Ejercicios - Propiedades de La DTF

 

]rgpnao`oas oa c` OJ_ Aiarknkngs rasuactgs pgr Xgornfg I`vnar Barrar` F`rk¿ F`rk¿İ`* ; oa gktuhra oa 8=02

Xasudae

C`s prgpnao`oas oa c` tr`esjgrd`kn¿g ge e onskrat` oa Jgurnar (OJ_0 ), `uequa sge sndnc`ras ` c`s prgpnao`oas oa gtr`s tr`esjgrd`kngeas oa Jgurnar, axbnhae egt`hcas onjaraekn`s oahnog ` su e`tur`caz` ffent`. Oahnog ` qua c` OJ_ as euastr` prneknp`c barr`dnaet` oa k¿`ckucg, oahadgs aetaeoar sus prgpnao`oas y su rac`kn¿ge ge 8 kge c`s gtr`s tr`esjgrd`kngeas utncnz`o`s ae ac prgkas`dnaetg oa sa˓e e`cas `cas onfnt`cas (O[] ). C` dat` oa astgs prgyaktgs y aiarknkngs as oas`rrgcc`r `cfue` j`dncn`rno`o y taear `cfue` noa` oac usg oa c`s prgpnao`oas oa c` OJ_. [ar¿`e `e aej`tnz`o a ej`tnz`o`s `s k`r`ktar¿İstnk`s İstnk`s uenk`s u ¿ enk`s oa c` OJ_ OJ_,, aspakn`cdae aspakn`cdaeta ta c` e`tur e`tur`caz` `caz` knrkuc`r oa tgog ac neoax`dnaetg t`etg ae ac ogdneng oac tnadpg kgdg ae ac oa jrakuaekn`.   C` OJ_ sa oaffea kgdg ue` gpar`kn¿g ge e sghra ue vaktgr oa tnadpg oa  E   puetgs x\=T, x\0T, . . . , x\e ∘ 0T 9 E ∘0

R \lT 4



el

x\eTV E 

 

p`r`  l  4 =, 0, 8, . . . , E  ∘   0

(0)

e4= ∘i 8ύ/E  ogeoa   V E . C` gp ar`kn¿ ge ge ae (0) as ue` tr`esjgrd`kn¿g ge e oa ue vaktgr oa   E   puetgs oa duastr`s E    4   a   x e  E  ae ac tnadpg \ T ` gtrg vaktgr oa  puetgs oa duastr`s ae ac ogdneng oa c` jrakuaekn`  R \lT. C` oaffenkn¿ ge ge (0) t`dhn¿ aae e puaoa sar netarprat`o` kgdg ue –duastrag ae c` jrakuaekn`— oa c` O_J_; (tr`esjgrd`kn¿ g ge e oa Jgurnar oa tnadpg onskratg). @c prnek prneknpng npng as eakas`rng eakas`rng oaknr ue` p`c`hr` `kark `kark` ` oa c` onja onjaraekn raekn` ` aetra aetra cgs t¿ aardnegs rdnegs OJ_ y JJ_. C` ? JJ_ (tr`esjgrd`kn¿ ge ge r¿ ` `pno` pno` oa Jgurnar) as prakns`daeta ue `cfgrntdg r¿` `pnog pnog p`r` ac k¿ ` `ckucg ckucg oa c` OJ_< eg as ue` tr`esjgrd`kn¿ ge ge sap`r`o`. Ae   D`tc`h, c` juekn¿ge ge   jjt  as snadpra utncnz`o` p`r` k`ckuc`r c` OJ_ y eg b`y juekn¿g ge e   ojt  oa tgo`s d`ear`s. [ndnc`rdaeta, c` juekn¿ge ge   njjt  as utncnz`o` p`r` k`ckuc`r c` OJ_ nevars`. nevar s`. ]gr kgesnfunaeta, as usu`cdaeta `kapt`hca utncnz`r cgs t¿ aardnegs rdnegs oa d`ear` netark`dhn`hca ku`eog sa raffarae ` cgs rasuct`ogs oac k¿` `ckucg. ckucg.

g ge e ]`c`hr`s ]`c`h r`s kc`va9 kc`va9  tr`esjgrd`kn¿

onskrat` oa Jgurnar, tr`esjgrd`kn¿ g ge e r¿ ` `pno` pno` oa Jgurnar.

* Nefaenarg

acaktr¿ g genkg, enkg, prgjasgr oac ¿`ra` `ra` oa prgkas`dnaetg onfnt`c oa sa˓ e e`cas `cas a nevastnf`ogr oac frup g NOA@[ oa c` J`kuct`o J`kuct`o oa Nefaenar¿ Nefaenar¿İ` oa c` Qenvarsno`o Onstrnt`c Jr`eknskg Igs¿a oa K`co`s. Kgrrag Kgrr ag Acaktr¿g genkg9 enkg9 rbarrar`Muonstrnt`c.aou.kg. 0 OJ_ as ac `kr¿ gendg gendg oa c`s p`c`hr`s nefcas`s   Onskrata Jgurnar _r`esjgrd. 8 O[] as ac `kr¿ g gendg endg oa c`s p`c`hr`s nefcas`s  Onfnt`c [nfe`c ]rgkassnef . ; O_J_ as ac `kr¿ gendg gendg oa c`s p`c`hr`s nefcas`s   Onskrata-_nda Jgurnar _r`esjgrd. ? JJ_ as ac `kr¿ g gendg endg oa c`s p`c`hr`s nefcas`s  J`st Jgurnar _r`esjgrd.

0

 

0.

Ai Aiad adpc pcgs gs oa c` OJ OJ_ _ oa sa˓ e e`cas `cas kgdueas

0. 0.0. 0.

