Olasılık Ve İstatistik - Ders Notu (Geniş Anlatım)

8/10/2019 Olasılık Ve İstatistik - Ders Notu (Geniş Anlatım)

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AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ ?

Jaerod @hezhc

9 ŕufot 6>><

?. Klrlő vh Khehc FojĿő

Fu dhrs slzh AcosĿcĿj Tharlsl“eh klrlől vh ľstotlstlđle th`hc oroëcorĿeĿ vhrhihjtlr. AcosĿcĿj,lcklcl fhclrchylilchrlel to` acoroj flc`hdlđl`lz duru`corĿ oëĿjcoyoe vh oeoclz hdhe flròthòtljshc yopĿdĿr. @adhre boyotto, bhpl`lz, tĿptoe sasyacamlyh jodor fótóe ocoecordo

yopĿcoe lstotlstljshc ëocĿőòcorĿe dózhecl `óőthrlchrlylz vh acosĿcĿj ùbojh`hsl hjaea`lvh gleoes ocoeĿedo sae d÷eh`dh yopĿcoe flr ëaj tortĿőòeĿe tojlp hdlc`hsledh acdujëo÷eh`cldlr.

Fu dhrsle lcj yorĿsĿedo, acoycordo fucueoe khrëhj rlsjl - vhyo jlőlshc flckl yhthrslzclđl`lzl-oëĿjcoòeĿe flr yacu acoroj acosĿcĿj ózhrledh jaeuőoiođĿz.

×rehj ?. Jrhdlflclthsl dóőój `óőthrlchrh vhrlche jrhdl dh (Yufprl`h chedlek), foejocor

earòc `óőthrlchrleh k÷rh khrl ÷dh`h acosĿcĿjcorĿ dobo dóőój acoe `óőthrlchrh yhel ehslc

hv jrhdlsl ( àrtkokh) vhrlrchrdl. Th`leot acoroj k÷sthrhflchihjchrl gozco őhychrl acòyoeacosĿ hv soblpchrleh farë vhr`h rlsjlel ldorh ht`hj lële, fu őhjlcd h vhrlche flechrih jrhdl

flr dh`ht bocleh khtlrlclp –àrtkokh dhsthjcl dhđhrcl jođĿt‐ acoroj yheldhe sotĿcĿrdĿ, yoel

jrhdlyl lcj vhrhe foejoyo eh jodor khrl ÷dh`h yopĿcĿrso yopĿcsĿe, foejo dhđhrcl jođĿdĿ

hcledh tutoecoro s÷z vhrdlđl `ljtorĿ ÷dh`hyl toobbót hdhrdl. Yaeuedo, dhđhrcl jæđĿtcor

bovuzueue flr joë ëhőlt tobvlch d÷eóőtóđó ëaj joròőĿj gleoesoc dózhech`hchr artoyo

ëĿjtĿ. Furodo lcj tobvlcl hcledh tutoecoro ÷dh`hdh ÷eihclj vhrlclrdl. \oel, dlyhcl` jl, tobvlcle

ea`leoc dhđhrl ?> `lcyae dacordĿr, khrl ÷dh`hchr tapco`Ŀ eh zoòe ?> `lcyae dacorĿ

oőorso ÷eih fu krufo ÷dh`h yopĿcĿyardu. Fu duru`do, eh zoòe ÷eihclđl acoecordoe

poro ortorso oeioj a zoòe ea`leoc dhđhrl dóőój krufo ÷dh`hchr yopĿcoflclyardu.

EosĿc acur do, fhclrtlche bhrboekl flr yhel ehslc hv jrhdlsl kóvhecl dhđlcjhe, flr ëaj rlsj

forĿedĿroe flr bovuzdoe acuőoe lcj krup kóvhedh acoflclyardu1 Dobo dóőój ÷eihclđl

acoecor dobo rlsjcl acuyarcordĿ. ’ Elyh1 Khrlyh dađru fojĿcdĿđĿedo, elyh plyosoeĿe

fhjchdlđlele ojsleh dobo kóvhecl acoe dhđhrcl jođĿt cor plyosodojl bhrjhsle fhjchdlđledhe

dobo rlsjcl acducor1 Fóyój YoyĿcor Joeueu bojjĿedo jaeuőurjhe, vh boekl jaőuccordo

ëocĿőĿp boekl jaőucordo ëocĿőòdĿđĿeĿ fhclrtljthe saero, fu jaeuyo khrl d÷ehihđlz.



Khehccljch, fu tór sarucorĿ ihvopcoedĿròj lële, saeuëcorĿe dođĿcĿ`Ŀ (elsplacoflclrclj klfl) bojjĿedo ëaj őhy flc`helz khrhjlr, gojot fozĿ duru`cordo dobo ozĿ lch dhldorh hdhrslelz= FozĿ duru`cordo flr saeuiue sodhih fhjchehe dhđhrl vhyo flr dođĿcĿ`Ŀedlđhr à`hetchrl klfl –tlplj‐ dhđhrchrl lch lcklchelrslelz. Dlđhr duru`cordo sodhih roskhchflr dhehyle ëaj soyĿdojl thjrorĿeĿe artocoòsĿ lch lcklchelyar acoflclrslelz vh f÷ych

duru`cordo bhr flr dhehyle gorjcĿ saeuëcorĿeĿe acoflclrclđl jaeusuedo ëaj őhy flc`hdhefóyój rojo`cor joeueu vhyo `hrjhzl cl`lt tharh`l fozhe slzh lyl flr tob`le vhrhflclr.

Dhrsle ljleil yorĿsĿ, vhrldhe jltch vh acosĿcĿj dođĿcĿ`corĿ bojjĿedo eosĿc flcklhdlelchihđl sarusuyco lcklcl acoiojtĿr. Bhrboekl flr o`plrlj flcl`dh, tó`hvorĿ` prafch`l lchjorőĿcoőoflclrslelz. Tó`hvorĿ` flrjoë (fhcjl flroz dobo ëaj) k÷zch`dhe khehc saeuëcorëĿjoròjtĿr. âcltlj yajcoòcordo (–saerojl foőjoecĿj shël`ledh jl`le lële ayjuccoeoiojsĿeĿz1‐), flr oejht glròsĿ tlplj acoroj yózflechrih patoeslyhc shë`heorosĿedoe he gozco flrjoë fle jlőlyh oejht uykucor. TĿffl dhehychrdh, flr lcoiĿe htjleclđljaeusuedo flrjoë dózleh jotĿcĿìĿdoe hcdh hdlche saeuëcordoe fótóe jltch lële

khehcch`hchr hcdh ht`hyh ëocĿőĿrĿz.

Hđhr lcklchedlđl`lz flr jltchele (÷rehđle flr khehc shël`dh ay juccoeoflchihj fótóeshë`hechr klfl) yoceĿzio flr oct-krufuedojl jlőlchrl (÷rehđle shë`hechrle roskhch flr÷rehjch`l klfl) k÷zch`chrshj, ihvopcoedĿròj lsthdlđl`lz saru oëĿsĿedoe (÷rehđle fhcclflr odoyĿe ay aroeĿ klfl) fu ÷rehjch`le khrëhjthe fótóe jltchyl th`slc hdlp ht`hdlđljaeusuedo flr ëhőlt fhclrslzclj acoiojtĿr. Fu fhclrslzclđl gar`óch hdlp juccoeĿ`ĿeĿ protljboch khtlr`hj acosĿcĿj tharlslele yađue juccoeĿ`ĿeĿ khr hjtlrlr.

×rehj 6 . @ort 6>>9 yĿcĿedojl lőkocdhe óë fuëuj yĿc saero, Lroj sovoőĿedo ÷che slvlcchr

ózhrledh yopĿcoe tortĿőòcĿ lcj Coeiht ëocĿőòsĿedo ócjh khehcledh?24< boeh bocjĿedoeacuőoe (tapco` acoroj ?62>? jlől) flr roskhch ÷rehjch`h oejht uykucoe`ĿőtĿr vh boeh

bocjcorĿ torogĿedoe fhyoe hdlche ÷có` soyĿcorĿ, 6< @lcyaecuj, tó` ócjhdhjl ÷có soyĿsĿeĿe

tob`lel lële juccoeĿc`ĿőtĿr. ËocĿőòeĿe yozorcorĿ, lőkocdhe saerojl lcj ?2 oy lële ??6 >>>

"gozco" ÷có` tob`leleh ucoőtĿcor vh "%<8'clj kóvhelrcljch" khrëhj rojo`Ŀe 7<>>> lch

?88>>> orosĿedo acduđueu lgodh httlchr. Fu lgodhele eh oecoò khcdlđlel dobo saero

dhrsth k÷rhihđlz. Tob`lele htrogĿedojl kóvhe orocĿđĿeĿe khelőclđl , ugoj flr oct-jltchdhe

fótóe ócjh lële bhsopcoòcor yopòeĿe jhedl lëledhjl vor acoe fhclrslzclđlele ÷cëósódór.

Fu pacltlj vh duykusoc htjlchrl acoe flr jaeu acduđu lële, ëocĿőò flcl`shc yoyĿecordo vh

fcakcordo yađue flr tortĿőò foőcot`ĿőtĿr - tortĿőòcorĿ ajuòj uykucoòdo lstotlstlđle

khrëhjthe –eosĿc‐ yopĿcdĿđĿ jaeusuedo slzh ëaj őhy ÷đrhthihjtlr.

6. Jó`h Tharlsl vh Acoycor

6.? Voskhch Dhehychr



ToeĿ` ?. Flr roskhch dhehy ’ he ozĿedoe tharlj acoroj - (o) sĿj sĿj vh oyeĿ jaőuccordo

jhygl acoroj thjrorcoeoflclr (f) ëaj lyl toeĿ`coe`Ŀő acosĿ saeuëcor jó`hsleh soblptlr .

Fueue stoedort ÷rehđl yozĿ (T) vh turo (B) klfl ljl saeuiu acoflche òdhel porootòdĿr (poroeĿe dlj flr őhjlcdh durò acosĿcĿđĿeĿ k÷r`hzcljthe khclyaruz). ľstotlstlj

ocoeĿeĿe flr dlđhr ÷eh`cl dhehy tóró flr shël` sĿrosĿedo yopĿcoe yajcoòdĿr. Dlyhcl` jlaycoò yopĿcoe yhrdhe ëĿjoecor orosĿedoe roskhch acoroj shëlche 6>>> jlőlyh jl`h ayvhrdljchrlel sarduj. Juroc acoroj, shël` kóeó ay juccoeoioj jltchdhe jhygl acoroj ëajsoyĿdo 6>>> jlőlclj ÷rehjch` shëhflclrlz. DacoyĿsĿyco jaőuc (o)“yĿ khrëhjchőtlr`lő acur uz.Fu dhehyle saeuë jó`hsl, shël`dh boekl odoy lële ay juccoedĿđĿeĿ fhyoe hdhe shë`hesoyĿsĿdĿr.

ToeĿ` 6. Flr ÷rehjch` uzoyĿ acoe Y, flr dhehyle fótóe acosĿ saeuëcorĿeĿe tapco`ĿdĿr .

Flrëaj oòëtoe ÷tóró, hsos ltlforlych acoycorĿe thj saeuiuyco dhđlc, acoycorĿefótóe saeuëcor krufu lch lcklchelyaruz. Fu ehdhech, lzchyhe f÷có`dh dhehyl jó`hchrilesledhe toeĿ`coyoiođĿz.

ToeĿ` 9. O acoyĿ bhrboekl flr saeuëcor tapco`Ŀ acoflclr (fu, thjcl saeuiu, faő jó`hyl

vhyo ÷rehj uzoyĿe tó`óeó lëhrlr).

Hđhr khrëhjchőhe saeuë O acoyĿeĿe flr hchòeĿ lsh, a zoòe O“eĿe khrëhjchőtlđlel

s÷ychyhflclrlz.

Flr ÷rehj acoroj, ől`dl khëhe shehjl foőjoecĿj yorĿőĿeĿ hch ococĿ`. He fosltőhjclych, ÷rehj uzoyĿeĿ (Y) ’ òetĿjsoc flr acoflclrclđl dóőóedóđó`ózdh- JosĿ` oyĿedofoőjoe acoroj shëlchflchihj jlőlchr acoroj toeĿ`coyoflclrdlj (×rehđle, lcj dheh`h acoroj÷e shël`chrle foőcĿio odoycorĿeo fojoflclrdlj).

Y 0 {Icletae, Buijofhh, @iIole, Afoò, ôuc, Vaèhy, Yibworzhehkkhr }

Fu lyl flr toeĿ`coò `Ŀ1 @ubth`hche dhđlc= Bhr eh jodor fuecor khrëhjchő`hslhe acosĿ saeuëcor acso do, òetĿjhe foőjo odoycorĿe (fođĿ`sĿz acoroj vhyo flr portlyh

fođcĿ acoroj) saerodoe yorĿőo jotĿcòsĿeĿ k÷r`hzcljthe khc`hylz. DacoysĿyco dobobotosĿz, roskhch dhehyle toeĿ`Ŀ ÷rehjch` uzoyĿeĿ fóyóthihjtlr.

0 {Icletae, Buijofhh, @iIole, Afoò, ôuc, Vaèhy, Yibworzhehkkhr, DlđhrDhàjrot Odoy, Dlđhr Iu`burlyhtël Odoy, Dlđhr FođĿ`sĿz Odoy }

Oeioj acoyĿ foslt tutòj lële, ől`dlclj fu acosĿcĿđĿ k÷r`hzcljthe khchcl`.



ľcklchelche fozĿ acoycor őuecor acoflclr =

–44eió O`hrljoe FoőjoeĿ flr iu`burlyhtël acoiojtĿr‐ 0 {Buijofhh, @iIole, ôuc,Yibworzhehkkhr, Dlđhr Iu`burlyhtël Odoy}

–44eió foőjoe 46eil foőjoe lch hvcl acoiojtĿr‐ 0 {Icletae}

6.6 Jó`h vh Acoycor BojjĿedo Dobo Gozco Flckl

6.6.? Jó`h Jopso`Ŀ –‐

Hđhr F“ele fótóe saeuëcorĿ O“yo olt lsh, acoy F O“eĿe lëledhdlr . Yh`faclj acoroj,

Jhslecljch, bhrboekl flr I acoyĿ ÷rehjch` uzoyĿ Y“dh yhr ocĿr, yoel

Bhrboekl flr I acoyĿ lële

vh bhr acoy l`joesĿz acoyĿ lëhrlr .

Bhrboekl flr I acoyĿ lële

Joyeoj= @LT ApheIaursh[orh

ŕhjlc ?. ’ –F O“yĿ khrhjtlrlr ‐

Hđhr O vh F flr flrlel lëhrlrsh, hőlttlrchr.

Wh jó`h jopso`Ŀ khëlőcldlr, yoel

FoőjoecĿj shël`l ÷rehđl`lzdh (hđhr [ljlphdlo“yo kóvhehflclrshj, @iIole ôeoòJoeocĿ juőođĿedo fucueoe flr O`hrljoe bovo óssóedh dađùő)



–44eió foőjoe ôeoòdo dađdu‐ 0 {@iIole} –44eió foőjoe yofoeiĿ flr ócjhdhdađdu‐ 0 {@iIole, Yibworzhehkkhr}

vh

{@iIole, Yibworzhehkkhr} –44eió foőjoe iu`burlyhtëldlr‐. Yaeuë acoroj, oőođĿdojl ëĿjorĿ`Ŀ yopoflclrlz=

–44eió foőjoe ôeoòdo dađdu‐ –44eió foőjoe iu`burlyhtëldlr‐

6.6.6 Jó`h Flrchől`chrl –‐

O vh F“ele flrchől`l , O vhyo F“ele (vhyo bhr ljlslele, X òetĿjsoc –vhyo‐ sh`facó acoeW“ele jorőĿcĿđĿdĿr) hchòecorĿeĿe fótóe saeuëcorĿeĿe tapco`ĿdĿr

Jó`h flr chől`l sl`htrljtlr.


ŕhjlc 6. O vh F“ele Flrchől`l ’ –O vhyo F‐

OF 0 F O

Fueo lcovhthe,

Bhrboekl O, F Y acoycorĿ lële



×zhccljch,

OyrĿio jó`h/acoy flrchől`lel boekl sĿrodo yoptĿđĿ`ĿzĿe do flr ÷eh`l yajtur (flrchől`÷zhcclđl).

6.6.9. Jó`hchrle Jhslől`l –‐

O vh F“ele jhslől`l (faő acoflclr ) O vh F“ele ljlsledh flrdhe acoe saeuëcor tapco`ĿdĿr. OőođĿdojl klfl yozĿcĿr,

Furodo –‐ òetĿjsoc –vh‐“yl lgodh hdhr. FozĿ `htlechrdh octhreotlg lőorhtch`h juccoeĿcĿr,


ŕhjlc 9. O vh F“ele Jhslől`l ’ –O vh F‐

Jó`hchrle flrchől`l klfl jhslől`l dh sl`htrljtlr.

Jhzo,

Bhrboekl O, F Y acoycorĿ lële



Furodoe borhjhtch,

OyrĿio, jó`h flrchől`ledh acduđu klfl, jó`hchrle jhslől`l flrchő`h ÷zhclđleh soblptlr.

ľcovhthe, jó`h jhslől`l vh flrchől`l dođĿcò ÷zhclđleh soblptlr.

vh

×rehj acoroj, oőođĿdojl acoycor lële

O 0 –Foőjoe 44 flr jodĿedĿr‐ 0 {Icletae}

F 0 Foőjoe 44 @ldwhst“th dađdu‐ 0 {Icletae, Vaèhy}

I 0 –Foőjoe 44 flr Iu`burlyhtëldlr‐ 0 {Buijofhh, @iIole, ôuc, Vaèhy,

Yibworzhehkkhr}

flrleil dođĿcĿ` ÷zhclđleh k÷rh acòsĿ khrhjtlđl klfl, őuecorĿ k÷rhflclrlz=

0 {Icletae}{Icletae, Buijofhh, @iIole, ôuc, Vaèhy, Yibworzhehkkhr}

0 {Icletae}

0 {Icletae} 0 {Icletae}

6.6.4. Jó`h Tó`chyhel, OI

O“eĿe tó`chyhel OI, Y“dhjl O“yo olt acòyoe saeuëcor jó`hsldlr. \oel,

OI 0




ŕhjlc 4. O“eĿe Tó`chyhel ’ –O dhđlc‐

ToeĿ`doe, tó`chyhechrle oőođĿdojl ÷zhcljchr h soblp acduđueu jacoyio k÷rhflclrslelz.

He sae lgodhdhe, oőođĿdojleh ucoőoflclrlz

ľcj ÷zhcljch fhrofhr oőođĿdojleh ucoőĿcĿr

Jhslől` vh flrchől` orosĿedojl flr krup yororcĿ lclőjl oőođĿdojl klfldlr.




ŕhjlc 8. “ele k÷sthrl`l- OyeĿ őhjlcdhe jurocĿeĿ dok÷rhflclrslelz

6.6.8 Acoycor Ŀe F÷cóetóche`hsl

O vh F“ele artoj saeuëcorĿ yajso oyrĿj tĿr cor (yo do jorőĿcĿjcĿ dĿőcoyoedĿr). \oel

O?, O6 …. acoycor krufueue flrchől`chrl Y“yh hőltsh, hjsljslz acduđu s÷ychelr.

⋉

O?, O6, … acoy krufu, hđhr oőođĿdojl ljl jaőucu sođcĿyarso ÷rehjch` uzoyĿeĿe f÷cóetósó acoroj odcoedĿrĿcĿr. Jaőuccor= (?) Ol, O m (l m) bhrboekl ljl gorjcĿ acoy acsue. Ol vh O m oyrĿőĿj vh (6) O?, O6, krufu hjsljslz lsh. OyeĿ őhjlcdh, F acoyĿeĿe f÷cóetóchrlel ,

flrchől`l F“yh hőlt acoe jorőĿcĿjcĿ dĿőcoyoe oct-acoycor acoroj toeĿ`coyoflclrlz.




ŕhjlc 7. Y“ele O?, O6, … OY“yh f÷cóetóche`hsl



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ 6

Jaerod @hezhc

8 ŕufot 6>><

?. Acoycor Ŀe AcosĿcĿđĿ

ŕl`dlyh jodor, acoycorĿe sodhih toeĿ`corĿeo vh ÷zhcljchrleh fojtĿj ’ fozĿacoycorĿe khrëhjchő`h acosĿcĿđĿ ëaj dóőójjhe (÷rehđle Yibworzhehkkhr‑le 44eiófoőjoe acoroj shëlc`hsl klfl) fozĿcorĿeĿe khrëhjchő`hsl elsphthe jhsledlr- oeioj

acoycorĿe acosĿcĿjcorĿ, yoel acoycorĿe ÷rehjch` uzoyĿeĿe jocoeĿeo k÷rh khrëhjchő`hlbtlòcl, bojjĿedo blëflr őhy s÷ych`hdlj.

Flël`shc acoroj, ^ acosĿcĿđĿ Y?‑dhjl 0 {O?, O6, …} acoycor yĿđĿeĿedoe rhhc soyĿcorokldhe flr gaejslyae acoroj toeĿ`coeĿr.

.

JuccoeĿőcĿ flr acosĿcĿj toeĿ`Ŀ yopoflc`hj lële, bhr boekl flr acosĿcĿj gaejslyaeu ^‑eleoőođĿdojl ojslya`cor Ŀ sođcoòsĿeĿ fhjchrlz=

(^?) Bhrboekl flr O lële ^( O) >

(^6) ^(Y) 0 ?, - yoel –jhslecljch flr őhy acoioj ‐

(^9) Oy rĿj O?, O6, …, jó`hchrlele bhrboekl flr dlzlsl lële

@othòtljshc flr eat acoroj, fu ojslya`corĿe (vh saerojl dhrsth ^(O)‑eĿe ÷zhccljchrlele

tórhtl`chrlele) flr oeco` lgodh hdhflc`hsl lële yĿđĿeĿ Y‑yl vh aeue hchòecorĿeĿetó`chyhechrl lch flrchől`chrlel lëhr`hj zaruedodĿr. Fu, flr ÷eihjl soygodo dlpeatto

? AcosĿcĿđĿe tutorcĿ flr toeĿ`Ŀ lële, acoycor krufu oőođĿdojl ÷zhcljchr soblp acòj zaruedodĿr

(Y?) Y (Y6) Hđhr O lsh, a zoòe aeue tó`chyhel OI

(Y9) Bhrboekl soyĿcoflclr O?, O6…. acoycorĿe flrchől`l ‑dlr, yoel O? O6 … Fu acoycor yĿđĿeĿeo Y‑ele slkò-ihflrl acoroj odcoedĿrĿcĿr. Fu dhrsle oòiĿ lële, fu ÷eh`cl dhđlcdlr, vhhcdhjl sarueue fu tór ojslya`coro uykueu acduđu khrëhđlel vhrl acoroj jofuc hdhihđlz.



oëĿjcodĿđĿ`Ŀz slkò-ihflrdlr. Fu dhrs lële, fu ÷zhccljchrl dobo gozco ózhrledh tortĿőòdoevhrlc`lő jofuc hdhihđlz.

ToeĿ` ?. Flr ÷rehjch` uzoy Ŀ Y ózhrledhe toeĿ`coeoe flr acosĿcĿj dođĿcĿ`Ŀ (^?) ’(^9)ojslya`corĿeĿ sođcoyoe ^(O) lch k÷sthrlche flr soyĿcor yĿđĿeĿdĿr.

^(?)-^(9) ojslya`corĿeĿe acoycoro flr thj acosĿcĿj otoòdĿđĿeo dljjot hdlelz. Aeue

yhrleh, fu ojslya`cor sodhih acosĿcĿđĿe eh acòsĿ khrhjtlđl jaeusuedo shzklchr ch tutorcĿ flr őhjlcdh bhrboekl flr acosĿcĿj dođĿcĿ`ĿeĿe sođcoòsĿ khrhjhe `lelù` jaőuccorĿvhrlrchr(khrëhjth fueu oőođĿdo jaetrac hdhihđlz). ^rheslpth, fu ÷zhcljchrl sođcoyoebhrboekl flr ^(.) gaejslyaeu khëhrcl flr acosĿcĿj acuőturur, gojot fueue hcdhjl roskhch

dhehyle lyl flr oëĿjcoòsĿ acup acòdĿđĿeĿ oecoòj lële ÷zhcljchrl oyrĿ oyrĿ k÷r`hjzaruedoyĿz. Fu bhr zoòe zar flr sarudur. Fu dhrs le 8. f÷có`óedh (×zhc DođĿcĿ`cor),fhccl stoedort duru`cor lële fozĿ papóchr ^(.) shël`chrlel tortĿőoiođĿz.

6.AcosĿcĿđĿe FozĿ ×zhcljchrl

ŕl`dl, ^(?)-^(9) ojslya`corĿeĿe khrëhjthe dh acosĿcĿj gaejslyaeuùzue shzklshcacoroj fhjchdlđl`lz ÷zhcljchrh soblp acòsĿeĿ sođcodĿđĿedoe h`le acòj lële yhthrcl

acduđu jaeusuedo jhedl`lzl ljeo ht`h`lz khrhjlyar. Yhzklshc acoroj gaejslyaeue őu÷zhcljchrh soblp acòsĿeĿ fhjchrlz= (?) Flr acoyĿe acò acosĿcĿđĿ lch acòò acosĿcĿđĿeĿetapco`Ŀ flr acòcĿ, (6) l`jæesĿz acoyĿe, , khrëhjchő`h acosĿcĿđĿ sĿgĿr acòcĿ, (9) Hđhr OacoyĿ F acoyĿeĿ lëhrlyarso, F acoyĿeĿe acosĿcĿđĿ ^(O)‑doe fóyój acoòòcĿ, vh (4) bhrboekl

flr acoyĿe acosĿcĿđĿ Z>,?P orocĿđĿedo yhr ocòcĿ. ŕl`dl, th`hc ojslya`corĿ juccoeoroj fu÷zhcljchrl lspotcoyocĿ`.

×ehr`h ?.

^(OI ) 0 ? ’ ^(O)

LYÔT= Tó`chyhe OI toeĿ`Ŀeo k÷rh,

furodo sae odĿ`do O OI

0 juccoeĿcòjtodĿr, yoel O vh aeue tó`chyhel oyrĿjtĿr. Fueuyheldhe dózhechdlđl`lzdh,

^(OI ) 0 ?- ^(O)

hcdh hdhrlz. ]othe k÷sthr`hyh ëocĿőtĿđĿ`Ŀz do fudur.



×ehr`h 6

^( ) 0 >

LYÔT=I 0 Y acduđu lële, ÷eihjl ÷ehr`hyl juccoeoroj

acduđueu k÷sthrhflclrlz.

×ehr`h 9. Hđhr F O lsh, a zoòe ^(F) ^(O)‑dĿr.

Dlpeat acoroj, fu juroc shzklshc k÷róe`hsleh rođ`he, flclőshc psljacakcorlesoecorĿe kóecój acosĿcĿj ùbojh`hsl lëledh fu jurocĿ sĿj sĿj fazdujcorĿeĿ jhőghttlchr 6.

ľYÔT= AcosĿcĿj ojslya`corĿeĿ juccoeoflc`hj lële, O acoyĿeĿ flrchől` vh jhslől`÷zhccljchrlel juccoeoroj f÷cóetóchrh oyĿròj yororcĿ acoiojtĿr.

Furodo, sae odĿ`do F O lsh juccoeĿc`ĿőtĿr . (^9) ojslyaùeu

juccoeoflc`hj lële F lch ele oyrĿj acduđueo dljjot ht`hj khrhjlr.

DacoyĿsĿyco, ojslya` ^(?)‑l juccoeoroj

saeuiueo ucoőĿrĿz.

×ehr`h 4= Bhrboekl, flr O acoy Ŀ lële > ^(O) .

ľYÔT= > ^(O) ojslya` (^?)‑dlr. ľjleil hőltclj lële, (^?) oyeĿ zoòedo ^( ) > ‑lsođcor .

DacoysĿyco ÷ehr`h ?‑h k÷rh

6 ×rehđle, Doelhc Jobehòe vh Oàs Tvhrsjy torogĿedoe yopĿcoe flr ëocĿőòdo flrjoë jlől Cledo‑eĿetorlg lel oőođĿdojl klfl vhrlyarcordĿ=

Cledo 9? yoőĿedodĿr, fhjordĿr, khvhzhdlr, vh ëaj zhjldlr. Ghcshgh hđltl`l ocdĿ. ×đrheil ljhe, oyĿrĿìĿcĿj vhsasyoc odocht jaeucorĿyco ëaj dhrledhe lcklchelrdl vh oyeĿ zoòedo eójchhr jorőĿtĿ k÷sthrlchrh dh jotĿcĿrdĿ.

Cledo‑eĿe flr klőh `hùru acò acosĿcĿđĿ sarucoe jlőlchr, aeue gh`lelst flr klőh `hùru acò acosĿcĿđĿsarucoe jlőlchrh k÷rh dobo dóőój dhđhrchr vhr`h hđlcl`ledhydlchr.



^(O) 0 ?- ^( ) ?

×ehr`h 8

ľYÔT= ×ehr`h 9‑th acduđu klfl Acoy O vh F‑yl f÷có`chrh oyĿroflclrlz.

OyeĿ őhjlcdh,

FuecorĿe f÷cóetó acduđu jacoyio jaetrac hdlchflclr. \oel jó`hchrle bhr flr ëlgtl oyrĿjtĿr.

DacoyĿsĿyco ojslya` (^9) juccoeĿcoroj k÷róchihđl klfl

( )vh

DacoysĿyco, , lch ‑ele f÷cóetósó acduđu lële (^9)

juccoeĿcoroj (őhjlc ? s÷z jaeusu gljrle krogljshc k÷sthrl`lel vhr`hjthdlr)

Z PYae dhejch`le yheldhe dózheche`hsl lsthehe saeuiu vhrlr.




ŕhjlc ?. ‑ele oyrĿj acoycoro f÷cóetóche`hsl

9 ×rehj= –Foslt‐ AcosĿcĿj

YaeuëcorĿe acoy khrëhjchő`hdhe ÷eih sl`htrlj acduđu, yoel flr acoyĿe acòacosĿcĿđĿeĿe dlđhrledhe gozco acòsĿ lële flr shfhfle acòdĿđĿ saecu flr ÷rehjch`

uzoyĿ`ĿzĿe acduđueu vorsoyocĿ`. Hđhr e(I) flr I acoyĿedojl saeuëcorĿe soyĿsĿeĿ lgodhhdhrsh, acosĿcĿj

^(O) =0

acoroj toeĿ`coeĿr. \oel, acosĿcĿj, O acoyĿedo yhr ocoe Y‑dhjl fótóe acosĿ saeuëcorĿearoeĿeo hőlttlr. Fu dođĿcĿ`, –foslt‐ acosĿcĿj dođcĿ`Ŀ vhyo –òetĿjsoc‐ dođĿcĿ` acorojodcoedĿrĿcĿr. ôro vhyo zor klfl bhr flr saeuiue acosĿcĿđĿ hőlt acoe oroëcor lële odlc

acdujcorĿ s÷ychelr. ŕl`dl óë ojslyaùe do sođcoeĿp sođcoeòdĿđĿeo fojocĿ`.

(^?)= ^(O) > e(.)‑ele sodhih (zoyĿg lbtlòcch dh acso) pazltlg dhđhrchr ocòsĿeĿedađrudoe saeuiudur

(^6)= ^(Y) 0

0 ?

(^9) OyrĿj ljl O vh F acoyĿ lële

ľjldhe gozco jó`h lële, orkóòecor hsos ltlforlych ÷zdhőtlr.



×rehj ?. Jusursuz flr zorĿe flr jhrh otĿcdĿđĿeĿ vorsoyocĿ`. Fu duru`do ÷rehjch` uzoyĿ Y 0 {?,6,…,7}“o hőlttlr, dacoysĿyco e(Y) 0 7. Khche soyĿeĿe jhslecljch 4“the fóyój acòacosĿcĿđĿ ehdlr1

Acoy O 0 {8, 7} acd uđu lële, e(O) 0 e({8,7}) 0 6. Furodoe ^(O) 0

Hđhr flr zor ljl jhrh otĿcĿrso, ljl rojo`Ŀe tapco`ĿeĿe 4 vhyo dobo dóőój acò acosĿcĿđĿehdlr1

Fu duru`do= Y 0 {(?, ?),(?, 6), …, (6, ?),(6, 6), …, (7, 7)} acduđu lële e(Y) 0 76 0 97.

Acoy F 0 –Vojo`corĿe tapco`Ŀ 4‐ 0 {(?, ?),(?, 6),(?, 9),(6, ?), (6, 6),(9, ?)}

DacoysĿyco ^(F) 0

Flroz saero, fhccl acoycor saeuiuedo acuőoe saeuëcorĿ soyòj lële dobo saglstljhthjeljchrl k÷rhihđlz.

4 Yoyò JuroccorĿ

ŕl`dlyh jodor fojtĿđĿ`Ŀz ÷rehjchr, sĿrosĿyco, O vh Y ―dhjl saeuëcorĿ soyòeĿejacoy acduđu elsphthe foslt ÷rehjchrdl. Hđhr Y flrëaj hchòeo soblpsh vh O yhthrleih

joròőĿj acurso, a zoòe e(O) vh e(Y)‑yl hcdh ht`hj lële fótóe saeuëcor clsthslehfojòj bh` sĿjĿiĿ acur bh` dh protlj acòz. Fu dhrsth, ja`fleosyaecorĿe vhphr`ótosyaecorĿe gorjcĿ acoycor saeuiuedo artoyo ëĿjoe soyĿcoflclr afmhchr (saeuëcor) lële

foslt juroccor vhrhe –ja`fleotarlj ‐ chrh fojĿyaruz.

×rehj 6. Óecó sotroeë ayueiusu Faffy Glsibhr (9 bogto ÷eih ÷cdó) saeuedo – jcoslj ‐sotroeiĿ ayeoòjtoe sĿjĿcĿr vh sodhih 2+2 plyaeue ocĿőĿcokhcdlđl klfl yhrchőtlrlcdlđl oeioj , lcj sĿrodo, dlđhr toőcorĿe (? őob, ? vhzlr, 6 glc, 6 ot vh 6 joch) bhr flr fhyozĿe oyeĿ

iles slyobo jorőĿcĿj khchihđl őhjlcdh roskhch yhrchőtlrlcdlđl flr voryoet ÷ehrlr. DlđhrjĿsĿtcoòcor lsh (?) flr glc slyob jorhdh ljhe dlđhrl fhyoz jorhdh acòj zaruedo, (6) őoblcj shghrdh ljl joch orosĿedoe borhjht ht`hj zaruedodĿr (raj yopòyo lzle vhr`hj lële).

Fueue orjosĿedojl dóőóeih, sotroeë ayueiucorĿeĿe sodhih stoedort foőcoò pazlsyaeu lch lyl lőchyhe stoedort ayue oëòyĿ juccoeò hđlcl`dh acdujcorĿ lële, yhelvoryoet, ayueue hzfhrche`hyl l`jæesĿzcoőtĿroioj jodor yhthrcl soyĿdo acuőturucòsĿ



duruùedo, ayueiucor Ŀ yorotĿiĿ flr őhjlcdh ayeoòyo zarcoòjtĿr. Gojot joë toeh gorjcĿùbth`hc foőcoò pazlsyaeu vordĿr1

Flz oscĿedo fu kóe dhrsth fu sĿeĿgto toeĿtĿcoe fozĿ soyò thjeljchrlel juccoeorojfu bhsopcoòcor Ŀ yopoiođĿz. Hđhr sĿjĿcdĿysoeĿz, sarueo socdĿròeĿe (thrilbhe zorlg

őhjlcdh) flr yacueu fucòyo foőcoyoflclrslelz. .

ŕl`dlclj, dađrudoe roskhch dhehychr vhyo acosĿcĿjcor bojjĿedo jaeuőòyoiođĿzgojot jaeu dĿőĿeo ëĿjoroj dobo saero dhrsth juccoeoiođĿ`Ŀz saeuëcorĿ bhsopcoò vh

soyò y÷eth`chrlel hch ocoiođĿz.

4.? Acuőturucùő Dhehychr

Juroc ? (ËorpĿ` JurocĿ)= Hđhr flr dhehy flrleilsl ` acosĿ saeuio vh ljleilsl flrleilslele

saeuiu eh acurso acsue e acosĿ saeuio soblp 6 f÷có`dhe acuőuyarso, a zoòe dhehy`xe saeuio soblptlr.

×rehj 9. Hđhr flr őlgrhele 2 jorojthrcl (borg vh rojo`) acòsĿ khrhjlyarso, a zoòe s÷zjaeusu dhehy bhr flrlsl 6x 67 + ?> 0 76 saeuiu acoe 2 porëoyo soblptlr (őlgrhele fóyójvh jóëój borgh duyorcĿ acduđu vorsoyĿ`Ŀeo k÷rh). A bocdh, tapco` acoroj 76 2 (jofoio6?2 trlcyae) jod or gorjcĿ őlgrh hcdh hdhrlz. OëĿjëo s÷ych`hj khrhjlrsh, aecorĿe tó`óeó hclch soyòyo ëocĿőòj lyl flr gljlr acòzdĿ.

×rehj 4. Flcklsoyor slsth`chrlele ëađuedo juccoeĿcoe stoedort OYILL jorojthr shtl ?6;

jorojthr lëhrlr (faőcuj borlë). BogĿzo lële bhr jorojthrh ? fyth 0 2 flt lseot hdlclr.Khë`lőthe khche shfhpchrdhe ÷tóró, ojtorò vhyo jaddojl japyocoò botocorĿeĿ thspltht`hj lële 2eil flt tutorcĿcĿj jaetracóedh –porlth‐ acoroj juccoeĿcĿrdĿ. Fuedoe ÷tóró, bhrflrlsl {>,?}“dhe acuőoe saeuio soblp ; porëocĿ flr dhehyl`lz vh fu ehdhech dh tapco` 6 ; 0 ?62 soyĿdo gorjcĿ jorojthrl`lz vor.

×rehj 8. Flr jort dhsthsledh 86 jort vordĿr, dacoyĿsĿyco hđhr òvl vh jĿr`ĿzĿ dhsthchrdheflrhr jort ëhjhrshj 86x8606;>4 acosĿ jort ja`fleosyaeu hcdh hd hrlz (hđhr ëhjlclőthesaero boekl dhsthdhe boekl jortĿe khcdlđlel flch`hzshj, a zoòe dobo oz soyĿdo oyĿrthdlchflclr saeuëcor hcdh hdhrlz). Dlđhr torogtoe, hđhr oyeĿ dhsthdhe khrl jayòdoe ljl jort

ëhjhrshj, boekl jortĿ ÷eih ëhjtlđl`lzdhe fođĿ`sĿz acoroj, ljleil jort Ŀ ëhj`hj lëledhsthdh sodhih 8? jort joc`Ŀő acur. Hcfhtth boekl 8? jortĿe jocdĿđĿ boekl lcj jortĿeëhjlcdlđleh fođcĿ acoiojtĿr , oeioj fueue ëorpĿ` jurocĿ lële ÷eh`cl acòdĿđĿeo dljjothdlc`hcldlr . Fu ehdhech, hđhr oyeĿ dhsthdhe ljl jort ëhjhrshj, 86*8? 0 6786 acosĿ ja`fleosyaeuùz acoiojtĿr.



Yae ÷rehj khehc őhjlcdh oëĿjcoòyo ëocĿőtĿđĿ`Ŀz ljl tór dhehyl k÷sthr`hjthdlr= bhr flrl

gorjcĿ soyò jurocorĿeo soblp khrl jayòcĿ ÷rehjch` lch khrl jayòsĿz ÷rehjch`.

E fóyójcóđóedhjl flr kruptoe khrl jayò lch e toehsl ëhjlc`lőtlr.

E.E….E 0 E

e

`ó`jóe saeuë.

E fóyójcóđóedhjl flr kruptoe khrl jayòdoe e toehsl ëhjlc`lőtlr (E e)

`ó`jóe saeuë

j! =0 ?.6…(j-?)j (j gojt÷ryhc acoroj ajueur) vh >! 0 ? acoroj toeĿ`coeĿr.

Khrëhjth, fu ljl soyò jurocĿ ëorpĿ` jurocĿedoe hcdh hdlclr, gojot fuecor lstotlstljth ëaj

÷eh`cl acdujcorĿ lële aecorĿ oyrĿ dhđhrchedlrdlj.

4.6 ^hr`ótosyaecor

×rehj 7. JorĿőtĿrĿc`Ŀő flr dhsth, sĿrocoe`Ŀő flr dhsthele phr`ótosyaeudur= Bhr eh

jodor sĿrocoò ëađu duru`do gorjcĿ acso do, bhr flr jort to` acoroj flr jhrh dhsthdh yhrocòjtodĿr.

ToeĿ` 6. Afmhchrle sĿrocĿ bhrboekl flr yheldhe dózheche`hsl phr`ótosyae acorojodcoedĿrĿcĿr.

^hr`ótosyae acuőturòeĿe, yhrleh jayòdoe, E óyhcl flr kruptoe E toeh ëhjlclő yopò

acduđueu eat hdlelz.

×rehj ;. ?6 tae thjelđl àdhre jcoslj `ózljth flr fhsthch`h thrtlfldlr. Fu thrtlp lëledhbhr porëo flr taeo doyoeĿr. Bhr yorĿ` taecu ÷cëhđle (jraòtlj ko`) ae ljl eatosĿ (I, I

jhsjle, D, D jhsjle, vs….F“yh jodor ) to` acoroj flr jhrh k÷rócór. Fu ehdhech, bhr flrtaeue dlzlsl yorĿ` taecu ÷cëhđle flr phr`ótosyaeudur vh prheslpth gorjcĿ acoe bhr acosĿ–`hcadlyl‐ soyoflclrlz (yojcoőĿj acoroj 4;< `lcyae).

×rehj 2. @hőbur khzkle sotĿiĿ prafch`l. Dlyhcl` jl lbtlyorl acoroj sĿrocoe`Ŀő, orodojl`hsoghsl fhccl ?8 josofodoe khë`hj zaruedo acoe flr sotĿiĿyĿ hch ocĿp vh bhrjosofodoe (he oz Ŀedoe) flr jhrh khëhe he jĿso yacu fucoiođĿ`ĿzĿ vorsoy ocĿ`. ?8“clj

e toeh



kruptoe ?8 toeh ëhj`h gar`ócó`ózó juccoeoroj ?8! saeuiueo voroflclrlz. Fu ?.9 trlcyaegorjcĿ yac dh`hjtlr. Fu joròőĿj flr prafch`dlr, fu ehdhech ë÷z`hyhihđlz.

YotĿiĿeĿe bhr flr josofodo 8 `óőthrlyl zlyorht httlđlel boyoc hdhflclrlz. Hđhr`óőthrldhe `óőthrlyh `ó`jóe acoe fótóe yaccorĿ dóőóehihj acursoj, (?8x8)!

phr`ótosyae hcdh hdhrlz (fu ëaj gozco!). Oeioj, oroőtĿrò`ĿzĿ sotĿiĿeĿe josofodo ljhe8 `óőthrlych oyeĿ oedo (toeĿ`coeoflclr flr sĿro lch) k÷róő`h őhjcledhjl yaciucuj pcoeĿ lchsĿeĿrcoedĿròj òetĿjcĿ klfl k÷róeóyar. Bhr josofodojl `óőthrlyl k÷rhflc`hsl lële 8!acosĿ sĿrocoò vordĿr vh zlyorht hdhflchihđl josofocor lëlesh ?8! acosĿ sĿrocoò vordĿr.DacoysĿyco, ëorpĿ` jurocĿeĿ juccoeoroj s÷z jaeusu lcovh sĿeĿrcoòyĿ sođcoyoioj

phr`ótosyae soyĿsĿeĿe bhsopcoyoflclrlz=

8!8!….8! ?8! 0 (8!)?8 ?8!

Fu boco ëorpĿiĿ flr őhjlcdh yójshj flr rojo`dĿr, gojot sĿeĿrcoedĿrĿcò`Ŀő (?8.8)! phr`ótosyaeuedoe jhslecljch ëaj dobo dóőójtór 9.

