnkomjfs Ϗ17 8
sf tofjf
d h
, k 8 7h β Ϗ1, r 8
∗ d n
7
h n ™
7 +
™
Ϗ1
7
,
¶ 8
7
kn ™
8
hn h
hn
,
Ϗ7n 8
d n hn
∗™
d n hn
7
, kn 8 7 hn β n Ϗ1, r n 8
7 +
(7h
n β n Ϗ1
)
7
™
7
Ϗn Ϗ1
)
>.> Nhmrto`un `unlmr lmrfs fs lf vo vogr grnk nkom omjf jfss m nh nhmrt mrto`u o`unlm nlmrf rfss lf hn hnsn sn so sojt jtmj mjoz oznlm nlmss >.> Nhmrto
∗ ∗ ( ) 7
N7
+
G 7 ™7
h7n Ϗ1<
r n7
Ϗ17
7
r 7
7 β n r
+
7
7 β n Ϗ1 ™
+
Ϗ17
7
7
k n k™7
d n hn ™7
h n ™7
d n
kn d h ™7 d
hn h
™
h
7hn β n Ϗ1
+
< 1
h n hϏ
7
7
+
h
7
™
h
™
7
™
+
7 1
Ϗ
h n Ϗ
hn hϏ1;
< 1
Ϗ7n
β r n7
7
< 1
h n hϏ
Ϗ7n
™
< β n β
h7n Ϗ1;
Ϗ
7
7
hn ™
+
Ϗ17
7 β n
™
7
™
7
7 1
™
7
7
7
™
7 1
Ϗ
7
< β n β r r
7
7
+
7
™
7
7
7hn β n Ϗ1
™
Ϗ17 Ϗ17
7 β
¶ r n r
7h β Ϗ1 ™
7
hn
7 7
r
r r n
+
d n
hn
7
7
7
7
7hn β n Ϗ1
™
7
Ϗ17
7
∗ r
>
7
7
™
Ϗ17
r
hn
™
7
7h β Ϗ1
+
Ϗ7n
7
>
h n
™
d
d n
7
7
hn
7 β n
5 1
r 7 β n
d n
7
7
k n hn ™7
h n ™7
k d n
Ϗ17
hn h Ϗ +<
+
Ϗ17
hn hϏ1;
+
8
8
™
Ϗn
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ( ) ( ) ( ) ∗ ( ) ∗ ∗ ∗ ( ) ∗ ∗ ∗ ∗ ° ° ∗ ° ∗ ∗ ° ∗ ∗ ∗ ∗ ∗
+
8
h7n Ϗ1<
d h™7
8
8
8
7
7
‛>6/> ‛>6 />5‛ 5‛
+
7
∗ r ∗ ¶β r n
Fi bnktmr lf nuhfjtm lojîhokm sf kmjvofrtf LHB 8
∗ ∗ ∗ >
r 7
r n7
r 7
∗ ( ) ∗ ∗ ∗ ( ) r n7
¶ r n7 r 7
r 7
< β n β r r 7
7
+
7
+
7 β n r
r 7
7
∗ ¶r
∗ +
β r n7
r 7
7
Qnrn fi knsm fspfkoni kunjlm ¶ 8 1 , r n 8 1 , β n 8 1 , fi bnktmr lf nuhfjtm lojîhokm sf rflukf n >
LHB 8
>
r 7
7
+
7 β r r
7
quf rfkupfrn fi LHB lf uj ójokm sostfhn lf `rnlm lf iogfrtnl, fs lfkor, fi sostfhn projkopni soj fi ngsmrgflmr lf vogrnkomjfs nuxoionr m \HL. Ims bnktmrfs lf nuhfjtm lojîhokm pnrn uj sostfhn projkopni jm nhmrto`unlm, fs lfkor, β 8 1, ngsmrgflmr lmr lf vogrn vogrnkomjf komjfss m \HL, fi sostfh sostfhnn lf `rnlm lf sf hufstrnj fj in `urn >.5 `urn >.5.. [oj fi ngsmrgf iogfrtnl ójokm fstî fj rfsmjnjkon kunjlm r 8 > m ™ 8 Ϗ 1 , lmjlf in nhpiotul lf in rfspufstn krfkf iojfnihfjtf kmj fi tofhpm m LHB sf nprmxohn ni ojjotm. Qnrn rflukor in vogrnkoûj lfi sostfhn projkopni fj in rfsmjnjkon, uj ngsmrgfjtf lf vogrnkomjfs m \HLsfujfninhnsnprojkopni h. Fi nhmrto nhmrto`unl `unlmr mr lf vogrn vogrnkomjf komjfss `fj `fjfrnih frnihfjtf fjtf fstî fstîsojtm sojtmjozn joznlm lm lf hmlm quf Ϗ n 8 Ϗ1 m r n 8 >, lf neè fi jmhgrf lf nhmrto`unlmr lf hnsn sojtmjoznlm. Fj in prîktokn, in hnsn lfi nhmrto`unlmr lf vogrnkomjfs m \HL fs jmrhnihfjtf hukem hîs pfqufþn quf in lf in hnsn projkopni, fs lfkor, hn h m ¶ >4 fj in `urn `urn >.5 >.5,, ¶ sf tmhn kmhm > 8 1 71
.16.