]ucs ]ucsgs gs

B`y sa˓ e`cas e`cas qua kgetnaeae uenk`daeta u ¿ enk`daeta uegs y kargs. ]`r` cgs snfunaetas aiarknkngs, ustao puaoa onhui`r c`s p`rtas ra`c Ae asta prgyaktg oas`rrgcc`radgs c` OJ_ oa knart`s sa˓e e`cas `cas a nd`fne`rn` oa c` OJ_ parg puaoa sar daigr onhui`r c` kgdueas (p.a. pucsgs, geo`s saeg, snek kge `cn`s g `snek, atk.). d`fentuo y c` j`sa. Kge   D`tc`h, ustao puaoa onhui`r sa˓ e`cas e`cas y sus tr`esjgre`c ndpucsg uent`rng 9   x n 4 \ 0 = = = = = = = T  kgrrasd`kngeas j¿`kncdaet `kncdaetaa `s `s¿¿İ qua ac ghiatnv ghiatnvgg as vnsu`c vnsu`cnz`r nz`r ue` `.   [a˓  pgeoa ` c` oaffenkn¿ge ge d`tad¿`tnk`9 `tnk`9 v`rnao`o oa p`ras oa tr`esjgrd`kngeas. Ae ac prgkasg, ustao oaha pgear `taekn¿gge e ` c`s sndatr¿İ`s İ`s qua pgor pg or¿¿İ`e ast`r 0   e 4 = prasaetas. ο \eT 4 (8)   0 =   e 4 0, 8, . . . , E ∘ [ufaraekn`s.   _go`s c`s sa˓ e e`cas `cas utncnz`o`s kge c` OJ_ sge ]`r` asta prghcad` k`ckuca ue` OJ_ oa 2 puetgs (as onskrat`s, onskrat `s, `s¿İ qua acc`s ac c`s oahae oah ae sar dgstr`o`s dgst r`o`s utncnz`eog utn cnz`eog  stad<  E  4  E    puetgs oaknr, E  oaknr,  4 2). Ae faear`c, ·qu¿ a as c` OJ_ oa  E  sndnc`rdaeta, c` tr`esjgrd`kn¿gge e as ue vaktgr oa v`cgras onsoa ο  oa  ο \eT7 kratgs, `s `s¿¿İ qua t`dhn¿ae ae oaha sar onhui`o onhui`o`` kge  stad. ]uastg qua c` OJ_ as oa v`cgr kgdpcaig, ustao taeor¿` qua onhui`r c`s p`rtas p`rtas ra`c a nd`fne`rn` nd`fne`rn` ae c` d`ygr d`ygr¿¿İ` oa cgs k`sgs. k`sgs. [n h.   _gogs _gogs ueg 9   x0 x0 4 \0 0 0 0 0 0 0 0 0T T. J¿İiasa qua asta ustao oasa` var onhuigs snduct¿``eags eags oa c`s p`rtas ra`c a nd`aiadpcg iuetg kge c` p`rta (`) ncustr` ac prneknpng oa  ou`fne`rn` oa cgs vaktgras ae ac ogdneng oac tnadpg y ae ac ogcno`o  oa   oa c` OJ_. dneng oa c` jrakuaekn`, utncnka cgs kgd`eogs  suhpcgt(88x) `etas oa k`o` kgd`eog  stad  p`r` jgrz`r qua cgs ku`trg onoaspc`z`og 9   x n s b 4 \ = = = 0 = = = = T . Onhuia Onhuia huigs sa`e kgcgk`ogs sghra c` dnsd` p`et`cc`, kge c`s p`rtas k.   Ndpucsg oaspc`z`og  c` d`fentuo oa cgs v`cgras oa c` OJ_. _r`ta kge gtrgs ra`c a nd`fne`rn` ue` aeknd` oa c` gtr`, kgdg ae c`   Jnfur` oaspc`z`d oaspc `z`dnaet naetgs‟·b gs‟·b`y `y ue oaspc`z`dna oaspc`z`dnaetg etg onjaraet onjaraetaa oa 0. Ac prgfr`d` qua prgouka c`  Jnfur` 0  snfua9 karg oa  xn  ogeoa c` OJ_ sa` pur`daeta ra`c7

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o.  ]ucsg rakt`efuc`r oa tras puetgs 9   xh 4 \0 0 0 = = = = =T< tr`ta t`dhn¿ ae ae ac k`sg oa ku`trg puetgs.

0=

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     r      g       c      ` 1       Z

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rakt`efuc`r uc`r snd¿ aatrnkg  trnkg 9   x h s y 4 \ 0 0 = = = = = 0 T. a.   ]ucsg rakt`ef Oaduastra qua ast` OJ_ sar¿` pur`daeta ra`c. Kgdp`ra c`s d`fentuoas oa c` OJ_ oa  xh  y   xhsy.

-0 -1 =

1

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äeonka e

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äeonka l

]`rta nd`fne`rn` oa x\eT

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]`rta nd`fne`rn` oa OJ_ oa x\eT

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0. 0.8. 8.

=      r      g       c      `       Z

=

Qe` sneusgnoa oa v`cgr ra`c as oaskrnt` pgr tras p`r¿`datrgs. `datrgs. C` jgrd` d`tad¿`tnk` `tnk` oa c` sa˓ e`c e`c oa tnadpg as

-1

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Geo` Geo`ss [a [aeg eg

-0 -0= =

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äeonka e

01

=

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s\eT 4  @ kgs(8 ύj g e + ϊ) kgs(8ύj 

01

äeonka l

 e 4 p`r`  e  4 =, 0, 8, . . . , E  ∘ 0 (;)

E    as c` cgefntuo oa c` sa˓ ogeoa   E  ogeoa  e`c, e`c,   @  su `dpcntuo,   j g   su jrakuaekn` y ϊ y  ϊ  c` j`sa rac`tnv` ae e ae  e  4 =.

Jnfur` 09  Onhuigs oa c`s p`rtas ra`c a nd`fne`rn` oa ue` sa˓ e`c oa tnadpg e`c onskratg y su OJ_ oa 0: puetgs kge   suhpcgt(88x)   p`r` kra`r ue`

vnsu`cnz`kn¿ g ge e oa ku`trg p`eacas p`eacas.. C` OJ_ ast¿ ` `c c`og oarakbg y c` sa˓ e e`c `c oa tnadpg oa v`cgr kgdpcaig ast¿` `c c`og nzqunarog.

`. K`cku K`ckuca ca c` OJ_ oa 80 puetgs oa ue` sakuaekn` sakuaekn` rapra raprasaesaet`eog ax`kt`daeta ue knkcg oa ue` geo` kgsaeg. Oatardnea c` jrakuaekn` oa ast` sneusgnoa. @saf¿ urasa urasa oa tgd`r ax`kt`daeta ue knkcg, eg ue knkcg d¿`s `s ue puetg (as oaknr, eg rapnt` c` prndar` duastr`). [n sa b`ka kgrrakt`daeta, c` raspu raspuast` ast` sar¿` axtrad`o`daeta sndpca.

ee 4 =9 =901< 01< xx 4 axp(i*e axp(i*ee/;) e/;)< < RR 4 jjt(xx) jjt(xx)< < ll 4 ee< suhpcgt(880), suhpcgt (880), stad(ee, stad(ee, ra`c(xx ra`c(xx)) )) tntca(„]@X_A tntca(„ ]@X_A XA@C oa x\eT„), xc`hac(„¿ xc`hac(„¿ İeonka İeonka \eT„) suhpcgt(88;), suhpcgt (88;), stad(ee, stad(ee, nd`f(xx nd`f(xx)) )) tntca(„]@X_A tntca(„ ]@X_A ND@FNE@XN@ oa x\eT„), xc`hac(„¿ İeonka İeonka \eT„) suhpcgt(888), suhpcgt (888), stad(ll, stad(ll, ra`c(RR ra`c(RR)) )) tntca(„ tntc a(„]@X_ ]@X_A A XA@C XA@C oa c` OJ_„), OJ_„), xc`hac(„ xc`hac(„¿ İeonka ¿ İeonka \lT„) suhpcgt(88?), suhpcgt (88?), stad(ll, stad(ll, nd`f(RR nd`f(RR)) )) tntca(„]@X_A tntca(„ ]@X_A ND@FNE@XN@ oa c` OJ_„), xc`hac(„¿ İeonka İeonka \lT„)

h. Xapnt` c` p`rta (`) p`r` ue` sa˓e`c e`c saeg, cuafg p`r` ue` kgsaeg kge ue oaspc`z`dnaetg oa j`sa oa ?1 . Ghsarva ◪

kuno`ogs`daeta oa cgs kgaffknaetas oa c` OJ_ (y kgdp`ra c`s c`s d`fentuoas j`sas).