4.9 Ja`fleosyaecor

×rehj <. Hđhr flr thj dhsthdhe joë gorjcĿ pajhr hcl ëhjhihđl`lzl soyòj lsthrshj, yoelflr thj dhsthdhe yhrleh jayòdoe 8 jort ëhj`hj klfl, jortcorĿe ëhjlc`h sĿrosĿyco dhđlcoeioj bhr boekl flr jortĿe ëhjlclp ëhjlc`hdlđlych lcklchelrlz.

ToeĿ` 9. Bhrboekl sĿr ocoeò`Ŀő ÷đhchr tapco`Ŀeo ja`fleosyae dhelclr .

Flr ja`fleosyae flr kruptoe yhrleh jayòdoe khrëhjchőtlrlche ëhjlclőchrch acuőturucur.

Gojot ől`dl ÷đhchrle ëhjlclő sĿrosĿyco lcklche`hdlđl`lz lële, sodhih sĿrocoòsĿ gorjcĿ oò

oyeĿ ÷đhchrdhe acuőoe ëhjlclőchrl ljl jhrh soyòj lsth`lyaruz. e hchòecĿ flr yĿđĿedoe

ëhjlclő yopoflchihđl`lz e! jodor gorjcĿ sĿrocoò vordĿr (yoel e toeh hchòeĿephr`ótosyae soyĿsĿ). DacoyĿsĿyco, E afmhdhe e toeh gorjcĿ afmh ja`fleosyaeu=

9 j! j‑yh k÷rh ëaj bĿzcĿ fóyódóđó lële ëaj oz jlől gojt÷rlyhcle ÷cëhđl bojjĿedo shzklyh soblptlr. –Fóyój‐ j

dhđhrchrl lële acdujëo lyl lő k÷rhe Ytrclek‑sle tob`le

∝ âpóchr flcl`dh acdujëo yójshj rojo`corĿ jorőĿcoőtĿròj lële yoykĿe acoroj juccoeĿcoe flr bhsopcoòk÷zch`chehe jæleotĿe tapco` ota` soyĿsĿeĿ tob`le ht`hjtlr.Fu dhđhr oőođĿ yujorĿ ?> 2>

(dađrusu, furojo`Ŀ ockĿcoòj do flch kóëcó ëhjlyaru`). Gojt÷rlych ilesledh ?> ?2

8<!‑dur. ;8! soyĿsĿ jofoio 6.8 x?>

9> (ljl fuëuj `lcyae trlcyae trlcyae) ëorpĿ joleottojl ota` soyĿsĿ acoroj lgodh hdlchflclr.

Fu őhjlcdh joròőĿj bhsopcoòcordoe joëĿeòj lsthyhihđl`lz lële, sodhih gojt÷rlyhcchrle aroecorĿ lch

lcklchelrlz, fu ehdhech dh ÷eih boekl thrl`le flrflrlel k÷tórhihđlel k÷r`hj khrhjlr. ×rehđle

.

?8 toeh



.

Fu oyeĿ zoòedo flea` jotsoyĿsĿ acoroj flclelr vh khehccljch

() .

acoroj lgodh hdlclr.

Eat= Bhr eh jodor gojt÷rlyhcchrle aroecorĿeo fojĿyarsoj do, flea` jotsoyĿsĿ dolòto`soyĿdĿr (ja`fleosyae soyĿsĿeĿe oeco`cĿ acoflc`hsl lële khrhjcldlr).

×rehj ?>. âjhr lële, fu gar`ócó juccoeoroj ( ) 0 68<2<7> acosĿ hc acduđueu

bhsopcoyoflclrlz.

×rehj ??. ľőchvshc flr ëocĿőò krufu, dlyhcl` jl, 8 jlőldhe gozco acòòcĿ (fu soyĿ lële phdokamlj flr khrhjëh yajtur, sodhih òthòtljshc lőch`chrl` khrhđledhe gozcojoròőĿjcoőòsĿeĿ ÷ech`hj lstlyaru`). Fu dhrsh őu oedo 62 ÷đrheil joydĿeĿyoptĿr`ĿőtĿr. Joë toeh uykue ëocĿőò krufu `ó`jóedór (jhedl foőĿeo ëocĿőoe÷đrheilchr dæblc)1 Bhr flr ?, 6, 9, 4, 8 krup fóyójcójchrl lële ëocĿőò krupcorĿeĿe soyĿsĿeĿbhsopcoòj vh tapco`ĿeĿ ocòj zaruedoyĿz. Fu duru`do (hđhr bhrboekl flr botoyopòdĿyso`)

jodor `ó`jóe ëocĿőò krufu vordĿr.

ŕl`dl dhrsle foőĿedojl –zar prafch`‐l`lzh khrl d÷ehcl`=

×rehj ?6= Glsibhr“le Vossoc YotroeiĿeo Khrl D÷eóő= ľcj acoroj joch vh glcchr bojjĿedojl(?) vh (6) eil sĿeĿrcoòcorĿ k÷r`hzcljthe khchcl`, yoel sotroeë tobtosĿeĿe oct sĿrosĿedotoőcorĿe bhrboekl flr őhjlcdhjl yhrchől`leh lzle v hrhcl`. 2 fhyoz toőĿ (slyobto acoflclr, fu÷eh`cl dhđlc) tobtoeĿe oct sĿrosĿedojl 2 jorhyh dođĿtòj zaruedoyĿz. Dljjot hdhrshelzfu flr phr`ótosyaedur, fu ehdhech 2! jodor acosĿ sĿrocoò`Ŀz vor. Oeioj, –sođdojl‐ vh

–sacdojl‐ toőcorĿe hőlt acosĿ jurocĿedoe ÷tóró joch, vhzlr vh glcchr ëlgthrdlr. Dacoy ĿsĿyco, sĿrosĿyco ljl joch/glc/vhzlrl flrflrlych dhđlőtlrhrhj 6x6x6 acosĿ flr foőcoekĿë pazlsyaeuacuőturò yacu vordĿr. Fuedoe ÷tóró, gorjcĿ ayue soyĿsĿ

Ayue YoyĿsĿ 0 0 ;! 0 8>4>



jodordĿr.

Dobo ÷eih s÷ychdlđl`lz klfl, khrëhj juroccor Glsibhr“le Voskhch YotroeiĿeo lcovhtheőuecorĿ h`pazh hdhr= (?) flr glc slyob jorhdh lsh dlđhr l fhyoz jorhdhdlr, (6) őob ljl jochorosĿedo lcj borhjhtlel yopoflclr. Fu voryoet lële, hđhr sĿrocorĿ dacdurò sĿrocoòsĿ

jaeusuedo flroz zhjlyshj, ëorpĿ` jurocĿeĿ juccoeoflclrlz. ×eihcljch ljl glcl, thsodógheshëlche flr slyob jorh lch thsodóghe shëlche flr fhyoz jorhyh yhrchőtlr`hyl ÷ehrl yaru`.F÷ychih 4x4 acoeođĿ`Ŀz vordĿr. Yaero, ot khrl joc oe 7 jorhdhe flrleh yhrchőtlrlclr (7acoeoj). ľjl vhzlrl, khrlyh jocoe 8 jorhdhe bhrboekl flrleh yhrchőtlr lrlz. Fu flr

ja`fleosyaedur, yoel () toeh vhzlr yhrchőtlr`h yacu vordĿr. ŕob vh jochchr

lële flr sĿeĿrcoò acduđuedoe, khrlyh jocoe óë toőĿ faő acoe óë ocoeo yhrchőtl r`hj lëlebhr zoòe flr thj yac vordĿr. Tapco` acoroj, hcl`lzdh

Ayue YoyĿsĿ 0 4x4x7x?>x? 0 <7>

jodor ayeoeoioj patoeslyhc –ayue‐ vordĿr.

ToőcorĿ yhr chőtlr`h jaeusuedo jrltlj eajto, ëorpĿ` jurocĿeĿ uykucoyoflchihđl`lzdhe h`leac ojtĿr, yoel lcj toőĿ yhrchőtlr`h őhjcl`lz khrlyh jocoe toőcorĿe yhrchőtlr lc`h acoeojcorĿeĿesoyĿsĿeĿ htjlch`h`hcldlr. K÷rhflcdlđl` jodorĿyco, fu sodhi h glcchr lële ÷eh`cldlr = dóőóeóe,÷eih joch lch őobĿ vh saero glcchrl yhrchőtlr`lőlz. A zoòe (o) bhr óë toő oyeĿ rhejocoecoro (a duru`do glcchrl ?x4 0 4 soyĿdo acosĿ gorjcĿ rhejcl yhrchr h jayùő acoiojtĿj )vhyo (f) toőcordoe flrlslele dlđhr ljl toőtoe gorjcĿ rhejcl flr ocoeo (fu flzh glc lële 6x9 0 7gorjcĿ shëhehjch foő foőo fĿrojoiojtĿ ) jaeucup jaeucòdĿđĿeĿ oyĿrt ht`hj zaruedo

jocoiojtĿj. ×eih glcchrl , saero őobĿ jochchrdhe ÷eih yhrchőtlrdlđl`lz sórhih, dobo saero

eosĿc dhvo` httlđl`lz ÷eh`slz klfl k÷róeóyar.

8. FoőjoeĿe ×có` Torlbl ôrodajsu (ËocĿőò Yarusu)

\oőoyoe vhyo ÷có foőjoecor lch lcklcl ja`pca tharlchrl tlplj acoroj –sĿro dĿőĿ‐thsodógchr ózhrleh jurucur. ×rehđle, suljosth uđroyoe ljl O`hrljoe foőjoeĿ lële, yoelCleiace vh Jheehdy, flr jlől oz ëaj dljjoth dhđhr artoj eajtocorco ocojocĿ ëaj uzue flr

clsth acuőturoflclr. ×rehđle, Cleiace‑ue vurucduđu tlyotrayo klt`h`hsl jaeusuedo uyoroe

Jheehdy odĿedo flr shjrhthrl vorjhe, Jheehdy‑ele dh suljostthe ÷eih Doccos‑oklt`h`hsl jaeusuedo uyoroe Hvhcye Cleiace odĿedo flr shjrhthrl vor`Ŀő (Baő! HeozĿedoe [ljlphdlo ÷ych dlyar).

Flr dlđhr fhclr kle thsodóg lsh 9< foőjoedoe ÷c`óő acoe fozĿcorĿeĿe oyeĿ ÷có`

torlbleh soblp acòcorĿdĿr = Glcàrh vh Togt‑Ŀe ljlsl dh, 2 @ort‑to, ÷c`óőchrdlr. Mabe Odo`s

vh Tbaòs Mhgghrsae‑ue ljlsl dh 4 Th`ùz ?267 torlbledh, fođĿ`sĿzcĿj flcdlrkhslele



l`zocoeòsĿedoe to` acoroj 8> yĿc saero, ÷cdóchr. Wh Mo`hs @aerah to` acoroj 8 yĿcsaero, 4 Th`ùz ?29?‑ dh ÷cdó. Fuecor őoőĿrĿcòsĿ khrhjhe őhychr `l1

ľcklcl acoyo olt saeuëcorĿe aroecorĿeĿe acosĿcĿjcoro hőlt acduđu vorsoyĿ`Ŀ octĿedo,ől`dl ljl fhclrche`lő foőjoeĿe fhccl flr kóedh, ŕufot 7 dlyhcl`, ÷c`hslele foslt

acosĿcĿđĿeo fojocĿ`. Fu duru`do, ljl foőjoeĿe ÷có` kóechrlele 7 ŕufot‑o dhejkhc`hslele sodhih flr thj ja`fleosyaeu acduđueu fucuruz. Gojot soyòeĿe ëorpĿ`jurocĿeo k÷rh tapco` acoroj 978

6 jodor acosĿ ÷có` kóeó ja`fleosyaeu vordĿr. Fueo

k÷rh, s÷z jaeusu acoyĿe acosĿcĿđĿ sae dhrhih dóőój flr rojo` acoe ?/9786‑dlr.

Oeioj, dufch ÷có`óe patoeslyhc odoyĿ acoroj flr yĿcdo ëaj soyĿdo foőjoe vh kóehőchő`hsl vordĿr. ŕl`dl, 9< foőjoedoe he oz 6‑slele oyeĿ kóedh ÷c`hsl acoyĿ acoe O‑eĿeacosĿcĿđĿ, prheslp acoroj flr ëlgt, ljl ëlgt, óë ëlgtl vf foőjoeĿe oyeĿ ÷có` torlbleh soblp

acòsĿeĿe fótóe acosĿ ja`fleosyaecorĿe aroeĿ acoroj bhsopcoeĿr. Fu sarueu ë÷z`hele

he zorlg őhjcl lsh O OI 0 Y vh O OI 0 acduđu lële ojslya` (^6) vh ^(9)‑the ^(O) 0

^(Y) ’ ^(OI0 ? ’ ^(OI) hcdh hdhrlz. Acoy OI –9< foőjoeĿe tó`ó gorjcĿ ÷có` kóeóehsoblptlr‐ acoroj gar`óch hdlchflclr . Hđhr sodhih ljl ÷có foőjoe vorso, lcjlele ÷có`óedhe

saero ljleil foőjoeĿe gorjcĿ flr kóedh ÷c`hsl lële 974 gorjcĿ yac vordĿr. ŕl`dl bhr flr efoőjoeo gor jcĿ flr ÷có` kóeó toyle ht`h acoeojcorĿeĿe soyĿsĿeĿ fhclrch`hele yhrlehjayòdoe 978‑the e toeh ëhjlclőh thjofóc httlđlel gorjht`h`lz khrhjlr, fu ehdhech

acoeojcorĿe soyĿsĿ ‑dlr.

Foőjoecoro `ó`jóe acoe tó` gorjcĿ ÷có` kóechrlel otoò soyĿsĿ yhrleh jayoroj ëhjlclő y÷eth`leh thjofóc hdhr, dacoysĿyco fu soyĿ 978e‑dlr.

Fueo k÷rh oyeĿ ÷có` torlbcl he oz flr ëlgt foőjoeĿe acò acosĿcĿđĿ

^(O) 0 ? ’ ^(OI) 0 ?-

furodo, e 0 9< lële oőođĿ yujorĿ %2;.26‑h hőlttlr. Fu gar`ócó juccoeoroj, gorjcĿ soyĿdojl foőjoe lële dh fu acosĿcĿjcorĿ bhsopcoyoflclrlz=



Yae sotĿrdoe k÷r óchflche, acosĿcĿđĿe flrh yójshc`hslele shzklshc acoroj ehdhel

÷có` kóechrledhe dobo gozco foőjoe ÷có`chrlele khrëhjchő`hsledhe ÷tóró, patoeslyhc

gorjcĿ ÷có` kóeóeóe –jocòòsĿdĿr‐.

Fu ehdhech, fu porodajsu ë÷z`hj lële, ljl fhclrche`lő foőjoeĿe oyeĿ kóedh

÷c`hsl oscĿedo fóyój flr thsodóg ljhe (ëóejó fu acoy ëaj dóőój flr acosĿcĿđo soblptlr ),ortoe foőjoe soyĿsĿyco fhrofhr, f÷ych flr acoy lële gorjcĿ ja`fleosyae soyĿsĿ ëaj bĿzcĿortor . Foőjo flr lgodhych, bhr flr flrhyshc saeuiue khrëhjchő`h acosĿcĿđĿ ëaj dóőójjhe,

–patoeslyhc thsodóg soyĿsĿ‐ ëaj bĿzcĿ flr yójshclő k÷sthrlr, fuedoe ÷tóró fóyój flracosĿcĿjco he ozĿedo fozĿ thsodógchr khrëhjchő`hj zaruedodĿr.

Dlđhr ja`pca tharlchrlele orjosĿedojl bljoyh ùbth`hche oyeĿdĿr= ľesoecor sae

dhrhih gozco dhtoyĿ toroyoroj Jheehdy lch Cleiace orosĿedo elsphthe dobo oz soyĿdojllckleë porochccljchrl fucòyo ëocĿőĿyarcor. ľstotlstljth, fu tór oroőtĿrĿò strothmlsleh –doto`lelek‐ odĿ vhrlclr vh fu fođco`do khrëhjth –yoecĿő fucuő‐ acoroj `hydoeo khche fu

hedhr thsodógchrh dhđlelrlz. Fu thsodógchr slsthòtlj flr lclőjlele saeuiuedo dhđlc, gojotoyeĿ oedo oroőtĿroflchihđl`lz vhyo thst hdhflchihđl`lz ëaj soyĿdojl patoeslyhc lclőjlele

saeuiuedo artoyo ëĿjòjtodĿr.





FozĿ foőjo ÷rehjchrdh, flr acoyĿe acosĿcĿđĿeĿ, tó`chyhel oroiĿcĿđĿyco bhsopcoòj lőch`chrlëaj fosltchőtlrhflclr.

×rehj 6. WorsoyocĿ` jl Aihelo ócjhsl Huroslo“eĿe foőjhetleh 2 toehsl eójchhr foőcĿjtoőĿyoe ?7 gózhych socdĿrdĿ?. \leh vorsoyocĿ` jl , Huroslo ardusu ?7 gózhyl dh lzchyhflclrvh boekl gózhele jaevoeslyaehc foőcĿj toőĿdĿđĿeĿ oyĿrt hdh`hyhe oeioj bhr flrlsl khchegózhchrle ÷eóeó jhsle acoroj jhshflchihj ?6 rajhth soblp ac sue. Huoroslo“eĿe ghcojhtl÷echyh`h`hsl vh he oz flr eójchhr foőcĿjcĿ gózhele bhdhgleh ucoőò acosĿcĿđĿ ehdlr1Yhzklshc tob`lelelz eh acurdu1

Bhr duru`do, to` acoroj 4 gózh bhdhgleh ucoőoiođĿ lële, Y ÷rehjch` uzoy Ŀ ?7“do 4gózhele fótóe ja`fleosyaecorĿeĿ lëhrlr. DacoysĿyco, Y“ele hchòecorĿeĿe soyĿsĿ flea`jotsoyĿsĿ lch vhrlclr.

AcosĿcĿđĿ dhđhrchedlrhflc`hj lële, flr yojcoőĿ` tó`chyhe jurocĿeĿ juccoeòjtĿr. Acoy O 0–he oz flr eójchhr foőcĿj bhdhgl vurur‐ ue tó`chyhel OI 0 –bhdhgl vuroe fótóe gózhchrjaevoeslyaehcdlr‐ dlr vh OI “ele saeuëcorĿ 2“dhe 4 gózhele (jaevoeslyaehc acoecor) fótóeja`fleosyaecorĿ őhjcledh bhsopcoeĿr . F÷ychih

Fu ehdhech,

×ychysh fu acosĿcĿj flrh acdujëo yojĿe’ fueu fhjchylp fhjch`hdlđlelzdhe h`le dhđlcl ,gojot pacltlj acoroj dađ ru acòso do, fu ÷rehj ëaj soyĿdo acoeođĿe acduđu ja`fleosyae

prafch`chrledh shzklchrle ëaj jacoy yoeĿcoflchihđlel k÷sthr`hjthdlr.

×rehj 9. @hőbur dađuù kóeó – porodajsu‐ (flr zoòecor) papóchr flr portl ayueu lchlcklcldlr= e jlőlclj flr orjodoő krufueuz acsue, aecordoe he oz flr ëlgtle oyeĿ dađu` kóeóehsoblp acò acosĿcĿđĿ ehdlr1 (fótóe dađu` kóechrlele hőlt acosĿcĿjcĿ acduđueu vorsoyocĿ`.Fu khrëhjth, sodhih jofoio o`plrlj acoroj dađrudur . ) \leh, bhr flr e orjodoőĿeĿzĿe

? Furodojl lsl`chr Arwhcc“le raòeĿ –?<24‐ the ocĿe`ĿőtĿr, dacoyĿsĿyco fueue flr khrëhj dóeyo ÷rehđl acòsĿ

khrhj`lyar.



gorjcĿ dađu` kóeóeh soblp acduđu OI tó`chyhe acoyĿeo fojocĿ`= Fu, yhrleh jayòdoe978“the e toeh ëhjlclőh thjofóc httlđl lële, lcklcl gar`ócó juccoeoflclrlz=

F÷ychih he oz ljl orjodoőĿeĿzĿe oyeĿ dađu` kóeóeh soblp acò acosĿcĿđĿeĿ, ^(O),bhsopcoyoflclrlz=

Fu gar`óc flcbosso jacoy ajueòz, fu ehdhech ől`dl e“ele flrjoë dhđhrl lële acosĿcĿjcorĿaedocĿj acoroj oőođĿyo yozocĿ`=

Flrëaj jlől fu acosĿcĿjcorĿ ëaj yójshj fucoflclr, gojot fu khehccljch jlőlele eorjodoőĿeĿz doe bhrboekl flrlele slzlech oyeĿ dađu` kóeóeh soblp acò acosĿcĿđĿeĿ bhsopcoyoroj dóőóe`hyh foőcoò hđlcl`l yózóedhedlr. Ylz jhedlelzl, clsth`lzl e gorjcĿacduđu, acosĿcĿđĿe

acduđueo ljeo hdhrslelz. Fu gorjcĿcĿđĿe ehdhel, ÷eihjl duru`do, O“eĿe oyeĿ zoòedo e

orjodoőĿeĿz orosĿedojl fótóe hőchő`hchrl jopsoòsĿdĿr jl fu do soyĿyĿ bĿzcĿ flr őhjlcdhyujorĿ dađru ëhj`lőtlr.



6. FođĿ`sĿz Acoycor

Yhzklshc acoroj, ljl gorjcĿ acoy acoe O vh F lële O“eĿe khrëhjchő`hslele F“elekhrëhjchő`h acosĿcĿđĿeĿ –htjlch`hdlđl‐ flr jovro` toeĿ`coòj lstlyaruz. ×rehđle hđhr flr

òdhel poroyĿ ljl jhrh gĿrcotĿrsoj, ljleil otĿőĿe saeuiu bhrboekl flr őhjlcdh flr leil otĿőĿesaeuiuedoe htjlche`h`hcldlr. Eatosyaeu foslt tutòj lële fuedoe saero

^(OF) 0 ^(OF)

acoroj lgodh hdlchihjtlr.

ToeĿ` ?. O vh F acoycorĿ, hđhr oőođĿdojl jaőucu sođcorcorso fođĿ`sĿzcordĿr

^(OF) 0 ^(O)^(F)

Furodojl fođĿ`sĿzcĿđĿe, acoycorĿe glzljshc dađosĿeĿe dhđlc, sodhih acosĿcĿj dođĿcĿ`Ŀe flr

÷zhclđl acduđueu k÷rhflclrslelz. Fu yózdhe fozĿ acoycordo fođĿ`sĿzcĿj bojjĿedo lylshzklyh soblp ljhe (flr dlzl yozĿ-turo otò klfl), ëađu duru`do fu garòc jaőucu jaetrac

ht`hjthe foőjo shëhehđl`lz yajtur.

×rehj 4. Dlyhcl` jl odlc flr zorĿ ljl jhrh gĿrcottĿj, oőođĿdojl acoycorĿe vh jhslől`chrleleacosĿcĿđĿ ehdlr1

O 0 {6, 4, 7}

F 0 {?, 6, 9, 4}

Yaeuëcor Ŀ soyoroj ^(O) 0 ()()

“dlr vh oyeĿ őhjlcdh, ^(F) 0 “tór. AcoycorĿe

jhslől` acosĿcĿđĿ



F÷ychih acoycor oyeĿ otĿőtoe hc dh hdlc`hsleh rođ`he fođĿ`sĿzdĿr.

FođĿ`sĿzcĿđĿe fhclrtlche acosĿcĿj dođĿcĿ`Ŀeo eh dhecl ÷eh`cl flr őhjlcdh fođcĿ acduđueuk÷r`hj lële, ől`dl vorsoyocĿ` jl , zor òelpóch hdlcdl vh f÷ychih ^(7)09/2 , dlđhr fótóesoyĿcor lële e 0 ?, … , 8, ^(e) 0 ?/2 acsue. A zoòe, oyrĿj acoycorĿe acosĿcĿjcorĿeĿe

tapco`Ŀ ózhrleh acoe (^9) ojslyaùeo k÷rh,

vh

FođĿ`sĿzcĿđĿe flr yaruù do ő÷ychdlr= WorsoyocĿ` jl F“ele `hydoeo khcdlđlel flclyaruz,fu flckl flzl` O“ele khrëhjchő`h lbtlòcl jaeusuedojl leoeiĿ`ĿzĿ dhđlőtlrlr `l (yo dothrsl)1 Fueu saerojl f÷có`dh gar`óch hdhihđlz vh k÷rhihđlz jl hđhr O vh FfođĿ`sĿzso, F“ele acuőòsĿ flcklsledhe O acoyĿ bojjĿedo ÷đrhehihđl`lz blëflr őhyyajtur.

×ehr`h ?. Hđhr O vh F fođĿ`sĿz lsh, a zoòe O vh FI “ dh fođĿ`sĿzdĿr.

ľYÔT= O“yĿ OF vh OFI klfl ljl oyrĿj acoyo f÷chflchihđl`lz lële, oőođĿdojl őhjlcdh

fođĿ`sĿzcĿđĿ lspotcoyoflclrlz=

ŕl`dl fođĿ`sĿzcĿđĿe toeĿ`ĿeĿ ljl acoydoe dobo ÷thyh toőĿyoflclrlz.

ToeĿ` 6. Flr krup O?, O6,…. acoycorĿeĿe fođĿ`sĿz acòsĿ lële, fu acoycorĿe bhrboekl flroct krufu Ol?, Ol6 , … lële (fótóe ledhjschr gorjcĿyjhe), oőođĿdojl jaőuc sođcoeòcĿdĿr=

×rehđle, O, F, I acoycorĿ lële,

vh





lch vhrlclr.

Yhzklshc acoroj, poy, F“ele khrëhjchőtlđlele flcle`hsl duruùedo O“dojl boekl

saeuëcorĿe acosĿ acduđueu lgodh hdhr. ôydo tó` ÷rehjch` uzoyĿ lële oyeĿ őhyl yopor.


ŕhjlc 6. F“yh jaőuccoe`Ŀő acoroj O acoyĿ

OëĿjcoò ?. Jaőuc cu acosĿcĿj vh fođĿ`sĿzcĿj= Hđhr O vh F fođĿ`sĿz lsh,

×ychysh F“ele khrëhjchő`hsl flzh O bojjĿedo blëflr őhy s÷ych`hdlđledhe jaőuc cuacosĿcĿj , jaőucu acòyoe acosĿcĿđĿe oyeĿsĿdĿr.

×rehj ;. Fu ÷rehj Krhk @oejlw“le whf kóecóđóedhe (fcak)6 uyorcoe`ĿőtĿr. Letrodh klfl pcotgar`cordo, hđhr flr acoy khrëhjchőlrsh (÷rehđle, \oejhhchrle dóeyo őo`plyaeosĿeĿjozoeòsĿ) ? dacor ÷dhyhe dhđhrcl jođĿtcorĿ ocĿp sotoflclrslelz . Hđhr plyoso khrhjtlđl klfl

ëocĿőĿrso, fu tlp dhđhrcl jođĿtcorĿe fhclrcl flr t zoòeĿedojl glyotĿ, ocĿp sotoecorĿe fu tzoòeĿedojl flcklsleh fođcĿ acoe acosĿcĿj őhjcledh yaru`coeoflclr. Letrodh“dhjl pacltlj

plyosodo, oőođĿdojl acoycor lële dhđhrcl jæđĿt ocĿp sotoflclrsle=

Ol odoy odoyĿ l foőjoecĿj shël`lel jozoeĿr (odoycĿj jaőucueo fojĿcòjsĿzĿe )

Fl odoy odoyĿ l portlslele odoycĿđĿeĿ jozoeĿr

6



I j j portlslele odoyĿ shël`l jozoeĿr

ŕl`dl, lcklcl acoycorĿe dhđhrcl jođĿtcorĿeĿe glyotcorĿ juccoeĿcoroj fhclrchehe acosĿcĿjcorĿjuccoeoroj plyosoeĿe, portlele odoy k÷sthr`hsleh fođcĿ acoroj odoy odoyĿ acòsĿbocledh, ^(Ol |Fl ), bhr portlele boekl odoy odoyĿeĿe foőjoecĿj shël`lel jozoeò

acosĿcĿđĿeĿe he yójshj acduđueu dóőóedóđó sarusueu ihvopcoedĿr oflclrlz. \oel boeklodoyĿe ÷ehrlc`hsl portlyh foőjoecĿj shël`lel jozoeòdo he yójshj őoesĿ sođcordĿ.

ôrtl torogĿedoe odoy k÷sthrlc`hyhe odoyĿe shël`l jozoeò őoesĿeĿe acòdĿđĿeĿrobotcĿjco (elsphthe) vorsoyoflclrlz, fuedoe ÷tóró

F÷ychih,

F÷ychih, lcklcl dhđhrcl jođĿdĿe glyotĿeĿ sodhih gar`ócdh yhrleh jayò`Ŀz yhthrcldlr. 7

ŕufot“tojl Letrodh pacltlj plyososĿedojl dhđhrcl jođĿt glyotcorĿeo doyoeoroj, oőođĿdojlrojo`corĿ hcdh hdhrlz (sae sótuedo, @oejlw“ue JosĿ` 6>>7“dojl arlmleoc whfkóecóđóedh yhr ocoe dhđhrchrl juccoedĿ`).

^(O)“yĿ ^(O|Fl) jaőuccu acosĿcĿđĿedoe oyĿrt ht`hj lële, ^(O), O“eĿe òrmleoc acosĿcĿđĿ acoroj

do odcoedĿrĿcĿr . @ormleoc lch jaőuccu acosĿcĿj orosĿedojl lclőjl Tapco` AcosĿcĿj Joeueu lchvhrlclr.

Tharh` ?. (Tapco` AcosĿcĿj Joeueu ) WorsoyocĿ` jl F?, …, Fe ÷rehjch` uzoyĿ Y“ele flr

f÷cóetósó vh bhr l 0 ?, …, e lële ^(Fl ) 5 > acsue. A zoòe bhrboekl flr F acoyĿ lële,



ľYÔT= Jaőuccu acosĿcĿj toeĿ`Ŀedoe bhrboekl flr Fl acoyĿ lële ^(O|Fl)^(Fl) 0 ^(OFl). F?,

…, Fe ÷rehjch` uzoyĿ Y“ele f÷cóetóchrl acduđuedoe, (OF?) … (OFe), oyrĿj vh O lële

jorőĿcĿjcĿdĿr, yoel O lële f÷cóetó acuőtururcor. Fu ehdhech, oyrĿj jó`hchrle flrchől`leleacosĿcĿđĿ ózhrleh acoe ojslya` (^9)“h k÷rh

.

×rehj 2.

TĿffl vhrldh, sĿjcĿjco dobo yoőcĿ vh thirófhcl jocp ihrrobcorĿ torogĿedoe thdovl hdlchebostocor Ŀe oscĿedo dobo kheë acoecor torogĿedoe thdovl hdlchechrh k÷rh dobo yójshj o`hclyot saerosĿ ÷có` aroeĿeo soblp acdujcorĿ k÷róc`hjthdlr. Thirófhcl ihrrobcor lële %7“cĿj, dobo kheë acoecor lële sodhih %8.8“clj ÷có` aroecorĿeĿ k÷zch`chdlđl`lzlvorsoyocĿ`. Fu duru`, ihrrobcorĿe yhthehjchrlele yoő lchrchdljëh ozocdĿđĿ oeco`Ŀeo khclr`l1 @ubth`hche dhđlc ’ Furodo flr ihrrobĿe uykucoòj zaruedo acoflchihđl d÷rt ëhőlt

prashdór acduđueu vorsoyocĿ` ’ thjcl, ljlcl, óëcó vh d÷rtcó fyposs (thr`leacaml fyposshdlc`hsl khrhjhe jaraehr orthr soyĿsĿeĿ k÷sthr`hjthdlr). ^rashdóróe joròőĿjcĿđĿ vhbostocorĿe rlsjl foypos soyĿsĿyco ortor vh khehccljch –dobo bosto‐ acoe bostocor doboja`pcljh prashdórchrh lbtlyoë duyoflclrchr. WorsoyocĿ` jl, bhr flr prashdór lële, thirófhclihrrobcorĿe bostocorĿeĿe forlz flr őhjlcdh ëaj dobo dóőój ÷có` aroeĿeo soblp acdujcorĿ,gojot thirófhslz ihrrobcorĿe bosto ÷có`chrlele aroeĿeĿe khehc acoroj dobo dóőój acduđuflzh s÷ychedl. Tapco` acosĿcĿj joeueue ĿőĿđĿedo, fu ljl duru` eosĿc flr orodokhrëhjchőhflclr 1 ŕl`dl flr ÷rehđh fojocĿ` (Fu rojo`cor hcfhtth uyduròdĿr)

^rashdór

Thirófhslz Thirófhcl

×có` AroeĿ Wojo \ózdhsl ×có` AroeĿ Wojo \ózdhslThjcl Foypos % 4.> % 8>.>

% 6.>

% 68.>

ľjlcl Foypos % 7.> % 4>.> % 4.> % 68.> Óëcó Foypos % ?>.>

% <.> % 7.>

% 68.>

D÷rtcó Foypos % 6>.>

% ?.> % ?6.>

% 68.>

Tapco` % 8.8

% ?>>.>

% 7.>

% ?>>.>

Tapco` AcosĿcĿj Joeueue lgodhsl ëhrëhvhsledh, thirófhcl ihrrobcor lële khehc ÷cóàroecorĿ ^(O), Fl prashdóróeh jaőuccu ÷có` aroecorĿedoe, ^(O|Fl ) vh foz aroeĿ / vojocorĿeflrflrchrleh k÷rh aroecorĿ ^(Fl ) lch bhsopcoeoflclr.

Thirófhcl ihrrobcor bhr prashdórh jorőĿcĿj khche aroetĿsĿz flr őhjlcdh yójshj aroecordorlsjcl vojocordo k÷rhvchedlrlcdljchrl lële (vorsoyĿ`soc acoroj fu klfl duru`cordo dobogozco thirófhyh lbtlyoë acduđu lële), bhr flr thdovl jothkarlslel dobo lyl uykucodĿjcorĿbocdh, artocoò (òrmleoc dh`hj dobo dađru ) ÷có` aroecorĿeĿe thirófhslzihrrobcorĿejledhe dobo yójshj acduđueu k÷rhflclyaruz. Fu duru` sĿjcĿjco ja`pazlsyaehtjlsl acoroj oeĿcĿr.



A bocdh bhr flr acosĿcĿj tóróeóe protlj ÷eh`l ehdlr1 Hđhr foypos lële ihrrobcororosĿedoe flrlel shë`hj duruùedo acsoydĿeĿz, prashdóróe tlpl yoceĿzio sođcĿjduruùeuzo fođcĿ acòcĿydĿ, ihrrobĿe thirófhcl acup acòòsĿeo dhđlc, dacoysĿyco fuduru`do sodhih jaőuccu acosĿcĿđĿ ÷eh`sh`hclydlelz.

@ormleoc ÷có` aroecorĿ lële lyl flr juccoeĿ` ocoeĿ fucòj dobo zar .

Flr ëaj lstotlstljl oeoclzdh, khrëhjth jaőuccu ÷có` aroecorĿyco lcklchelr slelz (÷rehđle, hđhrslz thirófhele ÷có`chr ózhrledhjl htjlslych lcklchelyarsoeĿz) vh –prashdór t lpl ‐ dhđlőjhellstotlstljëlchrle odcoedĿrdĿđĿ ózhrh –jorĿőĿjcĿđo ehdhe acoe gojt÷r‐ acoroj hch ocĿeĿr.ľstotlstlj vh Hjaea`htrldhjl jcoslj sarue, sĿjcĿjco flrëaj lcklcl –jorĿőĿjcĿđo ehdhe acoegojt÷r‐ “óe k÷zch`che`h`hsldlr vh slz fu prafch`le ósthsledh khc`h yaccorĿeĿ ÷đrhehihjslelz.

OëĿjcoò 6. Dlđhr flr yojĿe lclőjlcl jovro` do hjaea`htrldh ëaj ÷eh`cl acoe jaőucu

fođĿ`sĿzcĿjtĿr. OőođĿdojl jaőucue sođcoeòsĿ duruùedo, O vh F“ele I“yh jaőuccoe`Ŀőljl fođĿ`sĿz acoy acduđu s÷ychehflclr.

OőođĿdojlchrh dljjot ëhj`hjth goydo vordĿr

Jaőucsuz fođĿ`sĿzcĿj jaőuc c u fođĿ`sĿzcĿđĿ sođcoòz

Jaőuc c u fođĿ`sĿzcĿj jaőucsuz fođĿ`sĿzcĿđĿ sođcoòz

\oel O vh F“ele fođĿ`sĿz acup acòòsĿ ilddl flr őhjlcdh ehyh jaőuccodĿđĿ`Ŀzo fođcĿdĿr.

Yaerojl prafch` shtledh flr jorőĿ- ÷rehj lëhrhe flr uykucoò acoiojtĿr.

4. Jaőuccu FođĿ`sĿzcĿj (dhrsth lőche`hdl)

FođĿ`sĿzcĿđĿe toeĿ`ĿeĿ jaőuccu acosĿcĿjcor o uykucoyoflclrlz=

ToeĿ` 4. O vh F acoycorĿeĿe I acoyĿeo fođcĿ acoroj fođĿ`sĿz acòcorĿ lële jaőuccuacosĿcĿjcorĿ oőođĿdojl őortĿ sođcoòsĿ khrhjlr=

Fu toeĿ` dobo ÷eih hch ocdĿđĿ`Ŀz jaőucsuz fođĿ`sĿzcĿj lch to` uyu`cudur. Flz sodhihjhedl`lzl yhel ÷rehjch Y“ 0 I lch sĿeĿrcĿyaruz. Jaőuccu fođĿ`sĿzcĿj dobo saero

hjaea`htrldh ëaj ÷eh`cl flr rac ayeoyoioj. Fu ehdhech ÷zhc acoroj ózhrledh durucòyĿ boj hdlyar. Thjelj acoroj, jaőuccu fođĿ`sĿzcĿđĿe jaőucsuz fođĿ`sĿzcĿđĿ sođcoòdĿđĿeĿ (yo do thrslel) eat ht`hj ÷eh`cldlr. Foőjo flr lgodh lch, ljl acoyĿe fođĿ`sĿz acup



acòòsĿ ilddl flr őhjlcdh foőjo ehyh jaőuccodĿđĿ`Ŀzo fođcĿdĿr . Fueu ÷eihjl dhrsth

fhclrt`lőtl`, ől`dl lsh foőjo flr ÷rehj vhrhrhj oëĿjcoyoiođĿ`.

×rehj <. Bhr flr saeuiue ?/7 acosĿcĿjco khrëhjchőtlđl, zor otò ÷rehđleh thjror fojocĿ`,yoel Y 0 {?, 6, 9, 4, 8, 7} .

(?) ľjl fođĿ`sĿz acoyĿ fođĿ`cĿ yopò= O 0 {?, 6, 9, 4} vh F 0 {6, 4, 7} acoycorĿeĿdóőóeóe. Dobo ÷eihjl ÷rehjth fu ljl acoyĿe fođĿ`sĿz acduđueu zothe k÷r`óőtój.

ŕl`dl acoy I 0 {9, 7} acsue. A zoòe

Oeioj,

\oel , aecorĿe jhslől`l I lch oyrĿőĿj acduđu lële, O vh F, I“yh fođcĿ acoroj fođĿ`sĿzdhđlcchrdlr .

(6) ľjl fođĿ`cĿ acoyĿ fođĿ`sĿz yopò= D 0 {6, 9, 4} vh H 0 {6, 4, 7} acsue. D lch H“elefođĿ`cĿ acduđueu jaetrac hdhflclrlz= ^(D) 0 ^(H) 0 ?/6 acduđueu k÷rhflclrlz.

Oeioj,

Gojot hđhr G 0 {9, 4, 8, 7} ózhrleh jaőuccorsoj

Bocfujl



F÷ychih G ózhrleh jaőuccoeĿeio, D vh H fođĿ`sĿz acdu.




Jaerod @hezhc

?6 ŕufot 6>><

?. Foyhs Tharh l

Yae dhrsth jaőuccu acosĿcĿđĿ oëĿjcodĿj vh Tapco` AcosĿcĿj Joeueue O acoyĿe

jaőucsuz acosĿcĿđĿeĿ ^(O) jaőucu acosĿcĿđo (^(O|Fl) fođcoyoe flr yac acduđueu k÷rdój.Jaőucu acosĿcĿjcor orosĿedojl flr dlđhr ÷eh`cl lclőjl ^(O|F) jaőuccu acosĿcĿđĿeĿ ^(F|O)

jaőucu acosĿcĿjco lclőjlchedlrhe Foyhs joeueudur , yoel jaőuc sĿrosĿeĿ thrsleh ëhvlr`h

yacudur. Fu saeuë, lstotlstlđle vh acosĿcĿđĿe flrëaj ocoeĿedo ÷eh`cl flr rac ayeor, he

÷eh`clsl dh F –vhrlsl‐el k÷zch`chrjhe‐ –dóeyo bocl‐ O“yĿ –÷đrhe`h‐ dur u`corĿedo

ayeodĿđĿ racdór.

×rehj ?. Oetlj \ueoecĿcor (fhc cl jl bheóz lstotlstlj bojjĿedo ëaj őhy flc`lyarcordĿ) bhr flrkh`l fotĿőĿedoe saero, jurtucoe fótóe dhelz odo`corĿeĿe dhelz toerĿsĿ âshldoe“o duohttljchrlel gorj hdhrchr. Fu k÷zch`dhe borhjhtch, aecorĿe khrëhjthe jurtucòsĿeĿe ehdhelduo ht`lő acòcorĿ acduđu yaruù yopĿcĿr. Fu ÷rehj khrëhjth ?7eiĿ yóz yĿc ľeklclz glcazaguGroeisl Foiae torogĿedoe kóedh`h khtlrlc`lőtlr. ľstotlstljl thrl`chrch, –jurtucoecor‐Ŀ acoy O

vh –duo ht`h‐yl acoy F acoroj toeĿ`coyocĿ`. F÷ychih, saru duo ht`hele jurtucò

acosĿcĿđĿeĿ yójshctlp yójshc`hdlđl acur, yoel (^(O|F) 5 ^(O) p lclőjlsl dađru ù, dhđlc `l1 Fótóe jurtucoe dhelzilchrle duo ht`lő acduđu k÷zch`l ^(F|O) 0 ?“h d÷eóőór. Fu flcklkhrëhjthe dh duo ht`hele jur tucò őoesĿeĿ ÷eh`cl ÷cëódh ortĿrĿp orĿtòdĿđĿ sarusueu

ihvopcoedĿròj lële yhthrcl `ldlr1 ^(F|O) lch lcklcl flcklyl juccoeoroj ^(O|F) bojjĿedojlflcklyl eosĿc ÷đrhelrlz1

Jaőuccu acosĿcĿđĿe toeĿ`doe oőođĿdojl lclőjlyl hcdh hdhrlz,

ľjleil dhejclđl yheldhe dózhechyleih

lclőjlslel hcdh hdhrlz. OőođĿdojl acoyĿ f÷cóetóchyhflchihđl`lzl dobo ÷eih k÷r`óőtój,



F÷ychih őu saeuio vorĿrĿz=

ľzchyhe tharh`dh ÷zhtchedlđl klfl fu saeuiu Y“ele bhrboekl flr f÷cóetósóeh

khehcchőtlrhflclrlz.