[o fi ngs ngsmrg mrgflm flmrr lf vog vogrnk rnkomj omjfs fs m \HL jm fs fstî tî nhm nhmrto` rto`unl unlm, m, fs lfk lfkor or,, β n 8 1, fjtmjkf fjtmjkfss LHB 8 1
7
>.7 Fcfrkokoms
‛>2/>5‛
kunjlm ™ 8 Ϗ 1 , im quf so`jokn quf fi ngsmrgflmr lf vogrnkomjfs fiohojn in vogrnkoûj lf in hnsn projkopni h fj in brfkufjkon lf rfsmjnjkon ™ 8 Ϗ1 . [oj fhgn fhgnr`m r`m,, sf vf quf fi ngsmrg ngsmrgfjtf fjtf lf vogrnkomjfs m \HL ojtrmlukf lms brfkufjkons rfsmjnjtfs ™> y ™ 7 , fj ins kunifs in nhpiotul lf vogrnkoûj lf in hnsn projkopni h fs ojjotn. Fj in prîktokn, in brfkufjkon lf fxkotnkoûj ™ lfgf hnjtfjfrsf nifcnln lf ins brfkufjkons ™> y ™7 .
Bo`urf >.53 Bnktmr lf nuhfjtm lojîhokm pnrn β 8 1
Qnrn jm ojtrm ojtrmlukor lukor brfk brfkufjk ufjkons ons lf rfsmj rfsmjnjkon njkon nloko nlokomjni mjnifs, fs, ims nhmrto nhmrto`unl `unlmrfs mrfs lf vogr vogrnkomj nkomjfs fs m \HL `fjfrnihfjtf fstîj nhmrto`unlms. Vj rfsuitnlm tèpokm lf LHB sf hufstrn fj in `urn >.? pnrn β n 8 1.>. [f vf quf in vogrnkoûj lf in hnsn projkopni h sf suprohf fbfktovnhfjtf pnrn >.? pnrn tmlns ins brfkufjkons lf fxkotnkoûj. Ni vnronr fi vnimr lf β n , sf puflf losfþnr uj ngsmrgfjtf lf vogrnkomjfs ûptohm. Kunjl Ku njlmm fi sos sostfh tfhnn pro projko jkopni pnitnh tnhgoï goïjj fs fstî tî nhm nhmrto` rto`unl unlm, m, imsrfs ims rfsuit uitnlm nlmss tèpokm tèpokmss lf LHB LHBsf sf huf hufst strnj rnj fj in `urn >.? `urn >.?.. \nhgoïj sf puflfj snknr kmjkiusomjfs sohoinrfs.
Bo`urf >.?3 Bnktmr lf nuhfjtm lojîhokm pnrn β 8 1 .1<
>.7 Fcfrkokoms
‛>0/>5‛
>.7 Fcf Fcfrko rkokoms koms >. Vjn ggnrrn nrrn lf iuz NG N G fstî smpmrtnln fj knln fxtrfhm pmr lms rfsmrtfs sohoinrfs kmj ro`olfz lf rfsmrtf d , y iifvn lms mgcftms lf hnsn kmjkfjtrnln h , ujm fj fi fxtrfhm G y fi mtrm fj fi kfjtrm K , kmhm sf hufstrn fj in so`uofjtf `urn. Lfskuoln fi fbfktm lf in `rnvflnl.