8

 

8.0. 0. [akua [akuaek ekn` n` sae saek k kge kge `cn` `cn`ss k. Xapnt` Xapnt` c` p`rta p`rta (`) p`r` tras knkcgs oa ue` sneu sneusgnoa, sgnoa, `ue 8. utncnz`eog ue` OJ_ oa 80 puetgs. ·Ku¿`c `c as c` jrakuaekn` Qe` vaz aetaeonog, ac prneknpng oa ou`cno`o as h`st`eta pgoa ast` sneusgnoa sneusgnoa (ae r`on`eas pgr dua duastr`) str`)77 oargsg. Kgdg ue huae aiadpcg oa su usg ae ac k¿gdputg, gdputg, kgesnoara c` OJ_ oa c` sa˓ e`c e`c saek kge `cn`s. Oa `kuarog kge c` ou`cno`o, ac pucsg rakt`efuc`r y c` juekn¿ge ge `saek sge o. _r`ta kge ue vaktgr qua as ;.0 knkcgs oa ue` sneusgnoa. –tr`esjgrd`kngeas p`ras—. ·]gr qu¿a as c` OJ_ t`e onjaraeta7 `. C` OJ_ oa ue pucsg rakt` rakt`efuc`r efuc`r as ue` juekn¿ juekn¿ ge ge saeg kge `cn`s ae ώ ae  ώύe/E ))  sae(>ύe/E    ύe/E ) sae(ύe/E  sae(

 e  4 =, 0, . . . , E  ∘   0 p`r` e p`r`

@sud` qua E  qua E  as  as d`ygr qua >, onf`dgs E  onf`dgs  E  4  4 0: g E  g  E  4  4 80. k. Aekua Aekuaetra etra c` OJ_ OJ_ oa E  oa  E  puetgs  puetgs oa c` snfunaeta sakuaekn` `saek oaspc`z`o`9

h. Ac ou`c oa c` axpgeaekn`c kgdpcai` as ac ndpucsg oasύ/E . Aeu uctnpcg ctnpcg aetarg oa 88ύ/E  pc`z`og, ku`eog ώ ku`eog  ώ g  as ue d¿ kuaetra c` sakuaekn` oa 0: puetgs kuy` OJ_ as   R \lT 4 ύl/ 0: ai :ύl/0: .

`0 \eT 4

 sae(>ύ (e + 0)/E   sae(>ύ 0)/E )   ύ(e + 0)/E  sae(ύ sae( 0)/E )

 e 4   0 p`r` e p`r`  4 =, 0, . . . , E  ∘

]uastg qua c` juekn¿gge e `saek as parn¿gonk`, gonk`, asta oaspc`z`dnaetg as ue oaspc`z`dnaetg knrkuc`r.

 ώ g  4 1ύ/E  y   y k`ckuca c` OJ_ oa E  oa  E    4 0: k. @bgr` tr`ta tr`ta kge kge ώ

puetgs. Axpcnqua pgr qu¿ a eg b`y kargs ae c` OJ_ ae asta o. !Egta qua ac j`k j`ktgr tgr ae ac eudar`ogr eudar`ogr oa ` oa ` = \eT g  `0 \eT oaha k`sg. sar ndp`r! _r`ta oa raadpc`z`r ac > kge 0= ae ku`cqunar` oa c`s p`rtas p`rtas pravn`s pravn`s y k`ckuca k`ckuca c` OJ_. ·]gr ·]gr qu¿ a b`y t`et` t`e t` onjaraekn` aetra cgs k`sgs p`ras a ndp`ras7 ndp`ras7 o. C` j¿grduc` grduc` oa Aucar rac`knge` c` axpgeaekn`c kgdpcai` gdg kgeskge saeg y kgsaeg. [a` ώ [a`  ώ g  4 :ύ/E . Duastra k¿gdg trunr c` OJ_ oa sae(ώ sae(ώg e) oac rasuct`og oa c` p`rta (`).

8. 8.8. 8.

_rae oa ndpu ndpucs csgs gs

Oas`rrgcca ac snfunaeta axparndaetg9

8.

`. Faea Faeara ra ue trae oa ndpucsgs qua kgetaef` kgetaef` 8=6 duastr` duastr`s. s. Aetra cgs ndpucsgs oaha b`har 88 kargs. C` `ctur` oa k`o` ndpucsg oaha sar ue` kgest`eta. Cc`da ` ast` sa˓e`c p e`c  p\\eT.

OJ_s onj onj¿ ¿İkncas

C`s snfunaetas OJ_s sge onj¿İkncas İkncas oa k`ckuc`r ` d`eg, parg cgs rasuct`ogs eg sge onj¿ onj¿İkncas oa vnsu`cnz`r.   D`tc`h  b`ka

2

 p  p\\eT 4

j¿ `knc `knc gge k`ckuc`r c`s tr`esjgrd`kngeas p`ras, `s¿ `s¿İoac qua`e¿ c``cnsns vnsu`cnz`kn¿ e qua vnaea oac prneknpng oa ou`cno`o `cnsns oa Jgurnar as rajgrz`og pgr cgs snfunaetas aiarknkngs. ;

 4=

ο \e ∘ D g T

kge  D g  4 8;

 

ΰ  4   08 , asta as ue k`sg aspakn`c ogeoa c` tr`esjgrd`kn¿ge ge oa Jgurnar oa tnadpg kgetneug prgouka ue` tr`esjgrd` R ((ώ) 4 axp( 08 ώ8 ). kn¿gge e qua as oa c` dnsd` jgrd`, jgrd`, R 

 p\\eT. Ghsarva qua ae ac h. K`cku K`ckuca ca c` OJ_ oa 8=6 puetgs puetgs oa oa p ] \\lT t`dhn¿aae ogdneng oa c` OJ_,  OJ_,   ]  e tgd` u uenk`daeta ¿enk`daeta cgs v`cgras oa kargs y ue` kgest`eta. Oatardnea ac asp`kng aetra cgs ndpucsgs ae ac ogdneng oa  l.  l .

E    puetgs oa c`s duastr`s oa   f\eT. h. Jgrda ue vaktgr oa   E  E    sar¿` nfu`c ` 8C Egta qua   E  8C  + 0. Kgcgqua c`s duastr`s ae ue vaktgr oa d`ear` qua prasarvae sndatr¿ sndatr¿İ` p`r. Astg puaoa sar `kgdp`˓ e`og e`og kge c` rgt`kn¿ge ge oa c` duastr` d¿``ss  f\=T fr`eoa f fr`eoa \=T `c kgdnaezg oac vaktgr.

k. Ac par p ar¿¿İgog oa c` sa˓ e`c e`c oa aetr`o`,   D g , as ue onvnsgr oa c` cgefntuo oa c` JJ_. Qsa asta j`ktgr p`r` axpcnk`r c` jgrd` d`tad¿`tnk` `tnk` oa ]  oa  ] \\lT. Faear`cnka asta rasuct`og. Ae p`rtnkuc`r, praonf` c` OJ_ sn   p\eT kgetnaea 8; ndpucsgs sap`r`ogs pgr > kargs< cuafg varnffqua kge   D`tc`h.