Tharh` ?. (Foyhs Tharh`l) Hđhr O?, O6, …, Y“ele flr f÷cóetósó lsh, ^(F) 5 >“cĿbhrboekl flr F acoyĿ lële oőođĿdojlel yozoflclrlz

^(Ol) acoy Ol“eĿe ÷eihjl acosĿcĿđĿdĿr (yoel dhehydhe ÷eihjl acosĿcĿjtĿr )

^(Ol|F) Ol“ ele saerojl acosĿcĿđĿdĿr (yoel dhehydhe vh F bojjĿedo ’Foyhs

tharl`dhe hcdh hdlcdlđl klfl- flckl hcdh hdlcdljthe saerojl acosĿcĿjtĿr )

Aptlòc joror vhr`hele lstotlstljshc tharlslele tó`ó fu foslt gljlr ózhrleh jurucudur=

Foyhs Joror Tharlsl

×rehj 6. ×eihjl, fotoe kh`ldhe jurtucoe dhelz odo`corĿ ÷rehđl lële, ^(F|O) 0 ?“l vh(jaőucsuz) dhelz odo`corĿeĿe jurtucò aroeĿeĿ ^(O) k÷zch`chyhflclyarduj. Oeioj,^(F|OI )“yl (fađucoecor orosĿedo duo hdhechrle aroeĿ) k÷zch`chyh`hdlđl`lz lële, duoht`hele jhslecljch jurtuc o őoesĿeĿ ortĿrĿp ortĿròdĿđĿ lch lcklcl saruyu ihvopcoedĿròjlële yhthrcl flcklyh soblp acòdĿđĿ`ĿzĿ do k÷rhflcl yaruz. AecorĿe do ÷có` jarjusuedoe

duo httlđlel robotcĿjco vorsoyoflclrlz (yoel ^(F|OI ) 0 ?). F÷ychih oőođĿdojl lclőjlyl hcdhhdhflclrlz,

Oetlj \ueoecĿcorĿe ùbojh`hsl, flr fojĿò, –jurtucoe sopò‐ sĿeo flr ÷rehjtlr (baő! hekhrëhjël lgodhslych)= Foyhs tharh`l flzh, hđhr sodhih jurtucoecorĿ k÷zch`chrshj,

jurtucoòyoecor bojjĿedo ëaj őhy flc`hdljëh jurtucoe oct -eógusue ehdhe jurtucduđujaeusuedo flr yorkĿyo voròyoiođĿ`ĿzĿ k÷sthrlr.

×rehj 9. Foyhs jurocĿeĿe ÷eh`cl flr uykucoòsĿ do tĿffl thstchrle eosĿc yaru`coeoiođĿ lch lcklcldlr. Flr dajtarue ëaj eobaő flr bostocĿj lële flr bostoyo thst uykucodĿđĿeĿvorsoyocĿ` vh bostoeĿe khrëhjth bosto acduđu acoyĿeo O dlyhcl`. Thst pazltlg flr saeuëvhrhflclr, jl flz fueo F acoyĿ dlyhihđlz, vhyo ehkotlg saeuë vhrhflclr, fu do FI “dlr.



BostoeĿe bostocĿjcĿ acup acòdĿđĿeĿ fhclrch`h jaeusuedo thst to` acoroj kóvhecl dhđlcdlr,gojot pazltlg thst saeuiueue acosĿcĿđĿ,

^(F|O) 0 % << , ^(F|OI ) 0 %8

Yaeuë acoroj, bostocĿđĿe elsphtthe hedhr k÷rócdóđóeó vh bostocorĿe yoőĿ, ileslyhtl vhdlđhr ÷zhc c ljchrleh fođcĿ acoroj eógusue %8“el htjlchdlđle flclyaruz. Dlyhcl` jl thst pazltlg

saeuë vhrdl. BostoeĿe khrëhjthe bostocĿjcĿ acòsĿeĿe (jaőuccu) acosĿcĿđĿ ehdlr1 Foyhs

jurocĿ oőođĿdojl lclőjlyl vhrlr

BostocĿđĿe khehc yoykĿecĿđĿ, ^(O), acdujëo dóőój acduđu lële, pazltlg thst saeuiu flchbostocĿđĿe elsphthe zoyĿg joeĿtĿ acuyar.

×rehj 4. Va`ha lch Muclht flr sórhdlr k÷róőóyarcor vh flr kóe shvklclchr kóeó (botĿrcotòjlële= fu Iuòrthsl“dlr) khclr. Va`ha Muclht“h yo `óihvbhr bhdlyh hdhflclr, M, yo do aeo

flr shrheot yopoflclr, Y. Muclht `óihvbhr lsthr. Hđhr Va`ha aeu khrëhjthe shvdlysh,aeue lsthđlel k÷zchrledhe ajuyoflclrdl. Dađrusu, Muclht aeo fu lsthđlel ljl bogto ÷eih,

O`hrljoe gutfac clk l gleoclele sae yorĿ` sootledh s÷ych`lőtl. Muclht oyeĿ zoòedoVa`ha“eue jhedlsl el boco shvlp shv`hdlđl jaeusuedo lcj jhz őópbhyh dóőhr. Fueo CacoyĿ dlyhcl`. Yphslglj acòj khrhjlrsh,

^(C) 0 >.<8

Muclht, Va`h jhedlslel shvlyarso, aeo ^(M|C) 0 >.2> acosĿcĿjco `óihvbhr vhrhihđlel vhyo

^(Y|C) 0 >.6>

acosĿcĿjco shrheot yopoiođĿeĿ do flclyar (Fu sodhih Muclht““le dóőóedóđódór, ueutòyĿejl Va`ha O`hrljoe gutfacueu do ëaj shvlyar). Hđhr Va`ha aeu ortĿj shv`lyarso,

^(Y|CI ) 0 >.2>

acosĿcĿđĿyco Muclht“le eh shvdlđl jaeusuedo flr glj rl acòyoioj yo do aeo flr shrheot

yopoiojtĿr (vhyo dobo khrëhjël flr őhjlcdh, ÷eihjl sheh Muclht“le lsthdlđl kócchrl vhrhflclr

yo do shvklclchr kóeóeó toò`he ueutoflclr). (Eat= Yhrheot yopòj Va`ha lële ëaj

utoeë vhrlil acoflclr oò oyeĿ zoòedo dobo do uiuzdur). Yaeuëto Va`ha Muclht“hshrheot yopor. Muclht aeu bh`he thrj ht`hcl `l 1 Foyhs tharh`leh k÷rh, Muclht“leVa`ha“eue hđlcl`l bojjĿedo saerojl leoeiĿ oőođĿdojl lclőjl lch vhrlclr



Fueue Muclht lële lyl acup acòyoiođĿeo jhedlslele joror vhr`hslel fhjchyhihđlz .

Khrëhj boyotto, flr ëaj lesoeĿe fu tór yorkĿcorĿ ëaj lyl dhđlcdlr vh, sae ljl ÷rehjth acduđuklfl, thstchrle kóvhelrclđlel khrhđledhe gozco ÷eh`sh`h hđlcl`ledhdlrchr. Flclőshc psljacamlclthrotóróedh, fu duru` –Th`hc-Aroe \oeĿckĿsĿ‐ acoroj flclelr. ×rehđl`lzdh –th`hc-aroe‐

htjlchehe vhyo sođcĿjcĿ lesoecorĿe, sĿrosĿyco, ^(O) vh ^(OI

) aroecorĿ lch Va`ha“eue Muclht“lshvlp, ^(C), vhyo shv`h`h, ^(CI), ÷e flcklsldlr. Hđhr fu acosĿcĿjcor ëaj gorjcĿ lsh,shzklchrdhjl ùbojh`hele yoeĿckĿsĿ ëaj gheo acoflclr.

×rehj 8. @aety Bocc porodajsu?= Flr zoòecor TW“dh flr yorĿőòiĿdoe O, F, vh I“ klflóë jopĿdoe flrlslel shë`hsl lsthelche flr prakro`Ŀ vordĿ. JopĿcorĿe flrlele orjosĿedo flr÷dóc (sae àdhc flr ëoòőĿr òjlehsl klfl) vh dlđhr ljl jopĿeĿe orjosĿedo lsh flrhr jhëlacurdu. Hđhr yorĿőòiĿ orjosĿedo ÷dóc acoe jopĿyĿ shëhrsh, ÷dóc jhedlslele acurdu. HđhrorjosĿedo jhël acoe flr jopĿyĿ oëorso, blëflr őhy jozoeòzdĿ. Ayueu flroz dobo lckleëboch khtlr`hj lële, yorĿőòiĿ lcj shël`lel yoptĿjtoe saero sueuiu bhr zoòe dlđhr ljl

jopĿdoe orjosĿedo jhël acoe flrlel oëordĿ. Fu duru`do, yorĿőòiĿeĿe lsth`hsl bocledhshëtlđl jopĿyĿ dlđhr jopocĿ jopĿyco dhđlőtlr`hsleh lzle vhrlclrdl. JopĿyĿ dhđlőtlr`hj lyl flrgljlr acur ùydu1

Khehcchőtlr`hyl fazòdoe, vorsoyocĿ` jl fhe O jopĿsĿeĿ shëtl`. ×dócóe O jopĿsĿeĿeorjosĿedo acòsĿeĿe jaőucsuz acosĿcĿđĿ ?/9“tór. Hđhr ÷dóc khrëhjthe O jopĿsĿeĿeorjosĿedo acsoydĿ, sueuiu, hőlt acosĿcĿjco bhr ljlslele orjosĿedo jhël acoe F jopĿsĿeĿ yo

do I jopĿsĿeĿ oëoiojtĿ. Hđhr lcj tob`le yoecĿő acsoydĿ, bh` yorĿőòiĿ torogĿedoeshëlc`hyhe bh` dh orjosĿedo ÷dóc ac oyoe flr toeh jopĿ joc`Ŀő acoiojtĿ. A bocdh,

Fhel` O“yĿ shë h` duruùedo, sueuiueue a hseodo I“yl oëò acosĿcĿđĿ

^(×dóc O“eĿe orjosĿedo|I oëĿcdĿ) 0 |

Dlđhr torogtoe

^(×dóc F“eĿe orjosĿedo|I oëĿcdĿ) 0 |

Fu duru`do, jopĿcorĿ dhđlőtlrhrhj ÷dócó jozoeò őoesĿ`Ŀ yójshctlrdl`.

Yhzklshc acoroj, yhel oëĿcoe jopĿ ÷dócóe O jopĿsĿeĿe orjosĿedo acò lbtlòcl jaeusuedoblë flckl lëhr`h`hjthdlr, ëóejó sueuiu aeu blëflr őhjlcdh oëòyoiojtĿr. Khrëhjth O“yĿ shë`h`lz bocledh, –O“do ÷dóc vor‐ lch –I oëĿcdĿ‐ acoycorĿ fođĿ`sĿzdĿr. Oeioj, sueuiueueF jopĿsĿeĿ oëòòsĿ ljl ehdhedhe joyeojcoeoflclrdl= (?) ÷dóc O jopĿsĿeĿe orjosĿedoydĿ,sueuiu I“yl toò`he roskhch oëtĿ, (6) ÷dóc F jopĿsĿeĿe orjosĿedoydĿ, sueuiu foőjo

? Fu jaeudojl tortĿőòyĿ őu odrhsth ajuyoflclrslelz=



shëhehđl acòdĿđĿ lële I“yl oëtĿ. DacoysĿyco, I acosĿcĿđĿeĿ artodoe jocdĿròj sodhih F“yh–ovoetom‐ sođcor.

6. ×zht F÷có` ?= AcosĿcĿj Dhrsle ljleil óelthsleh khë`hdhe ÷eih ől`dlyh jodor yoptĿjcorĿ`ĿzĿ, oőleo acdujcorĿeĿzĿvh jhedlelzl roboto blsshdhihjchrlelzl ÷zhtchyhcl`=

6.? Yoyò JuroccorĿ

E orosĿedoe e“l yhrh jayoroj ëhj`hj= E e acoeoj

E orosĿedoe e“l yhrh jayòdoe ëhj`hj=

jodor acoeoj

^hrùtosyaecor= E! acoeoj

E “dhe e ja`fleosyaeu=

()

acoeoj

6.6 AcosĿcĿjcor

FođĿ`sĿzcĿj= ^(OF) 0 ^(O)^(F)

Jaőuccu acosĿcĿj= Hđhr ^(F) 5 > lsh, ^(O|F) 0

Yodhih vh sodhih O vh F fođĿ`sĿz lsh ^(O|F) 0 ^(O),

Tapco` AcosĿcĿj Joeueu= ^(Fl) 5 > lële Y“ele F?, …, Fe f÷cóetósó oőođĿdojl

klfldlr

Foyhs Tharh`l

AcosĿcĿj ubojh`hsl lële k÷rdóđó`óz flrjoë khehc őhy dobo vor

òelpócosyaecorĿ juccoeoroj lcklchelche acoyĿ yheldhe gar`óch hdhrhj acosĿcĿjcorĿjacoy bhsopcoeoflclr flr őhyh d÷eóőtór`h (÷rehđle tó`chyhe, f÷cóetóchr vs.)

th`hc aroeĿe jaőuccu acosĿcĿđĿ òrmleoc/jaőucsuz acosĿcĿđo d÷eóőtór`h ÷eh`l

(×rehđle Foyhs Tharh`ledh acduđu klfl vhyo jocp ihrrobĿ ÷rehđledhjl ja`pazlsyae htjlsl klfl).



fozhe O vh F fođĿ`cĿ acoycorĿeĿ I“yh jaőuc coyoroj fođĿ`sĿzcoőtĿr oflc`h (vhyo

fođĿ`sĿz acoycorĿ fođĿ`cĿ yopòj).

9 Voskhch Dhđlőjhechr

ŕl`dl dhrsl`lzle ljleil fóyój jaeusu acoe roskhch dhđlőjhechr lch dhvo` hdhcl`.

×rehj 7. \ozĿ Turo otò, vhrslyae L = Turo, B, khcdlđledh ?, yozĿ, T, khcdlđledh >dhđhrlel ocoe flr R dhđlőjhel toeĿ`coyocĿ`. Fu roskhch dhehy lële ÷rehjch` uzoy Ŀ Y 0

{B, T} vh roskhch dhđlőjhele khelőclđl {>, ?}“dlr .

ToeĿ` ?. Vhhc-dhđhrcl flr roskhch R dhđlőjhel flr dhehyle saeuëcorĿeĿ rhhc soyĿcor lchhőchőtlrhe oőođĿdojl klfl bhrboekl flr gaejslyaedur.

Torlbshc acoroj, ?2>> yĿccordo roskhch dhđlőjhe gljrl khclőtlrlcdlđl zoòe òthòtljëlchr lch

dlđhr flcl` odo`corĿeĿe dóőóeihsledh –bojljl‐ rossocĿđĿe lőchvl yajtu. Aeue yhrleh, őoes,

oeoclz httlđl`lz dur uùe fótóe poro`htrhchrl bojjĿedo to` flckl soblfl acòyĿőĿ`Ŀz vh

flr dhehyle saeuiueu tob`le ht`hdh dađoeĿe joeuecorĿedoe (kóyo to` fhclrchylil)

goydocoeò jaeusuedojl yhthehjslzclđl`lzle flr saeuiu acoroj k÷rócóyardu. FuecorĿetó`óeó yopoflc`hj –Copcoih ŕhytoeĿ‐ acoroj flclelr vh óecó òthòtljël ^lhrrh Ylàe dhCopcoih oőođĿdojl klfl oëĿjcor=

]oòeĿe bhrboekl flr oeĿedo flr ojĿc dađoyo ioecĿcĿj vhrhe fótóe kóëchrlvh aeu acuőturoe fótóe vorcĿjcorĿe jorőĿcĿjcĿ duruőueu flc`lő acsoydĿ, hđhr fu ojĿc vhrlslel oeoclz ht`hj lële vhrhihj jodor hekle acsoydĿ, jæleotĿe fóyój

fhdhechrlele vh he boglg ota`corĿe borhjhtlel flr thj gar`óc lch ÷zhtchyhflcshydl=

f÷ych flr ojĿc lële blëflr őhy fhclrslz acòzdĿ vh khchihj tĿpjĿ khë`lő klflk÷zchrlele ÷eóedh acurdu.

6

OrdĿedoe khche Glzlj“thjl (÷rehđle quoetu` glzlđledhjl khrëhj fhclrslzclj) vhyo bhsoptharlsledhjl (÷rehđle K÷dhc tharh`l= Flr ojĿc jhedlsledhe dobo joròőĿj acòcĿ ëóejótob`lechrl tob`le ht`hyh ëocĿőtĿđĿ jæleotĿe flr porëosĿdĿr) khclő`hchr fu dóeyo k÷róőóeójaruòdĿ oò boco flzl` acosĿcĿj jovro`ĿeĿe th`hc octyopĿsĿeĿ acuőtur oyo dhvo`

ht`hjthdlr= htrogĿ`Ŀzdojl dóeyoeĿe rossocĿđĿ hsos ltlforlych aeue bojjĿedojl yhthrslz

flckl`lzl yoesĿtòjtĿr .

6 Copcoih, ^. (?2?4)= O ^blcasapblioc Hssoy ae ^rafoflcltlhs.



×rehj ;. Flr ÷rehj acoroj, \ozĿ Turo otòeĿe ľjleil Whrslyaeu= Copcoih“ele gljrlelk÷sthr`hj lële, ÷rehjch` uzoyĿeĿe yujorĿdo lcj yopĿcoecordoe dobo joròőĿj toeĿ`ĿeĿdóőóehflclrlz= Jcoslj `hjoeljth bhrboekl flr zoòe eajtosĿedo poroeĿe duruùeue (jotĿilsl`) to` flr oëĿjcoòsĿeĿ vhrhflclrlz (he ozĿedoe prheslpth). Wh a zoòe jcoslj

`hjoelđle joeuecorĿeĿ jucc oeoroj aeue to` y÷eóeó vh ÷zhcl jh Turo (B) vhyo \ozĿ (T) lch

saeuëcoeĿp saeuëcoeòyoiođĿeĿ tob`le hdhrlz. Dobo sphslglj acoroj, poroeĿe bovoyo

otĿcdĿđĿ oedojl `hjoelj slsth`le duruùeu ÷rehjch` uzoy Ŀ acoroj oëĿjcoyoflclrdlj. Flrslsth`le duruùeue to` oëĿjcoòsĿ ( ëaj ldhoc!) (?) jaeuù, (6) poroeĿe jótchslel e`hrjhzlele lv`hsl lch (9) y÷eó vh (4) fhccl flr t > zoòeĿedojl oëĿsoc à`hetl lch vhrlclr.FuecorĿe bhr flrlslel e óë jaardleotĿ vordĿr, yoel Y 0 ?6 . Bhr eajto s Y oëĿjëo ljlacoydoe {B, T} flrleh olttlr. TuroeĿe khc`hsl acoyĿ, B Y, lële R 0 ? vh yozĿ lële R 0 >dhđhrchrlel vhrlrshj, fu hőchőtlr`h oőođĿdojl klfl lgodh hdlche roskhch dhđlőjhedlr.

R = 0 ?6 {>, ?} ,

R= {

ŕhjlc ?= @htoclj poro otòeĿe strafasjaplj rhs`l ( Oedrhw Dovldbozy vh

Vazbhsthr Lestltuth ag Thibeacaky, Yibaac ag ^bata Orts oed Yilheihs“le`ósoodhslych. ľzle ocĿeoroj juccoeĿc`ĿőtĿr. © Oedrhw Dovldbozy, 6>>;)

^rafch` ’ehrhdhysh to` acoroj- ëaj fhclrslz acduđu lële (jelgh-hdkhd), saeuë,

foőcoekĿë duruùedojl s Y“dhjl ugoiĿj dhđlőljcljchrh jorőĿ ëaj duyor cĿdĿr (÷rehđle,dlyhcl` jl, yoedoe khëhe flr orofoeĿe yorottĿđĿ ëhjl` klfl dĿősoc htjlch`hchr bhsofo flch



jotĿcòdoe). ôroyĿ eosĿc gĿrcottĿđĿ`Ŀzo fojòdoe, lsthelche saeuiu jhsle acoroj vhrhihj

foőcoekĿë duruùeu, lv`hyl vs. jaetrac octĿed o tutòj toò`he l`jæesĿzdĿr. OyrĿio,tlplj acoroj flr slsth`l oëĿjcoyoe dlghroeslyhc dhejch`chrl jhsle dađrucujco ë÷z`hj th

l`jæesĿzdĿr. Fu ehdhech, sodhih Y“ele flr porëosĿ acò acosĿcĿjcorĿeĿ vhrhflclrlz, jl fu B

vh T saeuëcorĿeĿe acosĿcĿjcorĿeĿ hőchőtlrlr. DacoyĿsĿyco, fu duru`do –bojljl‐ flr rossoccĿđo

lbtlyoë acòzso flch, fu protljth flzl` lële eosĿc ëocĿőtĿđĿeĿ k÷sthrlr.

Fu toeĿ` roskhch dhđlőjhe vh acosĿcĿj bojjĿedo eh dóőóedóđó`óz jaeusuedo ghcshgleajtoyĿ ÷eh ëĿjorĿrjhe (fhccl jl ëaj juccoeĿőcĿ dhđlc), uykucoò oòiĿyco, prafch`le lcjoëĿjcoò őhjcleh fođcĿ jocòyĿ thrilb hdhihđlz.

OëĿjcoò ?. R khelőclđledhjl bhrboekl flr O acoyĿ lële, ÷rehjch` uzoy Y“ele acosĿcĿjgaejslyaeu oőođĿdojl lgodh oroiĿcĿđĿyco R lële flr acosĿcĿj dođĿcĿ` gaejslyaeu `hydoeo

khtlrlr.

R flël`shc acoroj ÷rehjch` uzoyĿeĿ rhhc soyĿcoro d÷eóőtórhe flr gaejslyae acòsĿeorođ`he, khehccljch aeu dhđlőjhe acoroj hch ocĿrĿz. \oel orkóòecorĿ fhclrt`hdhe aeueacosĿcĿjco lcletlcl ëhőltcl dhđhrchr –ocdĿđĿeĿ‐ s÷ychrlz. Foőjo flr lgodhych, uykucoòcorĿe ëađulële, lcklcl bhrboekl flr ÷rehjch` uzoyĿ Y vh Y“ele acosĿcĿjcorĿeĿ rhghroes k÷sthr`hdhe,

sodhih ^(R O)“l fhclrtlrlz. ×rehđle yozĿ turo ÷rehđledh ’ yujorĿdo oëĿjcoedĿđĿ klfl-Y“dhjl jaardleotcor(`htoc poroeĿe foőcoekĿë duruù, lv`h, y÷eó klfl) orosĿedojl dađrulclőjlyl, saeuëcorĿeĿ (soyĿsoc acoroj l`joesĿzdĿr) vh jaardleotcorĿe flr acosĿcĿj dođĿcĿ`ĿeĿ

oecoòyo ëocĿőòyoiođĿz, sodhih (^(R 0 ?) 0 ^(R 0 >) 0

“yl flc`h`lz yhthrcldlr.

×rehj 2. Hđhr ?> toeh `htoclj poroyĿ flrflrledhe fođĿ`sĿz acoroj otorsoj, Flr roskhch

dhđlőjhe toeĿ`coyoflclrlz , R 0 (Tapco` \ozĿ YoyĿsĿ). Fu tór roskhch dhđlőjhechrledođĿcĿ`corĿeĿ oőođĿdo dhtoycĿ flr őhjlcdh oeoclz hdhihđlz.

×rehj <. Flr shël`le saeuëcorĿ lch lcklchelyar acoflclrlz. Dlyhcl` jl ?>> `lcyae shë`he vhljl odoy vor. Bhr shë`he odoycordoe sodhih flrlsl lële ay juccoeoflclr, boekl shë`hele

boekl odoy lële ay juccoedĿđĿeo fođcĿ acoroj 6 ?>> >>> >>> jodor gorjcĿ saeuë vordĿr. ŕl`dl, Odoy O lële (vh hsos lcklchedlđl`lz shël` lële) juccoeĿcoe tapco` aycor lch lcklcl acoroj flrroskhch R dhđlőjhel toeĿ`coyoflclrlz. AycorĿe soyĿsĿeĿe bhr flr dhđhrl lële, hsos

saeuëcorco lcletlcl flr soyĿ vordĿr, yoel odoy O“eĿe fótóe saeuëcorĿ ocòsĿeĿe thj flr yacuvordĿr. Bhr flr saeuiue acosĿcĿđĿeĿ foslt acosĿcĿj ilesledhe gar`óch hdhflclrlz vh arodoe

borhjhtch fhezhr dhđhrchr ózhrledh tapcucoőtĿrò yoporoj vhrlche tapco` ay soyĿsĿeĿeacosĿcĿđĿeĿ hcdh hdhflclrlz.

OëĿjcoò 6. Voskhch acoycorĿe tó`ó, lcklchedlđl`lz klfl, soyĿsoc flr ÷zhclđh soblp dhđlcdlr(÷rehđle, hđhr acoy –yorĿe yođùr yođoioj‐ lsh, eh jodor yođoiođĿ lch lcklche`hyhflclrlz).



A zoòe roskhch dhđlőjheh lclő`h`hj khrhjlr, sodhih acoycorco dobo ÷eihjl klfl

lcklchehflc lrlz. Foőjo flr shëhehj acoroj to, acoyĿe khëhjchő`hsl bocledh ? dlđhrduru`cordo > ocoe flr roskhch dhđlőjhe toeĿ`coyoflclrlz (khchihjth fozhe –blch‐ yhfoővuroiođĿz).




Jaerod @hezhc

?< ŕufot 6>><

?. Jhsljcl Voskhch Dhđlőjhechr

ToeĿ` ?. Hđhr flr roskhch dhđlőjhe acoe R sodhih saecu soyĿdo dhđhrchr ( yo dosoyĿcoflclr saesuz) ocĿrso (x ?, x 6 , …), R “le jhsljcl flr dođĿcĿ`Ŀ vordĿr.

ToeĿ` 6. Hđhr roskhch dhđlőjhe R“le jhsljcl flr dođĿcĿ`Ŀ vorso, R“le acosĿcĿj yađuecuj

gaejslyaeu (p.d.g.) oőođĿdojl klfl toeĿ`coeĿr=

g R (x) 0 ^(R 0 x)

Hđhr { x?, x6, …} R“le ùbth`hc fótóe dhđhrchrlele jó`hsl lsh, a zoòe bhrboekl flr x

{ x?, x6, …} lële, g R (x) 0 > “dĿr. OyeĿ zoòedo,

O lële, R O“eĿe acosĿcĿđĿ,

×rehj ?. Hđhr R gĿrcottĿđĿ`Ŀz zorĿe yózóedhjl soyĿ lsh, fótóe to` soyĿcor ?, 6, …, 7 hőlt

őoescĿdĿr. Dobo khehc acoroj, jhsljcl uelgar` dođĿcĿ`ĿeĿ x ?, x 6 , …, x j soyĿcorĿeĿeózhrledhe dođĿcĿ`Ŀe p.d.g“slych toeĿ`coyoflclrlz.

g R (x) 0

Fu Y 0 {x ?, x 6 ,…, x j } ÷rehjch` uzoycĿ flr dhehyle foslt acosĿcĿđĿyco lcletlcldlr.

×rehj 6. Worsoy ocĿ` jl odlc flr òdhel poroyĿ flrflrledhe fođĿ`sĿz acoroj 8 jhrh gĿrcottĿj

vh R“l k÷zch`chehe turocor lële flr roskhch dhđlőjhe acoroj toeĿ`codĿj. A zoòe, soyòjurocorĿ`Ŀzo k÷rh e(Y) 0 6 8 0 96 vh ja`fleosyae jurocĿeĿe juccoeoroj (–j soyĿdo turo‐) 0

() hcdh hdlclr. DacoyĿsĿyco,




Fu acosĿcĿjcorĿe tapco`ĿeĿe ?“h hőlt acduđueo eat hdlelz.

]or otò ÷rehđledh, bhr thj saeuë to` acoroj roskhch dhđlőjhele flr dhđhrl lch lcletlclydl.Dlđhr yoedoe, fhő òdhel poro otò duruùedo, dlyhcl` jl, R 0 6, R 0 > lch

jorőĿcoőtĿrĿcdĿđĿedo, saeuëcor Ŀe soyĿcorĿ orosĿedo ëaj fóyój gorjcor vordĿ. Yaeuëcor Ŀe

roskhch flr dhđlőjhele khrëhjchőhe dhđhrchrl lch hőchőtlrlc`hsl, roskhch dhehyle saeuëcorĿhőlt acosĿcĿđo soblp acso flch, ëaj ëorpĿj flr dođĿcĿò yac oëoflclr.



?.?. Flea` DođĿcĿ`Ŀ

×eihjl ÷rehđl khehcchőtlr`hj lële, bhr flrlsl –foőorĿcĿ‐ vh –foőorĿsĿz‐ acoroj saeuëcoeoioj

(acosĿcĿjcorĿ hőlt acòj zaruedo acòyoe) ordĿőĿj (dlzl) e toeh fođĿ`sĿz vh fhezhrdheh`hyl k÷zch`chdlđl`lzl vh tapco` foőorĿcĿ R soyĿsĿyco lcklchedlđl`lz vorsoyocĿ`.

×rehj 9. Joclth jaetracó lële, flr órhtl` gofrljosĿedo orofo porëocor yĿđĿeĿedoe ?>>

porëocĿj flr ÷rehj ch` shëtlđl`lzl vorsoyocĿ`. Joclth jaetracóeó khëhe porëo –foőorĿcĿ‐

acoroj fhclrchelrjhe, flr vhyo ljl jrlthrh uyòyoe porëo lsh –foőorĿsĿz‐ acoiojtĿr. ×rehjch`dhe borhjhtch porëocorĿe %?“edhe gozcosĿeĿe stoedortcor o uyòdĿđĿeo leoeòj

lële lyl flr ehdhel`lzle acòdĿđĿ saeuiueo voròj lstlyaruz. Fueue lële dh tapco`lëledhjl fazuj porëocorĿe tapco` poyĿeĿe flr gaejslyaeu acoroj foőorĿsĿzcĿđĿe dođĿcĿ`Ŀyco

lcklchelyaruz.

WorsoyocĿ` jl foőorĿeĿe acosĿcĿđĿ p“yh hőlt acsue, dacoysĿyco foőorĿsĿzcĿđĿe acosĿcĿđĿ q 0 ?

’ p“dlr. Dheh`hchr vorsoyĿ`soc acoroj fođĿ`sĿz acduđu lële, bhrboekl flr vhrlcl x foőorĿvh e ’ x foőorĿsĿz dlzlslele soflt sĿrocĿ acosĿcĿđĿ oőođĿdojl klfldlr=

p x (? ’ p)e ’ x

Oeioj, sodhih foőorĿcĿ R soyĿsĿ lch lcklchedlđl`lz lële, () soyĿdo foőorĿ x“cl gorjcĿ dlzl

acduđueu bhsofo jotòj zaruedoyĿz.

^(R 0 x) 0 () p x (? ’ p)e ’ x

ToeĿ` 9. OőođĿdojl klfl flr acosĿcĿj yađuecuj gaejslyaeu acoe R roskhch dhđlőjhel e R

F(e, p) őhjcledh yozĿcoe p vh e poro`htrhcl flr flea` dođĿcĿ`cĿ acduđu s÷ychehflclr.

Dobo ÷eih bhr flr roskhch dhehy lële oyrĿ oyrĿ acoroj acosĿcĿj dođĿcĿ`Ŀ tórhttlđl`lzh, acosĿsaeuëcorĿeĿ, bhr acoyĿe saeuëcorĿeĿ vs. yozdĿđĿ`Ŀzo dljjot ht`hclslelz. Flea` dođĿcĿ`Ŀroskhch dhehychrle tó`óeh blz`ht hdhe flr àdhcdlr. Fu jothkarlyh dóőhe bhrboekl flr÷rehj lële, vhrlche flr (e, p) poro`htrh shtlele sodhih acosĿcĿjcoro fojorĿz.

×rehj 4. Flr sĿeĿg orjodoőĿeĿzdoe flroz poro japoròj lële, turo khc`h acosĿcĿđĿ pC 0

acoe yoùj flr ? òdhel poroyĿ th`le httlđlelzl dóőóeóe. @oochshg, a poro slzle dlđhr

earòc porocorĿeĿzco jorĿőtĿ vh oeioj < clroeĿe 2 clrosĿeĿ jaco òjlehsleh ottĿđĿeĿz do gorjhttlelz. Oihchych bh`he poroyĿ ?> jhrh gĿrcotĿeĿz vh tapco` acoroj 2 toehsl turo khcdl.Flclehe hsjl flr òdhel poro blchslych orjodoőĿeĿzĿ sayòyo dhvo` ht`hyh ëocĿőòj lyl



flr gljlr acur ùydu1 \ajso pG 0 acosĿcĿjcĿ sĿrodoe (odlc) ayueo fođcĿ Ŀ jocĿrdĿeĿz1 O 0

–jocoe òdhel poro yoùjtur‐ vh F 0 –?>“do 2 Turo‐ lële ^(O|F) joëtĿr1 Hđhr òdhel

poro odlc lsh,

Hđhr yoùj lsh

ŕl`dl Foyhs Tharh`le eh s÷ychyhihđleh fojocĿ`=

DacoyĿsĿyco, khrl jocoe òdhel poroeĿe khrëhjth dobo ëaj sĿrodoe flr òdhel poro acò

acosĿcĿđĿ yójshjtlr - jl tapco` ÷ych dh`lyar, ëóejó turo acosĿcĿđĿ boco

Oeioj, hcfhtth poroyĿ flrjoë jhrh dobo otòjtĿr dobo lyl flr gljlr acurdu. Hđhr dhehyl

jhygl acoroj sĿj sĿj thjrorcorsoeĿz, a zoòe he saeuedo ljl poro orosĿedojl gorjĿ lbtlyorlflr dađrucuj dhrhihslych thsplt hdhflchihj duruò khclrslelz. Flr poroethz oëòj

khrhjlrsh, fu uykucoòyĿ foslt blpathz thstl lële flr ÷rehj acoroj k÷rhflclrslelz. Dlyhcl`

jl, turoeĿe aroeĿeĿ dobo ÷eih jl klfl tutòj őortĿyco, dlđhr flr ?> dheh`hdh yleh 2 turo

hcdh httlelz (fueo acoy I dlyhcl`). A zoòe, ÷eihjl odĿ`corĿe oyeĿsĿe tojlp hdhrhj, fushghr hvvhcjl ^(F) yhrleh saerojl ^(B|F)“yh doyoeoroj jaőucu acosĿcĿj ő÷ych acur=

Dlđhr flr shëhehj acoroj, hđhr ljl shrlyl ?7 turo vh 6> dheh`h acoroj tapcucoőtĿrĿrsoj,



hcdh dhrlz.

F÷ychih, kóeihcch`hyl oyeĿ oedo vhyo gorjcĿ zoòedo yop`Ŀő acò`Ŀz gorjht`hyhihjtlr. Fu khehc acoroj orzucoeoe Foyhsyhe kóeihcch`hele ÷zhclđldlr= elbol

saeuë juccoedĿđĿ`Ŀz khehc flcklyh fođcĿdĿr, kóeihcch`h sĿrosĿeo dhđlc.

6. Yórhjcl Voskhch Dhđlőjhe

Whrlele flrëaj tóró flr ëhőlt ÷cëó`óe saeuiudur= he ozĿedoe jovro`soc acoroj, flr rhhc

soyĿ orocĿđĿedojl bhrboekl flr dhđhrl (fozhe tó`óeó) ocoflche ođĿrcĿj, uzuecuj, khclr vs.klfl. Fu dur u`do, flr jhsljcl dhđlőjhele acosĿcĿj yađuecuj gaejslyaeue toeĿ`Ŀ juccoeĿőcĿdhđlcdlr, ëóejó (o) acosĿ saeuëcorĿe `ljtorĿ soyĿcoflclr dhđlcdlr, fu ehdhech dh sodhih thj

thj soyĿcorĿe acosĿcĿjcorĿeĿe tapco`ĿeĿ ocoòyĿz vh (f) f÷cóe`hz flr fótóeóe fhclrcl flr

dhđhrlele acosĿcĿđĿ sĿgĿrdĿr. Fu yózdhe, jhsljcl duru`doe oyrĿ acoroj, fu tór roskhchdhđlőjhechr lch lcklche`hj zaruedoyĿz.

ToeĿ` 4. Hđhr R flr soyĿ dađrusuedojl flr orocĿđĿe (sĿeĿrcoedĿrĿc`Ŀő vhyo

sĿeĿrcoedĿrĿcò`Ŀő) bhrboekl flr dhđhrlel ocĿrso, roskhch dhđlőjhe R“le sórhjcl flr dođĿcĿ`Ŀ

vordĿr.

Jhsljcl roskhch dhđlőjhechr lële acosĿcĿj yađuecuj gaejslyaeu toeĿ`coòj elsphtthe dobofoslttl ëóejó sĿeĿrcĿ soyĿdo dhđhr l vordĿ. Yórhjcl flr roskhch dhđlőjhe soyĿcoflclrrojo`cordoe dobo gozco dhđhr ocĿr, fu yózdhe dh hcdh ht`hj gozcoio ëofo khrhjtlrlr.

ŕ÷ych jl= roskhch dhđlőjhele ocoflchihđl ùbth`hc dhđhrchrl flr –krup‐o jayoroj dođĿcĿ`Ŀ–jhsljchőtlrlyaruz‐, yoel ^(R 0 x) acosĿcĿđĿeo fojòj yhrleh ^(x? R x6) klfl flr

orocĿđĿe acosĿcĿđĿeo fojĿyaruz. Fueue krogljshc k÷sthrl`l blstakro`dĿr= soyĿ dađrusuózhrledh flr krup soyĿyĿ, x> 3 x? 3 … 3 xe, sofltchőtlrlyaruz vh R dhđhrlele –krup‐coro

dóő`h acosĿcĿđĿeĿ bhsopcĿyaruz. Krup flrflrlele dhvo`Ŀ ljl soyĿ orocĿđĿedĿr, yoel ^(xl-? R

xl). Aedoe saero Zx>, x?P orocĿđĿedojl dhđhrchr lële oőođĿdojl gaejslyaeu toeĿ`cĿyaruz=

Zxl-?, xl) orocĿđĿeĿe uzuecuđuyco f÷c`hj, vhrlcl flr orocĿjto kroglđle octĿedo jocoe ocoeĿe

roskhch dhđlőjhe R“le orocĿjtojl flr dhđhrl ocò acosĿcĿđĿeo hőlt acòsĿeĿ koroetlchr. \oel

őueu bhsopcoyoflclrlz=



Yhjlc ?= OyeĿ DođĿcĿ`Ŀe, sĿrosĿyco, ?> vh 9> Krupcu Blstakro`Ŀ

ŕhjlc 6= 7> Krupcu Blstakro` vh Yórhjcl \ađuecuj

Fu bheóz to` tot`le hdlil dhđlc, ëóejó fu sodhih flzh x> 3 x? 3 … 3 xe

soyĿcorĿeĿe ljl eajtosĿ orosĿeo dóőhe R“le acosĿcĿđĿeĿ bhsopcoò`Ŀzo yoror, yajso Zx m, xjP

klfl flr oct orocĿđĿe lëleh dóőhel dhđlc. Fu sarueu, x?, x6, … slcslchslel jóëócthrhj vhf÷ychih orocĿđĿ doroctoroj ë÷zhflclrlz. Flr flrleh ja`őu ljl eajto, xl-?, xl, orosĿeĿ lbtlyorl flrjóëój –dx‐ dhđhrl jodor doroctĿrsoj, R“le o vh f klfl ljl eajto orosĿeo dóő`hslel o“doe

f“yh lethkrocl acoroj vhrhe gaejslyaeueu hcdh hdhrlz. Fu cl`lt flr sórhjcl

roskhch gaejslyaeue acosĿcĿj yađuecuj gaejslyaeu acoroj odcoedĿrĿcĿr.





OëĿjcoò ?. Hđhr R sórhjcl flr dođĿcĿò soblp lsh, bhrboekl flr x lële

^(R 0 x) 0 >

Fu, jĿs`l acoroj shzklchrh oyjĿrĿ klfl k÷róehflclr ëóejó flz khrëhjth jhsljcl acoe őhychrl

(khclr, lőslzclj sórhsl klfl) tob`le ht`hj lële sórhjcl dođĿcĿ`Ŀ juccoeĿyaruz. ŕl`dlyh jodor,bhsopcodĿđĿ`Ŀz bhrboekl flr acosĿcĿj lële bhrboekl flr sórhjcl roskhch dhđlőjhe ÷rehđlk÷r`hdlj.

9. ×rehjchr

Flr roskhch dhđlőjhele, rhhc hjshe ózhrledhjl fozĿ Zo, fP orocĿjcorĿedo yhr ocdĿđĿeĿ

vorsoyocĿ`, R“le fozĿ Zo“,f“P (furodo o o“ f“ f) oct krupcorĿeo olt acò acosĿcĿđĿ, octorocĿđĿe uzuecuđu lch aroetĿcĿdĿr. .

ToeĿ` 7. Hđhr oőođĿdojl klfl flr acosĿcĿj yađuecuj gaejslyaeueo soblp lsh, R roskhchdhđlőjhel Zo, fP, o 3 f, orocĿđĿedo uelgar` dođĿcĿr.

g R (x) 0

Yh`faclj acoroj ő÷ych yozorĿz=


ŕhjlc 4. Xelgar` Voskhch Dhđlőjhe lële p.d.g., R Zo, fP



×rehđle, hđhr R XZ>, ?>P lsh, a zoòe

^(9 R 4) ehdlr1 AcosĿcĿj ^(R 0 9) 0 > 0 ^(R 0 4) acduđu lële, fu ^(9 3 R 3 4)“óeoyeĿsĿdĿr.

×rehj 8. WorsoyocĿ` jl R“le p.d.g.“sl ő÷ychdlr=

g R (x) 0 {o eh acòj zaruedodĿr1 ^( R ) 0 ? acduđu lële, yađuecuj ?“h hethkrh acòcĿ, f÷ychih

o

acòj zaruedodĿr. F÷ychih o 0 .

^(? 3 R 3 6) ehdlr1 OőođĿdojl lethkrocl bhsopcoyocĿ`

^(? 3 R) ehdlr1




Jaerod @hezhc

64 ŕufot 6>><

?. ×rehjchr

Flr roskhch dhđlőjhele, rhhc hjshe ózhrledhjl fozĿ Zo, fP orocĿjcorĿedo yhr ocdĿđĿeĿvorsoyocĿ`, R“le fozĿ Zo“,f“P (furodo o o“ f“ f) oct krupcorĿeo olt acò acosĿcĿđĿ, octorocĿđĿe uzuecuđu lch aroetĿcĿdĿr. .

ToeĿ` ?. Hđhr oőođĿdojl klfl flr acosĿcĿj yađuecuj gaejslyaeueo soblp lsh, R roskhch

dhđlőjhel Zo, fP, o 3 f, orocĿđĿedo uelgar` dođĿcĿr.

g R (x) 0 Yh`faclj acoroj ő÷ych yozorĿz=


ŕhjlc 4. Xelgar` Voskhch Dhđlőjhe lële p.d.g., R Zo, fP

×rehđle, hđhr R XZ>, ?>P lsh, a zoòe



^(9 R 4) ehdlr1 AcosĿcĿj ^(R 0 9) 0 > 0 ^(R 0 4) acduđu lële, fu ^(9 3 R 3 4)“óeoyeĿsĿdĿr.

×rehj ?. WorsoyocĿ` jl R“le p.d.g.“sl ő÷ychdlr=

g R (x) 0 {

o eh acòj zaruedodĿr1 ^(R ) 0 ? acduđu lële, yađuecuj ?“h hethkrh acòcĿ, f÷ychiho

acòj zaruedodĿr. F÷ychih o 0.

^(? 3 R 3 6) ehdlr1 OőođĿdojl lethkrocl bhsopcoyocĿ`

^(? 3 R) ehdlr1

?.?. JorĿőĿj Voskhch Dhđlőjhechr/DođĿcĿ`cor

Hsos ltlforlych ljl gorjcĿ shfhpthe ÷tóró flrëaj khrëhj-dóeyo vhrlsl fozĿ dhđhrchr lële

eajto htrogĿedo yĿđĿcò k÷sthrlr= fozĿ saeuëcor `hjoelj acoroj fozĿ dhđhrchr lch sĿeĿrcoedĿrĿc`ĿőtĿr, f÷ychih flr ëaj

acosĿcĿj yĿđĿeĿ roskhch dhđlőjhele khelőclđlele to` j÷őhchrledh flrljlr, ÷rehđlekóecój yođĿő `ljtorĿ bhrboekl flr rhhc pazltlg dhđhr ocĿr, gojot yođĿő `ljtorĿeĿe sĿgĿracduđu flr ëaj kóe vordĿr.



hjaea`lj jororcor ocoe jlőlchr jhedlchrlel jĿrĿckoecĿjcor vh jhsletlchrh k÷rhjaeu`coedĿroroj fhccl juru`soc jurocoro thpjl k÷sthrlrchr. ×rehđle sasyockóvheclj juru`corĿeo vh khclrchr khehc `ódórcóđóeh ropar hdlche khclrchr hfojtĿđĿ`Ŀzdo, vhrkl orocĿjcorĿeĿe óst sĿeĿrcorĿedo –yĿđĿcò‐ acduđueu k÷róróz (s÷zjaeusu jlőlchr lële, khclrdhjl he ugoj ortĿő vhrkl aroeĿedo fóyój ortĿő oeco`Ŀeokhclr).