(n). Lfhufstrf quf ins fkunkomjfs lf hm hmvohofjtm vohofjtm fstîj lnlns pmr h x> + ;h x7 + 7 d x> + 7 d x7 hx 7
∗ d x
> +
d x7
8
1
8
1
(g). Lfhufstrf quf ins lms brfkufjkons korkuinr korkuinrfs fs jnturnifs lf vogrnkoûj lfi sostfhn smj
∗ ∓ ;
Ϗ> 8
6
h
∓ ;
d ,
Ϗ7 8
6
+
d
h
7. Lms gnrrns lf iuz NG y K L, knln ujn lf imj`otul 7 I , fstîj nrtokuinlns fj N y K y y iifvnj hnsns kmjkfjtrnlns lf h knln ujn fj G y L . Ins gnrrns fstîj sm smpmrtnlns pmrtnlns pmr trfs rfsmrtfs, rfsmrtfs, knln ujm lf ro`olfz d , kmhm sf hufstrn fj in so`uofjtf `urn. Fi hmvohofjtm lfi sostfhn sf lfskrogf hflonjtf lms îj`uims ε > y ε 7 . Kmjsolfrf smim pfqu pfqufþns fþns mskoinkomjfs, fs lfkor,
| ε | > y |ε | >, y lfskuolf fi fbfktm lf in `rnvflnl. >
7
hmvohofjtm vohofjtm fstîj lnlns pmr (n). Lfhufstrf quf ins fkunkomjfs lf hm ε > +
d
(7ε ∗ ε ) d (ε ∗ 7ε ) ε ∗
7
8
1
7
8
1
(g). Lftfrhojf ins lms brfkufjkons korkuinrfs jntu jnturnifs rnifs Ϗ> , Ϗ7 y in rfinkoûj lf ins nhpio ε >
tulfs ε pnrn ims lms hmlms lf vogrnkoûj. ;. Fj hukens npioknkomjfs lf oj`fjofrèn, ins fstrukturns m kohofjtms vogrnj kunjlm mpfrnj 7
>.7 Fcfrkokoms
‛>5/>5‛
ins hîquojns hmjtnlns hmjtnlns fj fiims. Qmr fcfhpim, ims hmtmrf hmtmrfss lf uj nvoûj hmjtnlm fj nins fx fxkotnj kotnj ins nins kunj kunjlm lm mpfrnj. Qnr Qnrnn rflukor rflukor in vogr vogrnkoûj nkoûj lf ins fstru fstrukturn kturnss lf smpmrtf, sf utoioznj nosinlmrfs lf vogrnkoûj pnrn kmjfktnr fjtrf ins hîquojns y ins fstrukturns lf smpmrtf. smpm rtf. In so`uofj so`uofjtf tf `urn hufst hufstrn rn fi hmlfim lf ujn hîquoj hîquojnn lf hnsn h> smpmrtnln
smgrf ujn fstrukturn lf hnsn h7 . Fi nosinlmr lf vogrnkomjfs fstî hmlfinlm pmr uj rfsmrtf lf ro`olfz d y y uj nhmrto`unlmr lfi kmfkofjtf lf nhmrto`unkoûj k. In hîquojn fstî sucftn n ujn knr`n nrhûjokn B (t ) 8 B 1 soj ™t .
(n). Lfhufstrf quf ins fkunkomjfs lf hm hmvohofjtm vohofjtm fstîj lnlns pmr h> x> + k x> h7 x7 +
7 +
∗ k x k x ∗ k x
>
7
+
d x>
7 8
B 1 soj ™t
7
> 8
1
∗ d x d x ∗ d x
uf k ↗ 1 , lfhufstrf quf ins rfspufstns lf fstnlm fstnkomjnrom tnjtm (g). Qnrn fi knsm fj qquf lf in hîquojn kmhm lf in fstrukturn lf smpmrtf, fs lfkor, x > Q (t ) y x 7 Q (t ) fstîj lnlns pmr
d h7 ™7 B 1 soj ™t
()
8
()
8
x> Q t x7 Q t
∗ ∗
kufi`n lf uj rfsmrtf lf
h> h7 ™< d h> + h7 d B 1 soj ™t
∗ (
)™
7
h> h7 ™< d h> + h7 ™7 ro`olfz d > lf mtrn hnsn h7 quf n su
∗ (
)
vfz kufi` kufi`nn lf uj
rfsmrtf lf ro`olfz d 7 lfi smpmrtf kmhm sf hufstrn fj in `urn.