E    puetgs oa   f \eT. Zarnffqua qua ac k. K`ckuca c` OJ_ oa   E  o. K`d K`dhna hna cnfar`daet cnfar`daetaa c` cgefntuo cgefntuo oa c` OJ_ y k`ckuca k`ckuca ue` rasuct`og as pur`daeta ra`c. [n eg cg as, ac vaktgr oa OJ_ oa 8== puetgs. ]uastg qua cgs ¿u uctndgs ctndgs 88 puetgs tnadpg eg jua j ua kgestruno kg estrunogg snd¿ sn d¿atrnk`daeta. atrnk`daeta. oa oa   p\eT sge karg, sgcg eakasnt`dgs acndne`r cgs kargs oac ffe`c. Axpcnqua p pgr gr qu¿ a cgs v`cgras oa c` tr`esjgrd`kn¿ ge ge sge t`e onjaraetas, parg egta qua c` OJ_ `ue tnaea ue o. Onhuia c` p`rta ra`c oa c` OJ_ y kgdp`ra ` ue` F`use¿ u udarg darg fr`eoa oa pnkgs asp`kn`ogs rafuc`rdaeta. sn`e`. ]uaoa sar eakas`rng rgt`r ac vaktgr OJ_ p`r` var qua sa va kgdg ue` F`ussn`e` (va`  jjtsbnjt). a. Ku`eog sa oghc` c` cgefntuo oa c` OJ_, oaha b`har `ue kargs ae c` tr`esjgrd`kn¿ge9 ge9 K`ckuca c` OJ_ oa ?0? puetgs oa oa   p\eT y onhuia c` d`fentuo. Oatardnea ku¿`cas `cas puetgs oa c` OJ_ sge ax`kt`daeta nfu`cas ` karg y axpcnqua pgr qu¿aa.. ·Pu¿a p`s`r¿İİ`` p`r` ue` OJ_ oa :80 puetgs7

a. Axparndaeta kge v v`cgras `cgras onjaraetas oa ΰ oa ΰ.. D`etaef` kgest`eta c` cgefntuo oa c` tr`esjgrd`kn¿ge. ge. _r`ta oa b`kar ac `ekbg oa c` F`ussn`e` ac dnsdg ae cgs ogdnengs oac tnadpg y c` jrakuaekn`. Egta qua ku`eog ac `ekbg oa c` F`ussn`e` oakraka ae ac ogdneng oac tnadpg, kraka ae ac ogdneng oa c` OJ_. Ast` as ue` oadgstr`kn¿ge ge oac prneknpng oac nekartnoudhra oac `e¿`cnsns `cnsns oa Jgurnar9 –Ac prgouktg oac `ekbg oac tnadpg y oac `ekbg oa c` jrakuaekn` as snadpra d`ygr qua ue` kgest`eta ffi`—. Aetgekas onsdneunr ac `ekbg oac tnadpg eakas`rn`daeta nekradaet`r¿` ac `ekbg oa c` jrakuaekn`.

Axpcnqua. j. K`ckuca K`ckuca c` OJ_ oa 0=8? puetgs puetgs oa p oa  p\\eT, oa euavg kge raccaeg oa kargs `etas oa c` JJ_. Egta qua c` tr`esjgrd`kn¿ge ge tnaea dukbgs pnkgs y p`rakae ast`r ae ue asp`kn`og rafuc`r, `c daegs `prgxnd`o`daeta. Dno` ac asp`kn`og y kuaeta cgs pnkgs. Ast`hcazk` c` rac`kn¿ge ge faear`c aetra ac par¿İgog İgog oa c` sa˓e`c e`c oa aetr`o`, p aetr`o`,  p\\eT, c` cgefntuo oa c` OJ_ y ac asp`kn`og rafuc`r oa cgs pnkgs ae c` OJ_.

8. 8.?. ?.

Axpge Axpgeae aekn kn`c `c ra ra`c `c

Gtr`` sa˓ Gtr e`c e`c kgd¿ kgdue u ¿ e as c` axpgeae axpgeaekn` kn`cc ra`c ra`c oakrak oakraknae naeta. ta. C` tr`esjgrd`kn¿ gge z e  z  oa ast` sa˓e`c e`c kgesnsta oa ue sgcg pgcg, `s `s¿¿İ Qe rasuct`og ras`ct`og jrakuaeta oa c` tagr tagr¿¿İ` oa Jgurnar as qua as duy saeknccg av`cu`rc`. [ne adh`rfg, ast¿` d`c pae-

8. 8.;. ;.

Qe` Qe` F`us F`ussn sn`e `e` `

qua –c` tr`esjgrd`kn¿gge e oa Jgurnar oa ue` F`ussn`e` as ue` s`r qua c` OJ_ as sgc`daeta ue` varsn¿ge ge duastra`o` oa c` F`ussn`e`—. Ast` saetaekn` as ax`kt`daeta varo`oar` p`r` tr`esjgrd`kn¿ gge z e  z . ac k`sg oa c` tr`esjgrd`kn¿ge ge oa Jgurnar oa tnadpg kgetneug, parg sgcg `prgxnd`o`daeta varo`oar` p`r` c` O_J_ y c` `. Faea Faeara ra ue` pgrkn¿ ge ge ffent` oa ue` sa˓ e`c e`c axpgeaekn`c9 x\eT 4 OJ_1 . E .. _gda ue e¿udarg (=, (=,>)e u\eT, p`r` =  =   ≯  e 3 E  udarg paqua˓ eg eg oa duastr`s, onf`dgs E  onf`dgs  E  4  4 ;8. `. Faeara ue` sa˓e`c e`c F`ussn`e` oa v`cgr ra`c9 f\eT 4  a

ΰe8

∘

∘C≯e≯C

h. K`cku K`ckuca ca c` OJ_ OJ_ oa E  oa E  puetgs   puetgs oa x oa x\\eT y onhuia c` d`fentuo oa c` OJ_, R \l T .

   

C` sa˓ e e`c `c sa truek` oa d`ear` snd¿aatrnk` trnk` `craoaogr oac grnfae (e (e  4 =). Askgi`  Askgi`   C  oa d`ear` qua c` F`ussn`e` sa` hnae duastra`o` b`st` sus kgc`s. C` sacakkn¿gge e ax`kt` oa C  oapaeoar¿` oa  oa   ΰ< qunz¿` ac axpgeaeta d¿``ss fr`eoa fr`eoa   ΰC8 oaha ast`r rastrnefnog ` sar daegs qua 0==. [n tgd`dgs

  

^ ((aiώ ) , c` d`fenk. Kgdp Kgdp`ra `ra R \l T  kge c`s duastr`s oa ^  tuo oa c` O_J_ oa   y \eT 4 (=, (=,>)e u\eT, ue` axpgeaekn`c oa cgefntuo neffent`.

1 As`s sa˓ e`cas e`cas p`r` c`s ku`cas  OJ_ {v \eT} ↘  Z   \ \lT sge cc`d`o`s cgs vaktgras prgpngs oa c` OJ_ (raff¿aar`sa r`sa `c snfunaeta kgeiuetg oa prgyaktgs,   OJ_ kgdg ue` d`trnz ))..