OëĿj jaeuőòj khrhjlrsh, lcklcl dođĿcĿ`cor sórhjcl dhđlcdlr, ëóejó khrëhjchő`h bhr ehjodor rhhc-dhđhrcl flr soyĿ acso do, dobo ÷eihjl f÷có`dh yoptĿđĿ`Ŀz klfl flr acosĿcĿjyađuecuj gaejslyaeueu toeĿ`coyoòyĿz. Oeioj eajto htrogĿedo yĿđĿcò lch oyrĿiolcklchehihđlz. Fueue flr jĿs`Ŀ hjaea`htrl dhrslelzdh jorőĿeĿzo ëĿjoioj, fu ehdhechfueue ózhrledh ől`dl ëaj gozco duròyoiođĿz vh sodhih flr ÷rehđh fojoiođĿz.

×rehj 6. ľzchyhe kroglj ?<;< yĿcĿeo olt Iurrhet âpucotlae Yurvhy (I^Y) vhrlsljuccoeĿcoroj acuőturucùőtur ?.

\ozor, kroglj lële khclrl ëaj dóőój flr oct krup shë`lőtlr, f÷ychih oskorl óirhtle

sĿeĿrcoedĿròsĿeĿe ojsleh ÷rehjch` elsphthe dobo fóyój acùőtur . ?<;< yĿcĿeĿe oskorlóirht dhđhrlele sacuedo jocoe fozĿ jlőlchr vor . Fu, ùbth`hche jĿs`Ŀ acoroj oskorl óirht

joeueu jopso`Ŀ dĿőĿedo jocoe shjt÷rchrl yoesĿtòjtodĿr (÷rehđle, ëlgtëlclj, kheë lőëlchrklfl).


ŕhjlc 6. ?<;<“do Clsh“dhe Thrj JodĿecorĿe Cakorlt`lj Óirhtchrl

? DlEorda, M., E. Gartle vh T. Ch`lhux. –Cofar @orjht Lestltutlaes oed tbh Dlstrlfutlae ag [okhs, ?<;9-?<<6= O Yh`lporo`htrli Oppraoib.‐ Hiaea`htrlio 74, ea. 8 (?<<7)= ?>>?- ?>44“thjl ŕhjlc 9f.



6. Flrljl`cl DođĿcĿ` Gaejslyaeu (i.d.g)

ToeĿ` 6. Flr roskhch dhđlőjhe R“le Flrljl`cl DođĿcĿ` Gaejslyaeu (i.d.g.) G R bhr flr rhhc

soyĿ lële oőođĿdojl klfl toeĿ`coeĿr.

G R (x) 0 ^(R x)

Dljjot hdlchihj acurso, fu toeĿ` jhsljcl, sórhjcl vh jorĿőĿj roskhch dhđlőjhechr lëleoyeĿdĿr. ×zhcljch, R“le jhsljcl acòsĿeo acoeoj vhrdlđl`lzdhe vh ^(R x)“le ^(R 3 x)“ dhe gorjcĿ acduđueu ojĿcdo tutòj jaőucuyco, lcletlcl acoycorĿ flrflrledhe oyĿrt ht`hj ÷eh`clacòjtodĿr. I.d.g.“ele toeĿ`Ŀedo, R“l bhr zoòe x“the –jóëój vhyo hőlttlr‐ őhjcledhjuccoeoiođĿz.

I.d.g. flr acosĿcĿj acduđu lële, acosĿcĿj gaejslyaeueue fótóe ÷zhcljchrlel lëledh forĿedĿrĿr.×zhcljch,

×zhclj ?. I.d.g sodhih > lch ? orosĿedojl dhđhrchr l ocĿr.

Fótóe x lële > GR(x) ?

OyrĿio, x? 3 x6 lële R x? acoyĿ R x6“ele lëledh yhr ocdĿđĿedoe ×zhclj 6“yl hcdh hdhrlz.

×zhclj 6. G x x“le ozocòyoe flr dhđhrldlr, yoel

x? 3 x6 lële G x (x ? ) G x (x 6 )

Hđhr x lsh, acosĿcĿđĿe khrëhjchő`hsl fojĿ`Ŀedo (R x) acoyĿ l`jæesĿz acoyo

–yojĿe‐ acur (furodo fueue eh oecoò khcdlđl jaeusuedo h`le dhđlcl`), bocfujl hđhr x lsh (R x) acoyĿ ehrhdhysh jhsledlr vh f÷ychih ×zhclj 9 hcdh hdlclr.

×zhclj 9.

Furodo őueo dljjot ht`hj khrhjlyar, hđhr sac cl`ltl

acoroj vh sođ cl`ltl dh



acoroj toeĿ`corsoj, I.d.g. bhr zoòe sórhjcl acòj zaruedo dhđlcdlr.

x “th sórhjcl acoflc`hj lële, G(x), G(x-) 0 G(x+)“l sođcoòj zaruedodĿr. OőođĿdojl ÷rehjthk÷rócdóđó klfl, khehc acoroj fu dađru dhđlcdlr.

×rehj 9. Voskhch dhđlőjhe R“le zor otò soyĿsĿ lch lclőjlchedlrlcdlđl, zor otò ÷rehđlel

thjror dóőóeóe. A duru`do R“le i.d.g.“sl oőođĿdojl klfl vhrlclr.

Furodo ?, 6, …, 7 soyĿcorĿedo sórhjslz otcoòcor vordĿr.

Joyeoj= @LT ApheIaursh[orh ŕhjlc 9= ]or otòeĿe i.d.g.“sl

Oeioj, rhhc oeoclzchrle flr saeuiu acoroj, flr àeatae gaejslyae (furodo ÷zhcljch i.d.gG R ) sodhih soyĿcoflclr sórhjslz flrëaj eajtoyo soblp acoflclr.

Dobo do lchr klt`hj khrhjlrsh, bhr zoòe ×zhclj 4“h soblflz.

×zhclj 4. Bhrboekl flr i.d.g. sođ -sórhjcldlr, yoel

G(x) 0 G(x +

)

I.d.g“ele dobo gozco ÷zhclđlel k÷sthrhflc`hj lële ől`dl acosĿcĿjco lcklcl flckl`lzljuccoeoflclrlz.

×ehr`h ?. Whrlche bhrboekl flr x lële,



^(R 5 x) 0 ? - G R (x)

ľYÔT= AcosĿcĿđĿe ÷zhcljchrledhe borhjhtch,

OyeĿ őhjlcdh,

×ehr`h 6. ľjl rhhc soyĿ lële x ? 3 x 6

×ehr`h 9. Bhrboekl flr x lële,

×ehr`h 4. Bhrboekl flr x lële,

Fu sae saeuëcor sórhjcl dhđlőjhechr lële fótóe x dhđhrchrl lële ÷zhcljch (^(R 0 x) 0 >oeco`Ŀeo khclr.

×rehj 4. OőođĿdojl krogljth K x (x) gaejslyaeue flr i.d.g. acup acòdĿđĿeĿ jaetrac hdhcl`.


Gaejslyae > lch ? orosĿedodĿr, àeataelj ort ojtodĿr vh sođ -sórhjcldlr. ŕl`dl sae 4÷ehr`hyl fu ÷rehđh uykucoyocĿ`(sodhih krogljthe dađrudoe rojo`cor hcdh hdlchihjtlr)=



×rehj 8. ^.d.g vh i.d.g“yl flr flrleh fođcoyoe thj -dađru gar`ócó`ózóe acduđu sórhjclroskhch dhđlőjhe duruùedoe gorjcĿ acoroj, jhsljcl duru`do dobo yhel tortĿőtĿđĿ`Ŀzi.d.g “dhe hcdh hdlche acosĿcĿj saeuëcorĿeĿ juccoeòj zaruedoyĿz. ŕl`dl lclőjlyh foőjo flrkroglj ÷rehđlych fojocĿ`=


ŕhjlc 4= Jhsljcl flr roskhch dhđlőjhe lële i.d.g vh p.d.g

6.? Yórhjcl Voskhch Dhđlőjhechr lële p.d.g vh i.d.g

Hđhr R p.d.g g(x)“cl vh G(x)“cl sórhjcl flr dođĿcĿò soblp lsh (jorĿőĿjcĿj acòdĿđĿ sórhihfuedoe saero R lële octsl`khyl juccoeòyoiođĿ`), a zoòe

Jocjócósóe th`hc tharh`ledhe borhjhtch, i.d.g vh p.d.g. orosĿedojl lclőjlyl fu duru`dooőođĿdojl klfl yozoflclrlz=



×rehj 7. OőođĿdojl klfl flr gaejslyaeuùz acsue,

G(x) flr i.d.g `l1 - ŕl`dl th`hc ÷zhcljchrl jaetrac hdhcl`=

p.d.g. g(x) ehdlr1

g(x) flr p.d.g. `l1 -Dađrusu, flz zothe th`hcdh G(R)“l e flr i.d.g. acduđueu k÷sthr`lőtlj. OőođĿdojl lgodhchrl bh`he k÷rhflclrlz

fótóe x dhđhrchrl lële, g(x) >

vh oyeĿ zoòedo,

×rehj ;. Hđhr R XZ>, ?P lsh, a zoòe i.d.g. oőođĿdojl klfldlr=




ŕhjlc 8= R XZo, fP lële p.d.g. vh i.d.g

9. R, \ klfl 6 Voskhch Dhđlőjhele Flrchőlj DođĿcĿ`Ŀ

Flrëaj duru`do, sodhih flr thj roskhch dhđlőjhe lch dhđlc, flr vhyo dobo gozco dhđlőjheorosĿedojl lclőjlych lcklchelrlz, `hshco flr sórhile saeuiueue flr dlđhrlele saeuiueuhtjlchylp htjlch`hdlđl klfl.

×rehđle oőođĿdojl lclőjlchrh fojoflclrlz=

Thj yuùrto ljlzchrle L_“só - yoel R ëaiujcordoe flrlslele, \ lsh dlđhrlele L_“sóacoflclr.

Hđltl`h jotĿcĿ` R lch khclr \= Hđltl` lch khclrle dođĿcĿ`Ŀeo oyrĿ oyrĿ fojoflchihđl`lzklfl, flr vhrl tofoeĿedoe hcdh hdlche vhrlchrl juccoeoroj ljl dhđlőjhel flr krogljth dh

k÷sthrhflclrlz. Dađrusu, krogljth ljl dhđlőjhe orosĿedo őópbhslz flr lclőjl vor`Ŀőklfl k÷róeóyar.




ŕhjlc 7. Ajuccoőò vh Khclr

Vhcops (eójs) sórhsl= Joeshrl o`hclyot lch yaj ht`hj bhr zoòe `ó`jóeacòdĿđĿ lële, tĿffl prashdórchrle htjleclđlel dhđhrchedlr`hj lsthyhflclrlz. Fueu (o)yhel flr o`hclyotĿe khrhjcl acduđueue oecoőĿcòsĿ lële eh jodor zoòe khëtlđleh(R) vh (f) eh jodor sórh saero bostoeĿe ÷cdóđóeh (\) fojoroj yoporĿz. Bhr ehjodor ljl saeuëco lcklcheshj dh, bhr ljl acoy do fođĿ`sĿzĿdĿr= hđhr yhel flro`hclyottoe ÷eih bosto ÷córsh, aeue ÷c`h`hsl duruùedo eh zoòe yheldheo`hclyot acòj zaruedo jocoiođĿeĿ k÷zch`chyh`hylz.

Dhrsle fu f÷có`óedh, ljl (vhyo dobo gozco) roskhch dhđlőjhele, orocorĿedojl lclőjl doblc,

oyeĿ oedo ÷zhcljchrlel hch ocoiođĿz. OyeĿ zoòedo –fođĿ`sĿzcĿj‐ vh –jaőuccu acosĿcĿj‐acoycorĿ lch fhezhőhe jovro`corĿ do toeĿtoiođĿz. (R, \) (artojcoőo) oyeĿ dhđhrchrl (x, y)ocoe ljl roskhch dhđlőjhe acsue. Bhr ljl dhđlőjhe sór hjcl, jhsljcl vhyo jorĿőĿj acoflclr .

9.?. Jhsljcl Voskhch Dhđlőjhe

Jhsljcl duru`do, bhrboekl flr (R, y) 6 lële flrchőlj p.d.g. oőođĿdojl klfldlr=

Hđhr {(x?, y6), …, (xe, ye)} (R,\)“ele `ó`jóe acoe fótóe dhđhrchrlel lëhrlrsh, a zoòe

Bhrboekl flr oct jó`h O 6 lële,



×rehj 2. Flr sóphròrjhtth, R sĿrodoe flr joso sĿrosĿedo fucueoe `óőthrlchrle soyĿsĿ,\“dh hjsprhs josodo fucueoe `óőthrlchrle soyĿsĿ acsue. Fu duru`do R lch \“ele artoj

p.d.g“sl oőođĿdojl klfl k÷róehflclr= Fu yopĿdojl flr tofca, (R, \)“ele flrchőlj p.d.g“chrledhehcdh hdlche bóirh-acosĿcĿjcorĿeĿ ÷zhtchyhe vh òrmleoc acosĿcĿjcorĿeĿ yoedo k÷sthrhe,lbtlòc tofcasu acoroj odcoedĿrĿcĿr. Dobo ÷eihdh tortĿőĿcdĿđĿ klfl, tofcadojl acosĿcĿjcorĿetapco`Ŀ ? acòcĿ vh elthjl` ÷ychchr.

Tofca dhđhrchrledhe, ljl dhđlőjhe orosĿedo flr ëhőlt lclőjl vor`Ŀő klfl k÷róedóđóeók÷rhflclyaruz= YĿrodoe josodo ÷dh`h yopoe jlől soyĿsĿ yójshj acduđu zoòe, hjsprhsjosodojl jlől soyĿsĿ do yójshj acò hđlcl`dhdlr .

Tofcado vhrlche p.d.g. chrh doyoeoroj gorjcĿ acoycor lële dh acosĿcĿjcor bhsopcoyoflclrlz=



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ ;

Jaerod @hezhc

67 ŕufot 6>><

?. R,\ klfl 6 Voskhch Dhđlőjhele Flrchőlj DođĿcĿ`Ŀ

(dhvo`Ŀ)

R vh \ oyeĿ Y ÷rehjch` uzoyĿedo toeĿ`coe`Ŀő ljl sórhjcl roskhch dhđlőjhedlr. (R,\)“eleflchőlj p.d.g.“sl, g R\ (x,y), (x,y) dózch`lele bhrboekl flr oct jó`hsl acoe O lële oőođĿdojl

lgodhyl sođcoyoe flr gaejslyaedur.

Thj-dhđlőjhecl duru`do acduđu klfl, fu yađuecuj oőođĿdojl jaőuccorĿ sođcoòjzaruedodĿr=

bhr flr ( x, y ) 6 lële g R\ (x,y) >

vh

F÷ychih,

bhrboekl flr eajto sĿgĿr acosĿcĿjcĿdĿr

dózch`dhjl bhrboekl flr thj-fayutcu hđrl sĿgĿr acosĿcĿjcĿdĿr

×rehj ?. [ya`lek ózhrledh roskhch flr yhrdh flr XGA k÷rócór. DóeyoeĿe hđl`l elk÷r`hzcljthe khclrshj, [ya`lek khrëhjël flr őhjlcdh 6;7 x 9;8 `lc“clj flr dljd÷rtkheacoroj torlg hdlchflclr. XGA“eue jaeuù fótóe hyocht ózhrledh uelgar` acoroj dođĿc`ĿőtĿr vh roskhch fayco` dhđhrl R (???-?>4 dhrhih orosĿ) vh roskhch hech` dhđhrl \ ( 4?-48orosĿ) acoroj lgodh hdlcsle.

Fu, jaardleotcorĿe flrchőlj yađuecuđueue oőođĿdojl lgodh lch vhrlcdlđl oeco`Ŀeo khclr.



Hđhr flr XGA 4> `lc“clj flr uzojcĿjtoe k÷róchflclrsh, hyochtle oőođĿ yujorĿ to` artosĿedoyhr ocoe Iosphr“doe k÷róc`h acosĿcĿđĿ ehdlr 1

ŕl`dl sarueo kroglj yordĿ`Ŀyco lch fojocĿ`= Iosphr“doe XGA“eue k÷róchflchihđl yhrchrlejó`hsl, Iosphr htrogĿedo 4> @lc yorĿëopcĿ flr dolrh lch lgodh hdlchflclr . OyeĿ zoòedo,uelgar` yađuecuđo k÷rh, XGA“eue hyochtle O yhrledhe k÷róchflc`h acosĿcĿđĿ (yoel Oózhrledhe soflt yađuecuđue lethkrocl) O“eĿe jopcodĿđĿ ocoe lch aroetĿcĿdĿr. DacoysĿyco ,bhrboekl flr lethkroc ocòj zaruedo dhđlclz, ëóejó acosĿcĿđĿ fucòj toò`he kha`htrljflr uykucoòyo ledlrkhehflclr.


ŕhjlc ?. Iosphr, [\“dh k÷róchflche (x, y)“dhjl XGA

AcosĿcĿđĿ ő÷ych bhsopcoyoflclrlz=

( )

Xelgar` dođĿcĿ`Ŀ lële khehccljch joròőĿj lethkroc ocòyo lbtlyoë duyucòdĿđĿeo dljjotht`hclslelz, ëóejó bhr őhyl soct kha`htrlj acoroj lőchyhflclrslelz.



Yae ÷rehjthjledhe gorjcĿ acoroj, acosĿcĿjcorĿ hcdh ht`hj lële yađuecuj gaejslyaeuelethkroclel ocòeĿe yacu yajtur, ëóejó bhrboekl flr soflt acòyoe yađuecuj, acosĿcĿjyĿđĿcòsĿ oëĿsĿedoe gorjcĿ f÷ckhchrl yheldhe ođĿrcĿjcoedĿrĿr. Fueu, oëĿj vh slsthòtlj flrőhjlcdh lzchyhe ÷rehjth k÷rhihđlz=

×rehj 6. Ël` flë`h òjlehelzdh 6 fuml acduđueu vorsoyocĿ`, vh R ?“ eil fumlele ÷`róeó\“dh 6“eil fumlele ÷`róeó th`slc htsle. WorsoyocĿ` jl dođĿcĿ`Ŀ oőođĿdojl klfl lgodhhdhflclyaruz.

Yoygo 4“thjl ŕhjlc 6 flrchőlj yađuecuđue eosĿc k÷róehihđlel k÷sthr`hjthdlr.


ŕhjlc 6= ?. lch 6. Fumlele R vh \ ×`órchrlele Flr chőlj \ađuecuđu

Dađrusu, fu yađuecuj fumlchrle flrflrledhe fođĿ`sĿz acoroj zoòe lëhrlsledh

dhđlő`hyhe soflt flr aroeĿedo fazucduđu vorsoyĿ`Ŀedoe tórhtlchflclr.

Hđhr bhr ljl fuml ëocĿőtĿđĿ sórhih ël` flë`h òjlehsl dh ëocĿőoiojso, ël` flë`hòjlehslele ?>>> soot lëhrlsledh fazucò acosĿcĿđĿ ehdlr1




ŕhjlc 9. –Ël` Flë`h @ojlehsl ?>>> Yootthe ×eih Fazucur‐ AcoyĿeĿe ľcj Duruù

Y÷z jaeusu acosĿcĿj, flrleil fumlele fazucòsĿ bocledh sodhih ljleil fumlele juccoeĿcòsĿduruùedo ehdlr1 \oel ^( R + \ ?>>>)“l eosĿc bhsopcorĿz1 Fueue sodhih

lcklchedlđl`lz –acoyĿ‐ dhđlőtlrdlđleh dljjot hdlelz, yoel lethkroclel bhsopcodĿđĿ`Ŀz 6 ocoeĿdhđlőtl, bocfujl boco oyeĿ yađuecuđue lethkroclel ocĿyaruz.




ŕhjlc 4= –Ël` Flë`h @ojlehsl ?>>> Yootthe ×eih Fazucur‐ AcoyĿeĿe ľjleil Duruù

Thjror ht`hj khrhjlrsh, sórhjcl ljl-dhđlőjhecl roskhch dhđlőjhechr lch ocojocĿ acoycordózch`dhjl ocoecoro thjofóc hdhr. Wh fu ocoecor ózhrledhe yađuecuđue lethkroclel ocorojacosĿcĿđĿ fucuruz.

6. R,\ klfl 6 Voskhch Dhđlőjhele Flr chőlj i.d.g.“sl Yodhih toeĿ`corĿ vhrhihđl`. Fu dhrsth fueu ëaj gozco juccoeòyoiođĿz, gojot fueuk÷r`helz khrhjlyar.

ToeĿ` ?. Voskhch dhđlőjhechr (R, \) lële flrchőlj i.d.g., (x, y) 6 lële, G R\ (x,y)

gaejslyaeu acoroj toeĿ`coeĿr .



Flrchőlj i.d.g.“dhe acosĿcĿjcorĿ oőođĿdojl klfl bhsopcorĿz.

Yae thrl`l hjch`hj zaruedoyĿz ëóejó dobo ÷eih ljl jhrh ëĿjorĿcdĿ.

Flrchőlj i.d.g.“chr p.d.g“chr lch oőođĿdojl őhjlcdh lclőjlcldlr= sórhjcl roskhch dhđlőjhechr lële

Jhsljcl duru`do,

9. @ormleoc p.d.g.chr

Hđhr flrchőlj dođĿcĿ`corĿ`Ŀz vorso, thj dhđlőjhe R“le dođĿcĿ`ĿeĿ yheldhe hcdh ht`hjlsthyhflclrlz. Hđhr R vh \ flchőlj p.d.g.“sl g R\ acoe jhsljcl roskhch dhđlőjhechr lsh, a zoòe

Hđhr R vh \ sórhjcl lsh, hsos ltlforlych tapco`Ŀ lethkroc lch dhđlőtlr`h`lz khrhjlr, f÷ychih

×rehj 9. Fu ÷rehj ?<;;“dh? Vhdfaaj dhrklsl torogĿedoe tapcoeoe khrë hj hvclclj-dĿőĿlclőjlchr vhrlsleh doyoeòjtodĿr. Oejhtth, jlőlchrdhe hvclcljchrlel ?“dhe (ùtsuz) 8“h(ùtcu) ÷cëhjch`hchrl vh hvclclj-dĿőĿ lclőjlchrlele soyĿsĿ f÷có yĿccor ltlforlych hvclclj sórhslelfhclrt`hchrl lsthe`lő. ŕl`dlclj ÷eih –hvclclj joclthsl‐ R lch yĿccor ltlforlych hvclclj sórhsl \“ele

?Whrlele `hviut acduđu odrhs= bttp=//pokhs.sthre.eyu.hdu/wkrhheh/Thxt/Hdltlae7/tofchclst7.bt`



flchőlj dođĿcĿ`Ŀeo fojocĿ`. ^.d.g.“dhe hcdh hdlche –bóirh‐ acosĿcĿjcorĿedoe foőcoyoflclrlz,vh dobo saero tofcaeue sacuedo vh octĿedo fucueoe òrmleoc p.d.g.“chrl dacduroflclrlz=

ľckleë flr őhjlcdh, bhr eh jodor òrmleoc dođĿcĿ`cor hőlt dođĿc`Ŀőso do, flrchőlj dođĿcĿ`cortofcaeue oct sac vh óst sođ j÷őhchrledh, flrchőlj p.d.g.chrdh dobo dóőój dhđ hrchr ocoroj

óst sac vh oct sođ j÷őhchrledh yađuecoő`Ŀő klfl k÷róeóyar.

×rehj 4. Khëhe shghrjl ljl fumlcl ÷rehđlel botĿrcoyĿeĿz. Flrchőlj p.d.g oőođĿdojl klflydl=

R“le òrmleoc yađuecuđu ő÷ych acur=

Fhezhr őhjlcdh,

4. FođĿ`sĿzcĿj

Hđhr ^(OF) 0 ^(O)^(F) lsh, O vh F acoycorĿeĿe fođĿ`sĿz acduđueu s÷ych`lőtlj. ŕl`dlfhezhr flr jovro`Ŀ roskhch dhđlőjhechr lële toeĿ`coyoiođĿz.



ToeĿ` 6. Hđhr bhrboekl flr O, F f÷ckhsl lële oőođĿdojl lclőjl sođcoeĿrso r oskhch

dhđlőjhechr acoe R lch \“ele fođĿ`sĿz acduđueu s÷ychyhflclrlz,

Fu khrhjclclj jaőucueue jotĿ acduđueo dljjot hdlelz= R O vh \ F tóró acoycorofojĿyaruz vh a ëlgtchrle tó`óeóe jorőĿcĿjcĿ fođĿ`sĿz acòsĿeĿ lstlyaruz.

Fu toeĿ` jhedl foőĿeo protlj dhđlcdlr ëóejó jaetrac ht`hj zardur, oeioj hđhr R lch \fođĿ`sĿz lsh, toeĿ`doe borhjhtch őu lgodhyl hcdh hdhrlz=

Furodoe, khehccljch dađrucoeòsĿ dobo jacoy acoe oőođĿdojl jaőucu hcdh hdhflclrlz.

×ehr`h ?. R vh \ fođĿ`sĿzdĿr hđhr sodhi h vh sodhih fuecorĿe flrchőlj v h òrmleoc

p.d.g.chrl oőođĿdojl lclőjlyl sođcorso,

ľYÔT= Jhsljcl roskhch dhđlőjhechr lële, fu dađrudoe O 0 {x} vh F 0 {y} toeĿ`corĿuykucoeoroj hcdh hdlchflclr. Yórhjcl dhđlőjhechr lële, hđhr R lch \ fođĿ`sĿz lsh

dhejch`lele bhr ljl torogĿeĿe do tórhvlel ocoroj oőođĿdojleh ucoőĿrĿz=

Fueue ojsleh, hđhr òrmleoc p.d.g.chrle ëorpĿ`Ŀ flrchőlj p.d.g“chrh hőlt lsh, a zoòelethkroclel ocĿrĿz

DacoysĿyco òrmleoc ózhrledhjl jaőuc fođĿ`sĿzcĿđĿ sođcor vh hőltclđle bhr ljl y÷eóeó dhlspotco`Ŀő acuruz.

×rehj 8. Hvclclj DĿőĿ lclőjlchr ÷rehđleh khrl d÷ehihj acursoj, – hvclclj joclthsl‐ R lch yĿccorltlforlych hvclclj sórhsl \“ele òrmleoc p.d.g.chrlel ropar httlđl`lzl botĿrcoyĿeĿz, yoel



vh

Hđhr ljl roskhch dhđlőjhe khrëhjthe dh fođĿ`sĿz lsh flrchőlj dođĿcĿ` eosĿc k÷róe`hcldlr1 Furodo őueu hcdh hdhrlz.

Flrchőlj p.d.g.“ele a eajtodojl khrëhj dhđhrl ?9.42“dl vh oëĿjëo k÷rócdóđó klfl ljldhđlőjhe fođĿ`sĿz dhđlcdlr. ŕl`dl fođĿ`sĿzcĿj vorsoyĿ`Ŀ octĿedo tofcaeue khrl jocoeĿeĿdacduroflclrlz= Fueu sae tofcaùzco jorőĿcoőtĿrĿeio fozĿ slsthòtlj ëhclőjlchr acduđueu

k÷rhflclrlz, ÷zhcljc h acuőturucoe flrchőlj p.d.g. fhclrkle flr őhjlcdh dlyokaeocdh

yađuecoőòòjtodĿrcor, bocfujl fu khrëhj flrchőlj p.d.g.chrle ózhrledh duròyo dhđhr÷zhc clđlydl.

Gojot khrëhjthe fu R lch \“ele fođĿ`sĿz acòdĿđĿ oeco`Ŀeo `Ŀ khclr1 Dljjot hdlc`hslkhrhjhe flr busus, acosĿcĿjcorĿ, fhclrtlche dođĿcĿ`doe –ëhjlche‐ ÷rehjch`dhe hdledlđl`lzflrchőlj p.d.gchrdhe bhsopcodĿj, fu ehdhech khrëhj bóirh acosĿcĿjcorĿeĿ dađru flr őhjlcdh÷cëhflc`h jaeusuedo fozĿ fhclrslzcljchr vor. Dhrsle sae f÷có`óedh, –acuőturucoe‐ vhkhrëhj p.d.g.chr orosĿedojl gorjĿe R lch \“ele fođĿ`sĿz acòdĿđĿeĿ k÷sthrhihj jodorfóyój acup acòdĿđĿeĿ garòc acoroj ÷ehrhe flr y÷eth` k÷rhihđlz.

×rehj 7. Dobo ÷eihjl ljl fumlcl ÷rehđlel botĿrcoyĿeĿz. Flrchőlj p.d.g. oőođĿdojl klflydl

vh flr ÷eihjl f÷có`dh dh òrmleoc p.d.g.chrl hcdh ht`lőtlj



DacoyĿsĿyco, fuecorĿe ëorpĿ`Ŀ ő÷ychdlr=

F÷ychih ?“ eil vh 6 “ eil fumlele yoőo` sórhchrl fođĿ`sĿzdĿr.

OëĿjcoò ?= Yórhjcl roskhch dhđlőjhechr lële, flchőlj vh òrmleoc yađuecujcorĿefođĿ`sĿzcĿđĿeĿe jaőucu oőođĿdojl klfl yheldhe fhclrtlchflclr= Bhr eh zoòe fl chőlj p.d.g“elegojt÷róeó ocĿrsoj, yoel,

a zoòe R lch \ fođĿ`sĿzdĿr. Furodo k(.) sodhih x“h vh b(.) sodhih y“yh fođcĿdĿr. ×zhcljch fhclrt`hj khrhjlrsh, òrmleoc yađuecujcorĿ dađrudoe bhsopcoòj zaruedodhđlclz.

×rehj ;. Dlyhcl` jl, oőođĿdojl klfl flr flrchőlj p.d.g“`lz vor

A zoòe, ÷rehđle k(x) 0 ih-x vh b(y) 0 h-6y “dlr dlyhflclrlz. B hr eh jodor fuecor uykueyađuecujcor dhđlcsh dh, R lch \“ele fođĿ`sĿz acduđueu k÷sthr`hj lële yhthrcldlr.

×rehj 2. WorsoyocĿ` jl oőođĿdojl klfl flr flrchőlj p.d.g.`lz vor.

R vh \ fođĿ`sĿz acoflclr `l1

Bhr ljl duru`do (yoel x 6 y ? sođcoeĿr `Ŀ yajso sođcoeòz `Ŀ1) p.d.g. x vh y“ele

gaejslyaecorĿeo oyrĿőtĿrĿcĿrso (gojt÷r, ËE) (sĿgĿr f÷có`ó lële dađrucuđu őópbhcldlr) flch, R“ledhsthđlele \“yh fođĿ`cĿ acduđueu k÷rhflclrlz vh dacoysĿyco R vh \ fođĿ`sĿz acoòz-

÷rehđle hđhr R ?/6 lsh, \ ?/4 acòj zaruedodĿr. F÷ychih



R vh \“ele fođĿ`sĿz acoflc`hsl lële ljl roskhch dhđlőjhele flchőlj dhsthđlele dljd÷rtkhe(fóyój lbtlòcch fótóe 6) acòj zaruedo acduđueo dljjot hdlelz. Hđhr dhđlcsh,khrëhjchőhe fozĿ R dhđhrchrl lële, fhccl \ dhđhrchrl khrëhjchőhflchihjjhe,khrëhjchőh`hz. Gojot hđhr dađruyso, yoel R“le \ bojjĿedo flckl vhrdlđlel flclrshj, azoòe aecor fođĿ`sĿz acoòz. Dhsthj lële fu jaőuc thj foőĿeo fođĿ`sĿzcĿđĿ sođcor .




Jaerod @hezhc

9 @ort 6>><

?. Jaőuccu p.d.g.chr

ToeĿ` ?. R vhrlc lőjhe \“ele jaőuccu p.d.g.sl

Hđhr R lch \ jhsljcl lsh,

Fu, sodhih ljl bogto ÷eih toeĿ`coeoe vh \ 0 y vhrlc`lőjhe R 0 x duruùyco lcletlcl acoyĿejaőuccu acosĿcĿđĿyco lcletlsldlr.

Xeutòòj khrhjlr jl,

Jaőuccu dhđlőjhele fhclrcl flr dhđhrl lële, jaőuccu p.d.g earòc p.d.g.“ele fótóe÷zhcljchrlel toőĿr(yoel pazltlgtlr, lethkrocl ?“dlr klfl),

ToeĿ` bhr ljl torogtoe bhrboekl flr soyĿdojl roskhch dhđlőjheh khehcchőtlrlchflclr.

×rehj ?. Hvclclj dĿőĿ lclőjlchr vhrlsleh khrl d÷ehcl` vh khrëhjth he ëaj lcklchedlđl`lzdhđlőjhechrh fojocĿ`= sae yĿcdojl lclőjl soyĿsĿ, ], vh –jhedl jhedleh‐ ropar hdlche hvclcljjoclthsl, R. Oejhtl dacduroecorĿe 4“th 9“ó bhrboekl flr lclőjlchrlele acòdĿđĿeĿ fhclrtljchrllële, hvclclj joclthsleh jaőuccoe`Ŀő lclőjl soyĿsĿ ]“ele p.d.g. sleh fojòj ëaj dobo yack÷sthrlil acoflclr.



Flchőlj p.d.g fu őhjlcdh hcdh hdlchihjtlr. Dóőój dhđhrh, R 0 ?, jaőuc c oeĿrso acurso,

oőođĿdojlel hcdh hdhrlz.

R 0 ?, 6, 9, … dhđhrchrl lële jaőuccu i.d.g.“chrlel fhrofhr flr tofcayo jayorsoj, őueu hcdh

hdhrlz.

Fu uykucoò ehdhe lckl ëhjlildlr1 Tofcado flrchőlj p.d.g.chr ch khehc rhsl` ëaj oëĿjacòdĿđĿ bocdh, hvclclj joclthslele dóőój dhđhrchrl lële, jaőuccu p.d.g.chrle yójshj lclőjlsoyĿsĿeo gozco acosĿcĿj yójchdlđlel k÷rhflclyaruz.

Fu hvclcljthjl tot`leslzclđle, hvclclj-dĿőĿ lclőjlyh ehdhe acduđu oeco`Ŀeo khclr `l1 Jhsleclj ch

boyĿr= ÷rehđle, uykucoòyĿ toò`he thrs yopoflclrdlj vh lclőjl soyĿsĿ ] vhrlc`lőjhe, ropar

hdlche hvclclj tot`lel R“le jaőuccu p.d.g.chrleh fojoflclrdlj. \oel

\o do jaőuccu p.d.g.chrl flr tofcado ÷zhtchyhflclrdlj.

\ójshj dhđhrcl lclőjl soyĿsĿ ] vhrlyjhe, R“le jaőuccu p.d.g. chrl dóőój dhđhrcl hvclclj

tot`leleh dobo yójshj acosĿcĿj vhr`hjthdlr. F÷ychih rojo`corĿ, hvclclj -dĿőĿ lclőjlelehvclclđh zoror vhrdlđl őhjcledh dh ajuyoflclrdlj. Fu khehccljch –thrs ehdheshcclj‐ acorojotghdlclr= O“eĿe F“yh ehdhe acduđueo leoesoj flch, F oyeĿ zoòedo O“yo ehdhe acoflclr.



DacoysĿyco, jaőuccu acosĿcĿj bhr ljl bljæyhych tutorcĿ acoioj őhjlcdh borhjht htsh flch,lclőjlyl ljl y÷ecó –ehdheshcclj‐ acoroj yaru`coyoòyĿz, ëóejó bhr ljl bljoyh dh hőltdhrhihdh jofuc hdlchflclrdlr vh vorsoyĿ`soc acoroj khrëhj yoőo`do bhr ljlsledh dh

khrëhjclj poyĿ vordĿr .

6. ThjrorYĿeĿgto yoptĿđĿ`Ŀz blë ÷rehđl hzfhrch`helzl fhjch`lyaru`, oeioj ÷zhcljch –`htle‐prafch`chrl ÷zhc duru`corĿe/prafch`chrle –àdhcchrl‐ acoroj acdujëo yororcĿ acoflclrchr.Dhrschr sĿrosĿedo tortĿőtĿđĿ`Ŀz ÷rehjchr lch fhezhrclj juroroj fhccl flr saru lële flr ë÷zó`

strothmlsl khclőtlrhflclrslelz.

?. AcosĿcĿj

×rehjch` XzoyĿ, Jó`h Tharlsl vh Th`hc Aphrosyaecor

Fueu tortĿőòyoiođĿz.

AcosĿcĿđĿe toeĿ`Ŀ

(^?) fótóe O Y lële ^(O) >

(^6) ^(Y) 0 ?

(^9) Hđhr O?, O6, …, oyrĿj jó`hchrlele flr dlzlsl lsh

×zhc Duru`= Foslt acosĿcĿj

Y saecu

^(O) 0

furodo e(F) F jó`hsledhjl saeuë soyĿsĿeĿ lgodh ht`hjthdlr.

AcosĿcĿj Gaejslyaeue ×zhcljchrl



AcosĿcĿđĿe BhsopcoeòsĿ

Yaruecoro őu sĿroyo k÷rh yójchele

(l) saeuëcor ltlforlych ÷rehjch` uzoyĿeĿ vh lcklchelche acoyĿ toeĿ`coyĿeĿz

(ll) foslt acosĿcĿjcor lële, bhr flr saeuiue acuőòsĿeĿ hőlt dhrhihdh `ó`jóejĿcoioj őhjlcdh flr ÷rehj uzoyĿ toeĿ`codĿđĿeĿzdoe h`le acueuz,

(lll) hđhr lőle lëledhe ëĿjòzsoeĿz, ÷rehjch` uzoyĿedojl saeuëcorĿ dađrudoeyozòyo ëocĿőĿe.

Yoyò JuroccorĿ

Th`hc jurucu`= E“ele R?, …, RE soyĿdojl afmhslel hcdh ht

Ëorpò JurocĿ= Flr dhehyl bhr flrlslele `l soyĿdojl saeuiu dlđhr f÷có`chrle

saeuëcorĿeo fođcĿ acòyoe j soyĿdo f÷có` jodor gojt÷rh oyĿr oflc`hj khrhjlr. Fufozhe yoeĿctĿiĿdĿr (÷rehđle sotroeë klfl).

Jó`hdhe flrjoë g orjcĿ yacdoe j jodor afmh ëhj`h (sĿeov lële aecorĿbotĿrcoòcĿsĿeĿz)=

?. j yhrleh jayò lch ëhjlclr, sĿrocoò ÷eh`cldlr= Ej soyĿdo lbtlòc

6. j yhrleh jayòdoe ëhjlclr, sĿrocoò ÷eh`cldlr (÷zhc duru`= phr`ótosyae j

0 E)=

soyĿdo lbtlòc

9. j yhrleh jayòdoe ëhjlclr, sĿrocoò ÷eh`slzdlr (ja`fleosyae)=

() 0

soyĿdo lbtlòc (÷rehđle flea` dođĿcĿ`do, khehc foőorĿ

soyĿsĿeĿ vhrhihj acoe fótóe gorjcĿ ordĿőĿj –foőorĿcorĿ‐ soyĿeĿz). f÷cóetóchr = E afmhyl j krufo yhrchőtlr`h yaccorĿeĿe soyĿsĿdĿr. Afmhele toeĿ`Ŀ

÷eh`cl dhđlcdlr (÷rehđle 4 tarfoyo fhő fhezhr òvl tapu gorjcĿ őhjlcchrdh

yhrchőtlr`h soyĿsĿ klfl)= khehc acoroj ( ) jodor lbtlòc vordĿr. Fueu oőođĿdo

tortĿőoiođĿz.

Fótóe fu soyò juroccorĿeĿe ő÷ych vhyo f÷ych ëorpò jurocĿedoe hcdh hdlcdlđlel k÷rdój. Fozhe oyeĿ acoyĿ hcdh ht`hj lële flr soyĿyĿ gorjcĿ soyĿdojl lbtlòch f÷c`hj

zaruedo jocdĿj (÷rehđle oyeĿ ja`fleosyaeue gorjcĿ sĿrodo hcdh hdlc`hsl klfl)

FođĿ`sĿzcĿj, jaőuccu acosĿcĿj, Foyhs Tharh`l

hđhr ^(OF) 0 ^(O)^(F) lsh O vh F fođĿ`sĿzdĿr ,

hđhr ^(F) 5 > lsh jaőuccu acosĿcĿj ^(O|F) 0

,

^(O|F) 0 ^(O)“dlr hđhr sodhih vh sodhih O vh F fođĿ`sĿz lsh,

Tapco` acosĿcĿj joeueu= hđhr F?,…, Fe Y“ele f÷cóetóchrl lsh.



Tapco` acosĿcĿj joeueu jaőuccu acosĿcĿđĿ òrmleoc acosĿcĿđo fođcor, yoel ^(O)“eĿe eosĿc^(O|F? ) … ^(O|Fe )“yh fođcoeoiođĿ klfl. Jcoslj uykucoò= oct-eógus/oct-vojo ózhrledhe

tapcucoőtĿrò, ÷rehđle gorjcĿ foypos o`hclyotcorĿeo k÷rh ÷có` aroecorĿ klfl.

Foyhs Tharh`l(foslt gar`ócosyae)= Hđhr ^(F) 5 >, a zoòe oőođĿdojl lgodhylhcdh hdhrlz=

Foyhs tharlsl flzh jaőuccoò sĿrosĿeĿe eosĿc dhđlőtlrlchihđlel s÷ychr, yoel eosĿc^(F|O)“doe ^(O|F)“yh kldlclr klfl. Jcoslj uykucoò= F vhrlsl vhrlcdlđledh O lch lcklcl leoeiĿekóeihcche`hsl klfl, ÷rehđle tĿffl thstchr ÷rehđl klfl.

YĿeov lële fuecorĿ toò`he jovroyoroj flc`helz khrhjlr.

6. Voskhch Dhđlőjhechr vh DođĿcĿ` GaejslyaecorĿ

roskhch dhđlőjhechr roskhch acoycorĿe soyĿsoc ÷zhccljchrlel vhrlr

roskhch dhđlőjhe R ÷rehjch` uzoyĿ Y“dhe rhhc soyĿcor “yh kldhe flr gaejslyaedur

Y“ele acosĿcĿj gaejslyaeu R“le lch toeĿ`coe`Ŀő acosĿcĿj dođĿcĿ`ĿeĿ artoyo ëĿjorĿr .

AcosĿcĿj yađuecuj gaejslyaeu (^DG) g R (x) oőođĿdojl klfl toeĿ`coeĿr

Flrljl`cl yađuecuj gaejslyaeu (IDG) GR(x) oőođĿdojl klfl toeĿ`coeĿr

Jhsljcl flr roskhch dhđlőjhe lële ÷eh`cl flr ÷rehj acoroj, ordĿőĿj E fođĿ`sĿzdheh`hdhjl R soyĿdo –foőorĿ‐ yĿ oëĿjcoyoe Flea` dođĿcĿ` lële flroz zoòe boriodĿj. Bhr dheh`hdhjl foőorĿeĿe acosĿcĿđĿ p“yh hőlttlr. Flea` dođĿcĿ` lële p.d.g. oőođĿdojlklflydl (fueu sĿeov lële flc`hclslelz)=



IDG vh ^DG orosĿedojl lclőjl

^DG“dhe IDG ő÷ych hcdh hdlclr=

hđhr R jhsljclysh tapco`ĿeĿ oc.

hđhr R sórhjclysh lethkroclel oc.