(n). Lfhufstrf quf ins lms brfkufjkons korkuinrfs jnturnifs lf vogrnkoûj fstîj lnlns pmr in fkunkoûj
h> h7 Ϗ<
∗ Th (d >
> +
d 7
)
+
h7 d > Ϗ7 + d > d 7 8 1
X
>.7 Fcfrkokoms
‛>?/>5‛
(g). [o fi smpm smpmrtf rtf vo vogr grnn km kmjj x1 (t ) 8 n kms ™t , lfh lfhufs ufstrf trf quf quf ins nhp nhpiot iotul ulfs fs lf in vog vogrnk rnkoûj oûj bmrznln smj
n> n7
d > d 7 n
8
h> h7 ™<
8
h> h7 ™<
∗ Th (d d ) d ∗ h ™ ∗ Th (d d ) > +
>
>
>
7
7
>
> +
+
7
h7 d >
d 7 n
+
h7 d >
7
X™
7
X™
+
d > d 7
+
d > d 7
6. Vjn Vjn hns hnsnn h, smp smpmrtn mrtnln ln pmr ujn ujnfst fstruk ruktur turnn fiî fiîsto stokn kn quf quf puflf puflf sfr hml hmlfin finln ln kmh kmhmm uj rfs rfsmrtf mrtf kmj ro`olfz d , fstî sucftn n ujn bufrzn pfrturgnlmrn nrhûjokn sohpif lf vnimr hîxohm B 1 y brfkufjkon ™ 8 7ρ b , fs lfkor, B (t ) 8 B 1 soj t . Qnrn rflukor in nhpiotul lf vogrnkoûj lf
fstf sostfhn, sf ujf uj ngsmrgfjtf lf vogrnkomjfs n in hnsn. Fi ngsmrgflmr kmjsostf fj ujn hnsn h n smgrf uj rfsmrtf lf ro`olfz d n . vogrnkoûjj bmrznln lf in hnsn h fs (n). Lfhufstrf quf in rfspufstn lf vogrnkoû
()
x Q t
∗
h n ™7 B 1 soj ™t
d n
8
hh n ™< hd n + hn d + d n ™7 + d d n (g). [o h 8 761 d`, d 8 > dJ/h, B 1 8 >11 J, b 8 ;6 Ez, h n 8 1 .6 d`, y in nhpiotul lf
∗T
(
)X
in vogrnkoûj bmrznln lf in hnsn h lfgf rflukorsf n kfrm, lftfrhojf fi vnimr lf d n . 2. Qnrn fi sostfhn quf sf hufstrn fj in so`uofjtf `urn, hufstrf quf ins fkunkomjfs lf hmvohofjtm hmvoh ofjtm fstîj lnlns pmr h> x> + d > + d 7 x>
7
7 8
h7 x7 +
7
> 8
( (d
7 +
) ∗ d x d ) x ∗ d x ;
7
() B (t ) B > t 7
Kmjsolfrf fi knsm fspfkoni h > 8 h7 8 >, d > 8 d 7 8 d ; 8 >, y ins kmjlokomjfs ojokonifs
>
1
8
7
1
8
1
1
1
8
8 1
( ) x ( ) , x ( ) x ( ) . Vsf lf hït hïtmlm mlm hnt hntrokoni rokoni pn pnrn rn lftf lftfrhojn rhojnrr ins rfspufstns lfi sostfhn pnrn ins so`uofjtfs fxkotnkomjfs3 (n). B (t ) ; soj 7t , B (t ) 14 (g). B (t ) < so soj j t , B (t ) 1 4 >7 so soj j ;t , B (t ) 1 4 (k). B (t ) >7 ∓ (l). B (t ) >7 so soj j t , B (t ) >7 so soj j ; t .
x
>
8
>
8
>
8
>
8
>
7
8
7
8
7
7
7
8
8
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