   ^  ^ ((aiώ )  4

?

0 0 ∘ =,>a

iώ

∘

(:)

 

Onhuia c`s d`fentuoas sghra c` dnsd` fr¿`ffk`‟axpcnqua `ffk`‟axpcnqua as trukucaet`. B`y juekngeas netare`s oa   D`tc`h  cc`d`o`s c` onjaraekn o njaraekn`` ae t¿ardnegs ardnegs oac vaet`eag. jcnpcr y  jcnpuo p`r` o`r c` vuact` oarakb`-nzqunaro` sghra c`s ffc`s y `rrnh`-`h`ig sghra c`s kgcude`s. Enefue` oa ast`s as cg qua oasa`dgs p`r` c` OJ_, pgrqua ast`s juekngeas o. Gtr` sa˓ e`c e`c rac`knge`o` puaoa sar kra`o` duastra`eog c` b`kae cg snfunaeta9 O_J_ oa ` oa  ` e u\eT. Kraa ue` OJ_ pgr p gr daong oac duastra duastragg ygut \eT 4  x  x\\E  ∘  ∘ 0 ∘ eT   0 p`r`  e  4 =, 0, . . . , E  ∘ oa c` j¿grduc` grduc` (:)

 

Z  Z \\lT 4  ^   ^ ((aiώ )

E ∘ ∘ 0T,  x\0T E ∘ ∘ Aetgekas  x\=T Aetgekas x \=T as netark`dhn`og kge x kge  x\\E  0T, x \0T kge x kge x\\E  8T y `s `s¿¿İ sukasnv`daeta. sukasnv`daeta. ]gr gtrg c`og, c `og, c` c ` vuact` knrkuc`r eakas`rn` p`r` c` OJ_ s`tnsj`r¿İİ``

ώ 4(8 4(8ύ/E  ύ/E )l

E  puetgs Z  \lT p`r` ghtaear  _gda c` NOJ_ oa  oa   E   puetgs oa  oa   Z \ ghtaear   v \eT. yknr \eT 4  x  x\\∘e   d¿ggo E   x\=  ∘ eT o  E TT 4  x \=,, E  ∘ Axparndaeta kge onjaraetas cgefntuoas oa tr`esjgrd`kn¿ge ge  E ,, pgrqua ku`eog E  p`r` E  p`r` ku`eog  E    ↘ ∕, ac rasuct`og oaha ghta- p`r`   e   4 0, 8, . . . , E   ∘  0. Ae asta k`sg,   x\=T   eg sa duava  earsa duy kark`eg ` ` `  ` e u\eT. dnaetr`s qua  qua   x\0T as netark`dhn`og kge   x\E   ∘ 0T, 0T,   x\8T kge x\E  ∘  ∘ 8T y ``ss¿İ ssukas ukasnv`daet nv`daeta. a. ]uastg qua c` OJ_ jua jgrd`o` ` tr`v¿ aass oa duastr`s ae ac ogdneng oa c` jrakuaekn`, oaha b`har  `cn`snef  ae ac ogdneng oac tnadpg. Oarnva ue` j¿grduc` grduc` p`r`   v \eT ae t¿aardnegs rdnegs oa y \eT, h`s`o` ae ast` noa` oac  `cn`snef  oa tnadpg.

`. Askrn Askrnh` h` ue `rkbnvg `rkbnvg .d p`r` c` gpar`kn¿ ge ge o`r c` vuact` knrkuc`rdaeta< cc¿`dac` `dac` kjcnp. ]`r` ue vaktgr ffc` oa E  oa  E  pue puetgs  kjcnp  as saekncc`daeta \   x(0), x(E9-098)   T , parg ustao oaha b`karc` tr`h`i`r p`r` ffc`s, kgcude`s y d`trnkas (as oaknr, o`r c` vuact` knrkuc`r oa k`o` kgcude`).

;.

h. Zarnffqua c` prgpnao`o oa c` OJ_9 ‒O`r ue` vuact` knrkuc`r ae ac tnadpg as o`r ue` vuact` knrkuc`r ae c` jra-

[ndatr [ndatr¿ ¿İ`s ae c` OJ_

Asta prgyaktg ravns` onjaraetas `trnhutgs oa c` sa˓e`c e`c kgdg kuaekn`—. Qsa vaktgras oa pruah` parg b¿`f`cgs `f`cgs oa v`cgr kgdpcaig p`r` prgh`r ast`saeknccgs prgpnao`o kgdpcat`p`r, ndp`r, pur`daeta ra`c y pur`daeta nd`fne`rn` y axpcgdaeta. r` su ndpcnk`kn¿ge ge p`r` ue` astruktur` aspakn`c oa c` OJ_ rasuct` rasu ct`eta. eta. _`dhn¿ae ae sa netrgouka netrg ouka ac `t `trnh rnhutg utg o oaa snd sndatr¿ atr¿İİ`` kg kgee iuf`o` oahnog ` qua as ac ou`c ndpgrt`eta oa pur`daeta k. Zarnffq arnffqua ua c` prgpnao`o oa c` OJ_9 –Qe` kgeiu kgeiuf`o` f`o` ae ac ra`c9 –Qe` sa˓ e`c e`c oa tnadpg pur`daeta ra`c tnaea ue` OJ_ tnadpg o` ue` kgeiuf`o` d¿`s `s ue` vuact` knrkuc`r ae c` snd¿ aatrnk` trnk` kgeiuf`o` y vnkavars`—. Ae duk dukbgs bgs k`sgs, ast`s jrakuaekn`—. Ast`hcazk` c` ou`c oa ast` prgpnao`o< cuafg sndatr¿¿İ`s puaoae sar utncnz`o`s p`r` sndpcnffk`r cgs k¿ sndatr ``ckucgs, ckucgs, varnj¿¿İquac` kge ue` sa˓e`c varnj e`c oa pruah` oa v`cgr kgdpcaig. aspakn`cdaeta ae v`rn`kngeas onjaraetas oac `cfgrntdg JJ_. [ufaraekn`s   C` neoax`kn¿ ge ge oa ue vaktgr oa E  oa  E  puetgs  puetgs ae

;. ;.8. 8.

oa 0 ` E  `  E .. [n [ne e adh`rfg, adh`rfg , dukb`s oa cc`s `s ssndatr¿ ndatr¿İ`s İ`s oa c` OJ_ sge axpras`o`s ae t¿ aardnegs rdnegs oa vuact`s kge ras x\∘e   d¿ paktg `c grnfae (as oaknr, c` rastrnkkn¿ge x ge x\\eT 4  x\ go E  go  E TT oaffea ue` sa˓ e e`c `c p`r). Astg prasaet` prghcad`s ku`eog sa tr`h`i` ae   D`tc`h. Qe` sgcukn¿ge ge saekncc` sar¿ sar¿İ` kra`r ue euavg `rkbnvg .d p`r` oas`rrgcc`r ac vgcta`og knrkuc`r. Astg ast`rr¿İ` h`s`og ast` h`s` og ae c` `rntd¿ `rn td¿aatnk` tn k` d¿ggoucg E  oucg  E ,, c` ku`c sa ndpcadaet` ` tr`v¿aass oac `rkbnvg .d   dgo  oaskrnt  oaskrntgg ae ac @p¿aaeonka eonka @. Adpaz`eog oasoa  kjcnp, ustao puaoa askrnhnr aetgekas `rkbnvgs .d `onknge`cas p`r` axtr`ar c`s p`rtas p`r a ndp`r oa ue vaktgr oa sa˓ e`c, e`c, kgdg t`dhn¿ae ae sus p`rtas snd¿aatrnk` trnk` kgeiuf`o` y `etnsnd¿atrnk` atrnk` kgeiuf`o`. kgeiuf`o` . D`tc`h v`

;.0.