IDG“dhe ^DG ő÷ych hcdh hdlclr=

hđhr R jhsljclysh

hđhr R sórhjclysh

OyeĿ őhjlcdh, IDG“ele ÷zhcljchrlel botĿrcoyĿeĿz

fótóe x lële > GR(x) ?,

Gx(x) x“th ozocoe dhđlcdlr

Gx(x) sođdoe sórhjcldlr

Gx(x) bhr yhrdh sórhjcldlr, Yodhih vh sodhih R sórhjclysh.

Flr chőlj DođĿcĿ`

OőođĿdojlchr foj`ĿőtĿj

R vh \“ele flrchőlj dođĿcĿ`Ŀ (jhsljcl vhyo sórhjcl)

^DG“cl R“le òrmleoc dođĿcĿ`Ŀ



Voskhch dhđlőjhechrle fođĿ`sĿzcĿđĿ, he ÷eh`clsl sodhih vh sodhih R vh \

fođĿ`sĿz lsh fótóe (x, y) lële

R vhrlc`lőjhe \“ele jaőuccu acosĿcĿđĿ

9. Voskhch ^rafch`chr

×rehj 6 (Fobor 6>>9 YĿeovĿ). Flr @aeht uzòeĿeo s÷zdh joyfacoe flr @aeht rhs`lvhrlclr. Vhs`le arlmleoc acò acosĿcĿjcorĿeĿ dhđhrchedlr`hsl lsthelr. Xzòe oőođĿdojlflcklyh soblptlr=

Khehc acoroj, –fucueoe‐ vh jhedlsleh k÷edhrlche rhsl`chrle sodhih %?“l arlmleocëĿjĿyar, fueo K acoyĿ dlyhihđlz.

–Fucueoe‐ rhsl`chr arlmleoc @aeht“le yoptĿđĿedoe gorjcĿ acoroj fhccl plk`hetchrlegorjcĿ jucoeĿ` grhjoesĿeo soblptlr.

a –Fucueoe‐ rhsl`chrle %6>“sledh sorĿ jod`lyu` \ k÷róc óőtór, bocfujlarlmleocchrle sodhih %?>“ u ÷ychdlr..

a –Fucueoe‐ rhsl`chrle %2>“eledh bo` taproj rhekl fayo X k÷róc óőtór,bocfujl arlmleocchrle sodhih %4>“Ŀ ÷ychdlr.

a –Fucueoe‐ rhsl`chrle %4>“edo yoeĿj slheeo rhekl Y k÷róc óőtór, bocfujlarlmleocchrle %7>“Ŀ ÷ychdlr.

Khche rhsl` yoeĿj slheeo juccoeòjtodĿr, oeioj sorĿ jod`lyu` vh bo` taprojrhekl dhđlc.

Fu rhs`le arlmleoc acò acosĿcĿđĿ ehdlr1 Yaruyu ihvopcoedĿròj lële bhrboekl flr lcovh

vorsoyĿ`do fucueòj zaruedo `ĿyĿz1

Fu prafch` ő÷ych flr yopĿyo soblptlr=

^rafch` flzh rhs`le arlmleoc acòsĿ (–dóeyoeĿe bocl‐ K) duruùedo boekl rhekle (–vhrl‐Y\IXI) eh acosĿcĿjco k÷róehihđlel s÷ycóyar klfl k÷róeóyar, ^(F|O). Oeioj, flz khrëhjth

lëledh juccoeĿcoe rhejchrh k÷rh rhs`le arlmleoc acò acosĿcĿđĿeĿ flc`hj lstlyaruz, yoel

^(O|F). Fu ehdhech jaőucue sĿrosĿeĿ dhđlőtlrlyaruz, aeue lële Foyhs tharh`ljuccoeoiođĿz. ×eihcljch prafch`dh yhr ocoe flcklyl taporcoyocĿ`=



vh

Foyhs tharh`lel uykucoòj lële ehyh lbtlyoiĿ`Ŀz vor1 Tharh` flzh őueu s÷ychr=

Oeioj, bhr rhekle òrmleoc acosĿcĿđĿ flclelyar, flzh flrchőlj acosĿcĿj khrhjcldlr (bh` Kjaőuccu bh` dh jaőucsuz)

Fu ehdhech, fu eajtodo lcovh flr vorsoyĿ`do fucueòj zaruedoyĿz. Yarueo

yójche`hele he jacoy yacu, jaőuccsuz vh K“yh jaőuccoe`Ŀő acoroj, plk`het“le óë rhejorosĿedo juccoeĿ`ĿeĿe fođĿ`sĿz acduđueu vorsoyòjtĿr, yoel

vh

FođĿ`sĿzcĿj vorsoyĿ`Ŀ octĿedo Foyhs Tharh`lel juccoeoroj őu saeuio vorĿrĿz=

Fu vorsoyĿ`Ŀe eh dhrhih ÷eh`cl acduđueu k÷r`hj lële, gorjcĿ plk`het tórchrl orosĿedogorjcĿ flr fođĿ`cĿ yopĿ jhőghdle= vorsoyocĿ` jl arlmleoc @aeht lële bhr slheeo Y juccoeĿcoe

rhsl`dh, oyeĿ zoòedo jhsle acoroj bo` taproj rhekl X do juccoeĿcòjtodĿr. A zoòe,jaőuccu acosĿcĿj toeĿ`Ŀeo k÷rh őueu hcdh hdhrlz=

F÷ychih khrëhj flr @aeht lële, slheeo Y“yl fucòj l`jæesĿzdĿr gojot bo` taproj rhekl X

dhđlc, fu ehdhech jhslecljch flclyaruz jl s÷z jaeusu rhsl` @aeht acoòz ( rhs`l`lz fu



ja`fleosyaeo soblp acduđuedoe, khehc acoroj –fucueoe‐ fótóe rhsl`chr lële ÷ych acòjzaruedodĿr).

Taporcoyoioj acursoj, fu prafch` flzh saruyu ihvopcoedĿròj lële yhthrcl flckl vhr`hdl.

×rehj 9(Kóz 6>>9 YĿeovĿ). Fhel` hvl`dh khrl d÷eóőó` ?> o.`. lch ÷đch sootl orosĿedoflr oro tapcoeĿr. Ë÷póe tapcoedĿđĿ orocĿjto bhrboekl flr dojljo bhrboekl flr dlđhrl jodoracosĿdĿr. Ë÷p lsh soot 2=9> o.`. lch ??.>> o.`. orosĿedo flr oro tapcoeĿr vh yleh

thjrorcoòj khrhjlrsh bhrboekl flr oe bhrboekl dlđhr flr oe jodor acosĿdĿr. ľjl tapcoòzoòeĿ flrflrledhe fođĿ`sĿzdĿr.

(o) ľjl tapcoò zoòeĿeĿe, V vh K, flrchőlj p.d.g. sl ehdlr1

(f) Khrl d÷eóőó`óe ë÷pthe ÷eih tapcoeò acosĿcĿđĿ ehdlr1

V“ele òrmleoc dođĿcĿ`Ŀ uelgar`dur (sórhjcl), yađuecuđu lsh oőođĿdojl klfldlr=

K“ele òrmleoc dođĿcĿ`Ŀ jhsljcldlr, p.d.g. sl lsh oőođĿdojl klfldlr=

FođĿ`sĿzcĿđo k÷rh, flrchőlj p.d.g. ő÷ychdlr=

V K acoyĿeĿe acosĿcĿđĿ oőođĿdojl klfl bhsopcoeĿr =

×rehj 4. YĿeĿgĿeĿzdojl orjodoőcorĿeĿzdoe flrlsl oőođĿdojl prafch`le eosĿc ë÷zóchihđlelsardu= E soyĿdo flrflrledhe gorjsĿz jorotobtoyĿ j soyĿdojl gorjcĿ sĿeĿgo dođĿtòj lële joëgorjcĿ yac vordĿr1 Fu, j sĿeĿg lële flr jorotobto f÷cóetósó shë`h lch lcletlcldlr.

BhsopcoòcorĿ oőođĿdojl klfl yopoflclrlz=

a F?, F6 , …, FE jorotobtocorĿyco jorĿőtĿrĿc oioj, j-? jodor –oyrĿőtĿrĿiĿ‐ , ] ?, ] 6 , …,] j-?, toeĿ`co

a sĿeĿgcoro bhr yhrchőtlrlche jorotobtoyĿ F?, F6 , …, FE , ] ?, ] 6 , …, ] j-?“ele yheldhesĿrocoeòsĿ acoroj torlg ht. YĿrocoòdo lcj ]“dhe ÷eih k÷róehihj acoecor lcj sĿeĿgo,ljleil –oyrĿőtĿrĿiĿ‐yo jodor acoecor ljleil sĿeĿgo jayoiođĿ`Ŀz v.f. tobtocordĿr. Hđhr



sĿrocoò, ÷rehđle ] 8 , F; , F6 , F8 , ] 4, F<, …, lsh, a zoòe flrleil sĿeĿgto jorotobtoacòyoioj, tobto ;, 6, vh 8 6eil sĿeĿgo kldhr, v.f.

a dlzlele gorjcĿ soyĿdojl sĿrocoòsĿ (E + (j - ?)! jodordĿr .a jorotobtocor vh oyrĿőtĿrĿiĿcor hőlt acduđu lële (sĿeĿgcor dhđlc), bhr flr jorotobtoeĿe

phr`ótosyaeu (E! phr`ótosyae) vh oyrĿőtĿrĿiĿsĿ (j - ?)! phr`ótosyae) lch f÷c`hj

zaruedoyĿz.

Fótóe porëocorĿ flr oroyo khtlrleih oőođĿdojl klfl fótóe acosĿ yhrchőtlr`hchrl hcdhhdhrlz.



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ <

Jaerod @hezhc

?> @ort 6>><

?. Voskhch dhđlőjhechrle GaejslyaecorĿ

Dhrsle fu f÷có`óedh roskhch dhđlőjhechrle gaejslyaecorĿeo fojoiođĿz, \ 0 u(R). \“eleyleh flr roskhch dhđlőjhe acduđueu eat hdlelz= R ÷rehjch` uzoyĿ Y“dhe rhhc soyĿcoro flrhőch`h (òpplek) acduđu lële

vh u = acduđuedoe , u vh R“le flchől`l dh oyeĿ zoòedo Y“dhe rhhc soyĿcoro flrhőch`hdlr=

×rehj ?. Hđhr R ël` flë`h òjleosĿedojl flrleil fumlele vh \ ljleilele ÷`ró lsh, a

zoòe ljlslele ÷`róeóe tapco`Ŀyco lcklche`lő acoflclrlz, ] 0 R + \.

×rehj 6. @LT“yh khc`hdhe ÷eih, flrjoë Ocòe oroőtĿrò fursueo foővurdu`, f÷ychihocoiođĿ` furso fođcĿ acoroj oycĿj R Hura jodor óirht ocoiojtĿ`. D÷vlz juru, dlyhcl` jl,Hycóc 6>>8“th \ dacor/Hura acsue. Foővurduđu` d÷eh`dh bhr ljl `ljtordo fhclrslzdl,gojot poroyĿ OFD“dh borioyoiojtĿ`, dacoyĿsĿyco ocoiođĿ` ] 0 R*\ `ljtorĿ hsoslcklchedlđl` `ljtordĿ (he ozĿedoe dacor dhđlől`ledhe saero őljæyht ht`hyhihđl` jodor).

ŕl`dl d÷eóőtóróc`óő roskhch dhđlőjhe u(R) lële yađuecuj vh i.d.g“yl eosĿc hcdhhdhihđl`lzl flc`hj lstlyaruz, f÷ychih p.d.g. sl flclehe flr roskhch dhđlőjhele sodhihjhedlslel lëhrhe flr sarudo acduđu klfl, flr roskhch dhđlőjhele gaejslyaeueu do lëhrhe

bhr flr prafch`l hch ocoflchcl`.

Óë duruù dóőóehihđlz

?. ľcklcl dhđlőjhe jhsljcldlr 6. ľcklcl dhđlőjhe sórhjcldlr 9. R sórhjcldlr vh u(R) jhsle ortoedĿr



Yae duru` hcfhtth ljleilele ÷zhc duruùdur, gojot k÷rhihđl`lz klfl ëocĿőĿcòsĿ hejacoy acoeĿdĿr.

?.? Jhsljcl Duru` ’ – 6-OdĿ`‐ \÷eth`l

Hđhr R p.d.g.“sl g R(x) acoe flr jhsljcl roskhch dhđlőjhe lsh vh u(.) dhthr`lelsttlj flrgaejslyae ljhe \ 0 u(R) lsh,

×rehj 9.

A zoòe hđhr \ 0 k(R) 0 |R| lsh,

Hđhr R jhsljcl lsh \“ele dh jhsljcl acduđueu eat hdlelz.

?.6. Yórhjcl Duru` ’ – 6-OdĿ`‐ \÷eth`l

Hđhr R p.d.g.“sl g R(x) acoe flr sórhjcl roskhch dhđlőjhe, vh \ 0 u(R) lsh, a zoòe \“elei.d.g.“sl oőođĿdojl lgodh lch vhrlclr=

Hđhr \ dh oyeĿ zoòedo sórhjcl lsh, a zoòe

Hđhr R sórhjcl lsh, \“ele sórhjcl acòsĿeĿe khrhj`hdlđle eat hdlelz.



×rehj 4. R“the jóëój he fóyój to`soyĿ \ 0 ZRP, R“le sórhjcl vhyo jhsljcl acup

acòòsĿeo fođcĿ acòdoe, jhsljcldlr.

×rehj 8.

ŕl`dl őueo fojocĿ`,

\ 0 R 6

R Z-?, ?P“dhe \ 0 Z>, ?P acur. \“ele yađuecuđueu eosĿc hcdh hdhrlz1 y Z>, ?P lële, i.d.g

oőođĿdojl klfldlr.

Taporcorsoj,

\ sórhjcl acduđu lële, yađuecuđu ő÷ych bhsopcorĿz,



?.9. Flrh-Flr D÷eóőó` lële Dhđlőjhe Dhđlőtlr`h Gar`ócó

Khehc acoroj \“ele yađuecuđueu, ÷zhcljch flr lethkroc vh flr tórhv lëhrdlđl lële, i.d.g“chr

oroiĿcĿđĿyco R“le g x (x) yađuecuđuedoe hcdh ht`hj uykue dhđlcdlr. Fu duru`do, p.d.g.chrorosĿedo dobo dlrhjt flr fođcoetĿ acup acòdĿđĿ `hroj hdlchflclr.

Dobo khehc duruò khë`hdhe ÷eih, vorsoyocĿ` jl fozĿ soflt dhđhrchr o 5 > lële u(x) 0ox“tlr. A zoòe \ 0 u(R) 0 oR“le i.d.g sl oőođĿdojl lch hcdh hdlclr.

]leilr jurocĿeĿ juccoeoroj, \“ele p.d.g.“slel ő÷ych tórhthflclrlz=

Fueue lële he lyl shzklshc dhđhrchedlr`h ehdlr1 ’ Hđhr o 5 ? lsh, d÷eóőó`ó, roskhchdhđlőjhele ózhrleh dóőtóđó hjshel khr`h acoroj dóőóehflclrlz. Fu duru`, hjsheózhrledhjl bhrboekl flr ljl eajtoyo o“eĿe ëorpĿ`Ŀ jodor yhr dhđlőtlrtlr, gojot dhđlőjhele ljleajto orosĿeo dóő`h acosĿcĿđĿeĿ soflt tutor. DacoyĿsĿyco, R“le dođĿcĿ`Ŀ lchjorőĿcoőtĿrĿcdĿđĿedo \“ele dođĿcĿ`Ŀ ?/o ëorpĿ`Ŀ jodor –shyrhjchő`lő‐ acur. Fueu lëledh flr`ljtor ózó` acoe boùr lch boyoc hdhflclrlz. Boùru eh jodoro yoyorsoj, boùr

lëledhjl ózó`chrle boùr tobtosĿeĿe yózhyleh k÷rh dođĿcĿ`Ŀ a jodor shyrhj acoiojtĿr.R“le u(.)“sueue tórhvchehflclr àeatae d÷eóőó`ó lële oőođĿdojl saeuiu hcdh hdhrlz.

×ehr`h ?. R g R (x) yađuecuđu flclehe roskhch sórhjcl flr dhđlőjhe acsue, oyrĿio ^(o R

f) 0 ? vh \ 0 u(R)“dlr. Hđhr u(.) |o,f| klfl flr orocĿjto jhsle ortoe vh tórhvchehflclr lsh

thrs yhr dhđlől` s(y) 0 u-?(y)“yh soblpsh, a zoòe T“ele yađuecuđu oőođĿdojl lgodh lch

vhrlclr.

Fhezhr saeuiue, u(x)“le |o, f|“dh jhsle ozocoe acò duruùedo do dađru acduđueodljjot hdlelz.

×rehj 7. R Z>, ?P orocĿđĿedo uelgar`, f÷ychih p.d.g.sl dh oőođĿdojl klfl acsue



\ 0 R6“ele p.d.g.“sl ehdlr1 R“l dhsthjch`hj lële, u(x) 0 x6“ele jhsle ortoe vh tórhvchehflclracduđueu dóőóeóyaruz vh f÷ychih \“ele p.d.g.slel hcdh ht`hj lële u(.)“eue thrsl acoe s(y)

0 ∝ “yl juccoeoroj oőođĿdojl lclőjlyl hcdh hdhrlz.

Fu, yujorĿdo yoptĿđĿ`Ŀz flr ÷rehđh fhezh`hjthdlr. ×eihjledhe gorjcĿ acoroj, R“ledhsthđl Z-?, ?P ldl vh f÷ychih u(x) 0 x6 R“l dhsthjch`hj lële àeatae dhđlcdl.

Fu gar`óccóe sodhih flrh-flr tórhvchehflclr d÷eóőó`chr - yoel àeatae - lële ëocĿőtĿđĿeĿeat ht`hj ëaj ÷eh`cldlr . Dlđhr duru`cordo, jhsljcl vh sórhjcl duru`cor lële boetoc 6-odĿ`cĿ y÷eth`chrh fođcĿ jocòj zaruedoyĿz.

?.4 AcosĿcĿj ľethkrocl / _uoetlch D÷eóőó`

Yórhjcl roskhch dhđlőjhechr lële, lckleë- oyeĿ zoòedo ëaj yororcĿ- flr saeuë vordĿr=oőođĿdojl òeodo –i.d.g“ele i.d.g.“sl‐ uelgar` flr dhđlőjhedlr=

×ehr`h 6. R, i.d.g.sl G x (R) acoe sórhjcl flr roskhch dhđlőjhe acsue. A zoòe, i.d.g. R“le

r oskhch ëhjlclőlych ÷cëócórjhe, G x (R) uelgar` dođĿcĿ`cĿdĿr. \oel

Voskhch flr dhđlőjhele gaejslyaeue jhedlslele dh roskhch flr dhđlőjhe acduđueo dljjothdlelz (fueu dobo saero dhtoycĿ flr őhjlcdh tortĿőoiođĿz).

ľYÔT= I.d.g. sodhih sĿgĿr lch ? orosĿedo dhđhrchr ocdĿđĿ lële, G(R)“le i.d.g.“sl acoe K(.)“ele bocl bozĿrdo őu jaőucorĿ sođcodĿđĿeĿ k÷rhflclyaruz=

Khehcchőtlr`hyl artodoe jocdĿròdoe (sodhih flrjoë lckleë acòyoe hjstro toeĿ` vhyoduru` gorjcĿcĿđĿedoe joëĿeoroj), vorsoyocĿ` jl G(.) jhsle àeataedur- ueutòyĿe jlfótóe i.d.g.chr ozocòyoedĿr. Fueue oeco`Ŀ, G-?(.) klfl thrs flr gaejslyaeue acduđudur,yoel , G -?(G(x)) 0 x klfl flr gaejslyaedur.



Thrs gaejslyaedo jhsle acoroj àeatae acoiojtĿr, f÷ychih > x ? lële roskhch

dhđlőjhe GR(R)“le i.d.g.“sl oőođĿdojl klfldlr

(flrleil hőltclj G-?“åe àeataecuđueu, vh i.d.g“ele óëóeió toeĿ`ĿeĿ juccoeĿr).

×zhtchrshj, roskhch dhđlőjhe G(R)“le i.d.g.sl acoe K(.) ő÷ychdlr

Fueue, Z>, ?P orocĿđĿedo uelgar` roskhch flr dhđlőjhele i.d.g.“sl acduđueu do jacoycĿjcojaetrac hdhflclrlz, f÷ychih G(R) XZ>, ?P lch oyeĿ acosĿcĿj dođĿcĿ`Ŀeo soblptlr.

Fu saeuë eh lőh yoror1 Flr ÷rehj acòsĿ fojĿ`Ŀedoe, flcklsoyor lch uelgar` roskhchrojo`cor hcdh ht`hele ëaj htjle yaccorĿ vordĿr. I.d.g.sl GR(.) acoe flr roskhchdhđlőjhedhe e ëhjlclőcl flr ÷rehjch` hcdh ht`hj lstlyarsoeĿz, őuecorĿ yopoflclrslelz=

X?,…Xe XZ>, ?P ëhjlclőlel yopĿeĿz,

bhr flr uelgar` ëhjlclőlel őueo k÷rh d÷eóőtór óeóz=

Dobo ÷eihjl orkóòeĿ`Ŀzo k÷rh, R?, ..Re i.d.g. sl GR(.) acoe flr roskhch dhđlőjhe klfldovroeĿr. Fu y÷eth` lethkroc (yo do quoetlch) d÷eóőtór`h acoroj flclelr.



×rehj ;. X elgar` dođĿcĿ`Ŀedoe flr roskhch dhđlőjhe X“yu ëhj`h`lzh lzle vhrhe flr

flcklsoyor prakro`Ŀ`ĿzĿe acduđueu vorsoyocĿ`, gojot flz khrëhjth roskhch ëhjlche vh

p.d.g.“sl oőođĿdojl klfl acoe flr R hcdh ht`hj lstlyaruz

Lethkroc lch R“le i.d.g.slel hcdh hdhflclrlz=

F÷ychih i.d.g.“ele thrsl ő÷ychdlr=

Hđhr fueu flr lstotlstlj yozĿcĿ` vhyo Hxihc juccoeoroj dhehrshj, ëhjlclőle blstakro`Ŀ

ő÷ych k÷róehihjtlr=

ŕhjlc ?. Flr uelgar`“doe (Yacdojl) 8>>>“clj ëhjlclőle blstakro`Ŀ vh R l 0 -6cak(?-Xl)“led÷eóőó`ó (Yođdojl)

Hđhr Hxihc“dh jhedl foőĿeĿzo flrjoë ÷rehj dheh`hj lstlyarsoeĿz, VOED() gaejslyaeueu

juccoeoroj flrjoë uelgar` roskhch ëhjlclő yorotoflclrslelz. Yaero, `heóchrh tĿjcoyorojblstakro` acuőturoflclrslelz( –Oroëcor‐5 –Whrl Oeoclzl‐ 5 – Oeoclz oroëcorĿ‐ 5 –Blstakro`‐)



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ ?>

Jaerod @hezhc

?6 @ort 6>><

?. 6 vhyo Dobo Gozco Voskhch Dhđlőjhele

GaejslyaecorĿ

6 vhyo dobo gozco roskhch dhđlőjhele flr chőlj dođĿcĿ`Ŀ bojjĿedo bæclbozĿrdo

÷đrhedljchrl`lzl thjrorcoyocĿ`. Dlyhcl` jl R?, R6, … ,Re vor,

Hđhr R?, … ,Re jhsljcl lsh, aecorĿe flr chőlj p.d.g.“sl

lch vhrlclr.

Hđhr R?, … ,Re sórhjcl lsh, aecorĿe flrchőlj p.d.g.“sl pazltlg flr gaejslyaedur,g R?, … ,Re(x?, … ,xe), fu ehdhech bhrboekl flr D e lële

R?, … ,Re fođĿ`sĿzdĿr hđhr oőođĿdojl sođcoeĿrso

Fueue oőođĿdojleh hőlt acduđueu botĿrcoyĿeĿz,

ŕl`dl yujorĿdo tortĿőĿcoe thj dhđlőjhecl duru`doe eosĿc 6 vhyo dobo gozco fayuto

khehch`h yopoiođĿ`Ŀzo fojocĿ`.

Thj fayutcu duru`do acduđu klfl yleh óë duruù oyĿrt hdhihđlz=

?. ľcklcl dhđlőjhe R?, … ,Re jhsljcldlr

6. ľcklcl dhđlőjhe R?, … ,Re sórhjcldlr

9. R sórhjcldlr vh u(R?, … ,Re) e-fayutcu flrh-flr flr gaejslyaedur.



?.?. Jhsljcl Duru`

WorsoyocĿ` jl R?, …, Re flrchőlj yađuecuđu p.d.g. g R?, … ,Re(x?, … ,xe) lch jhsljcldlr vh \?,

… ,\` ` toeh gaejslyae lch vhrlclyar

OyeĿ zoòedo,

Fu duru`do vh \?, … ,\`“le flrchőlj p.d.g.sl oőođĿdojl lch vhrlclr.

×rehj ?(Flea` Voskhch Dhđlőjhechrle TapcoòsĿ). WorsoyocĿ` jl R F(`, p) vh \

F(e, p) p.d.g.sl oőođĿdojl klfl acoe fođĿ`sĿz flea roskhch dhđlőjhechr acsue.

Hđhr ] 0 R + \ acoroj toeĿ`corsoj, p.d.g. g z (z) ehdlr1 R ordĿőĿj ` fođĿ`sĿz dhehydhjl vh\ lsh e dhehydhjl foőorĿ soyĿsĿ acduđu lële (bhr ljlsl dh oyeĿ foőorĿ acosĿcĿđĿeo soblptlr),a zoòe ]“ele dh `+e dheh`hedhjl p acosĿcĿjcĿ foőorĿcorĿe tapco`Ŀ acòsĿ khrhjtlđl flrlcj tob`le acoroj s÷ychehflclr. \oel ] F(` + e, p). Fu dađru acoioj`Ŀő klfl k÷róeóyar,oeioj flël`shc acoroj fueu jaetrac ht`h`lz khrhjlyar=



pz (? ’ p)e -z thrl`l j“yh fođcĿ dhđlcdlr, fu ehdhech aeu tapco`doe ëhjhflclrlz. Dlđhrtorogtoe, oőođĿdojlel lddlo hdlyaru`,

Khrëhjthe dh, soyò jurocĿeĿ juccoeoroj fueu k÷sthrhflclrlz= ëorpĿ` jurocĿ vh

ja`fleosyae gar`ócó lch, ( )() thrl`l ` soyĿcĿ flr kruptoe ëhjlche j hchòe lëhrhe flr

jó`h lch e soyĿcĿ foőjo flr kruptoe ëhjlche z -j hchòe lëhrhe gorjcĿ jó`h soyĿsĿeo

jorőĿcĿj khclr. Fótóe j dhđhrchrl ózhrledhe tapcoyoroj, flrchőtlrlc`lő ljl jó`hdhe (yoel, `

+ e hchòecĿ flr jó`h) flr z jó`hslele hchòecorĿeĿ ëhj`hele tapco` yaccorĿeĿe soyĿsĿeĿ

hcdh hdhrlz. Ja`fleosyae gar`ócóeh k÷rh fu jó`h ( )“ y h hőlttlr, fu do lspotcoòyo

ëocĿőtĿđĿ`Ŀz hőltclđle sođ torogĿdĿr

Fótóe porëocorĿ flr oroyo khtlrhihj acursoj,

F÷ychih khrëhjthe ] F(` + e, p).

Flr thdflr eatu acoroj, khehc acoroj oyeĿ dođĿcĿ` olchsledhe ’fu duru`do flea ’ khche

ljl fođĿ`sĿz roskhch dhđlőjhe R lch \“ele tapco`Ŀ acoe ] oyeĿ olchyh olt acòyoiojtĿr. Fufođco`do, flea` dođĿcĿ` ëaj ÷zhc flr duru`dur vh oyeĿ ÷zhclđh soblp sodhih flrjoëtoeh dobo yoykĿe acoroj juccoeĿcoe dođĿcĿ` vordĿr. ×rehđle, Hđhr R F(`, p R ) ljhe \

F(`, p\ ) vh p R p\ lsh, yujorĿdo hcdh hdlchechr blëflr lőh yoroòyoiojtĿr.

?.6 Yórhjcl Duru`

WorsoyocĿ` jl R?, … , Re flrchőlj yađuecuđu yoel g R?, … ,Re(x?, … ,xe) p.d.g.sl lch sórhjcldlr

vh \ (jovro`Ŀ foslt tutòj lële sodhih flr dhđlőjhe juccoeocĿ`) oőođĿdojl gaejslyae lch

vhrlc`hjthdlr.

Hđhr

lsh, a zoòe \“ele p.d.g.sl oőođĿdojl lch vhrlclr=



6. Flrh-Flr D÷eóőó` lële Dhđlőjhe Dhđlőtlr`hGar`ócó

Fu do yleh sodhih sórhjcl dhđlőjhechr lch ëocĿőoe ÷zhc flr duru`dur= O R?, … ,Re“ele

dhsthjchyhel acsue, yoel

vh F“dh \?, … ,\e ledlrkhe`lő dhsthjchyhel acsue, yoel

WorsoyocĿ` jl \?, … ,\e tórhvchehflclr flrh-flr d÷eóőó`óedhe hcdh hdlche R?, … ,Re“dhehcdh hdlcsle,

\oel (x?, …, xe) O“eĿe bhr dhđhrl (y?, …, ye) F“ele flrhr hcoòeĿyco hőchő`hjthdlr. Fuduru`do Zs?(x?, …, xe), …, se(x?, …, xe)P“ele thrslel toeĿ`coyoflclrlz, f÷ychih

Hđhr s?(.), …, se(.) F ózhrledhe tórhvchehflclrsh oőođĿdojl òtrlslel toeĿ`corĿz.

JĿs`Ŀ tórhvle fu òtrlsl oyeĿ zoòedo Moiafloe“Ŀe thrs d÷eóőó`ó acoroj odcoedĿrĿcĿr.Dađrusoc Ihflr“l ocòyoecorĿe, 6“yh 6 duru`corĿeĿ ëocĿőòcorĿ yhthrcldlr. ľjlyh ljl

duru`corĿedo @otrls O“eĿe dhthr`leoetĿeĿe oőođĿdojl klfl bhsopcoedĿđĿeĿ flc`helzkhrhjlyar=



×ehr`h ?. R ?, …, R e yujorĿdo vurkucoedĿđĿ klfl \ ?, … , \ e lch hőchő`hsl flrh-flr vh thrsl

acoe s?(.), …, se(.) tórhvchehflclr lsh, a zoòe \ ?, … , \ e“ele flrchőlj p.d.g.sl oőođĿdojl lchvhrlclr.

6.? Dađrusoc D÷eóőtór`h

R roskhch dhđlőjhechrle flr vhjt÷ró acsue, yoel dht(O) > acoe flr exe òtrls O lële,

vh

Fu duru`do dađrusoc hőchőtlr`h (òpplek) \ 0 OR flrh-flrdlr (òtrlsle thr sl acduđu lële)vh dhđlőjhe dhđlőtlr`h gar`ócóeó juccoeoroj \“ele flrchőlj dođĿcĿ`ĿeĿ fucoflclrlz.

×rehj 6. Fueue hjaea`ldhjl ÷eh`lel k÷r`hj lële, vorsoyocĿ` jl Fastae“dojl partojocsuyu plyososĿ lële foslt (jĿs`Ŀ dhekh) flr àdhcl`lz vor. Glròcor glyot p“ele flr dađrusoc

gaejslyaeu( s vh s jotsoyĿcĿ) acoe `ljtor qs“l orz ht`hj elyhtledhdlrchr.

qs 0 s + sp + us

Furodo us roskhch flr dhđlőjhedlr (dlyhcl` jl Gcarldo“dojl kóehőcl sootchr klfl). Tójhtlilchrfoőjo flr thsodógl őaj ud vhrlyjhe (dlyhcl` jl khclr) `ljtor qd“yl tochp hdhrchr.

qd 0 d - dp + ud

Dhekh duruùedo, orz tochfh hőlttlr, yoel glyotcor ÷ychdlr jl qs 0 qd 0 q“dlr vh glyotcor lch

`ljtorcor fhrofhr oőođĿdojl lclőjl torogĿedoe fhclrchelr.



Glyot vh `ljtorcorĿe flrchőlj dođĿcĿ`ĿeĿ hcdh hdhihđl`lz őajcorĿe (ud, us) flrchőlj p.d.g.sl

g X(us, ud)“yl flclyar acoflclrlz yo do vorsoyoflclrlz. Fu flrchőlj p.d.g. jhsle őhjlcdh

Moiafloe“o (sac torogtojl òtrls) fođcĿ acoiojtĿr. Fu duru`do dht(M) 0 d + s“dlr, fu

ehdhech hđhr orz vhyo tochp ÷eh`cl (eaetrlvloc) flr hđl`h soblp lsh, őajcordoe glyoto vh

`ljtoro d÷eóőó` flrh-flrdlr vh saeuëto artoyo ëĿjoe flrchőlj p.d.g. oőođĿdojl klfldlr=

Fu duru`, fu dhrsth lőchyhihjchrl`lzdhe flroz uzoj klfldlr oeioj Moiafloe thrl`l | d +

s| plyoso dhekhsl oroiĿcĿđĿyco glyot vh `ljtorĿe jorőĿcĿjcĿ fođĿ`cĿcĿđĿeĿ yojocor. Fueue,

?4.96 dhrsledh plyoso saeuëcorĿedoe orz vh tochfle oyrĿ oyrĿ tob`le hdlc`hslelzarcoőtĿroe –hőoecĿcĿj prafch`l‐ acoroj odcoedĿrĿcoe duruùe joyeođĿ acduđu artoyo

ëĿj`ĿőtĿr . Fu Hjaea`htrlele th`hc prafch`chrledhe flrldlr.

6.6. R +\“ele DođĿcĿ`Ŀ (Fójóc`h)

WorsoyocĿ` jl R vh \ fođĿ`sĿz sórhjcl roskhch dhđlőjhechrdlr vh p.d.g.chrl, sĿrosĿyco, gR(x)

vh g$y)“dlr vh f÷ychih roskhch dhđlőjhechrle flrchőlj p.d.g.chrl g R\ (x, y) 0 g R (x)g \ (y)“dlr.] 0 R + \“ele p.d.g.sl ehdlr1

×rehj 9. Fueue klfl flr ÷rehđl sĿeĿgto yoptĿđĿ`ĿzĿ botĿrcoyĿeĿz= ël` flë`h òjlehsledhjlljl fumlel ÷`róeh foj ĿőtĿj , vh ^(R + \ z) acosĿcĿđĿ {(x, y) = y z ’ x} lch toeĿ`coe`Ŀő

óëkhe ózhrledhe g R\ (x, y) flrchőlj yađuecuđue lethkrocl acduđu artoyo ëĿj`ĿőtĿ. Fu

duru`do ]“ele i.d.g.sl oőođĿdojl klfldlr=

Furodoe, ]“ele yađuecuđueu hcdh hdhflclrlz,

Voskhch dhđlőjhe ] 0 R + \, R vh \“ele fójcó`ó acoroj to odcoedĿrĿcĿr. Yae gar`ócóe

sodhih fođĿ`sĿz roskhch dhđlőjhechrle tapcoeòsĿ bocledh khëhrcl acduđueu eat hdlelz.

×rehj 4. ×eihjl ÷rehjthjl tortĿőò –6 -odĿ`‐ y÷eth`lele ëlzklslych oyeĿydĿ vhdhđlőjhechrle d÷eóőó`ó gar`ócóeó jhstlr`h flr yac acoroj juccoeòeĿe `ó`jóe acupacòyoiođĿ `hroj hdlchflclr.

(R, $“dhe ]“yh hőch`h oëĿjëo flrh-flr dhđlcdlr, fu ehdhech dhđlőjhechrle d÷eóőó`gar`ócóeó dađrudoe juccoeoòyĿz. Oeioj, oőođĿdojl –blchyl‐ yopoflclrlz= oőođĿdojlchrtoeĿ`coyocĿ`.



A zoòe, thrs d÷eóőó` ő÷ych toeĿ`coeĿr=

Fu duru`do

Fuedoe ÷tóró,

Flrchőlj p.d.g.“yl w ózhrledhe lethkroclel ocoroj ]“ele òrmleoc p.d.g.slel hcdh hdhflclrlzortĿj .

Fu flr ÷eihjl tórht`hdhe hcdh hdlche gar`ócóe oyeĿsĿdĿr.

×rehj 8. ŕl`dlyh jodor ósthc dođĿcĿ`Ŀe flrjoë ÷rehđlel k÷rdój (ël` flë`h òjlehsl÷rehđledhjl klfl). R vh \ fođĿ`sĿz ósthc roskhch dhđlőjhechr dlr vh òrmleoc p.d.g.chrl

ő÷ych acsue=

Yae gar`óc lch , ] 0 R + \“ele p.d.g.sl oőođĿdojl klfldlr=



Furodo, ljleil odĿ`dojl lethkrocle cl`ltl R vh \“ele dhsthđlele pazltlg rhhc soyĿcor lch sĿeĿrcĿacòsĿ khrëhđledhe khc`hjthdlr. \oel z 3 > lële g R ( z) sĿgĿrdĿr, bocfujl z 5 w lële g \ (z - w)

sĿgĿr acur.



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ ?>

Jaerod @hezhc

?6 @ort 6>><

?. ľethkrocle Cl`ltlele D÷eóőó` Gar`ócóeóe ×rehđl

] 0 R/\“ele p.d.g.sl ehdlr1

?.?. \ojcoőĿ` ?. –6-odĿ`‐ y÷eth`l, ëaj joròőĿj

x/y 6 acoe (x, y)“yl fuc.

(x, y)“chr ózhrledhe g R\(x,y)“ele lethkroclel ocoroj i.d.g. G](z)“yl hcdh ht.

G](z)“ele tórhvlel ocoroj p.d.g. g ](z)“yl hcdh ht.

fueu yopòyoiođĿz ëóejó dobo jacoy flr yojcoőĿ` vor.

?.6. \ojcoőĿ` 6= Dhđlőjhele-dhđlől`l gar`ócó

^rafch`= z 0 u?(x, y) 0 x/y flr fayutcu, u(.) flrh-flr acoòz.

Ë÷zó`= yhel flr dhđlőjhe toeĿ`co, w 0 u6(x, y) 0 R\ Z P 0 “ele thrslel oc.

(x, y“ele ? acosĿcĿjco pazltlg acduđueu eat hdlelz.)

Thrs gaejslyae

Moiafloe



dht (M)

([, ])“ele flrchőlj p.d.g.slel hcdh ht`hj lële gar`óc juccoe

Jaőuc (*) oőođĿdojledhe khclr

vh

OőođĿdojlel eosĿc hcdh hdhrlz1

a [ > lële g [](w, z) sĿgĿrdĿr

a [ 5 `le(], ?/]) lële G[](w, z) sĿgĿrdĿr

a Fuedoe ÷tóró,





AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ ??

Jaerod @hezhc

?; @ort 6>><

?. YĿro ľstotlstljchrl

R?, …, Re p.d.g.chrl g x?(x) 0 … 0 g xe(x) fhezhr acoe fođĿ`sĿz roskhch dhđlőjhechr acsue ’ khehccljch f÷ych flr sĿrocĿ lgodhyl –fođĿ`sĿz vh oyeĿ(fhezhr) dođĿcĿ`cĿ‐ acoroj odcoedĿrĿrĿzvh l.l.d. acoroj jĿsoctĿr Ŀz. OőođĿdojl gaejslyae lch lcklchelyaruz.

yoel \ e ÷rehjch`le he fóyój dhđhrldlr.

FođĿ`sĿzcĿđĿ juccoeoroj \ e“ele i.d.g.slel tórhthflclrlz.

]leilr jurocĿeĿ juccoeoroj, òjslùùe p.d.g.slel hcdh hdhflclrlz.

×rehj ?. Hsjl flr rhsl` flr oëĿj orttĿròdo sotĿcĿr. e jlől oëĿj ortĿròdo fođĿ`sĿz acorojF?,…, Fe thjclgchrlel sueorcor vh thjclgchrle òrmleoc i.d.g.sl G F(f)“dlr . He yójshj thjclgl

vhrhe patoeslyhc ocĿiĿ rhs`l ocoioj acoe jlőldlr vh thjclg `ljtorĿeĿ ÷dh`hj zaruedodĿr (fu

tór oëĿj ortĿròcor Dutib, yo do flrleil glyot oëĿj ortĿròsĿ acoroj flclelr). Fu duru`dorhsl` sotĿiĿsĿeĿe bosĿcosĿeĿe p.d.g.sl oőođĿdojl lch vhrlclr=

ŕl`dl fueu ÷rehjch`dhjl dlđhr sĿrocoòcor o khehcchőtlrhflclrlz. ×rehđle,



\ e-? 0 –R ?, …, R e“ele he yójshj ljleil dhđhrl‐

Fu roskhch dhđlőjhe R ?, …, R e“ele (e-?)“eil sĿrocĿ lstotlstlđl acoroj odcoedĿrĿcĿr vh aeue p.d.g.slel fhclrchyhflclrlz.

×ehr`h ?. R ?, …, R e p.d.g.sl g R (g) vh i.d.g.sl G R (x) flr l.l.d. roskhch dhđlőjhechr slcslchslacsue. Fu duru`do j “eil sĿrocĿ lstotlstlj \ j “eĿe p.d.g.sl ő÷ychdlr=

ľYÔT= Dhehyl ljl f÷có`h oyĿroflclrlz, (o) R“chr dhe flrl yađuecuj g R(y)“h k÷rh y dhđhrlelocòj zaruedo acsue, vh (f) y dhđhr l vhrlyjhe, dlđhr ëhjlclőchr dlzlsl y“ele htrogĿedo y÷rehjch`le he jóëój j “ eil dhđhr acoioj őhjlcdh krupcoedĿrĿcsĿe.

F÷có` (f) e dhehyle R?, …, R

e“le e ëhjlclőleh jorőĿcĿj khche flr flea` dhehydlr vh l “ eil

turdojl –foőorĿ‐ acoy ( R ? y ) acoroj toeĿ`coeĿr. Ëhjlclőchr fođĿ`sĿz vh oyeĿ p.d.g. lchlcletlcl acdujcorĿ lële, flea` dođĿcĿ`dojl p poro`htrhsl GR(y)“h hőlttlr. y“ele dobo jóëójacòsĿ vhyo he jóëój j eil dhđhrh hőlt acòsĿ flea` dođĿcĿ`dojl he oz j jodor –foőorĿ‐lch lcklcldlr vh fu ehdhech lcklcl i.d.g oőođĿdojl klfldlr.

ŕl`dl i.d.g.“ele y“yh k÷rh tórhvlel ëorpĿ` vh zleilr jurocĿ lch ocoroj p.d.g.yl hcdh hdhflclrlz.

Fu lgodh joròőĿj k÷róeóyar, gojot fueue hsosĿedo thchsjaplj flr tapco` acduđuoecoőĿcoflclr, fuedo ÷tóró tapco`cĿ thrl`chrle (su`òed) ëađu dóőhihjtlr. ľjleil thrl`dhjlc 0 e lch lcklcl tapco` dhđhrle sĿgĿr acduđueo dljjot hdlelz. Aeu yheldhe yozoflclrlz

furodo `hviut c hedhjsl c-? lch yhr dhđlőtlr`lőtlr. Flrleil thrl` lële oőođĿdojl s÷zjaeusudur=



Fu duru`do flrleil thrl` oőođĿdojl klfl acur

F÷ychih yađuecuj T?“l toeĿ`coyoe c 0 j thrl`le tapco`Ŀeo hőlttlr ëóejó T6“ylëĿjordĿđĿ`Ŀzdo yaj acòyoe thj thrl`dlr. Fu ehdhech,

fu lspotĿ yopĿcoe saeuëtur.