]`rta `rtass p`r p`r a nndp dp`r `r

C`s sndatr¿İİ`s `s prnd`rn`s prnd`rn `s ast¿`e `e h`s`o`s ae c` p`rno`o y onsp`rno`o. ]gr supuastg, ast`s oahae sar oaffeno`s –knrkuc`rdaeta—. `. ]`r ]`r`` usg ae ac rastg oa asta aiarknkng, aiarknkng, askrn askrnh` h` ue` juekn¿ gge e oa   D`tc`h  qua axtr`nf` c` p`rta p`r oa ue vaktgr< b`f` cg dnsdg p`r` c` p`rta ndp`r. Ast`s juekngeas oahae cc`d`r ` c` juekn¿gge e  kjcnp  askrnt` pravn`daeta. h. Faeara ue` sa˓ e`c e`c oa pruah` oa v`cgr ra`c v ra`c  v\\eT utncnz`eog utncnz`eog r`eo. [acakkngea ue` cgefntuo rac`tnv`daeta kgrt`, onf`E  4 E  4 dgs   E  dgs  4 01 g   E   4 0:. Kgdputa c` OJ_ oa   v \eT p`r`  Z \\lT y cuafg cg snfunaeta p`r` `dh`s cgefntuoas ghtaear Z  ghtaear p`r a ndp`r.

C`s sndatr¿ sndatr¿İ`s sge knrkuc`ras knrkuc`ras

_go`s c`s gpar`kngeas kge c` OJ_ sge bakb`s sghra ue ogdneng oa neoax`kn¿ge ge qua as knrkuc`r. ]uastg qua tgo`s c`s prgpnao`oass oa sndatr¿İ` pnao`oa İ` sa s a rao raoukae ukae u uenk`daeta ¿enk`daeta `  ogs  gpar`kng-

 v\\eT. k. K`ckuca c`s p`rtas p`r a ndp`r oa  v va \eT 4   08 v \eT + v \∘e   d¿ ggo E  o  E TT  0 vg \eT 4 8 v \eT ∘ v \∘e   d¿ ggo E  o  E TT



eas h¿``snk`s, snk`s, c` kgeiuf`kn¿ e y o`r c`prgvaa vuact`, as juekn¿ asaekn`c taear D`tc`h prgva juekngeas p`r` k`o` ue`.gge a ue` gge e kgeiuf`r netare`9  kgei. [ne adh`rfg, c` gpar`kn¿gge e –o`r c` vuact`— 1



 

   

Z \\l T  as p`r, pgr daong oa k`ckuc`r c` Cuafg kgdputa c`s OJ_s oa ast`s ogs sa˓e`cas e`cas p`r` c` o. Duastra qua  qua   a Z  snfunaeta p`rta. Z \\l T  c` ku`c p`rta ndp`r oa  oa  aa Z  ku `c oahar¿İİ`` sar karg. Duasd Z  Z \\lT  as ndp`r. tra qua  qua  d o. ]`r` c` OJ_ kgdput`o` ae c` p`rta (`), axtr`nf` sus p`rtas ra`c a nd`fne`rn` y b`f` c` snfunaeta kgdp`r`kn¿gge9 e9

   

Aetgekas pgoadgs kgekcunr qua c` OJ_ oa ue` sakuaekn` ra`c as snd¿ atrnk` atrnk` kgeiuf`o` kgeiuf`o`,, c` ku`c as c` prgpnao`o prgpnao`o ou`c aspar`o`. ·Ku¿`c `c sar¿İ` İ` ac rasuct`og rasuct `og p`r` p`r ` ue vaktgr oa aetr`o` pur`daeta nd`fne`rng7

   

OJ_ va \eT   vs.   a Z \ Z \lT

Z  \l T OJ_ vg \eT   vs.   d Z \

[n v [n  v\\eT as kgdpcai`, duastra qua ast`s dnsd`s rac`kngeas eg sa d`etnaeae.

;.?.

Ku`cqunar sa˓ e`c e`c oa v`cgr kgdpcaig puaoa sar oaskgdpuast` ae ku`trg ku`trg suh-s suh-sa˓ a˓ e`cas, e`cas, k`o` ue` oa c`s ku`cas ku`cas axbnha ue` knar knart` t` sndat s ndatrr¿İ`. Aspak¿ Asp ak¿İffk`daet İffk `daeta, a, cc`` sa˓ sa e`c e ˓`c kgdpcai`, v kgdpcai`, v\\eT, snadpra puaoa sar askrnt` kgdg

a. C`s egkngeas oa p`r a ndp`r puaoae puaoae sar axtaeono`s `c k`sg oa kgdpcaigs oaffenaeog ogs `trnhutgs cc`d`ogs  s  snd ndat atr¿ r¿İİ`  `  kgeiuf`o`   y   `etnsndat `etns ndatr¿ r¿ İİ` ` kgeiuf`o`  kgeiuf `o` . Faeara ue` sa˓ e e`c `c oa pruah` oa v`cgr kgdpcaig,   v \eT. Kgdputa sus p`rtas snd¿aatrnk` trn k` kgei kgeiuf`o uf`o`` y `etnsnd¿ `etn snd¿aatrnk` trn k` kgei kgeiuf` uf`o` o` ` tr`v¿as as oa vksy \eT 4   80 v \eT + v \∘e   d¿ go E  go  E TT

 

∖

 

_go`s c`s sndatr¿ sndatr¿İ`s pgsnhcas









v \eT 4  v r,a \eT + vr,g \eT +  i vn,a \eT + vn,g \eT + i

 

(2)

suh¿İeonkas İeonkas oaegt`e oaegt`e ra`c (r), nd`fne`rng nd`fne`rng (n), p`r (6) ogeoa cgs suh¿ v \eT ∘ v \∘e   d¿ vk`s \eT go E  go  E TT (a) a ndp`r ndp`r (g). (g). Aetge Aetgekas kas   vn,g \eT as c` p`rta ndp`r oa c` p`rta nd`fne`rn` oa v oa v\\eT. Ac dnsdg groae oa oaskgdpgsnkn¿gge e  Z \\l T. puaoa sar bakbg p`r` c` OJ_,  Z   D`tc`h Askrnh` juekngeas oa axtr`nf`e ast`s p`rtas oa ue vaktgr oa sa˓ e`c e`c oa v`cgr qua kgdpcaig. Z  Z \\lT 4  Z r,a +  i Z n,a ,g \lT n,a \l T + Z nn,g r,g \l T + i r,a \l T + Z r,g 4   80

∖

 

j. Duastra qua c` p`rta ra`c oa   vksy \eT as snadpra p`r y qua c` p`rta nd`fne`rn` as ndp`r. Zarnffqua astgs bakbgs ae   D`tc`h   p`r` c` p`rta snd¿ aatrnk` trnk` kgeiuf`o` oa ue` sakuaekn` sakua ekn` oa pruah` pruah` `ca`tgrn`. `ca`tgrn`. Ast`hcazk Ast`hcazk`` ue` rac`kn¿ ge ge varn j¿İquac` İqu ac`.. sndnc`r p`r` v p`r`  v k`s \eT< varnj

[ne adh`rfg, ast¿` d`c `sudnr qua c` OJ_ oa ue` oa c`s suh-sa˓ e`cas e`cas sa `iust` ` c` kgrraspgeonaeta suh-sakuaekn` oa c` OJ_. `. Askrn Askrnh` h` ue` juekn¿ juekn¿ge ge oa   D`tc`h  qua oaskgdpgef` ku`cqunar vaktgr ae sus ku`trg p`rtas, kgdg jua oaffenog ae (2).

f. Zarnffqua qua c` OJ_ oa ue` sa˓e e`c `c snd¿aatrnk` trnk` kgeiuf`o` kgeiuf`o ` as pur`daeta ra`c. ·Pu¿a sukaoa kge c` OJ_ oa ue` sa˓ sa e e`c ˓`c `etn-snd¿ `etnsnd¿aatrnk` trn k` kgeiuf`o kgei uf`o`7 `7

h. C`s sndatr sndatr¿¿İ`s oa c` OJ_ puaoae puaoae sar k`t`cgf`o` k`t`cgf`o`ss ae ue on`fr`d` qua duastr` c` kgrraspgeoaekn` oa c` sndatr¿ sndatr¿İ` ae ac ogd ogdnen nengg oa c` tr`esj tr`esjgrd grd`kn `kn¿ge ¿g e ` ue` oa c`s ku`;. ;.;. ;. OJ_ OJ_ oa ue` ue` sakua sakuaek ekn` n` ra`c ra`c trg suh-sa˓ e`cas e`cas ae ac ogdneng oac tnadpg. Kgdpcata ac snfunaeta on`fr`d` kgeakt`eog k`o` suh-sa˓e`c e`c ae ac og]gr ou`cno`o, c` OJ_ oa ue` sakuaekn` pur`daeta ra`c oaha dneng oac tnadpg • tnadpg  •  kge c` suh-sakuaekn` `prgpn`o` ae ac sar snd¿atrnk` atrnk` kgeiuf`o`. kge iuf`o`. Kgdnaeka K gdnaeka kge k ge ue` sakuaekn` saku aekn` ra` ra`cc oa  ◪.. ogdneng ogdne ng oa c` jrakuaekn jrakuaekn``  ◪ E   E   puetgs v puetgs  v\\eT, c` ku`c eg as en p`r en ndp`r.

   

[nda [n datr¿ tr¿İ` İ`ss oa c` OJ_

 Z \\l T 4 OJ_ v \eT . `. K`ckuca su OJ_9 OJ_9 Z 

 

      vn,g \eT  

vr,a \eT vr,g \eT vn,a \eT

Z \\l T   a   d Z \ Z \l T . @oad¿``s, h. Znsu`c Znsu`cnka nka   a Z  s, fr`ffqua c` d`fentuo y c` j`sa oa Z  oa  Z \\l T y egta ku`cqunar ku`cqun ar sndatr¿ s ndatr¿İ` İ` ghg hvn`.

 Z \\lT, c` ku`c k. oaha Axtr`nf` Axtr`nf c` p`rta `etnsnd¿atrnk` atrnk` kgeiuf`o` kgei uf`o` oa oa Z  sar` karg.

• • • •

◪ ◪ ◪ ◪

  Z  r,a \lT   Z  r,g \lT   Z  n,a \lT   Z  n,g \lT

ra`c y p`r ra`c ra`c a ndp` ndp`rr nd`f nd`fne ne`r `rng ng y p` p`rr nd`fn nd`fne` e`rng rng a ndp` ndp`rr

k. O¿ aaiadpcgs iadpcgs eud¿ akgrrakt`. rnkgs p`r` dgstr`r qua k`o` kgeaxn¿gge e aea ac on`fr`d` as arnkgs

:

 

?.

_ruk rukgs gs p p`r `r` ` c` OJ OJ_ _ ne nev vars ars` `

 ^ \\l T|l4e . ?. Kgdp`ra cgs v`cgras eud¿arnkgs arnkgs oa o a x\eT y v \eT 4  ^ 

Ae c` pr¿`ktnk`, `ktnk`,  b`row`ra  aspakn`c   aspakn`c puaoa sar kgestrunog p`r` k`ckuc`r c` OJ_. Ae asta k`sg, as vaet`igsg us`r ac dnsdg h. Oarnv Oarnvaa c` rafc` faear`c faear`c p`r` c` rac`kn¿ ge ge aetra cgs v`cgras b`row`ra  p`r` c` OJ_ nevars` (nOJ_). Asta prgyaktg duasoa x oa  x\\eT y   v \eT y pruaha pgr qu¿a accgs ast¿ `e `e rac`knge`ogs tr` tras onjaraetas jgrd`s qua c` nOJ_ sa puaoa kgdput`r oa d`ear` d`ear` t`e saekncc`. utncnz`eog ue `cfgrntdg oa c` OJ_ onrakt`. _gog ast¿` h`s`og ae ac bakbg qua c` j¿grduc` grduc` p`r` c` nOJ_ as kark`e`daeta no¿aetnk` aetnk` ` `quacc` `qua cc` p`r` c` OJ_ onrakt`, on rakt`, axkaptg axk aptg pg pgrr ue snfe snfegg k. ]rg ]rgfr` fr`da da ue `rk `rkbnv bnvgg .d qua ndpcadae ndpcadaeta ta c` nOJ_ oa /E  daegs ae ac axpgeaeta y ue j`ktgr oa 0/E  0 . `kuarog ` asta `cfgrntdg. ]ru¿ aahacg hacg sghra `cfuegs p`ras x\eT 4

 0 E 

OJ_ kgegknogs. kgegknogs.

E ∘0

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R \l TV E el ∘

 

(>)

l4=

?.8.

 e 4   0. p`r` e p`r`  4 =, 0, 8, . . . , E  ∘

nOJ_ ` tr`v¿ tr`v¿ as as oa kgeiuf`o`s kgeiuf`o`s

Asta aiarknkng tr`t` kge ac k¿`ckucg `ckucg oa c` OJ_ nevars` utncnv`rn`s kgeiuf`o`s kgeiuf`o`s hnae kgcgk`o`s. kgcgk`o`s. Qsa ue` sakuaekn` [ufaraekn`s.   _ras juekngeas juekngeas nOJ_ onjar onjaraet aetas as sge oaskr oaskrnn- z`eog v`rn`s pruah`  R \\lT faear`o` kgdg ae ac Aiarknkng ?.0. t`s ae asta prgyaktg. ]gr kgesnfunaeta, tras `rkbnvgs .d sap`- oa pruah` R  r`ogs oahae sar askrntgs y prgh`ogs. Oaetrg oa  D`tc`h, c` Oasa`dgs dgs dgstr dgstr`r `r qua cgs snfunaeta snfunaetass tras p`sgs sge aqunjuekn¿ ge ge OJ_ nevars`  njjt  as ra`cdaeta ndpcadaet`o` utn- `. Oasa` v`caetas ` c` nOJ_ oa R  oa  R \\lT. cnz`eog ueg oa astgs trukgs. Askrnh` c` cnst` oa c` juekn¿ge ge njjt  p`r` var ku`c as.  R \\lT. 0. Kgeiu Kgeiufua fua R 

?.0.

nOJ_ ` tr`v¿ tr`v¿ as as oa rgt`kngeas knrkuc`ras knrkuc`ras

∖

  onrakt`  l T. E  p 8. Kgdpu Kgdputa ta c` OJ_ oa R  oa  R  \oa E  Kgdg `etas, taeadgs ue  vaktgr oa  puetgs uetgs,, `s¿İ qua pgoadgs b`kar astg.