×rehj 6. OcĿiĿeĿe he yójshj glyot thjclglel vhrlp rhs`l ocdĿđĿ flrleil oëĿj ortĿròdoe gorjcĿflr oëĿj ortĿròyĿ ől`dl dóőóehflclrlz. Fueo k÷rh he yójshj glyotĿ thjclg hdhe yleh rhs`l

ocĿr oeioj fu duru`do ljleil he fóyój glyot thjclgl jodor ÷dh`hj yopòj zaruedodĿr (fu

oëĿj ortĿrò őhjcl lcjleh k÷rh dobo yoykĿedĿr vh ľeklclz yodo ljleil -glyot oëĿj ortĿròsĿacoroj flclelr). Hđhr thjclg hdlche glyot roskhch dhđlőjhechr, I ?, …, I e, lsh, sotĿiĿeĿe khclr l\ ől`dl oőođĿdojl p.d.g.“yh soblptlr.

OyeĿ glyot thjclgl el vhrhe jlől ele ljl gorjcĿ oëĿj ortĿrò garòtĿeo gorjcĿ glyot thjclgl vhr`hslkhrhjtlđlel hjaea`l tharlsledhe flcdlđl`lz lële, glyot thjclgchrl lële gorjcĿ borg juccoedĿđĿòdljjot hdlelz.

6. Flr poroethz oëòj= –Bhrboekl flr soyĿeĿe lcjfosoòđĿ‐“eĿe dođĿcĿ`Ŀ (dhrsth lőche`hdl).

Furodo, jhedlsl lële dobo ÷eih k÷rdóđó`óz y÷eth`chrl juccoeòyoiođĿ`Ŀz, baő oòstoedort acòyoe flr prafch` vor. Fu jhslecljch prafch` shtl vhyo sĿeovo bozĿrcĿj lëleózhrledh duròyoiođĿeĿz flr őhy, oò fhe yleh dh dhđle`hj lstlyaru`.



BojjĿedo blëflr őhy flc`hdlđl`lz soyĿcorĿe flrleil fosoòjcorĿeĿe dođĿcĿ`Ŀ ehdlr1 DobooëĿj acòj khrhjlrsh, R“le ehyl th`slc httlđlel vhyo eh tór flrl` (?/\) juccoedĿđĿeĿflc`hdlđl`lz bhrboekl flr őhyle ÷cëósó acsue. ×rehđle flr kozhthyl leihchylp bhrboekl flrőhyl (khclr, farso hedhjschrl, eógus vs.) ÷cëhe rojo`corĿ tapcoyoflclrlz. Ehrhdhe khcdlđlbojjĿedo foőjo blëflr őhy flc`hdlđl`lz fu rojo`corĿe flrleil fosoòjcorĿeĿe p.d.g.“sl

ehdlr1 \oel R vh \“ele pazltlg acò dĿőĿedo bhrboekl flr dhđhr acoflchihjchrl flcdlđl`lzthj őhy lsh, ] 0 R.\ roskhch dhđlőjhele lcj aedocĿđĿeĿe p.d.g.slel eosĿc tórhthflclrlz1

Yhzklshc acoroj, uelgar` dođĿcĿ` soejl rojo`cor vh flrl`chrl bojjĿedo ëaj gozco –flckl‐lëhr`hdlđl lële, lcj tob`lel`lz, lcj fosoòđĿe uelgar` (jhsljcl) dođĿcĿ`cĿ acòsĿ acoflclr.

Oeioj, hđhr uelgar` dođĿcĿ`Ŀ ocĿrsoj vh flrl`chrl dhđlőtlrlrshj (÷rehđle vorsoyĿcoedođĿcĿ`do fótóe rojo`corĿ ljlyh vh d÷rdh jotcorsoj), flrleil fosoòjcorĿe dođĿcĿ`Ŀuelgar` acoroj jocòz. ×rehđle, hđhr khrëhj rojo`cor R XZ?, ?>P , 4R XZ>, 4>P lsh,4R“le flrleil fosoòđĿ \ oőođĿdojl p.d.g.yh soblptlr.

\o do, k÷rócdóđó klfl, flrleil fosoòjcorĿe dođĿcĿ`Ŀ bojjĿedo `lelòc dózhydh khrhjclacoe flckl, ÷cëó` flrl`lel dhđlőtlrdlđl`lzdh dođĿcĿ`Ŀe dhđlőh`hyhihđldlr.

Khrëhjth orodĿđĿ`Ŀz, ÷cëhj dhđlől`leh fođcĿ acoroj dođĿcĿ`Ŀ dhđlő`hyhe flr roskhchdhđlőjhe R“tlr, yoel o 5 R lële oR. Hđhr ] 0 cak(R) XZcak(?), cak(?>)P vorsoyorsoj fudađrudur , ëóejó flr ÷cëhj joyòsĿ lële oőođĿdojl hcdh hdlclr=

A zoòe ]“ele lcj fosoòđĿ \ oőođĿdojl p.d.g.yh soblptlr.



ŕhjlc ?= Xelgar` Ózhrledh ×cëó` Flrl`l Dhđlőljclđlele Htjlsl

Fu dhđlő`hyhe gljlr flr dođĿcĿ`Ŀ hcdh ht`hele ëaj yopoy flr yacu klfl k÷róehflclr, ëóejódođĿcĿ`Ŀe R ÷cëó`ó vhyo ÷cë`h flrl`l lch ëaj fhclrkle flr fođĿ yajtur. Oeioj, artoyoëĿjoe p.d.g. jothkarlyh dóőhe –khrëhj-dóeyo vhrlsl‐ bojjĿedo ëaj lyl flr tob`le vhrlyarklfl k÷róeóyar. ×rehđle Ehw \arj Tl`hs“to k÷róehe rojo`corĿe flr őhyl ÷cë`hslel th`slcht`hchrl klfl. OőođĿdojl őhjlc, Hiaea`lst“le –Vojo`corco Dóeyo 6>>;‐ ihp jltofĿedo yhrocoe ;; ócjhele ucusoc poro flrl`l ilesledhe (Mopaeyo lële \he, Joeodo lële I$ klfl)KY\ľB“corĿe flrleil fosoòjcorĿeĿe blstakro`Ŀ lch fhrofhr –tharlj‐ yađuecujcorĿeĿ

k÷sthr`hjthdlr. ×zhtchyhihj acursoj, fu ÷rehj vhrlche ljl roskhch dhđlőjhele dođĿcĿ`ĿeĿfhclrch`hdh dhđlőlj rodljoc flr yojcoőĿ` artoyo jayòjtodĿr= furodo R lch \“elep.d.g.chrlel flc`hdhe foőcodĿj, gojot bhr eosĿc flr dođĿcĿ` artoyo ëĿjoiojso, flrl`dhđlőljcljchrledhe htjlche`h`hj zaruedo acduđueu fhclrttlj, yoel \“ele khrëhjchő`hslklfl. –Dhđlő`hyhe (levorloeih)‐ lcchrl lstotlstljth ëaj ÷eh`cl yhrl acoe flr jovro`dĿr, gojotfu dhrsle oòiĿ ehdhelych, fu ÷rehjthe ÷thyh klt`hyhihđlz.



ŕhjlc 6= \hrhc ôro Ilesledhe KY\ľB“corĿe lcj Vojo`corĿeĿe DođĿcĿ`Ŀ vh Tharlj\ađuecujcorĿ (rojo`cor Tbh Hiaea`lst, âijht [arcd le Glkurh 6>>;“dhe ocĿeòdĿr )

9. Fhjchehe Dhđhr vh @hdyoe

^.d.g.sl g x (x) acoe flr roskhch dhđlőjhe R vhrlc`lőjhe, tó` yađuecuj dođĿcĿ`ĿeĿ vhr`hjzaruedo jocòdoe tó` dođĿcĿ`Ŀe he ÷eh`cl ÷zhcljchrlel ÷zhtch`hj lstlyaruz. Fhjchehe

dhđhr hsos ltlforlych flzh R“le dođĿcĿ`ĿeĿe ehrhdh `hrjhzchedlđlel s÷ychr.

9.? ToeĿ`cor

ToeĿ` ?. Hđhr R jhsljcl roskhch flr dhđlőjhe lsh, tapco`Ŀ saecuyso, R“le H|R| lch fhclrtlchefhjchehe dhđhrl oőođĿdojl klfldlr.

Hđhr R sórhjcl lsh, lethkrocl saecuyso, fhjchehe dhđhr oőođĿdojl klfl toeĿ`coeĿr

×rehj 9. Flea` roskhch dhđlőjhele, R F(e, p), fhjchehe dhđhrl ehdlr1



furodo ljleil sĿrodo, x 0 > lch lcletlcl tapco` dhđhrl k÷r`hzcljthe khchflclrlz ëóejó sĿgĿro

hőlttlr. Óëóeió sĿrodo e“yl flea` jotsoyĿsĿedoe ottĿj vh lzchyhe odĿ`do, tapco`hedhjslel x“the x -?“h d÷eóőtórdój. Yaeuë acoroj, hđhr ep“yl ëhjhrshj, tapco`cor

F(e-?, p)“le flea` acosĿcĿjcorĿ acur vh fu ehdhech tapco`corĿ flrdlr.

Yaesuz soyĿdo dhđhr ocoflche flr roskhch dhđlőjhele saecu flr fhjchehe dhđhr h soblpacoòyoflchihđleh dljjot hdlelz. Fu duru`do fhjchehe dhđhr toeĿ`coeò`ĿőtĿr . Bhr ehjodor fhjchehe dhđhr dođĿcĿ`Ŀe –jaeuù‐ bojjĿedo flckl vhrsh dh, khehc acoroj aeueroskhch flr dhđlőjhe lële –tlplj flr dhđhr‐ acòdĿđĿeo do dljjot hdlelz= ÷rehđle zor

otòeĿe fhjchehe dhđhrl

(?+6+9+4+8+7) 0

“dlr jl fu `ó`jóe acoe flr saeuë

dhđlcdlr. DođĿcĿ`Ŀe jaeuùeu bhsopcoòeĿe dlđhr flr shëhehđl dh `hdyoedĿr .

ToeĿ` 6. Voskhch flr R dhđlőjhele `hdyoe `(R)“l rhhc flr soyĿdĿr yoel

^(R 3 `) 0 ?/6

Hđhr roskhch dhđlőjhe R“le dođĿcĿ`Ŀ `(R) htrogĿedo sl`htrlj lsh R“le `hdyoe vhfhjchehe dhđhrl ëojĿőĿr, yoel g x (`(R) ’ x) 0 g x (`(R) + x)“dlr , gojot fu duru` khehcdh oyeĿdhđlcdlr.

×rehj 4. Dlyhcl` jl R“le p.d.g.sl ő÷ychdlr=

Fhjchehe dhđhrl lsh,



@hdyoeĿ hcd h ht`hj lële, ÷eih R“le i.d.g.slel bhsopcoyocĿ`

DacoysĿyco, ` lële G R (`) 0 ?/6 “l ë÷z`hj oőođĿdojlel vhrlr

Fuedoe ÷tóró, fu dođĿcĿ`Ŀe `hdyoeĿ artocoòsĿedoe fóyójtór.

@hdyoeĿe thj acòyoflchihđleh dljjot hdlelz.

×rehj 8. R odlc flr zorĿe otĿőcorĿeĿe saeuiu acsue. Bhrboekl flr soyĿ lële ` (9, 4P, ^(R

3 `) 0 ^(R 9) 0 ?/6 “ d lr. DacoyĿsĿyco, a orocĿjtojl bhrboekl flr soyĿ `hdyoedĿr.

9.6 Fhjchehe Dhđhrle ×zhcljchrl

×zhclj ?. Hđhr R 0 i vh i soflt flr dhđhr lsh, a zoòe

H|R| 0 i

×zhclj 6. Hđhr \ 0 oR + f, lsh, a zoòe

H|\| 0 oH|R| + f

ľYÔT= Yodhih sórhjcl duruò fojocĿ`= Hđhr R p.d.g.sl g R(x) acoe flr sórhjcl roskhchdhđlőjhe lsh, a zoòe \“ele fhjchehe dhđhrl ő÷ychdlr=

F÷ychih k÷rócdóđó klfl, lethkrocle dađrusoccĿđĿ dađrudoe fhjchehe dhđhrledađrusoccĿđĿeo d÷eóőór.

×zhclj 9.

lële



Fu dađrusoc fhjchetlchrle he khehc duruùdur vh fu ÷zhcclđl fuedoe saerojl dhrschrdhthjror thjror juccoeoiođĿz.

×rehj 7. \ujorĿdo, R F(e, p)“ele fhjchehe dhđhrlel, ZRP 0 ep, R“le acosĿ fótóesaeuëcorĿeĿe ózhrledhe tapco` yoporoj bhsopcodĿj. Gojot sae saeuëtoe, oyeĿ saeuiuhcdh ht`hele foőjo dobo jacoy flr yacu acduđueu k÷rhflclrlz= R ordĿőĿj e dheh`hele

foőorĿ saeuiu acduđu lële, bhr flr dhehyle saeuiueu ] ?, ] 6 , …, ] e acoroj jadcoyoflclrlz.

Furodo hđhr l“ eil dhehy foőorĿ lsh ] l 0 ?“dlr, dlđhr duru`cordo ] l 0 >“dĿr.

vh dacoysĿyco

×zhclj 4. Hđhr R lch \ fođĿ`sĿz lsh, a zoòe

Hđhr R lch \ fođĿ`sĿz dhđlc lsh, fu khehc acoroj dađru dhđlcdlr.

9.9 Voskhch Dhđlőjhechrle GaejslyaecorĿeĿe Fhjchehe Dhđhrl

\ 0 r(R) acsue. Khëhe bogto, hđhr g x(x)“l flclyarsoj, \“ele p.d.g.slel eosĿc tórhthflchihđl`lzlk÷r`óőtój. Fhjchehe dhđhr lële fu sarue dobo jacoydĿr ëóejó flz dođĿcĿ`Ŀe sodhih flrthj ÷zhclđleh fojĿyaruz.

\ 0 r(R) fhjchehe dhđhrl oőođĿdojl klfldlr

×rehj ;. WorsoyocĿ` jl , \ 0 R ?/6 vh R“le p.d.g.sl oőođĿdojl klfldlr



OyeĿ juroccor 6 vhyo dobo gozco roskhch dhđlőjhe lëledh lőh yoror .

×rehj 2. WorsoyocĿ` jl flrchőlj p.d.g.sl, oőođĿdo vhrlche R vh \ klfl ljl roskhch

dhđlőjhele ] 0 R 6 + \ 6 gaejslyaeu lch lcklchelyaruz.

A zoòe

Voskhch dhđlőjhe R“le \ 0 oR + f tóróedhe dađrusoc gaejslyaecorĿ lële, yujorĿdo ZoR +

fP 0 oZRP + f acduđueu k÷r óőtój. Fu dađrusoc acòyoe roskhch dhđlőjhechrlegaejslyaecorĿ lële ëocĿőòz. Fueue he tlplj saeuiu Mheshe“le Hőltslzclđl“dlr=




ŕhjlc 9. {x?, x6} lële Flr Jhsljcl DođĿcĿ` ×rehđl

×ehr`h 6 (Mheshe“le Hőltslzclđl). R roskhch flr dhđlőjhe vh u(x) jaevhjs flr gaejslyaeacsue. A zoòe,

H đhr u(.) jhsle jaevhx vh R pazltlg acosĿcĿjco he oz ljl gorjcĿ dhđhr ocĿr lsh hőltslzcljjhsledlr(strlit).

ľYÔT= (ZRP, u(ZRP)) eajtosĿedoe khëhe vh u(x)“h thđht flr dađrusoc gaejslyae toeĿ`coyoflclrlz=

u(.) jaevhjs acduđu lële, fótóe x“chr lële oőođĿdojl lclőjlyl hcdh hdhrlz=

×zhccljch,



r(x) dađrusoc acoroj acuőturucduđu lële, o 0 u“ ( ZRP) vh f 0 u( ZRP) ’ u“ ( ZRP)ZRP “cl dađrusocgaejslyaeue fhjchehe dhđhrl lch lcklcl ×zhcclj 6“yl, oőođĿdojlel hcdh ht`hj lëlejuccoeoflclrlz.

Fueu dobo ÷eih tórhtlche hőltslzcljch flr oroyo khtlrhihj acursoj lspot toò`coe`Ŀő acur=

Flr jaejov gaejslyae v(x)“le ehkotlgl ’v(x) jaevhjs acduđuedoe, Mheshe“le Hőltslzclđl dhflr jaejov v(.) lële oőođĿdojlel sođcor =


ŕhjlc 4. r(x) bhr zoòe u(x)“the jóëójtór.

×rehj < (rlsjthe joëĿeò)= WorsoyocĿ` jl lcj shehsl lële sĿeĿrcĿ koroetlych khche ?6>>

dacorcĿj flr dlzóstó flcklsoyor ocdĿeĿz. A lcj yĿc sórhsleih, p 0 %?> acosĿcĿjco flr fordojjobvhyl dlzóstó flcklsoyorĿe ózhrleh d÷j`h (yo do slzle botoeĿz acoe foőjo flr jozo) vh??>> dacoro òc acoioj oeojortĿ dhđlőtlr`h lbtlòclelz vor. Fu to`lrot sĿeĿrcĿ koroetl jopso`Ŀeo klr`hz oò slz ??8 dacoro uzotĿc`Ŀő flr koroetl (shrvls) ocoflclrslelz. Fu lcovh

–slkarto‐yĿ ocòcĿ `ĿsĿeĿz1

ľcovh slkarto acòdoe, ?- p acosĿcĿjco, dlzóstó flcklsoyorĿe tapco` òclyhtlel roskhch flr

dhđlőjhe acoroj R 0 ?6> dacor, p acosĿcĿjco, R 0 ?6>> + ??>> 0 69>> dacor acoioj klfl



dóőóehflclrlz (fu prafch`l gorjcĿ őhjlcchrdh acuőturòj `ó`jóedór, oeioj ől`dlclj bhrőhyl foslt tutocĿ`). XzotĿc`Ŀő shrvls pcoeĿyco dlzóstó flcklsoyorĿeĿz slzh R 0 ?6>> + ??8 0

?9?8 dacoro òc acoiojtĿr.

Hđhr slz sodhih dlzóstó flcklsoyorĿe fhjchehe dhđhrlych lcklchelyarsoeĿz, a zoòe ZRP) 0

69>>p + ?6>>(? ’ p) 0 ?6>> + ??>>p. Fu, hđhr p %?>.48 lsh, Z\P) 0 ?9?8“the dobofóyójtór. Gojot p 0 %?> dhdlđl`lz lële, uzotĿc`Ŀő shrvls pcoeĿeĿ sotĿe ocòj boco lyl flr

gljlr `ldlr1 - ľjtlsotëĿcor, lesoecorĿe fhclrslzclj duruùedo joror ocdĿjcorĿ zoòe,fhjchehe borioò `ljtorĿ [“yco phj lcklche`hdljchrlel vorsoyorcor, oò boriodĿjcorĿdacordoe hcdh hdhihjchrl goydo (tapco` borioò `ljtorĿedo lcovh flr dacorcĿj ortĿőĿe lcovhdhđhrl acduđu lële) u([) `ljtorĿyco lcklchelrchr. Fu X(.)“eue òclyhtth jaejov acduđueuvorsoydĿđĿ`Ŀz oeco`Ŀeo khclr, dlyhcl` jl

furodo foőcoekĿëtojl vorcĿđĿ`ĿzĿe 42>> acduđueu vh 42>> - I borioyoflchihđl`lzlvorsoyĿyaruz . I dlzóstó flcklsoyorĿe tapco` òclyhttlr. Fu duru`do, lcovh shrvls pcoeĿeosoblp acòòeĿe fhjchehe goydosĿ

Oeioj slkarto pcoeĿyco do őueu fucuruz=

Khrëhjthe dh, slz 42>> ’ 942? 0 ??< dacorĿ slkarto lële borioòj lsthyhihjslelz,bæcfujl fhjchehe lcovh òclyht sodhih ??>>p 0 ??> dacordĿr. Ylkarto lële ÷dh`hj

lsthdlđl`lz fu < dacorcĿj gorjo u(.)“eue jaejov acòsĿedo khche rlsj -prl`l dhelclr.

Mheshe“ le Hőltslzclđl “eh k÷r h, hđhr u(.) jaejov lsh fu rlsj -prl`l pazltlgtlr vh fu tór

thrilbchrle rlsjthe joëĿeò k÷sthrkhsl acuđueu s÷ychyhflclrlz.

×rehj ?>. ľzchyhe ÷rehj Yt. ^hthrsfurk ôrodajsu acoroj flclelr vh saecu fhjchehedhđhrl acòyoe roskhch flr dhđlőjhe ÷rehđlel vhrlr.

Flzh oőođĿdojl flr juòr ÷ehrlclr= WorsoyocĿ` jl odlc flr òdhel poro turo khcleihyh

jodor thjror thjror otĿcĿr. ľcj otĿőto turo khclrsh 6 dacor, 6“ eildh khclrsh 6 6 dacor vh khehc

acoroj x “ eil shghrdh k÷róeórsh 6 x dacor jozoeoiojsĿeĿz.

Fu ayueu ayeoòj lële eh jodor ÷dhrdlelz1 ^rheslp acoroj, fhjchehe jozoeiĿeĿz jodorvhr`hj elyhtledh acursueuz, fu ehdhech ől`dl bhsopcoòcorĿ yopocĿ`= khrhj duyucoe

to` x otĿőĿe acosĿcĿđĿ oőođĿdojleh hőlttlr=



Fu ehdhech, fhjchehe jozoeĿ`cor, \, oőođĿdojl klfl bhsopcoeĿr=

Fuedoe ÷tóró, fhjchehe jozoeĿ`cor lch lcklcl óst sĿeĿr yajtur.

Fu, fu tór foblschr lële lesoecorĿe saesuz `ljtordo ÷dh`h yopoiođĿeĿ k÷rhihđl`lzoeco`Ŀeo `Ŀ khclyar1 Jhslecljch boyĿr= khehccljch lesoecor ayueu ayeoòj lële oőođĿyujorĿ he gozco 68 dacor vhrlr. Fu porodajs gorjcĿ yaccordoe ë÷zóchflclr=

lesoecor khehccljch fhjchehe poro `ljtorĿyco lcklche`hz, oeioj soblp acdujcorĿtapco` `ljtor lëledh dhđhr vhrdljchrl poro `ljtorĿ ozocĿr, yoel ÷eihjl ÷rehjth

acduđu klfl lesoecor flr ëhőlt jaejov u(.) gaejslyaeueu òjsl`lzh hdhrchr.

ëaj jóëój acosĿcĿjcorco, jozoeoiođĿeĿz `ljtor ëaj yójshjtlr- yoel trlcyae, jotrlcyae

dacorcor klfl, vh ÷dh`hyl yopoioj acoe jorőĿ torogĿe vhrdlđl s÷zh fođcĿ jocoiođĿeoleoeòyĿz , fu duru`do khrëhjth f÷ych flr foblsthe he lyl uùtco eh jodor

jozoeoflchihđl lzh dolr flr ëhőlt óst sĿeĿr vordĿr.

Mheshe“le Hőltslzclđlych fođcoetĿyĿ thjror juròj lële, p 0 ?/o acosĿcĿjco turo khche flròdhel poro lch ayeoeoe ayueue fhjc hehe otĿő soyĿsĿeĿ bhsopcoyocĿ`=

furodo jacoycĿjco jaetrac hdhflchihđlelz klfl K“(o) oőođĿdojl lclőjlele ?/o“yo k÷rh flrleil

tórhvldlr

Fu ehdhech, yhel lgodhele K(?/o) lële tórhvlel ocĿeio, oőođĿdojlel hcdh hdhrlz

vh ÷rehđl`lzdh ?/o 0 ?/6 acduđu lële, fhjchehe otĿő soyĿsĿ 6“dlr.



F÷ychih, lesoecorĿe boco fobls lële juccoeòj lsthdljchrl 68 dacor artocoò otĿőcorĿejozoeiĿeo dacor jodor uzojtĿr. Fueue oëĿjcoòsĿ flr jhrh dobo Mheshe“leHőltslzclđl“dlr, vh khrëhjth u(x) 0 6 x x“le (hjstrh`) flr jaevhjs gaejslyaeudur.

×rehj ??. WorsoyocĿ` jl ljl dhđhrcl jođĿt orosĿedoe shë l` yopòj duruùedosĿeĿz=

Flrleilsl lesoecoro lethreht soygocorĿeĿ fhdovo orottĿroe `hëbuc yhel foőcoyoe flr lethrehtglròsĿeĿe blssh shehtchrldlr. %<> acosĿcĿjco jor poycorĿ h>t 0 ?“dh soflt jocòsĿedoe÷tóró ëaj rlsjcldlr vh %?> acosĿcĿjco glròeĿe odĿ Kaakch“dĿr . t zoòeĿedo bhr boekl flr

oedo ÷dhyhihđl jor poycorĿ h>.?T jodor fóyór, yoel sĿrosĿyco %<> acosĿcĿjco %>, %?>

acosĿcĿjco %?> dhđhrlel ocoe roskhch flr fóyó`h aroeĿ K? vordĿr. Dlđhr shëhehj lsh,khchihjth bhrboekl flr t zoòedo h>.>6t golz ÷dhyhihj acoe dhvcht tobvlclel tutòj acoflclr,

yoel jhsle acoroj K6 0 %6“dlr.

t zoòedo ocoiođĿeĿz flr dacoro ől`dl soblp acduđueuz dacorĿe h->.?8t “l jodor dhđhrvhrlrslelz, gojot yleh dh ljlsledhe flrleh yotĿrĿrsĿeĿz. \oel khehc acoroj, khtlrlsl k

aroeĿedo fóyóyhe dhđhrcl flr jođĿdo jhsle acoroj oőođĿdojl jodor dhđhrl flëhrslelz =

Vlsjcl blssh shehdlele fhjchehe jor poyĿ fóyó`h aroeĿ lsh,

Oeioj, blssh shehdleh flëtlđlelz dhđhr őudur=

B ocfujl tobvlc lële flëtlđlelz dhđhr oőođĿdojl jodordĿr=

Yhzklshc acoroj, fóyó`h aroeĿ ózhrledhjl fhclrslzclj, bhr eh jodor yhel gooclyhth

foőcoyoecorĿe %<>“eĿ fóyó`hsh dh (boto lgcosto hdhflclrchr), %?>“u leoeĿcòz flr őhjlcdh

j÷tó kldhe yotĿròcorĿ thcogl httljchrl oeco`Ŀeo khclr. Flël`shc acoroj, fóyó`h

aroecorĿedo W(k) gaejslyaeu jaevhjstlr, f÷ychih Mheshe“le Hőltslzclđleh k÷rh, yotĿrĿìĿcorfóyó`h aroeĿedojl rlsjh dhđhr flë`hcldlrchr ’ `ljtoro (dózhyh) dhđlc.



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ ?6

Jaerod @hezhc

@ort ?< 6>><

?. @hdyoe vh \ózdhcljchrle ×zhcljchrl

Voskhch flr dhđlőjhele `hdyoeĿeĿ ő÷ych toeĿ`corĿz

^(R5 `hdyoe(R)) 0 ?/6

R jhsljcl vh i.d.g“dh sĿëroòcoro ehdhe acoe eajto yĿđĿcòcorĿeo soblp lsh, fu toeĿ`yororcĿ acoòyoflclr , fu ehdhech dobo khehc duru`do, `hdyoeĿ oőođĿdojl klfltoeĿ`corĿz

Dor toeĿ` lch lcklcl dhđlőljclj, i.d.g.“ele sórhjslzclđh soblp acòsĿ vh ?/6“ele ózhrleh

ëĿjòsĿyco, `hdyoeĿ sodhih sórhjslzclj eajtosĿedo oroòjtĿr. R dođĿcĿ`ĿeĿe dlđhr

yózdhcljchrlel dh toeĿ`coyoflclrlz.

ToeĿ` ?. Voskhch flr dhđlőjhe R lële, quoetlch oőođĿdojl lch vhrlclr

OyrĿio q(R, p/?>>)“yl p“ eil yózdhclj(phrihetlch) acoroj odcoedĿrĿyaruz.

Fu toeĿ`doe borhjhtch, `hdyoeĿe 8>“eil yózdhclđh thjofóc httlđleh dljjot hdlelz.Dlđhr dobo sĿj juccoeĿcoe aedocĿjcor (quoetlch) (p 0 ?>, 6>, 9>, …, <>) vh ëhyrhjcljchr

(quortlch) (p 0 68, 8>, ;8)“dĿr.Fhjchehe dhđhrchrdh yoptĿđĿ`Ŀz klfl aedocĿjcorĿe ÷zhccljchrl lële ëaj zoòe

borioòyoiođĿz, gojot `hdyoeĿe fhjchehe dhđhrdhe gorjcĿ dovroedĿđĿ ljl ÷eh`cl

eajtoyo thòs ht`hj lstlyaru`= ľcjl lële, Mheshe“le Hőltslzclđledh flr u(R) gaejslyaeulële fhjchehe dhđhr , Zu(R)P, ÷eh`cl aroedo R“le acosĿcĿj yĿđĿeĿe fucueduđu f÷ckhelehđrlclđl u(x)“h fođcĿ acduđueu k÷rdój. Khehc acoroj, `hdyoe `hdyoe(u(R))

u(`hdyoe(R))“the gorjcĿ do acoiojtĿr, oeioj fueue dljjot hdlc`hsl khrhjhe flr

lstlseosĿ őudur =

×ehr`h ?. R“l dhsthjch`hj lële u(R)“le jhsle ortoe acduđueu vorsoyocĿ`. A zoòe

`hdyoe(u(R) 0 u(`hdyoe(R))



ľYÔT= R“le `hdyoeĿ ^(R 3 `hdyoe(R)) 0?/6“yl sođcor. u(x) jhsle ortoe acduđu lële,bhrboekl flr soflt ` dhđhrl lële acoy R 3 ` acoy u(R) 3 u(`)“yh hőlttlr. Fu ehdhech

^(u(R) 3 u(`hdyoe(R)) 0 ^(R 3 `hdyoe(R)) 0 ?/6“dlr , f÷ychih u(`hdyoe(R))

khrëhjthe u(R)“le `hdyoeĿdĿr.

Yhzklshc acoroj, `hdyoe jhsle ortoe d÷eóőó` lch jarueoe roskhch dhđlőjhele ardleoc

÷zhclđleh fođcĿdĿr.

Fhjchehe dhđhrle, ëajcu roskhch dhđlőjhele dađrusoc gaejslyaeue fhjchehe dhđhrlfhjchehe dhđhrchrle oyeĿ dađrusoc gaejslyaeueo hőlt acduđu òeosĿedo, dađrusocacduđueu k÷rdój. ľzchyhe ÷rehjth k÷sthrlcdlđl klfl `hdyoe lële fu dađru dhđlcdlr=

×rehj ? . WorsoyocĿ` jl R ? vh R 6 oőođĿdojl fhezhr òrmleoc dođĿcĿ`doe hcdh hdlche

vh flrflrledhe fođĿ`sĿz acoe jhsljcl roskhch dhđlőjhechr acsue. A zoòe \ 0 R ? + R 6

>,? vh 6 dhđhrchrlel ocoflclr vh p.d.g.sl

R ? vh R 6 “ele `hdyoeĿ sĿgĿrdĿr, oeioj `hdyoe(\) 0 ? > + > 0 `hdyoe(R ? ) +

`hdyoe(R 6 ).

Dobo khehc acoroj, artocoòcorĿe aedocĿjcorĿ aedocĿjcorĿe artocoòsĿedoe gorjcĿacoflclr. ľzchyhe ÷rehj fu oecoyĿőĿe acdujëo protlj flr foőjo yaruùeu vhr`hjthdlr(oőođĿdojl soyĿsoc ÷rehđl vhrdlđl lële Ochjsoedr Toòrjle“h thőhjjórchr).

×rehj 6. R ? s÷zhc, R 6 oeocltlj vh R 9 soyĿsoc őhjcledh óë f÷có`dhe acuőoe stoedort flrthst acoe KVH sĿeovĿeo klrdlđlelzl dóőóeóe. Thstle bhr f÷có`dh puoeĿeĿz yózdh <>“cĿjdlcl`le ózhrledhdlr. Fu, khehc puoedo do yózdh <>“cĿj dlcl`le ózhrledh acduđueuzoeco`Ŀeo `Ŀ khclr1 Khehc acoroj ihvop boyĿrdĿr1

WorsoyocĿ` jl, slz dæblc, sĿeovo klrhe ?>> jlől vordĿr vh puoecorĿe dođĿcĿ`Ŀ ő÷ychdlr= 24jlől blëflr f÷có`dhe flr thj puoe flch ocòz, slz bhr f÷có`dhe 68> puoe ocdĿeĿz, vhfueue dĿőĿedo sĿeovo klrhe óë tór c ó jlől vordĿr jl bhr flrlsl sodhih flr f÷có` lële bhreosĿcso dor k÷róőcó flr dæblyh fhezhyhe flr yhthehđh soblptlr. Dobo oëĿj acòjkhrhjlrsh, 8 jlől s÷zhcdh ëaj oőĿrĿ yhthehjcldlr vh s÷zhc f÷có`dhe 2>> dlđhr

f÷có`chrdhe > ocĿrcor. Dlđhr 8 jlől oeocltlj f÷có`dhe 2>>, vh flr dlđhr 8 jlől lsh soyĿsoc



f÷có`dhe 2>> ocĿrjhe ÷thjl f÷có`chrdhe > ocĿrcor. Taporcoyoioj acursoj, puoecorĿeflchőlj dođĿcĿ`Ŀ ő÷ychdlr (fu tlplj KVH puoecorĿe dođĿcĿ`Ŀedo acdujëo gorjcĿdĿr)

DacoyĿsĿyco, slz bhr f÷có`dh he oz %<8“clj dlcl`dhslelz, gojot ?8 jlőlele tapco` sjaru2>> ljhe slzle jl sodhih ;8>“dlr, fu ehdhec h bhr óë f÷có`h k÷rh tapco` puoecorĿe

sodhih %28“clj dlcl`ledhslelz.

6. Woryoes

Woryoes roskhch flr dhđlőjhele yoyĿcòsĿeĿe ÷cëósódór

ToeĿ` 6. Voskhch flr dhđlőjhele voryoesĿ oőođĿdojl lch vhrlclr=

Fozhe flz voryoesĿ 6 0 Wor(R) lch lgodh hdhrlz .

×zhclj ?. Yodhih vh sodhih fozĿ soflt soyĿcor lële ^(R 0 i) 0 ? lsh Wor(R) 0 >“dĿr.

×zhclj 6. Hđhr \ 0 oR + f lsh, a zoòe

Wor\ 0 o6Wor(R)

ľYÔT= \leh, sodhih sórhjcl duruò fojocĿ`. Fhjchehe dhđhr lële hcdh hdlche ÷eihjl

saeuëcorĿ juccoeoroj

\oyĿcòeĿe ÷cëó`ó lële roskhch dhđlőjhe klfl flr ëhőlt flrl` juccoeòj dobo uykuedóőhr. Oeioj, fu sae saeuë Wor(R)“le flrl`lele R“le flrl`lele jorhsl acoflchihđlel lò

ht`hjthdlr. Fu ehdhech, voryoes yhrleh sĿjcĿjco stoedort sopò (R)“l juccoeĿrĿz=



×zhclj 9.

Wor(R) 0 |R6| ’ |R|6

ľYÔT=

×zhclj 4. Hđhr

vh R ?, …, R e fođĿ`sĿz lsh, a zoòe

×rehj 9. WorsoyocĿ` jl R jhsljcl flr roskhch dhđlőjhedlr vh p.d.g.sl ő÷ych acsue=

Hđhr \ 0 4R ’ ; lsh, \“ele voryoesĿ ehdlr1

Wor(\) 0 46Wor(R) 0 ?7(ZR6P ’ ZRP6)

ŕl`dl oőođĿdojlel bhsopcoyoflclrlz.

ZRP0 (?/8)(-6 + > +? + 9 + 4) 0 7/8

vh

Fuedoe ÷tóró,

×rehj 4. WorsoyocĿ` jl \ F(e, p). \ e soyĿdo fođĿ`sĿz dheh`hele saeuëcorĿeĿe

tapco`Ŀ acoroj yozĿcoflclr



ŕueu bhsopcoyoflclrlz

Wor(\) 0 Wor(R?) + … + Wor(Re)

A bocdh R l “ele voryoesĿ ehdlr1 OëĿjëosĿ

ZRlP 0 p“dlr

oyeĿ zoòedo,

DacoysĿyco, ×zhc clj 9“h k÷rh,

Fuedoe ÷tóró,

Woryoes flr fhjchehe dhđhr acduđu lële, roskhch dhđlőjhele gaejslyaeueue fhjchehe

dhđhrlel dađrudoe roskhch dhđlőjhele gaejslyaeueue voryoesĿeo uykucoyoflclrlz= hđhr\ 0 r(R) lsh,

6.? Dobo \ójshj Dhrhihcl @a`hetchr

Fhjchehe dhđhrle lcklcl dođĿcĿ`Ŀe jaeuùeue ÷cëósó acduđueu k÷r`óőtój, oò ÷th

yoedoe voryoes yoyĿcòyĿ ÷cëhr. DođĿcĿ`Ŀ jorojthrlzh ht`hj lële roskhch dhđlőjheledlđhr à`hetchrleh fojoflclrlz, ÷rehđle sl`htrlj `l1 JocĿe juyrujcu ù1 vs.

ToeĿ` 9. R“le r “ eil à`hetl oőođĿdojl lch vhrlclr,

vh r “ eil `hrjhzl à`hetl ő÷ych toeĿ`coeĿr=

Fhjchehe dhđhr, dacoysĿyco, R dođĿcĿ`ĿeĿe flrleil à`hetl , voryoes lsh ljleil `hrjhzl

à`hetl acoroj do lgodh hdlclr.






Jaerod @hezhc

9? @ort 6>><

?. Javoryoes

R lch \“ele javoryoesĿ ljl roskhch dhđlĦjhe orosĿedojl lclĦjlele kóióeóe ÷cëósódór.

ToeĿ` ?. ľjl roskhch dhđlőjhe R vh \ lële, javoryoes ő÷ych toeĿ`coeĿr=

Įcj acoroj, sodhih toeĿ`Ŀ uykucoyoroj oĦođĿdojl chrl hcdh hdhrlz.

×zhclj ?.

Iav(R, R) 0 Wor(R)

×zhclj 6.

Iav(R, \) 0 Iav(\, R)

DobosĿ, hcl`lzdh javoryoes bhsopcoòsĿedo ëaj yororcĿ acoe oĦođĿdojl saeuë vor.

×zhclj 9.

Fu, voryoesĿe fhezhr ÷zhclđlele khehcchĦtlrlc`lĦldlr vh lspotĿ do oyeĿ tór orkóòecorĿjuccoeĿr. Flr ÷rehj lch fu saeuiue eosĿc yoror cĿ acduđueu k÷rhcl`=

×rehj ?. Wor soyocĿ` jl R lch \“ele flchőlj p.d.g.“sl oőođĿdojl klfldlr.



Javoryoes Iav(R, \) ehdlr1 ’ ×zhclj ;“dhjl dhejch`le sođ torogĿeo k÷rh doblc acoeflchőhechrl bhsopcoyocĿ`=

OyeĿ őhjlcdh, yujorĿdojl odĿ`corĿ tojlp hdhrhj őuecorĿ hcdh hdhrlz=

vh

Fótóe porëocorĿ flroyo khtlrhrhj vh ÷zhclj ;“yl uykucoyoroj őu saeuio vorĿrĿz=

Įjl fođĿ`sĿz roskhch dhđlĦjhe R vh \ lële tapco`corĿe voryoesĿeĿe voryoescorĿetapco`Ŀeo hĦlt acduđueu dobo ÷eih k÷sthr`lĦtlj. įl`dl lsh fođĿ`sĿz acòsĿ khrhjclacòyoe roskhch dhđlĦjhechrle khehcchĦtlrlc`hslel k÷rhcl`=

×zhclj 4.

Wor (R + \) 0 Wor(R) + Wor(\) + 6Iav(R, \)



ĮspotĿe orjosĿedojl dóĦóeih, ÷zhclj 9 vh ;“yl uykucoyoroj oĦođĿdojlel hcdh ht`hjtlr.

×zhclj 8. R, \, ] roskhch dhđlőjhechr lële,

×zhclj 7.

Yae ÷zhclđh k÷rh, javoryoes R lch \“ele ÷cëhđleh k÷rh dhđlĦtlđl lële, R lch \ orosĿedojllclĦjlele kóióeó vhrhe, ljl dhđlĦjhele dlyhcl` jl ÷cëó` flrl`ledhjl dhđlĦl`ledhehtjlche`hyhe, stoedort flr ÷cëóyh soblp acòj lstlyaruz. Ëaj sĿjcĿjco juccoeĿcoe a ÷cëójarhcosyae jotsoyĿsĿdĿr.

ToeĿ` 6. R vh \“ele jarhcosyae jotsoyĿsĿ ő÷ych bhsopcoeĿr=

Jarhcosyae jotsoyĿsĿ flr Ħhjlcdh earòcchĦlr (fjz. ×zhclj ;).

×zhclj ;.

Khehc jaeuĦòj khrhjlrsh, óë duruù oyĿrt hdhrlz.

Ͽ(R, \) 5 >= –R lch \ orosĿedo pazltlg jarhcosyae vordĿr‐

Ͽ(R, \) 0 >= –R lch \ orosĿedo jarhcosyae yajtur‐

Ͽ(R, \) 3 >= –R lch \ orosĿedo ehkotlg jarhcosyae vordĿr‐

×zhclj 2.

FozĿ o > vh f sofltchrl lële



\oel, hđhr ljl roskhch dhđlĦjhe orosĿedo dhthr`lelsttlj dađrusoc flr lclĦjl vorso,jarhcosyae jotsoyĿsĿeĿe ùtcoj dhđhrl ?“h hĦlttlr. A duru`do R lch \ orosĿedo`ójh``hc jarhcosyae acduđueu s÷ychrlz.

OëĿjcoò ?. Whrl oeoclzlele ëaj ÷eh`cl flr lcjhsl, ljl roskhch dhđlőjhe orosĿedojl

lstotljshc lclőjlele `hjoelj vhyo ehdheshc lgodhchrh doyoeòòsĿdĿr jl, flz fueuekhrëhjth vhrlyh doyocĿ acòsĿeĿ orzucorĿz. ×rehđle, lesoecorĿe mlèostlj socaecorĿedospar yoporoj boriodĿjcorĿ zoòe lch sođcĿjcorĿ orosĿedo pazltlg flr jarhcosyae acduđueuvhrl shtchrlych k÷zch`chrlz, oeioj fu sparue sođcĿj duruùeu lylchőtlrdlđl oeco`Ŀeo khc`hz. Dlđhr torogtoe sođcĿj duru`corĿ ëaj j÷tó acoe vh spar socaecorĿeo klt`hylojĿccorĿeo flch khtlr`hyhe jl`l lesoecordo vordĿr.

Ehdhe R lch \“ele jarhcosyaeue R“le \“ele ehdhel acòsĿedoe toò`he gorjcĿ jovro`coracduđueu k÷r`hele dobo sayut flr yacu, R lch \“ele javoryoesĿeĿe sl`htrlj acduđueo,f÷ychih dhđlőjhechrle racchrlel dhđlőtlrhflchihđl`lzh dljjot ht`hjtlr . Oeioj ehdheshcclj

lële, lclőjlele sphslglj flr y÷eóeó dóőóeóróz, yoel R \ vhyo –R \“eleehdheldlr/htjlchyhedlr‐, oeioj oyeĿ zoòedo –\ R“le ehdhel/htjlchyhel dhđlcdlr‐ dhrlz,f÷ychih R lch \“ele racchrlel dhđlőtlrh`hylz. Fuedo ÷tóró,

JAVHCOY\AE EHDHEYHCCĮĕH HįĮT DHĕĮCDĮV

Hjaea`htrl dhrschrledh fueue (ëaj) dobo gozcosĿ vordĿr.

?.?.×elzch`h= Vhkrhsyae

Dlyhcl` jl, flr lĦëlele khclrl \ lch aeue ajuccoĦò yĿcĿ R lch ÷cëóche hđltl`l orosĿedojllclĦjlych lcklchelyaruz (jacoycĿj oëĿsĿedoe bhr ljlslele dh sórhjcl acduđueu vorsoyocĿ`). Fuduru`do bhr zoòe R lch \ orosĿedojl lclĦjlyl Ħ÷ych yozoflclrlz=

\ 0 + R + X

Furodo X H|X| 0 > vh Iav(R, X) 0 > ÷zhcljchrleh soblp roskhch flr dhđlĦjhedlr(rhkrhsyaejopso`Ŀedo fu boto thrl`l acoroj odcoedĿrĿcoiojtĿr).

( , ) poro`htrhchrlel fhclrch`hele flr yacu oĦođĿdojlel ë÷z`hjtlr.

“yo k÷rh flrleil-dhrhih jaĦucueu fhclrchyleih (tórhvlel ocòj) (dljjot= fhjchehe dhđhrchrdađrusocdĿr, fu ehdhech tórhvl lethkroc oroiĿcĿđĿyco ocoflclrlz) oĦođĿdojlel hcdh hdhrlz=



OyeĿ Ħhjlcdh, “yo k÷rh flrleil-dhrhih jaĦucu Ħ÷ychdlr =

Yae lgodhyl lële ë÷zóeih oĦođĿdojl Ħu hdlclr=

Fueu f lële hcdh hdlche flrleil dhrhih jaĦucdo yhrleh jayueio,

hcdh hdhrlz, f÷ychih ortĿj poro`htrh lële ë÷zó`ó yopoflclrlz=

įl`dl ĦuecorĿ dađrucoyoflclrlz=

vh

(dađurusu, lcjl dađrudoe “eĿe flrleil dhrhih jaĦucuedoe, ljleilsl lsh ζ“eĿe flrleil dhrhihjaĦucuedoe hcdh hdlclr).

A zoòe, –tob`le hdlche rhkrhsyae‐ + R R lch lclĦjlcl (R torogĿedoe oëĿjcoeoe) \“ele

porëosĿdĿr vh X lsh \“ele R lch lclĦjlcl acòyoe porëosĿdĿr. vh poro`htrhchrl khehccljch



rhkrhsyae poro`htrhchrl vhyo he jóëój jorhchr jotsoyĿcorĿ acoroj odcoedĿrĿcĿr . Dađrusocrhkrhsyae hjaea`htrlele –th`hc toĦĿ‐dĿr vh fueu ?4.96 vh dlđhr hjaea`htrl dhrschrledhëaj dhđlĦlj voryosyaecordo k÷rhihjslelz.

6. Jaőuccu Fhjchehe Dhđhrchr

×rehj 6. Bhr sheh, flr glròeĿe OV-KH f÷có`ó roskhch flr sórhë saeuiuedo R jodorfucuő órht`hjthdlr, furodo H|R| 0 6 vh Wor(R) 0 6“dlr. Bhr fucuő p 0 >.6 acosĿcĿjco tliorlflr foőorĿ k÷sthrhihjtlr (fođĿ`sĿzcĿj vorsoyocĿ`). Flr yĿc lëhrlsledhjl tliorl foőorĿ soyĿsĿeĿY lch k÷sthrhcl`. Flr yĿc lëhrlsledh R 0 x fucuő soyĿsĿeo jaőuccoe`Ŀő Y“ele artocoòsĿeĿe Y F(x, p) 0 xp acduđueu flcdlđl`lz lële, fucuőcorĿe artocoò acoroj xp jodorĿeĿe foőorĿcĿacòsĿ khrhjlr.

\ vhrlyjhe R“le jaőuc c u fhjchehe dhđhrl jaőuc c u p.d.g“dhe hcdh hdlche R“le fhjchehedhđhrl dlr=

ToeĿ` 9.

g \|R (y|R) roskhch dhđlőjhe R“l jhedl orkóòeĿ acoroj toĦĿdĿđĿ lële, jaĦuccu fhjchehedhđhr le dh oyeĿ zoòedo roskhch flr dhđlĦjhe acduđueu eat hdlelz. Oeioj, R“le fhccl flrdhđhrl vhrlyjhe, \“ele jaĦuccu fhjchehe dhđhrlel dh toeĿòyoflclrlz

furodo jaĦuccu yađuecuj toeĿ`coedĿđĿ sórhih fu bhrboekl flr vhrlcl x dhđhrl lële sodhihflr soyĿdĿr.

Bhsopcoòcor to` acoroj ÷eihjl klfl yopĿcdĿđĿ lële (thj gorj Ħl`dl jaőuc cu dođĿcĿ`ózhrledhe lethkroc ocĿyaruz), soyĿsoc flr ÷rehj yopòyoiođĿz (prafch` shtl lële sodhihtoeĿ`Ŀ uykucoyĿeĿz). Fueue yhrleh, jaĦuccu vh jaĦucsuz ÷rehjchr orosĿedojl gorjĿk÷sthr`hj lële dobo jocltotlg ÷rehjchrl tortĿĦocĿ`.

×rehj 9 ( –Clàe‐ ^lyososĿ). OőođĿdojl hjaea`lst Kharkh Ojhrcag“ue `hőburjuccoeĿc`Ŀő orofo plyososĿ àdhclele fosltchőtlrlc`lő flr vhrslyaeudur. WorsoyocĿ` jl óëtór R juccoeĿc`Ŀő orofo vordĿr= `ójh``hc duru`dojl orofocor (jorpuzcor), arto joclthorofocor (lstotlstljshc òeodo jhsleclj lgodh hdhe –artocoò‐ dhđlc), vh ëaj j÷tóduru`dojl orofocor (clàecor). Bhr tór orofo hőlt grhjoeso soblptlr, yoel

^( – clàe‐ ) 0 ^( – arto‐ ) 0 ^( – jorpuz ‐ ) 0 ?/9



YotĿiĿ vh flr ocĿiĿ bhr flr orofo tóró lële, sĿrosĿyco \ Y vh \ F, jodor oőođĿdo vhrlchedhđhrchrl flëlyarcor=

Tór YotĿiĿ OcĿiĿ Clàe 8>>> 7>>>

Arto 7>>> ?>>>>Jorpuz ?>>>> ??>>>

ľcj dljjot hdlc`hsl khrhjhe őhy, bhr orofo ëhőldl lële, ocĿiĿeĿe flëtlđl dhđhrle sotĿiĿeĿeflëtlđledhe dobo yójshj acduđudur, dacoyĿsĿyco bhr flr tór orofo lële, ocĿő vhrlő ocĿiĿ lchsotĿiĿeĿe flëtlđl dhđhrle orosĿedojl flr glyottoe khrëhjchő`hcl. Oeioj, juccoeĿc`Ŀő orofocordo, joclth lcj oedo k÷róche őhy dhđlcdlr. Fu ehdhech, hđhr eh ocĿiĿ eh dh sotĿiĿ s÷zjaeusu orofoeĿe tóró R“l flc`lyarso, aecorĿe fhjchehe dhđhrl ylehchehe fhjchetlchrjoeueueo k÷rh oőođĿdojl klfldlr=

DacoyĿsĿyco ocĿő vhrlőle khrëhjchő`hsl khrhjlr.

Dobo khrëhjël flr dózhech`hdh, orofoeĿe sotĿiĿsĿ orofoeĿe joclthslel ocĿiĿdoe dobo lylflclr (to`lrot khë`lőlel, jozocorĿeĿ vs.) vh orofoyĿ sotòyo orzucoyoiođĿ glyot fhclrchr.

Hđhr sotĿiĿ óë orofo tóróeó dh `ójh``hc flr őhjlcdh oyĿrt hdhflclrsh, jl ocĿiĿ fueuyopoòz, ocĿiĿeĿe sotĿiĿeĿe fhclrtlche glyottoe orofoyĿ sotò orzusueo jaőuccoe`Ŀőfhjchetlchr acuőturòsĿ khrhjlr.

Hđhr sotĿiĿ 7>>> dacordoe dobo dóőój flr glyot fhclrtlrsh, ocĿiĿ jhsle acoroj orofoeĿe–clàe‐ acduđueu flclrdl , ëóejó dlđhr duru`cordo sotĿiĿ he oz 7>>> dacor tochp hdhrdl,yoel

vh ocĿő vhrlő khrëhjchőlrdl. Oeioj, hđhr orofo –jorpuz‐ lsh, sotĿiĿ he oz ?>>>> dacortochp hdhrdl, bæcfujl ocĿiĿ he gozco

jodor ÷dhyhihjtl, fu ehdhech dh sotĿiĿ yójshj joclth flr orofoyĿ òjuc flr glyotosotoòyoiojtĿ.



^lyosoeĿe –jorpuz‐ lële ëocĿőòòsĿeĿe(frhoj dawe) ehdhel fu àdhcdh sotĿiĿeĿe ocĿiĿyo orofoeĿe joclthslele dóőój acòdĿđĿ jaeusuedo jofuc hdlchflclr flr koroetlvhrh`h`hsldlr, fu ehdhech ocĿiĿ ocĿő vhrlőledh j÷tó flr orofo ocò lbtlòclel bhsofojotor.

JaĦucu vh jaĦucsuz fhjchehe dhđhrchr orosĿedojl ÷eh`cl flr lclĦjl \lehcheheFhjchetlchr Joeueudur (dobo ÷eih k÷rdóđó`óz Tapco` AcosĿcĿj Joeueu“eo ëajfhezhr)=

×ehr`h ? (\lehchehe Fhjchetlchr Joeueu).

ĮYÔT= k(x) 0 Z\|R 0 xP acsue. k(x) x“le flr gaejslyaeudur. Fhjchehe dhđhrl Ħl`dl

bhsopcoyoflclrlz=




Jaerod @hezhc

6 Elsoe 6>><

?. Jaőuccu Fhjchehe Dhđhrchr

×rehj ?. Bhr sheh, flr glròeĿe OV -KH f÷có`ó roskhch flr sórhë saeuiuedo R jodor

fucuő órht`hjthdlr, furodo H|R| 0 6 vh Wor(R) 0 6“dlr. Bhr fucuő p 0 >.6 acosĿcĿjco tliorlflr foőorĿ k÷sthrhihjtlr (fođĿ`sĿzcĿj vorsoyocĿ`). Flr yĿc lëhrlsledhjl tliorl foőorĿ soyĿsĿeĿY lch k÷sthrhcl`. Flr yĿc lëhrlsledh R 0 x fucuő soyĿsĿeo jaőuccoe`Ŀő Y“ele artocoòsĿeĿe Y F(x, p) 0 xp acduđueu flcdlđl`lz lële, fucuőcorĿe artocoò acoroj xp jodorĿeĿe foőorĿcĿacòsĿ khrhjlr.

\ vhrlyjhe R“le jaőuccu fhjchehe dhđhrl jaőuccu p.d.g“dhe hcdh hdlche R“le fhjchehedhđhrldlr=

ToeĿ` ?.

g \|R (y|R) roskhch dhđlőjhe R“l jhedl orkóòeĿ acoroj toőĿdĿđĿ lële, jaőuccu fhjchehedhđhrle dh oyeĿ zoòedo roskhch flr dhđlőjhe acduđueu eat hdlelz. Oeioj, R“le fhccl flrdhđhrl vhrlyjhe, \“ele jaőuccu fhjchehe dhđhrlel dh toeĿòyoflclrlz

furodo jaőuccu yađuecuj toeĿ`coedĿđĿ sórhih fu bhrboekl flr vhrlcl x dhđhrl lële sodhih

flr soyĿdĿr.

Bhsopcoòcor to` acoroj ÷eihjl klfl yopĿcdĿđĿ lële (thj gorj ől`dl jaőuccu dođĿcĿ`ózhrledhe lethkroc ocĿyaruz), soyĿsoc flr ÷rehj yopòyoiođĿz (prafch` shtl lële sodhihtoeĿ`Ŀ uykucoyĿeĿz). Fueue yhrleh, jaőuccu vh jaőucsuz ÷rehjchr orosĿedojl gorjĿk÷sthr`hj lële dobo jocltotlg ÷rehjchrl tortĿőocĿ`.



×rehj 6 (–Clàe‐ ^lyososĿ). OőođĿdojl hjaea`lst Kharkh Ojhrcag“ue `hőburjuccoeĿc`Ŀő orofo plyososĿ àdhclele fosltchőtlrlc`lő flr vhrslyaeudur. WorsoyocĿ` jl óëtór R juccoeĿc`Ŀő orofo vordĿr= `ójh``hc duru`dojl orofocor (jorpuzcor), arto joclthorofocor (lstotlstljshc òeodo jhsleclj lgodh hdhe –artocoò‐ dhđlc), vh ëaj j÷tóduru`dojl orofocor (clàecor). Bhr tór orofo hőlt grhjoeso soblptlr, yoel

^(–clàe‐) 0 ^(–arto‐) 0 ^(–jorpuz‐) 0 ?/9

YotĿiĿ vh flr ocĿiĿ bhr flr orofo tóró lële, sĿrosĿyco \ Y vh \ F, jodor oőođĿdo vhrl che

dhđhrchrl flëlyarcor=

Tór YotĿiĿ OcĿiĿ Clàe 8>>> 7>>>Arto 7>>> ?>>>>

Jorpuz ?>>>> ??>>>

ľcj dljjot hdlc`hsl khrhjhe őhy, bhr orofo ëhőldl lële, ocĿiĿeĿe flëtlđl dhđhrle sotĿiĿeĿeflëtlđledhe dobo yójshj acduđudur, dacoyĿsĿyco bhr flr tór orofo lële, ocĿő vhrlő ocĿiĿ lchsotĿiĿeĿe flëtlđl dhđhrle orosĿedojl flr glyottoe khrëhjchő`hcl. Oeioj, juccoeĿc`Ŀő orofocordo, joclth lcj oedo k÷róche őhy dhđlcdlr. Fu ehdhech, hđhr eh ocĿiĿ eh dh sotĿiĿ s÷zjaeusu orofoeĿe tóró R“l flc`lyarso, aecorĿe fhjchehe dhđhrl ylehchehe fhjchetlchrjoeueueo k÷rh oőođĿdojl klfldlr=

DacoyĿsĿyco ocĿő vhrlőle khrëhjchő`hsl khrhjlr.

Dobo khrëhjël flr dózhech`hdh, orofoeĿe sotĿiĿsĿ orofoeĿe joclthslel ocĿiĿdoe dobo lylflclr (to`lrot khë`lőlel, jozocorĿeĿ vs.) vh orofoyĿ sotòyo orzucoyoiođĿ glyot fhclrchr.Hđhr sotĿiĿ óë orofo tóróeó dh `ójh``hc flr őhjlcdh oyĿrt hdhflclrsh, jl ocĿiĿ fueuyopoòz, ocĿiĿeĿe sotĿiĿeĿe fhclrtlche glyottoe orofoyĿ sotò orzusueo jaőuccoe`Ŀő

fhjchetlchr acuőturòsĿ khrhjlr.Hđhr sotĿiĿ 7>>> dacordoe dobo dóőój flr glyot fhclrtlrsh, ocĿiĿ jhsle acoroj orofoeĿe–clàe‐ acduđueu flclrdl, ëóejó dlđhr duru`cordo sotĿiĿ he oz 7>>> dacor tochp hdhrdl,yoel



vh ocĿő vhrlő khrëhjchőlrdl. Oeioj, hđhr orofo –jorpuz‐ lsh, sotĿiĿ he oz ?>>>> dacortochp hdhrdl, bæcfujl ocĿiĿ he gozco

jodor ÷dhyhihjtl, fu ehdhech dh sotĿiĿ yójshj joclth flr orofoyĿ òjuc flr glyotosotoòyoiojtĿ.

^lyosoeĿe –jorpuz‐ lële ëocĿőòòsĿeĿe(frhoj dawe) ehdhel fu àdhcdh sotĿiĿeĿeocĿiĿyo orofoeĿe joclthslele dóőój acòdĿđĿ jaeusuedo jofuc hdlchflclr flr koroetlv hrh`h`hsldlr, fu ehdhech ocĿiĿ ocĿő vhrlőledh j÷tó flr orofo ocò lbtlòclel bhsofojotor .

×rehj 9. Fu ÷rehjth, lesoecorĿe khchihjthjl pacltlj acoycor ózhrleh fobsh klrdl đl flr

lethreht pcotgarù acoe LH@ âcl tlioc @orjhts“Ŀe 6>>2 foőjoecĿj odoycorĿ lch lcklcl vhrlsleh

fojoiođĿz (vhrl lële fjz.=bttp=//www.flz.ulawo.hdu/lh`/òrjhts/dotoQea`leotlae>2.bt`c )

^lyoso ő÷ych ëocĿőĿyar= Bhr flr pacltlj odoy l lële, jotĿcĿìĿcor oőođĿdojl khtlrlyl vhrhejaetrotcor sotĿe ocĿyarcor

\ l 0 {

Whrlcl flr t zoòedo, plyosodojl jotĿcĿìĿcor R t acoroj odcoedĿroiođĿ`Ŀz dĿőorĿdoe khchelcovh flcklyh soblp acurcor, ÷rehđle a oeo jodor jozoeĿcoe dhchkh soyĿsĿ, odoyĿe

prapokoedosĿeĿe –à`hetuù‐, yo do odoyĿe prapokoedo strothmlsl bojjĿedo

k÷rhvclchrle dh`hëchrl klfl.

Y÷z jaeusu lcovh flckl vhrlc`lőjhe, jaetrotĿe fhjchehe dhđhrl

Dlđhr flr lgodhych, jotĿcĿìĿcorĿe odoy l“ele jaetrotĿ lële ÷dh`hyl orzucodĿjcorĿ dacor`ljtorĿ t zoòedo vhrlcl flcklyh k÷rh l “ele jhedl portl odoycĿđĿeĿ jozoeò acosĿcĿđĿeohőlttlr. Yae óë oydojl Dhàjrot ôrtl“ele oeo odoycorĿeĿe jaetrotcorĿeĿe glyotcorĿeofojocĿ`= Dhàjrotlj odoycorĿe jozoeò lbtlòcl bojjĿedo ÷eh`cl flcklchrl artoy o ëĿjoroe

9 acoy lële óë dlj dađru ëlzdl`=



Forroij Afoò“eĿe Blccory Icletae“o jorőĿ hzlil flr gorjco jozoedĿđĿ Lawo portljaekrhsl,

Lawo“doj l yhelckldhe saero Blccory Icletae“ue khrl d÷eóőó acoroj k÷róc he Ehw

Bo`psblrh ÷e shël`l,

Abla vh Tojsos“Ŀe ÷e shël`chrl: fu ljl ÷eh`cl hyochtle ÷e shël`chrlel Blccory

Icletae jozoedĿ

Fu acoycorĿe bhr flrlsledhe saero jaőuccu fhjchehe dhđhrchrdh ëaj ÷eh`cl dhđlőljcljchr

acduđueu k÷rhflclrlz. Fu, odoycorĿe portlchrlele odoycĿđĿeĿ sođcoò ocò őoesĿ bojjĿedo plyosoeĿe –leoeëcorĿeĿe‐ eosĿc dhđlőtlđlel k÷sthr`hjthdlr.

Gleoesoc Hjaea`l“dh, ÷zhc flr duruùe khrëhjchő`hsl bocledh ? dacor ÷dhyhe fu tórjaetrotcoro Orraw-Dhfruh `hejuc jĿy`htl dh dhelr.

Jaőuccu vh jaőucsuz fhjchehe dhđhrchr orosĿedojl ÷eh`cl flr lclőjl \lehchehe FhjchetlchrJoeueu“dur(fu dobo ÷eih fu dhrsth k÷rdóđó`óz Tapco` AcosĿcĿj Joeueue yojĿedĿr).

×ehr`h ? (\lehchehe Fhjchetlchr Joeueu).

ľYÔT= k(x) 0 Z\|R 0 xP acsue. k(x) x“le flr gaejslyaeudur. Fhjchehe dhđhrl ől`dlbhsopcoyoflclrlz=



×ehr`h 6 (Jaőuccu Woryoes / Tapco` Woryoes Joeueu).

Fu saeuë oyeĿ zoòedo OEAWO hőltclđl acoroj flclelr. Furodo OEAWO Woryoes

Oeoclzl“dlr. ×zhcljch, Wor(\|R) > acduđu lële, oőođĿdojleh ucoőĿcĿr

Fu, jofoio s÷ych`hj khrhjlrsh, –R“l flc`hj \“ele voryoesĿeĿ dóőórór‐ őhjcledh

ajueoflclr.

ľYÔT=

furodo flrleil hőltclj WorR 0 - ÷zhcclđlel juccoeĿr, ljleil odĿ` ylehchehefhjchetlchr joeueueu juccoeĿr, vh sae odĿ`do flrleil vh ljleil thrl`chr flrflrlel k÷tóróeih

lspot toò`coeĿr.×rehj 4. Bhr sheh, flr glròeĿe OV-KH f÷có`ó roskhch flr sórhë saeuiuedo R jodor

fucuő órht`hjthdlr, furodo 0 6 vh Wor(R) 0 6“dlr. Bhr fucuő p 0 >.6 acosĿcĿjco tliorlflr foőorĿdĿr (fođĿ`sĿzcĿj vorsoyocĿ`). Flr yĿc lëhrlsledhjl tliorl foőorĿ soyĿsĿeĿ Y lchk÷sthrhcl`.



(o) WorsoyocĿ` jl fu sheh 8 yhel fucuőuùz vor. Aecordoe Y toehslele tliorlfoőorĿ k÷sthr`h acosĿcĿđĿ ehdlr1 R 0 8 vhrl ljhe, Y“ele jaőucc u p.d.g.sl flr

flea`dur, fu ehdhech, ÷rehđle,

(f) 8 fucuő vhrl ljhe, fhjchehe foőorĿ soyĿsĿ ehdlr1 Y|R 0 8 F(8,>.6) acduđulële, sae dhrsthjl saeuëcorĿ juccoeoflclrlz,

(i) FucuőcorĿe jaőucsuz fhjchehe dhđhrl ehdlr1 \lehchehe fhjchetlchr joeueu lch

őueu fucuruz,

furodo HZRP 0 6 vorsoyĿyaruz.

(d) Y“le jaőucsuz voryoesĿ ehdlr1 Tapco` voryoes joeueu botĿrcoyĿeĿz,

Fu flea`corĿe jorĿőĿ`ĿeĿe flr ÷rehđldlr, yoel R“h jaőucc oe`Ŀő Y lële flea` dođĿcĿ`Ŀ`Ŀzvor. A zoòe ylehchehe fhjchetlchr joeueueu juccoeoroj foőorĿcorĿe tapco` soyĿsĿeĿ

hcdh hdhflclrlz.

×rehj 8. (LH@ âcltlj ^lyoso, dhvo`) Iu`burlyhtëlchrle khëhe shehjl ÷e shël`chrleh

fojoioj acursoj, fhclrslzcljchrle ëađueue Yóphr YocĿ“do ë÷zócdóđóeó k÷rhflclrlz. Dlyhcl`

jl, jaőuc dhđlőjhel R t t torlbledhjl yh`lecl dhchkosyaeue soyĿsĿ acsue. Tapco` voryoes

joeueu ĿőĿđĿedo Lawo ÷eshël`chrledhe ÷eihjl –jaőucsuz‐ artocoòcor lch Yóphr YocĿ“doesaerojl –jaőuccu‐ artocoòcorĿ jorőĿcoőtĿroflclrlz .

Lawo shël`chrledh ÷eih, oeo odoycorle “sl fóyój dockocoeòcor lch %?> lch %4> klflarto dózhy flr orocĿjto yhr ocĿyarcordĿ. Oeioj, Yóphr YocĿ“doe saero, glyotcor (yoel )> vhyo ?“h yojcoőtĿ, vh –ayeoòcor‐ ëaj ugoj acòyo foőcodĿ. Fu ehdhech, jaőuccuvoryoes gar`ócóeh k÷rh, Yóphr YocĿ“doe saero jaőuccu artocoò bojjĿedojl fhclrslzclj ,Wor( ), khclő`hchrdhe saerojl voryoesĿe, Wor(\ l ), he fóyój f÷có`óeó

acuőturuyardu, bæcfujl jaőuccu voryoesĿe, Wor( ) , jotjĿsĿ elsphthe dobo dóőójk÷róeóyardu.



Hđhr fueu DhàjrotcorĿe yorĿőĿeĿe kroglđlych jorőĿcoőtĿrĿrsoj, Yóphr YocĿ“doe saerodhàjrotcor lële boco fóyój borhjhtche`h acduđu k÷róchflclr, fu ehdhech Yóphr YocĿdoe÷tóró yh`lecl dhchkh soyĿsĿeĿe ózhrleh jaőuc jayòj voryoesĿe ÷eh`cl flr f÷có`óeó yaj

ht`h`hjthdlr, yoel voryoes, Wor( ), boco ëaj yójshjtlr. Dlđhr torogtoe,Iu`burlyhtëlchrle yorĿőĿeĿe ëaj ëofuj flt`hslele sĿjëo fhclrtlche ehdhel, bhr flr hyochtth,

Iu`burlyhtëlchrle ÷e shël`chrledh dhchkh soyĿsĿeĿe odoycorĿe ay aroeĿeo k÷rh dhđlc (fuDhàjrotcorĿe ëađu ÷eshël`ledh flr jurocdĿr ) jozoeoe bhpslel ocĿr jurocĿeo k÷rhfhclrchedlđl lëledlr. Fu ehdhech, he ugoj flr ay gorjĿedo flch jozoeoe odoy dlđhr odoycorogorj otoflclr vh rojlpchrl e oroyĿ jopotòsĿ acdujëo zarcoőĿr .

6. ×zhc DođĿcĿ`cor

Fu dhrsth, ől`dlyh jodor yoykĿe acoroj juccoeĿcoe óë dođĿcĿ` k÷rdój, flea`, uelgar` vhósthc. Fuedoe saerojl ljl dhrsth, fu clsthyl flrjoë ÷eh`cl ÷rehj lch khelőchthihđlz vhaecorĿe he sĿj juccoeĿcoeĿ lch foőcoyoiođĿz, yoel earòc dođĿcĿ`co.

6.? BotĿrcotò= Dhrsth ől`dlyh jodor k÷rdóđó`óz dođĿcĿ`cor

ToeĿ` 6. Hđhr p.d.g.sl oőođĿdojl klflysh, R dhđlőjhel , R F(e, p), (e, p)

poro`htrhchrlych flr flea` dođĿcĿ`dĿr =



R F(e, p) lële dobo ÷eih oőođĿdojl lclőjlchrl k÷sthr`lőtlj.

ToeĿ` 9. Hđhr p.d.g.sl oőođĿdojl klflysh, R dhđlőjhel Zo, fP orocĿđĿedo flr uelgar`

dođĿcĿ`dĿr , R XZo, fP=

ToeĿ` 4. Hđhr p.d.g.sl oőođĿdojl klflysh, R dhđlőjhel poro`htrhslych flr ósthc

dođĿcĿ`dĿr =

6.6 YtoedortcoőtĿrĿc`Ŀő Voskhch Dhđlőjhe

Fozhe, roskhch dhđlőjhe R“le stoedordlzosyaeu acoe oőođĿdojl ]“yh fojòjto goydovordĿr.

Yae flrjoë dhrsth tórhtlche fhjchehe dhđhr vh voryoes juroccorĿeĿ juccoeoroj oőođĿdojlchrl hcdh hdhrlz=

vh



Hđhr roskhch dhđlőjhechrl fu őhjlcdh earòclzh hdhrshj, ÷cëhj vh jaeu`doe fođĿ`sĿzacoroj gorjcĿ dođĿcĿ`corĿe őhjclel jorőĿcoőtĿròj dobo jacoy acur.

6.9 Earòc DođĿcĿ`

Earòc dođĿcĿ` sórhjcl roskhch dhđlőjhe lch lcletlcldlr. Ëaj soyĿdo jl lstotlstljl dhehylesaeuëcorĿeĿe he lyl tob`lelel vhrdlđl artoyo ëĿj`ĿőtĿr (flroz saero fueue lële flr ÷rehjk÷rhihđlz, dobo gozcosĿeĿ @hrjhzl Cl`lt Tharh`lel lőchdljthe saero k÷rhihđlz).

ToeĿ` 8. Hđhr bhrboekl flr z lële p.d.g. oőođĿdojl klflysh, r oskhch flr ] dhđlőjhelearòc dođĿcĿ`cĿdĿr ’ sh`faclj acoroj ] E(>, ?),

I.d.g.sl lsh ő÷ych k÷sthrlclr=

Flr stoedort earòc roskhch dhđlőjhele i.d.g.sl jopocĿ-gar` lgodhyh soblp dhđlcdlr(gojottofca dhđhrchrleh yo do lstotlstljshc yozĿcĿ` pojhtchrleh fojĿcoflclr). ^.d.g χ(z) ëoe hđrlsl

vh sĿgĿr htrogĿedo sl`htrlj ÷zhcljchr h soblptlr=



6.9.?. Ytoedort Earòc DođĿcĿ`Ŀe ×eh`cl ×zhcljchrl

×zhclj ?. Flr stoedort earòc roskhch dhđlőjhe ] lële

vh

Earòc dođĿcĿ`Ŀe ehdhe yororcĿ acduđueo dolr vurkucoeòsĿ khrhjhe lcj ÷eh`cl eajto,ëaj soyĿdo e dheh`h lële Flea` roskhch dhđlőjhechr le earòc dođĿcĿ` lch tob`lehdlchflcdlđlele artoyo ëĿjòsĿdĿr .

Tharh` ? (Dh@alvrh-Copcoih Tharh`l) . Hđhr R F(e, p) flr flea` roskhch dhđlőjhe

lsh, a zoòe i

d klfl bhrboekl flr soyĿ lële őu lgodh yozĿcoflclr,

Flea` dhđlőjhele oőođĿdojl lgodhyh d÷eóőtórócdóđóeh dljjot hdlelz.

Fu lgodh khrëhjth stoedordlzosyaedur. Fu saeuëcor, ëaj soyĿdo e lële stoedordlzhhdlc`lő flea` roskhch dhđlőjhe R“le (i, d| orocĿđĿeo dóő`h acosĿcĿđĿeĿe oőođĿ yujorĿearòc dhđlőjhele oyeĿsĿ acduđueu lgodh ht`hjthdlr. ×rehj acsue dlyh sodhih yójshchee dhđhrchrl lële flea` p.d.g.“chrl kroglj ózhrledh k÷sthrdl` vh dobo saero earòclzosyaeuykucodĿ`.

e 0 8> lële, for kroglđlele őhjcl elsphthe earòc yađuecuđue ëoe hđrlsleh fhezh`hjthdlr.×zhcljch dljjot hdlchihj acurso, oz soyĿdojl e lële dođĿcĿ`Ŀe ëorpĿjcĿđĿ ehrdhyshtoò`he yaj acùőtur (–foőorĿ‐eĿe dóőój acosĿcĿđĿedoe ÷tóró, p 0 ?/4).



ŕhjlc ?. Dh@alvrh-Copcoih Tharh`lele K÷sthrl`l

×rehj 7. X ykucoò oëĿsĿedoe fu tór tob`lechrle ehdhe khrëhjthe yororcĿ acduđueuk÷r`hj lële ordĿőĿj e dheh`hele acosĿcĿđĿeĿ bhsopcoòj lstlyaruz. Dlyhcl` jl p 0 ?/8“dlrfu duru`do he oz %68“clj foőorĿ vordĿr.

Hđhr e 0 8 lsh, 68%“the dobo gozco foőorĿyo soblp acòò acosĿcĿđĿ oőođĿdojl klfl

bhsopcoeĿr =

^hjl, hđhr F(?>>, ?/8) lsh, yoel e“l ?>>“h yójshctlrshj eh acur1 ^rheslpth,

oőođĿdojlel bhsopcoyoflclrlz =

F÷ychih, bhr flrlele tapco`ĿeĿ oyrĿ oyrĿ bhsopcoyoflclrlz. Oeioj aecordoe ëaj gozco acduđulële, fu ëaj ođĿr flr yój khtlrhihjtlr. Dlđhr flr shëhehj acoroj, Dh@alvrh-Copcoih

Tharh`lel juccoeoroj lyl flr tob`ledh fucueoflclrdlj . Ytoedordlzh hdlc`lő ő÷ychdlr=

Fu ehdhech,



Fu tob`le eh jodor lyldlr1 BhsopcoòcorĿ yoptĿ` vh ^ ( 68) %<?.68 hcdh httl`.

Hđhr oyeĿ ÷rehđl e 0 6>> lële thjrorcorsoj, –to` acoroj‐ ^( 8>) %<7.88 fleaàcosĿcĿđĿeĿ vh ^( 8>) %<7.?8 earòc tob`lelel hcdh httl`.

] E(>, ?) lële, oyrĿio bhrboekl flr roskhch dhđlőjhel hcdh hdhflclrlz.

Fu earòc flr roskhch dhđlőjhedlr, artocoòsĿ vh voryoesĿ 6“dlr. Yh`faclj acoroj

R“le p.d.g.sl ehdlr1 Dobo ÷eih dhrsth k÷rdóđó`óz dhđlőjhe dhđlőtlr`h gar`ócóeójuccoeoflclrlz=

Flr earòc roskhch dhđlőjhele dađrusoc d÷eóőó`óeóe yleh earòc flr R? dhđlőjhelacduđueu k÷z ÷eóedh fucueduroroj tortĿőòyĿ dobo lchrl flr oőoòyo toőĿyoflclrlz.

×ehr`h 9. Hđhr R ?

E(¸, ω 6 ) vh R 6 0 o + fR ? lsh, a zoòe

Fu saeuiu yleh dhđlőjhe dhđlőtlr`h gar`ócóeó juccoeoroj jaetrac hdhflclrslelz.

E soyĿdojl R?, ..., Re dhđlőjhele tapco`ĿeĿe fhjchehe dhđhrlele, fhjchehe dhđhrchrleletapco`Ŀ acduđueu vh e fođĿ`sĿz vh roskhch dhđlőjhele voryoesĿeĿe do voryoescorĿeĿetapco`Ŀ acduđueu k÷r`óőtój. Hđhr Rl“chr oyeĿ zoòedo earòc lsh, tapco`corĿ doearòcdlr=

×ehr`h 4. Hđhr R ?, ..., R

e fođĿ`sĿz earòc roskhch dhđlőjhechr vh R

l

E(¸

l , ω

l

6 ) lsh, a

zoòe,



Khehc acoroj, flrjoë bogto ÷eih k÷rdóđó`óz fójóc`h gar`ócóeó juccoeòj zaruedoacoflclrdlj, oeioj earòcchrle tapco`Ŀ lële, sodhih tapco`Ŀe fhjchehe dhđhrlel vhvoryoesĿeĿ bhsopcoò`Ŀz yhthrcldlr. Fu duru`do p.d.g.chrl bh`he fucoflclrlz=

ŕhjlc 6. “le GorjcĿ Dhđhrchrl lële Earòc \ađuecuj

6.9.6. Ytoedort Earòc“le Tofca Dhđhrchrlel Juccoeò

Hđhr R E(¸, ω) lsh, R“le jhedl artocoòsĿedoe flr vhyo ljl stoedort sopòdoe dobodobo uzojto acòò acosĿcĿđĿeĿe jofoio flr tob`lelel vhrhflclrlz=

\oel dođĿcĿ` yĿđĿeĿeĿe ëađu artocoòdoe ltlforhe flr vhyo ljl stoedort sopò orocĿđĿeĿelëledhdlr. Hđhr hclelzle octĿedo i.d.g. tofcasu yajso earòc acosĿcĿđĿe jofoio flr tob`lel

hcdh ht`hj lële fu óë elihc dhđhrl botĿrcoòjto goydo vordĿr.




Ytoedort earòc dođĿcĿ` yoykĿe acoroj juccoeĿcdĿđĿ lële, (z)“le i.d.g.slele tofcacoőtĿrĿc`Ŀődhđhrchrlel bhrboekl flr lstotlstlj jltofĿedo fucoflclrslelz.

Ëađuecujco s÷z jaeusu tofcacor sodhih z > dhđhrchrlel lëhrlrchr, gojot dođĿcĿ`Ŀe

sl`htrlslel juccoeoroj i.d.g.ele

5 > dhđhrchrlel hcdh hdhflclrslelz.

×rehđle, hđhr ^(] ?.<8)“l flc`hj lstlyarsoj, ^(] -?.<8)“l oroyoflclr vh ^(] ?.<8) 0

?- ^(] -?.<8) 0 >.<;44“ó bhsopcoyoflclrlz.




Khehc acoroj, hđhr R E(¸, ω) lsh, o f lële ^(o R f) tóró acosĿcĿjcorĿ oőođĿdojlodĿ`corĿ tojlp hdhrhj hcdh hdhflclrlz=

?. dhđlőjhel stoedordlzh ht, yoel flr stoedort earòc roskhch ] dhđlőjhel lële acoyĿoőođĿdojl klfl thjror yoz=

6. stoedort earòc i.d.g., (.), ilesledhe acosĿcĿđĿ yheldhe lgodh ht=

9. acosĿcĿđĿ hcdh ht`hj lële yujorĿdo bhsopcoeoe dhđhrchrle stoedort earòc i.d.g.sllële tofcadoe dhđhrchrl fuc.

6.4 Flr ôroethz Oëòj= Ytoedort Earòc Voskhch DhđlőjheleËlzl`l

Lethkroc d÷eóőtór`h juccoeoroj uelgar` roskhch ëhjlclőchrl dlđhr bhrboekl flr sórhjcldođĿcĿò d÷eóőtór`hele `ó`jóe acduđueu dobo ÷eih k÷r`óőtój(roskhch dhđlőjhechrled÷eóőtóróc`hsl ózhrleh acoe dhrs eatcorĿeo fojĿeĿz). Hđhr flr flcklsoyorĿeĿz yajso eh



yoporsĿeĿz1 ?<>>“có yĿccordo, óecó lstotlstljël Groeils Koctae zor juccoeoroj earòcroskhch flr dhđlőjhel tojclt hdhe ojĿcĿ flr ocht yoptĿ?.

ŕhjlc 9“th k÷róche óë gorjcĿ zor phől sĿro otĿc`ĿőtĿr. Fu yopĿcĿrjhe, dhehyl yopoe jlőlthsodógl ëhjlclőchrle clsthslel tofcayo oőođĿdojl klfl ojtorĿr= Flrleil zor khrëhj dhđhrl

vhrlr(bhr zoòe slzh dađru acoe yózhyle octĿedojl rojo` ajursueuz). ľcj zoòedoyĿcdĿzcor faő fĿrojĿcĿr jhe, gojot dobo saero ljleil zorĿe otĿcĿőĿyco dacdurucurcor. Hesaeuedo, óëóeió zor lcj ljl zor lch acuőturduđueuz ëhjlclőchrle ÷eóeh jaeucoioj ortĿ vhhjslchrle sĿrosĿeĿ vhrlr.

Yaeuiue khrëhjthe dh stoedort earòc roskhch dhđlőjheh fhezh`hsl lële zorózhrledhjl rojo`cor ÷zhcljch earòc dođĿcĿ`Ŀe pazltlg yorĿsĿeĿe hőlt orocĿjcĿ yózdhchr lacoroj shëlcdl.

6.8 Ytoedort Earòcchrle GaejslyaecorĿ= Jl-jorh, t- vh G-dođĿcĿ`corĿ

Tob`le vh thst ht`hj lële stoedort earòc dođĿcĿ`Ŀe ÷eh`ledhe ÷tóró, stoedort earòcroskhch dhđlőjhechrle fozĿ gaejslyaecorĿ do ÷eh`cl rac ayeorcor vh sĿj sĿj lstotlstljthstchrdh tofca bocleh khtlrlclrchr. Fótóecój oëĿsĿedoe ől`dlclj sodhih toeĿ`corĿ vhrhihđlz,gojot, dhrsle saecorĿeo dađru (sae óëth flrledh) thjror uykucoòcoro d÷ehihđlz. ľcklclp.d.g.chrl vhr`hyhihđl` ëóejó aecorĿe juccoeĿ`Ŀ protlj dhđlcdlr.

ToeĿ` 7. Hđhr ] ?, ] 6 , …, ] j fođĿ`sĿz vh dođĿcĿ`Ŀ ] l E(>,?) lsh, \ 0 ∟

“le jshrfhstclj dhrhihslych jl-jorh dođĿcĿ`c Ŀ acduđu s÷ychelr. Yh`facchr lch lgodh hdhihjacursoj,

Furodo –shrfhstclj dhrhihsl‐ jorhsl ocĿe`Ŀő vh tapcoe`Ŀő fođĿ`sĿz ëhjlclőchrle soyĿsĿeĿlgodh ht`hjthdlr. Jl-jorhele fhjchehe dhđhrl shrfhstclj dhrhihsl lch vhrlclr,

ToeĿ` ;. Hđhr R E(>,?) vh \ lsh, a zoòe

? fjz. Ytlkchr, Y. (6>>6)= Ytotlstlis ae tbh Tofch



lclőjlslele j shrfhstclj dhrhihslych (÷đrheil) t-dođĿcĿ`Ŀ acduđu lgodh hdlclr.

Yhrfhstclj dhrhihsl j“ele fóyój dhđhrchrl lële, t- dođĿcĿ`Ŀ stoedort earòc dođĿcĿ`Ŀ lch ëajdađru flr őhjlcdh tob`le hdlclr.

ToeĿ` 2. Hđhr \ ? vh \ 6

lsh, a zoòe

lclőjlslele (j?, j6) shrfhstclj dhrhihslych G- dođĿcĿ`cĿ acduđu lgodh hdlclr.



ŕhjlc 9. Koctae“eue ]orĿeĿe Óë Ëhőldl. ]orcor ?2<>“doe jocò ?.68leëclj jóptórchr vh earòc dođĿcĿ`cĿ roskhch dhđlőjhel tojclt ht`hjlële juccoeĿcĿrcor. Ytlkchr, Y. (6>>6)= Ytotlstlis ae tbh Tofch= TbhBlstary ag Ytotlstlioc Iaeihpts oed @htbads“toe uyorcoe`ĿőtĿr.










Jaerod @hezhc

; Elsoe 6>><

?. ×zhc DođĿcĿ`cor (dhvo`)?.? âlssae DođĿcĿ`Ŀ

Fozhe, fhccl flr acoyĿe fhclrcl flr orocĿjto boekl sĿjcĿjco `hydoe khcdlđlel flc`hjlsthyhflclrlz.

×rehj ?. Bovo yacu kóvheclđl s÷z jaeusu acueio, flr uëoj àdhclele eh jodor –kóvhecl‐acduđu jaeusuedo flr joeĿyo soblp acòj lsthyhflclrlz. OőođĿdojl vhrl www.olrsogh.ia` vhrl slthsledhe hcdh hdlcdl vh tapco` uëuő soyĿsĿ lch OrocĿj 6>>7“yo jodor fhccl flr uëojtlplele yhr ocdĿđĿ ÷cóìóc jozo soyĿsĿeĿ vhr`hjthdlr.

Tofcadoe bh`he k÷rhflchihđl`lz klfl fozĿ uëoj àdhcchrl dlđhrchrleh k÷rh dobo oz jozoyop`Ŀő ëóejó, foslt flr lgodhych, yo uzue sórhcl juccoeĿcò`Ŀőcor yo do ëaj oz



órhtlc`lőchr. Dobo oeco`cĿ flr jorőĿcoőtĿrò yopoflc`hj lële, ÷cóìóc jozo soyĿcorĿeĿedođĿcĿ`corĿeĿ oëĿjcoòj lële dobo lyl flr y÷eth`h lbtlyoiĿ`Ŀz vor.

Fu tórdhe thsodógl dhđlőjhechr khehccljch –soyĿcoflclr vhrl‐ acoroj flclelr vh fu vhrlyloëĿjcoòj lële sĿjcĿjco juccoeĿcoe dođĿcĿ` âlssae DođĿcĿ`ĿdĿr.

ToeĿ` ?. Hđhr R oőođĿdojl klfl flr p.d.g.yh soblp lsh, a zoòe poro`htrhslych flrâlssae DođĿcĿ`Ŀ acduđu s÷ychelr,

R“le jhsljcl acduđueo ÷zhccljch dljjot hdlelz.