Asta aiarknkng tr`t` kge ac k¿``ckucg ckucg oa c` OJ_ nevars` utncnz`eog ue` prgpnao`o oa c` tr`esjgrd`kn¿gge e qua as kgegkno` kgdg  ou`cno`o . Asta d¿ atgog atgog as netaras`e netaras`eta ta pgrqua aej`tnz` c` e`tur`caz` knrkuc`r oa tgo` c` neoax`kn¿ge ge oa c` OJ_. Ac `rkbnvg .d  kjcnp  oas`rrgcc`og ae ac prgyaktg ; sar¿` u utnc ¿tnc `qu¿İ.

;. Kgeiu Kgeiufua fua c` OJ_ oa s`cno` rasuct`eta rasuct`eta.. ^ \\lT, `bgr` [n ac rasuct`og oa c` OJ_ pravn` juar`   ^   ^  \l T. taeadgs ^  taeadgs ∖

`. Faeara Faeara ue` sakuaekn` sakuaekn` `ca`tgrn` `ca`tgrn`   x\eT utncnz`eog ue` o`r c` vuact` knrkuc`r. C` cgefntuo oa c` sakuaekn` puaoa sar arnkgs oa   x\eT y cgs oa   v \eT 4 ku`cqunar`, parg as jrakuaeta askgfar ue` pgtaekn` oa 8< h. Kgdp`ra cgs v`cgras eud¿arnkgs l | ^  \ T . Egta qua ac ajaktg oa c`s gpar`kngeas kgeiuf`l4e  E  4 sn as `s¿ `s¿İ, sacakkngea sacakkn gea E   4 0:. Qtncnka  jjt  p`r` k`ckuc`r c` o`s b`e snog k`dhn`r ac snfeg oa c` axpgeaekn`c kgdpcai`. OJ_ oa x oa  x\\eT y cc`da astg R  astg  R \\l T. Ast` sakuaekn` sar¿` c` saOarnva c` rafc` faear`c p`r` c` rac`kn¿ge ge aetra cgs v`cgras kuaekn` oa pruah` p`r` c` nOJ_. Oas`rrgcca cgs snfunaetas  x\ e   v e oa x oa \ T y \ T y pruaha pgr qu¿ a accgs ast¿ `e `e rac`knge`ogs p`sgs kge   D`tc`h9 oa ue` d`ear` t`e saekncc`. ∖

 R \\lT utncnz`eog ue` o`r c` 0. Oa c` vuact` vuact` ` c` sakuaekn` sakuaekn` R  vuact` knrkuc`r. ]uastg qua R  qua  R \\lT as iustg ue vaktgr oa k. Kraa ue `rkbnvg .d oa   D`tc`h  qua ndpcadaeta asta `c0: v`cgras v`cgras kgdpcaigs, kgdpcaigs, puaoa sar tr`t`og tr`t`og kgdg sn juar` fgrntdg p`r` c` nOJ_. Kgdp`ra kge c` ndpcadaet`kn¿ge ge ue vaktgr oa duastr`s oa tnadpg. @s¿ @s¿İ c` gpar`kn¿gge e oa ra`cdaeta us`o` ae   D`tc`h< ac cnst`og oa c` juekn¿ge ge oa o`r c` vuact` oaffea ue euavg vaktgr oa tnadpg pgr D`tc`h njjt puao  puaoaa sar ghtaenog ` tr`v¿as as oa o a  typa njjt.  R \\lT|l4( e) d¿go daong oa y oa  y\\eT 4  R  go  E  . ∘

8. ]uastg ]uastg qua c` sakuaekn` y sakuaekn` y \eT as iustg ue vaktgr oa 0: ?.;. ;. Gtrg Gtrg tru trukg kg oa oa c` nOJ nOJ_ _ v`cgras kgdpcaigs, puaoa sar utncnz`og kgdg c` aetr`o` ?. ` ue` OJ_  onrakt`  (as oaknr, `pcnqua   jjt  oa euavg). Asta aiarknkng aiarknkng duastr` `ue gtr` jgrd` jgrd` oa kgdput`r c` OJ_ Cc`da `c rasuct`og ^  rasuct`og  ^ \\l T. nevars`. [u pruah` t`dhn¿ae ae sa h`s` ae c` prgpnao`o kgeiuf`o` oa c` OJ_. Qtncnka c` dnsd` sakuaekn` oa pruah` kgdg  ^ \\l T as iustg ue vaktgr ae cgs ogs aiarknkngs prakaoaetas. ;. Qe` vaz oa euavg, euavg, c` sakuaekn` sakuaekn` ^  oa 0: v`cgras, `s¿ `s¿İ qua puaoa sar kgesnoar`og kgdg ue vaktgr oa tnadpg sn l sn  l  as raadpc`z`og pgr e pgr  e..

`. Oasa` Oasa`dgs dgs dgstr dgstr`r `r qua cgs snfunaeta snfunaetass tras p`sgs sge aqunv`caetas ` c` nOJ_ oa R  oa  R \\lT (va` c`  Jnfur` 8). 6

 









ℓa R \lT



E  ℓa x\eT



JJ_ ℐd R \lT





E  ℐd x\eT

Jnfur` 89  nOJ_ ` tr`v¿ aass oa k`dhng oa c` c`ss p`rtas ra`c a nd`fne`rn`.

 R \\l T y oa0. Netark`d Netark`dhna hna c`s p`rtas ra`c a nd`fn nd`fne`rn` e`rn` oa oa R  ffe` ac rasuct`og kgdg ue vaktgr oa tnadpg< astg as, oaffe` v oaffe`  v\\eT `s¿ `s¿İ qua su p`rta ra`c as cc`` p`rta nd`fne`rn` oa R  oa  R \\l T y su p`rta nd`fne`rn` as c` p`rta ra`c oa R  oa  R \\l T. 8. K`ckuca K`ckuca c` OJ_ onrakt` onrakt` oa v oa  v\\eT y cc`da ` astg Z  astg  Z \\l T. Z \lT y ;. Netar Netark`d k`dhna hna c`s p`rtas p`rtas ra`c ra`c a nd` nd`fne fne`rn `rn`` oa   Z \ oaj¿¿İe`c`s kgdg ue vaktgr oaj vaktgr oa tnadpg tnadpg