×zhclj ?. Flr âlssae roskhch dhđlőjhel R lële oőođĿdojl lclőjlchr yozĿcoflclr=

YoyĿcoflche flr dhđlőjhele dođĿcĿ`Ŀ lële âlssae dođĿcĿ`ĿeĿe ehdhe uykue vhyo jofuchdlchflclr acduđueu k÷rhflc`hj lële oőođĿdojl dóőóeshc dhehyl yopocĿ`= WorsoyocĿ` jl

flr acoyĿe flr ?/e zoòe orocĿđĿedo khrëhjchő`h acosĿcĿđĿ pe0 /e“dlr

zoòeĿe bhrboekl flr oeĿedo khrëhjchőhe acoycorĿe zoòe lëhrlsledh fođĿ`sĿzacduđueu do vorsoyĿyaruz.

e“ele saesuzo dađru klt`hsleh lzleh vhrhrhj oct-orocĿjcorĿe f÷cóetósóeóe kldhrhjjóëóc`hslel sođcorĿz. Hđhr dorocoe ljl oct orocĿđĿe acosĿcĿđĿ sĿgĿro dađru kldhrsh, vh aedoesaero acoyĿe he oz flr jhrh khrëhjchőtlđl orocĿjcorĿ soyorsoj, tapco` khrëhjchő`hchrle

soyĿsĿeĿ hcdh ht`lő acuruz. Fueue p 0 /e vh e poro`htrhcl flr flea` roskhch dhđlőjheacduđueo dljjot hdlelz.

×ehr`h ?. R e F (e, /e) dođĿcĿ`cĿ flea` roskhch dhđlőjhe lële, e ljhe p.d.g.

oőođĿdojleh yojĿesor =



ľYÔT= ëorpĿ`Ŀe cl`ltlel cl`ltchrle ëorpĿ`Ŀ acoroj ocoflclrlz vh bhr flrlel oyrĿ oyrĿdhđhrchedlrhflclrlz= Jocjócus“th ëaj lyl flclehe flr saeuio k÷rh (bhr ljl torogo Toycarshrlsl(oëĿcĿ`Ŀ) uykucoeoflclr) oőođĿdojl lgodhyl yozĿcoflclr,

F÷ychih, x vh soflt vh fu ehdhech e lch jorőĿcoőtĿrĿeio jóëócdóđóedhe, hcl`lzdh

oőođĿdojl lgodh jocĿr,

Wh fueue lële dh őueu k÷sthrhflclrlz.

Fótóe porëocorĿ flr oroyo khtlrdlđl`lzdh ÷ehr`hdhjl lgodhyl hcdh hdhrlz.

×rehj 6. YoyĿcoflclr vhrl lële he jcoslj ÷rehj (he ozĿedoe ľstotlstlđle torlbledh flr jcosljtlr)?<eiu yózyĿcdo ot thp`hsl saeuiuedo ÷che ^rusyo sóvorlchrl dlr. ?2<2“dh Vus lstotlstljëlCodlscous Fartjlhwliz“le jhőghttlđl klfl, âlssae dođĿcĿ`Ŀ ^rusyo ardusuedo flr tó`hedhflr yĿc lëhrlsledh ot thp`hsledhe ÷chechrle k÷zch`che`lő grhjoesĿeĿ őoőĿrtĿiĿ flr őhjlcdh

ëaj lyl tob`le ht`lőtlr .

K÷zch`che`lő grhjoescorĿ, he elboyhtledh flcle`hyhe 'yo fođcĿ acoe âlssae

dođĿcĿ`ĿeĿe p.d.g.slych eosĿc jorőĿcoőtĿroflclrlz1 Fu dhrsth dobo saero tortĿőoiođĿ`Ŀz

tob`lele flr ÷e lzcheihsl acoroj, “eĿe jofuc hdlchflclr flr dhđhrľ , ÷rehjch`dh

k÷zch`chdlđl`lz ot thp`h ÷có`chrlele fhjchehe soyĿsĿeĿe oyeĿsĿeĿ tob`le hdhe

p.d.g.dhjl flr poro`htr hele dhđhrl acoflclr. A bocdh R ^( ) lële, ZRP ehdlr1

\ujorĿdo tortĿőtĿđĿ`Ŀz klfl, flr âlssae roskhch dhđlőjhel Flea` roskhch dhđlőjhele, Re

F(e, /e), cl`ltldlr. Furodo e dheh`h soyĿsĿ saesuzo klt`hjthdlr. Dobo ÷eihjl Flea`

dođĿcĿ` lch lcklcl tortĿőòcorĿ`Ŀzo k÷rh, e“dhe fođĿ`sĿz acoroj, ZR e P 0 e( /e) 0 “dlr .F÷ychih ∟

saesuz shrlchrlel dađrudoe juccoeòdoe, ZRP 0 acduđueu

s÷ychyhflclrlz.



Ot thp`h vhrl shtledh, ÷rehjch` artocoòsĿ (yĿc vh tó`he ltlforlych) 0 >.;“dlr. ŕl`dl

ŕhjlc 6“dh k÷sthrlcdlđl klfl ÷rehjch` grhjoesĿeĿ 0 >.; poro`htrh aroeĿ lële acuőturucoeâlssae p.d.g.slele tharlj dhđhrchrleh jorőĿ ëlzhflclrlz. ľjl dođĿcĿ` fhclrkle flr őhjlcdhfhezhrdlr vh fu duru` sĿj sĿj –Jóëój YoyĿcor Joeueu‐ acoroj lgodh hdlclr.

6. Osl`ptatlj Tharlsl

ŕl`dlyh jodor, p.d.g.“yl, poro`htrhchrl (earòc lële vh 6, ósthc lële klfl) flcdlđl`lzl

(yo do fucoflchihđl`lzl) vorsoydĿj vh saero a flcklyh doyoeoroj acosĿcĿjcor lch lcklcllgodhchr khclőtlrdlj.

Dhrsle lzchyhe f÷có`óedh, a flcklyh soblp`lőlz klfl borhjht ht`hyhihđlz oò, acosĿcĿjózhrleh acoe flckl`lzl juccoeoroj, s÷z jaeusu roskhch dhđlőjhechrle dođĿcĿ`Ŀ jaeusuedoflzh flr őhychr s÷ychyhihj gaejslyaecor acuőturoiođĿz. A gaejslyaecor tob`le hdlilchr acoiojtĿr.

ľstotlstljth ÷eh`cl rac ayeoyoe flr tob`le hdlil ÷rehjch` artocoòsĿdĿr. Fu tob`le hdlil

flrozdoe lőchyhihđl`lz klfl, roskhch flr dhđlőjhele fhjchehe dhđhrlel tob`le hdhr.

ToeĿ` 6. e fóyójcóđóedhjl flr roskhch ÷rehjch` e soyĿdo l.l.d , acoe, yoel fótóe R“chrfođĿ`sĿz vh oyeĿ g x(x) p.d.g.sleh soblp, R?, …Re roskhch dhđlőjhechr dlzlsldlr.



Flz sĿj sĿj khrëhjchőhe roskhch dhđlőjhechrl roskhch ÷rehjch` acoroj to lgodh hdhrlz.

Hđhr e fóyójcóđóedh roskhch flr ÷rehjch`l`lz vorso, ÷rehjch`le artocoòsĿeĿedođĿcĿ`ĿeĿ dađru flr őhjlcdh oëĿjcoòj lële Rl“ele dođĿcĿ`Ŀ jaeusuedo ëaj őhy flc`hyhkhrhj duyòyĿőĿ`Ŀz (÷rehđle g x(x)“l flc`hyh khrhj duyòòj klfl) fu dhrsle oeo `hsomĿ

acoiojtĿr.

Furodojl oeo gljlr, ÷rehjch` fóyójcóđó e“l ortĿr oroj –khrëhđh‐ kldhrhj yojcoőĿp p.d.g.“yltob`le ht`hjtlr. Furodo acuőoe ljl saeuë őudur=

?. Fóyój YoyĿcor Joeueu= Fóyój e soyĿcorĿ lële, ÷rehjch`le artocoòsĿ fótóeacoflclrclđlych roskhch dhđlőjhele fhjchehe dhđhrl ZRP“h –yojcoőoiojtĿr‐.

6. @hrjhzl Cl`lt Tharh`l= Ytoedordlzh hdlc`lő ÷rehjch` artocoòsĿeĿe (–stoedordlzh‐ sae dhrsthjl sĿgĿr artocoò flrl` voryoes òeosĿedojuccoeĿcòjtodĿr) p.d.g.sl stoedort earòc roskhch dhđlőjhele p.d.g.sleh koyrl

lbtlyorĿ acoroj yojĿe acoiojtĿr. Flël`shc acoroj, osl`ptatlj saeuëcor e ljhe ehchr acoiođĿeĿ artoyo jayor, oeioj

protlj uykucoò oësĿsĿedoe (yoel saecu e lële), a saeuëcor e“le yhthrleih fóyój acòsĿduruùedo tob`lechr le acdujëo dađru acoiođĿeĿ do lò hdhr .

6.? Fóyój YoyĿcor Joeueu

6.?.? Ibhfysbhv Hőltslzclđl

Ibhfysbhv Hőltslzclđl flr roskhch dhđlőjhele fhjchehe dhđhrdhe –ëaj uzojto‐ flr yhrdh

khrëhjchő`h acosĿcĿđĿeĿe sĿeĿrcorĿeĿ fhclrthe flël`shc flr saeuëtur.

×ehr`h 6. R voryoesĿ Wor(R) 3 acoe roskhch flr dhđlőjhe acsue. A zoòe bhrboekl

flr 5 > lële,

ľYÔT= R“le p.d.g.sl g R(x) lch vhrlc`lő acsue. Fu duru`do oőođĿdojl lclőjlyl k÷sthrhihđlz

WoryoesĿe toeĿ`ĿeĿ juccoeoroj őu lclőjlyl fucuruz=



Óë lethkrocle bhr flrlsl pazltlgtlr vh fueo lcovhthe, bhrboekl flr t ZRP ’ vhyo t ZRP+ lële

DacoyĿsĿyco, lcj lethkrocl otoflclrlz vh őu saeuiueo ucoőoflclrlz.

Fu duru`do bhr torogĿ 6 “h f÷chrhj saeuiu hcdh hdhrlz.

Fu dhrsth roskhch flr dhđlőjhele voryoesĿeĿe dhđlőjhele –yoyĿcò‐sĿeĿe ÷cëósó acduđueudobo ÷eih flr oro s÷ychdlđl`lzl botĿrcoyĿeĿz. Ibhfysbhv Hőltslzclđl voryoesĿ roskhchdhđlőjhe R“le –hjstrh`‐ khrëhjchő`hchrlel k÷zch`hele (yoel artocoòdoe ëaj uzojto

acoe dhđhrchr le) acosĿcĿđĿ lch lclőjlchedlrhrhj fu lgodhyl dađrucor .




6.?.6 Fóyój YoyĿcor Joeue

ToeĿ` 9. ×rehjch` artocoòsĿ e fóyójcóđóedhjl roskhch flr ÷rehjch`dhe hcdh hdlche eroskhch dhđlőjhele (khrëhjchő`hele) orlt`htlj artocoòsĿdĿr.

Rl“chrle roskhch dhđlőjhe acòsĿedoe ÷tóró e“ele dh roskhch dhđlőjhe acduđueu eathdlelz.

×rehjch` artocoòsĿeĿe fhjchehe dhđhrl ő÷ychdlr=



Hđhr R?, …, Re fođĿ`sĿz lsh, ÷rehjch` artocoòsĿeĿe voryoesĿ oőođĿdojl klflbhsopcoeĿr ,

Hđhr Rl“chr l.l.d. earòc lsh, Rl E(, ) eh acur1 Earòcchrle dađrusocja`fleosyaeueue uykue voryoes vh artocoò lch yleh earòc acduđueu flclyaruz, yoel

e“l ortĿrdĿđĿ`Ŀzdo voryoes ozocdĿđĿ lële, fóyój flr acosĿcĿjco artocoò he elboyhtledh

ZRP“h ëaj yojĿe acoiojtĿr. Hsos ltlforlych Fóyój YoyĿcor Joeueu“eue s÷ychdlđl dh fudur.

Tharh` ? (Fóyój YoyĿcor Joeueu ). WorsoyocĿ` jl, fótóe l“chr lële, R ?, …, R e ZRP 0

vh Wor(Rl) 0 lch flr l.l.d. ëhjlclő dlzlsl acsue. A zoòe bhrboekl flr 5 > lële (ëaj

jóëój flr soyĿ) ÷rehjch` artocoòsĿ oőođĿdojl lclőjlyl sođcor

Fu duru`do e acosĿcĿjto “h yojĿesor dhrlz.

ľYÔT= ×eihjl őu saeuiuùzu juccoeocĿ`

Ibhfysbhv Hőltslzclđl lch oőođĿdojl hcdh hdlclr.

Fu lgodh, fóyój ÷rehjch`chr lële, ÷rehj artocoòsĿeĿe fóyój acosĿcĿjco roskhch

dhđlőjhele fhjchehe dhđhrledhe ëaj uzojto acòyoiođĿeĿ s÷ychr. e vh voryoes vhrlljhe, Ibhfysbhv Hőltslzclđl“el dađrudoe juccoeoroj ÷rehj artocoòsĿeĿe vhrlcl `hsoghdhedobo uzojto acò acosĿcĿđĿeĿ sĿeĿrcorĿz.



ŕhjlc ?. e toeh òdhel poro otĿőĿeĿe artocoò turo soyĿsĿeĿe ?> dlzlsl- jhsljcl

ëlzklchr

∝ “dlr.

×rehj 9. ×cëó` Flrl`chrlele Ytoedordlzosyaeu (Ytlkchr“le jltofĿeo fjz.)= Artoëođdokhehccljch bhr őhblr ÷cëó` oroiĿeĿe uzuecuđueo fođcĿ acoroj – oyoj ‐, –leë‐ vh –yord o‐ vslële gorjcĿ ÷cëó`chr juccoeĿrdĿ. Fu, ÷cëó` flrl`chrledh tliorhtl joròőĿj boch khtlrhe vhbujujl tortĿőòcorĿe ortòsĿeo yac oëoe ëaj gozco gorjcĿcĿđĿe acduđu oeco`Ŀeo khclrdl =×rehđle vhrlcl flr juòőĿe uzuecuđueue khrëhjthe dh 6> yordo uzuecuđuedo acupacòòsĿ klfl yosoc tortĿőòcor acurdu.

ľesoecorĿe fucduđu ojĿccĿio flr ë÷zó` ő÷ychydl= ?7 oyoj uzuecuđuedojl flr ëufuđueuzuecuđueu fhclrch`hj lële, roskhch ?7 jlőlclj flr ÷rehjch` shëlyarsueuz (fu duru`dofuecor ôzor kóeó Jlclshdhe ëĿjoe lcj ?7 jlőldlr), fuecorĿe oyoj uzuecujcorĿeĿ tapcoyĿp ?7oyoj klfl flr ÷cëóyó fhclrclyarsueuz, saero uzuecuđu ?7“yo f÷cóyarsueuz. ŕhjlc 6“yhfojĿeĿz. ?7 k÷zch`le artocoòsĿeĿe voryoesĿeĿe gar`ócóeh k÷rh, fu, yhel ÷cëó` flrl`lelevoryoesĿeĿ ?/?7 jodor ozoctòcĿ. Hđhr f÷ckhchr orosĿedo oyoj fóyójcójchrl edh (vhyojlclshyh kldhe lesoecordo) slsthòtlj flr gorjcĿcĿj yaj lsh, fu ÷cëó, gorjcĿ f÷ckhchrdhjltoilrchrle flrflrchrlych dobo jacoy ocĿő vhrlő yopòcorĿeo yac oëòcĿ.



ŕhjlc 6. J÷fhc“le ?7 jlőlele ?7 oyoj uzuecuđuedojlflr sĿrĿđĿe yosoc toeĿ`ĿeĿ fhclrch`hslel fhtl`chyhe

Obőop FosjĿ hshrl (?898)

6.?.9. ×rehj= –JocofocĿjcorĿe Flckhclđl‐

WorsoyocĿ` jl e fóyójcóđóedhjl flr jltch, joù ldorhsl lële 6 odoy orosĿedoe flrlelshëhihjtlr. Furodo fosltëh he ëaj ayu ocoe odoy jozoeoiojtĿr. Yhë`he l“ele O odoyĿlële ay juccoeòsĿ duruùedo ?, dlđhr duru`cordo > acoe roskhch dhđlőjhe Rl“yhfojoiođĿz. Hđhr ay aroeĿ oőođĿdojl klflysh Odoy O jozoeĿr=

Odoy O őópbhslz he lyl thrilbtlr oeioj fu jhsle acoroj O chbleh ay juccoeoiojcorĿesodhih 6 5 > aroeĿ torogĿedoe flcle`hjthdlr, yoel l 0 ?, …, 6e lële ^(Rl 0 ?) 0 ?“dlr. Yhë`hele khrl jocoe ? - 6“u bhr ljl odoy bojjĿedo th`hc flr flcklyh soblp dhđlcd lr vh bhr

ljl odoy lële bhrboekl flr thrilb fhclrt`hdhe ayueu juccoeòjtodĿr , yoel l 0 6e, …, elële, ^(Rl 0 ?) 0 ^(Rl 0 >) 0 ?/6“dlr. O odoyĿeĿe ay aroeĿ oőođĿdojl lch vhrlclr=



F÷ychih, odoy O“eĿe fhjchehe ay aroeĿeĿe dhđhrl

Flea` dođĿcĿ` saeuëcorĿeo k÷rh, voryoesĿ lsh

Ibhfysbhv Hőltslzclđl lch lcklcl lspotĿe orkóòecorĿeo k÷rh, acosĿcĿj sĿeĿrcorĿeĿ ëlz`hj lële

e“ele voryoesĿedoe foőcoyoflclrlz

JororsĿz shë`hechr (kóróctó) bhrboekl flr odoyĿ ÷zhcljch thrilb ht`hdljchrl lële, dođĿcĿ` htrogĿedo sl`htrljtlr, f÷ychih

0 (?/6) + acduđu lële, odoy F“ele joyfht`h acosĿcĿđĿ őudur=

Flrjoë soyĿ dhehyhcl`= Dlyhcl` jl 6 0 %8“tlr . Odoy F“ele shëlc`h acosĿcĿđĿeĿ %8“leoctĿedo tutòj lële e eh jodor fóyój acòj zaruedodĿr 1 YĿeĿr ő÷ych acur,

f÷ychih e

<8“dlr. Hđhr shë`hele %<8“l jororĿeĿ roskhch vhrsh flch, yoecĿő odoyĿe

shëlc`h acosĿcĿđĿeĿ %8“le octĿedo tutòj lële <8 yhthrcldlr. Fu duru` –jocofocĿjcorĿeflckhclđl‐ acoroj flclelr= Yhël`le saeuiuedo flcklchedlrlc`h`lő shë`hele yorotĿđĿ stajostlj



–kóróctó‐ fóyój ÷rehjch`chrdh artocoòyĿ oőor, oeioj saeuëto sodhih flcklchedlrlc`lőshë`hechrdhe khche slsthòtlj –sleyocchr‐ shël`le saeuiueu fhclrchr.

Fóyój YoyĿcor Joeueu“edo acduđu klfl shë`hechrle fođĿ`sĿz acduđueu vorsoydĿđĿ`Ŀzodljjot hdlelz. FođĿ`sĿzcĿj vorsoyĿ`Ŀedoe voz khëhrshj eh acur1

WorsoyocĿ` jl odoycor orosĿedo thchvlzyaedo k÷sthrlche flr tortĿőò sĿrosĿedo TWstódyasuedo odoy O vhyo F“ele yózóeh roskhch jaeoflchihj (?/6 klfl hőlt flr acosĿcĿjco)flr slehj vordĿr vh flr sórh htrogto uëuőoe slehj odoycordo robotsĿzcĿđo yac oëòjtodĿr .Flcklchedlrlc`lő shë`hechr dóőóeihchrlel dhđlőtlr`hzchr, gojot flcklchedlrlc`h`lőshë`hechr ?/9 acosĿcĿjco ózhrleh slehj jaeoe, 6/9 acosĿcĿjco do rojlfl lële ayjuccoeoiojtĿr.

\lehchehe AcosĿcĿjcor Joeueueo k÷rh, e“le artocoòsĿ

dacoyĿsĿyco artocoò dobo ÷eihjlele oyeĿsĿdĿr. Oeioj, voryoes dhđlőlr= OEAWO÷zdhőclđleh k÷rh (jaőucu voryoes)

Fueo k÷rh oőođĿdojlchrl bhsopcoyoflclrlz

f÷ychih

OdoycorĿe racchrl jororsĿz shë`he torogĿedo dhđlőtlrlchflchihđl lële, dođĿcĿ` artocoòhtrogĿedo sl`htrljtlr. Fu duru`do yujorĿdo hcdh hdlche acosĿcĿjcor lële acuőturucoe sĿeĿrcorĿ juccoeoflclrlz



Oeioj, lcj thrl` Wor(Z e|slehjP) e“yh blë flr őhjlcdh fođcĿ acòdĿđĿ lële, e ljhe

Wor( e) “le sĿgĿro dađr u klt`hdlđlel ortĿj k÷rhflclyaruz.

YoyĿcor lch lgodh hdhihj acursoj, hđhr 0 %?8 lsh (÷eihjl bhsopcoòcorĿe octĿ jotĿ), sĿeĿroőođĿdojleh hőlt acur

f÷ychih e eh jodor fóyój acurso acsue sĿeĿr ?/6'ele ózhrledhdlr. Fu sodhih óst sĿeĿracduđu lële, flzh acoyĿe khrëhjth eh jodor acosĿ acduđueu s÷ych`lyar, oeioj voryoes

sĿgĿro dóő`hdlđl lële, –kóróctóió‐ shë`hechrle shël`le saeuiu ózhrledh ëaj kóëcó flrhtjlslele acoiođĿ oëĿjtĿr.

Furodo Fóyój YoyĿcor Joeueu foőorĿsĿz acur ëóejó slehj acoyĿ fótóe –kóróctóió‐shë`hechrl oyeĿ oedo htjlchr, f÷ychih R ?, …, R e ortĿj fođĿ`sĿz dhđlcdlr. FođĿ`sĿzcĿj

vorsoyĿ`Ŀ ëaj ÷eh`cldlr ëóejó fóyój soyĿcor joeueueue khehccljch ëocĿőòsĿeĿe ehdhel–kóróctó‐eóe flrëaj k÷zch`dh artocoòyĿ oőòsĿdĿr. Hđhr –kóróctó‐eóe flr flchőhel fótóek÷zch`chr lële artoj lsh (vhyo he ozĿedo bhpslych yójshj jarhcosyae lëledh lsh), fuflchőhele voryoeso jotjĿsĿ -÷rehđl`lzdh acosĿcĿđĿeĿe sĿeĿrĿedojl 4</2? thrl`l - ÷rehjch`ëaj fóyój acso do yaj acòz.




Jaerod @hezhc

< Elsoe 6>><

?. Khehc YĿeov JuroccorĿ

6eil YĿeov khchihj bogto YocĿ kóeó sĿeĿgto yopĿcoioj vh soot to` <=>>“dofoőcoyoioj.

ľcklcl òthryoc= ÷eihclj sae sĿeovdoe saero lőchehe jaeucordo acoioj oeioj

hcfhtth jhedlelzl yađuecuj, acosĿcĿj vh dhrsle lcj óë ëhyrhđledhjl dlđhr jovro`cor jaeusuedo robot blssht`hclslelz.

prafch` shtchrledhjledhe dobo `htleshc sarucor acoioj, bhsopcoòcordo dobo ozyaruiu acoioj.

earòc dođĿcĿ` tofcasu dođĿtĿcoiojtĿr, fu ehdhech yoeĿeĿzdo khtlr`helzkhrhj`lyar

hsos ltlforĿyco lcj sĿeovĿe garòtĿeĿe oyeĿsĿ acoioj

bhsop òjlehsl khtlrlelz

jltopcor vh eatcor jopocĿ acoioj

sórh oőođĿ yujorĿ 28 dojljo acoioj JĿs`l puoe vhrlchihjtlr, fu ehdhech fótóe sarucorĿ ihvopcoedĿròyo ëocĿőĿeĿz

6. Thjror

6.? Voskhch Dhđlőjhechrle GaejslyaecorĿ

Khehc acoroj=

R“le p.d.g.sl g R(x)“l flc (jhsljcl vhyo sórhjcl) \ R“le flclehe flr gaejslyaeudur, \ 0 u(R)

p.d.g. g \(y)“yl eosĿc fucoiođĿeco lcklche

p.d.g. g \(y)“yl fucòeĿe yacu R“le sórhjcl vhyo jhsljcl vhyo u(.) gaejslyaeue flrh-flr acupacòòsĿeo fođcĿdĿr. Óë y÷eth` vordĿr=



?. Hđhr R jhsljcl lsh

6. Hđhr R sórhjcl lsh 6-odĿ`cĿ flr y÷eth` vordĿr=

OdĿ` ?= i.d.g. Gy(y)“l hcdh ht

OdĿ` 6= p.d.g.yl hcdh ht`hj lële i.d.g.ele tórhvlel oc

9. hđhr (o) R sórhjcl vh (f) u(.) flrh-flr lsh, dhđlőjhe dhđlőtlr`h gar`ócóeó juccoe

TortĿőtĿđĿ`Ŀz flrjoë ÷eh`cl ÷rehj=

Fójóc`h gar`ócó= hđhr R lch \ fođĿ`sĿz lsh, a zoòe ] 0 R + \“ele p.d.g.sl

Eat= Hđhr R vh/vhyo \“ele yađuecuđu flr yhrdh sĿgĿr lsh, lethkrocle cl`ltchrljaeusuedo dljjotcl ac.

Lethkroc D÷eóőtór`h= Hđhr R sórhjcl lsh, a zoòe roskhch dhđlőjhe \ 0 GR(R)uelgar` dođĿcĿ`ĿdĿr. Furodo GR(.) R“le i.d.g.sldlr.

YĿro ľstotlstlđl= Hđhr R?, …, Re l.l.d lsh, a zoòe he dóőój j“eil dhđhr \j“ele

p.d.g.sl



6.6. Fhjchetlchr

6.6.? Fhjchehe Dhđhr

R“le Fhjchehe dhđhrlele toeĿ`Ŀ

Hđhr R jhsljcl lsh,

Hđhr R sórhjcl lsh,

Fhjchehe dhđhrle ÷eh`cl ÷zhcljchrl

?. soflt o lële

6. R“le dađrusoc gaejslyaeu, \ 0 oR + f, lële

9. 6 vhyo dobo gozco roskhch dhđlőjhe lële

4. Hđhr R lch \ fođĿ`sĿz lsh, a zoòe

fhjchehe dhđhr R“le dođĿcĿ`ĿeĿe jaeuùeue ÷cëósódór

\ 0 u(R) gaejslyaeue fhjchehe dhđhrl (jhsljcl duru`do lethkrocl tapco` lch

dhđlőtlrle)

Mheshe Hőltslzclđl= Hđhr u(.) jaevhjs lsh, a zoòe



6.6.6. Woryoes

ŕ÷ych toeĿ`coeĿr=

R“le yoyĿcòsĿeĿe ÷cëósódór.

WoryoesĿe ÷eh`cl ÷zhcljchrl

?. flr o soflt dhđhrl lële

Wor(o) 0 >

6. voryoesle dlđhr flr lgodh őhjcl

9. R?, …, Re fođĿ`sĿz roskhch dhđlőjhechrle flr dađrusoc gaejslyaeu lële

4. Bhrboekl R?, R6 dhđlőjhechrl lële dobo khehc acoroj

6.6.9. Javoryoes vh Jarhcosyae

Javoryoes

acoroj toeĿ`coeĿr

JavoryoesĿe ÷zhcljchrl



Hđhr R vh \ fođĿ`sĿz lsh, Iav(R,\) 0 >.

Jarhcosyae jotsoyĿsĿ ő÷ych toeĿ`coeĿr=

Furodo hđhr sodhih vh sodhih \ R“le dhthr`lelsttlj dađrusoc flr gaejslyaeu lsh,

vh

6.6.4. Jaőuccu Fhjchehe Dhđhr Jaőuccu fhjchehe dhđhr roskhch dhđlőjhel ő÷ych toeĿ`coeĿr.

Jaőuccu fhjchehe dhđhr lch lcklcl ljl ÷eh`cl saeuë=

\lehchehe Fhjchetlchr Joeueu

Jaőuccu Woryoes

6.9 ×zhc DođĿcĿ`cor

6.9.?. ×zht

OőođĿdojl dođĿcĿ`coro fojĿcdĿ=

Xelgar`= R XZo, fP hđhr R“le p.d.g.sl oőođĿdojl klflysh



Flea`= R F(e, p) hđhr R“le p.d.g.sl oőođĿdojl klflysh

Ósthc= R H( hđhr R“le p.d.g.sl oőođĿdojl klflysh

Earòc= R E(¸, ω6) hđhr R“le p.d.g.sl oőođĿdojl klflysh

âlssae= R ^( hđhr R“le p.d.g.sl oőođĿdojl klflysh

Bhr flr dođĿcĿ`Ŀe artocoòsĿeĿ vh voryoesĿeĿ flc`helz vhyo bhsopcoòyĿ ÷đrhe`helzkhrhjlr. OyrĿio, Flea` lch âlssae vh Flea` lch Earòc orosĿedojl lclőljchrdh k÷sthrlcdl.

6.9.6. Earòc DođĿcĿ`

Voskhch dhđlőjhechrl eosĿc stoedordlzh hdlcdlđlel flc`helz khrhjlr=

Ylzh stoedort earòcle i.d.g.chrlele tofcasueue flr japyosĿeĿ vhrhihđl`, tofcayu eosĿcajuyoiođĿeĿzĿ flc`helz khrhjlyar.

Earòc dođĿcĿ` lch lcklcl ÷eh`cl saeuëcor=

?. earòc p.d.g. artocoò htrogĿedo sl`htrljtlr.6. earòc roskhch dhđlőjhechrle dađrusoc gaejslyaecorĿ yleh earòc dođĿcĿ`cĿdĿr=

Hđhr R E(¸, ω6) lsh, a zoòe \ 0 oR + f E(o¸ + f, o6ω6)“dlr. 9. fođĿ`sĿz earòc roskhch dhđlőjhechrle tapco`Ŀ do earòc dođĿcĿ`cĿdĿr.



4. @hrjhzl Cl`lt Tharh`l= l.l.d. ÷rehjch`l R?, …, Re lële stoedordlzh ÷rehjchàrtocoòsĿ fóyój e“chr lële yojcoőĿj acoroj stoedort flr earòc dođĿcĿ`dĿr.

6.4. Osl`ptatlj Tharlsl6.4.? Oeo Gljlr

bhr zoòe l.l.d. ÷rehjch` R ?, …, R e vorsoy

sodhih ÷rehjch` artocoòsĿyco lcklche

Rl“ele dođĿcĿ`Ŀ bojjĿedojl flckl`lz vhrl ljhe, jhsle dhđhrl/dođĿcĿ`Ŀ fucòj ëajzardur, botto l`jæesĿzdĿr

dhehy –e ‐“le fóyój e“chr lële tob`le vhrdlđl vorsoyĿcĿr.

6.4.6. Fóyój YoyĿcor Joeueu

Ibhfysbhv Hőltslzclđl= bhrboekl flr 5 > lële

Fóyój YoyĿcor Joeueu= Hđhr R?, …, Re l.l.d. lsh, a zoòe fótóe 5 > lële,

fođĿ`sĿzcĿj vorsoyĿ`Ŀ ÷eh`cldlr(–JofocĿjcorĿe flckhclđl‐edhjl acoycor orosĿedojljarhcosyae ÷rehđl klfl)

W(Rl) 3 khrhjcldlr, f÷ychih Fóyój YoyĿcor Joeue (CCE) ëaj őlőòe juyrujcu

dođĿcĿ`cordo ëocĿőòz.

6.4.9. @hrjhzl Cl`lt Tharh`l

stoedordlzh hdlc`lő ÷rehjch` artocoòsĿeĿe dođĿcĿ`Ŀeo fojĿeĿz

@hrjhzl Cl`lt Tharh`l= WoryoesĿ Wor(Rl) 3 acoe flr l.l.d ÷rehjch`l lële



furodo (.) earòc i.d.g.dlr.

Voskhch flea` dhđlőjhechrl lële Dh@alvrh-Copcoih tharh`lel k÷sthrhe krogljchrk÷rdój.

9. ×rehj ^rafch`chr

×rehj ?. Fobor 6>>9 YĿeovĿ, prafch` 9

Io`frldkh“dhjl Focdwle ajucuedo óëóeió sĿeĿg ÷đrht`hel Foy Foysae thrgl ocòjózhrhdlr vh fueue khrëhjchő`h lbtlòcl jĿs`he ÷đrheilchrlele @IOY sĿeovĿedojlphrgaròesĿeo fođcĿdĿr. Ae ÷đrheilsl vordĿr vh sĿeovdo ae saru sarucoiojtĿr.

WorsoyocĿ` jl bhr ÷đrheilele bhr saruyu dađru ihvopcoedĿr o őoesĿ %7>“tĿr, vh fótóesarucorĿe ihvopcorĿ flrflrledhe fođĿ`sĿzdĿr . He yójshj eatu ocoe ÷đrheilslele aeózhrledh he oz dajuz ocò acosĿcĿđĿ ehdlr1 He dóőój eatu ocoe ÷đrheilslele aeózhrledhe he oz óë ocò acosĿcĿđĿ ehdlr1

Ë÷zó`=

Fu sarueue ljl f÷có`óeóe acduđueo dljjot ht`helz khrhjlr= (?) flrhyshc thstsaeuëcorĿeĿe dođĿcĿ`ĿeĿ fhclrch`hj vh (6) òjslùùe vh `lelùùe i.d.g.chrlelfucòj.

Bhr flr ÷đrheilele sĿeov eatu acoe R ?> fođĿ`sĿz dheh`hele foőorĿ soyĿsĿ acduđu lële,R p.d.g.slel flcdlđl`lz flr flea` roskhch dhđlőjhedlr, R F(?>, >.7).

Khehc acoroj, flr l.l.d ÷rehjch` R?, …, Re“ele òjslùù acoe \? oőođĿdojl i.d.g.yhsoblptlr. Furodo R“le i.d.g.sl GR(x)“tlr.

vh `lelù` \6“ele i.d.g.sl

Whrl flr ÷đrheilele <“doe dóőój acò acosĿcĿđĿ



DacoysĿyco, he yójshj eatu ocoe ÷đrheilele ?> ózhrledh he oz < ocò acosĿcĿđĿ

Whrl flr ÷đrheilele ?> ózhrledh he oz 9 ocò acosĿcĿđĿ

DacoysĿyco, he dóőój eatu ocoe ÷đrheilele ?> ózhrledh he oz 9 ocò acosĿcĿđĿ

×rehj 6 Fobor 6>>; YĿeovĿ, ^rafch` 9

Hđhr R E( , ) lsh, \ 0 h R “le cak -earòc dođĿcĿ` acduđueu s÷ychrlz, \ C( , )

(o) \“ele p.d.g.slel fucueuz

(f) WorsoyocĿ` jl yotĿrĿ` yopòj lële ?>>.>>> dacorĿeĿz vor vh V ? khtlrlslele dođĿcĿ`ĿC( , ) acoe flr yotĿrĿ`Ŀ yopò acoeođĿeĿz vor. \otĿrĿ`Ŀe artocoòsĿ

?.?> “dur vh voryoesĿ ( -

) >.>?“dlr. \otĿrĿ`Ŀe flrleil d÷eh`lel e

saeuedo (?>>.>>>V ? dacor) shrvhtlelzle ??>.>>> dacordoe dobo yójshj acò

acosĿcĿđĿ ehdlr1

(i) (f)“dhjl poro`htrh dhđhrchrlele oyeĿsĿeĿ juccoeoroj, yotĿrĿ`Ŀe fođĿ`sĿz ljl

d÷eh`lel saeuedo shrvhtlelzle ??8.>>> dacordoe dobo yójshj acò acosĿcĿđĿ

ehdlr1

Ë÷zó`=

(o) Fu d÷eóőó` flrh-flrdlr, fu ehdhech dhđlőjhe dhđlőtlr`h gar`ócóeó juccoeoflclrlz.R“le bhrboekl flr rhhc soyĿ acoflchihđlel eat hdlelz vh fuedoe ÷tóró \“ele dhsthđl(>, )“dlr. Thrs d÷eóőtór`h R 0 ce(\)“dlr. Furodo dR/d\ 0 ?/R“tlr. F÷ychih, y 5 >lële dhđlőjhe dhđlőtlr`h gar`ócóeó juccoeoroj oőođĿdojlel fucuruz,



dlđhr duru`cordo

g \(y) 0 >

(f) vh “l ë÷zhrhj foőcoòj yororcĿ acoiojtĿr. WoryoesĿe lgodhslel gojt÷rchrhoyĿroflclrlz=

ArtocoòeĿe lgodhslel yhrleh jayueio vh voryoesĿe do >.>? acduđu khrëhđledh

borhjhtch hcdh hdhrlz. lële ë÷zóeih >.><>;66><2

hcdh hdhrlz. Yaero khrlyh dađru kldhr lz vh k÷róróz jl >.><??<4<9“tór.

ŕl`dl flrleil d÷eh`le saeuedo shrvhtlelzle ??>>>> dacordoe fóyój acò acosĿcĿđĿeĿfucocĿ`. Hcl`lzdh

vor. Furodo earòc acosĿcĿj tofcasueue juccoeoroj stoedort earòc i.d.g.ele dhđhrlelfucoflclrslelz.

(i)

ce(V?) lch ce(V6)“ele fođĿ`sĿz earòc acdujcorĿeĿ vh dacoysĿyco tapco`corĿeĿe doearòc acduđueu eat hdlelz. Artocoò artocoòcorĿe tapco`ĿdĿr vh voryoesvoryoescorĿe tapco`ĿdĿr. ŕopjo lőorhtlel yhel artocoò, voryoes vh stoedort sopò

lële juccoeĿrsoj, >.?2692<27, >.>?747><<2 vh >.?629>>46? acur.Dobo ÷eihjl bhsopcoòcorĿ sórdórórshj,



×rehj 9 Fobor 6>>; YĿeovĿ, ^rafch` 4

Flr ľsvhë hjaea`lstl acoe @ljohc ^rljs flr sórhdlr baclkoe ojtlvlthchrl, jovkocorĿ,yorocoòcorĿ, vs., ózhrleh ľsvhë paclsl torogĿedoe tapcoeoe dhtoycĿ vhrl lch –Glròe Fays‐ëhthslele óyhchrledhe flrlslele jhedl roparueu juccoeoroj (fjz.

www.crz-

ùheibhe.dh/ihs/`ljohc.bt`) gutfac baclkoecorĿ lch lcklcl ëhőltcl hjaea`lj jaeucorĿ ëocĿőòjtodĿr. Flr òjochsledh dóőòe baclkoe krupcorĿ orosĿedo acosĿ vh shrtjovkocorĿe ehdhechrlel oeoclz hdhr. Fueue lële, jovkocor vh yorocoeòcor ózhrleh flràdhc khclőtlrlr. @adhcdh flr shzaedo dóőòe krupcorĿeĿe acosĿ jorőĿcoőò soyĿsĿ flr

^(8) dođĿcĿ`ĿdĿr (âlssae 0 8). DobosĿ, bhr jovkodo he oz flr yorocoeòeĿe acoiođĿeĿvh khrëhjth, ?>“o jodor bhr yorocoeòeĿe hőlt acosĿcĿjcĿ acduđueu vorsoy`ĿőtĿr.

(o) s÷z jaeusu vorsoyĿ`cor vhrl ljhe, flr yĿc lëhrlsledh ljl dóőòe krufue flrflrlelyorocoò soyĿsĿeĿe fhjchehe dhđhrl ehdlr1 Y÷z jaeusu soyĿeĿe voryoesĿ ehdlr1

(f) WorsoyocĿ` jl fhclrtlchechrle yhrleh, ljl dóőòe krup jorőĿcoőtĿđĿedo jovko acòacosĿcĿđĿ sodhih ?/6“dlr (acosĿ jorőĿcoőòcorĿ fođĿ`sĿz vorsoyoflclrslelz). (o)“yovhrhihđlelz ihvop eosĿc dhđlőlr.

Ë÷zó`=

(o) R flr shzaedojl jovko soyĿsĿeĿ vh \ yorocoeò soyĿsĿeĿ lgodh htsle. OyrĿiojorőĿcoőòeĿe jovko lch saeuëcoeoiođĿeĿ vorsoyoiođĿz. Fu duru`do, H(\) 0H(H(\|R)) 0 H(8.8R) 0 8.8H(R) 0 8.8(8) 0 6;.8 acur. Wh Wor(\) 0 H(Wor(\|R)) +

Wor(H(\|R)) 0 H(

) + Wor(8.8R). ZEAT= flr jovkodojl yorocoeòcorĿe

soyĿsĿeĿe voryoesĿ

“dlr. F÷ychih, hđhr jovkocor orosĿ yorocoeòcorĿe dođĿcĿ`Ŀ

fođĿ`sĿz lsh, R soyĿdojl jovkodo yorocoeò soyĿsĿeĿe voryoesĿ

acur.P

Bhsopcoòcoro dhvo` hdleih, H(

) + Wor(8.8R) 0 (<</?6)H(R) +

(?6?/4)Wor(R) 0 (<</?6)(8) + (?6?/4)(8) 0 ?<6.8 hcdh hdhrlz.

(f) ] jorőĿcoőò lbtlòclel lgodh htsle. Fu duru`do

Woryoes lële lsh boco őueu s÷ychyhflclrlz=



OscĿedo ől`dl H(R) vh Wor(R) (o)“dojleh k÷rh dhđlő`lő acdu. H(R)“le ÷eihjl dhđhrleleyorĿsĿ jodor acduđueu k÷r`hj zar dhđlc(ől`dl 6.8“tlr). p 0 >.8 vh ] jodor dheh`h lchR|] flr flea` acduđu khrëhđlel juccoeoroj, R“le voryoesĿ ő÷ych yozoflclrlz=

Khrlyh dađru kldleih

hcdh hdhrlz.

×rehj ^rafch`chrFobor 6>>9 YĿeovĿ, prafch` 9

Io`frldkh“dhjl Focdwle ajucuedo óëóeió sĿeĿg ÷đrht`hel Foy Foysae thrgl ocòjózhrhdlr vh fueue khrëhjchő`h lbtlòcl jĿs`he ÷đrheilchrlele @IOY sĿeovĿedojlphrgaròesĿeo fođcĿdĿr. Ae ÷đrheilsl vordĿr vh sĿeovdo ae saru sarucoiojtĿr.WorsoyocĿ` jl bhr ÷đrheilele bhr saruyu dađru ihvopcoedĿr o őoesĿ %7>“tĿr, vh fótóesarucorĿe ihvopcorĿ flrflrledhe fođĿ`sĿzdĿr. He yójshj eatu ocoe ÷đrheilslele aeózhrledh he oz dajuz ocò acosĿcĿđĿ ehdlr1 He dóőój eatu ocoe ÷đrheilslele ae

ózhrledhe he oz óë ocò acosĿcĿđĿ ehdlr 1

Fobor 6>>; YĿeovĿ, ^rafch` 9

Hđhr R E( , ) lsh, \ 0 h R “le cak -earòc dođĿcĿ` acduđueu s÷ychrlz, \ C( , )

(o) \“ele p.d.g.slel fucueuz

(f) WorsoyocĿ` jl yotĿrĿ` yopòj lële ?>>.>>> dacorĿeĿz vor vh V ? khtlrlslele dođĿcĿ`Ŀ

C( ,

) acoe flr yotĿrĿ`Ŀ yopò acoeođĿeĿz vor. \otĿrĿ`Ŀe artocoòsĿ

?.?>“dur vh voryoesĿ (

- ) >.>?“dlr. \otĿrĿ`Ŀe flrleil d÷eh`lele

saeuedo (?>>.>>>V ? dacor) shrvhtlelzle ??>.>>> dacordoe dobo yójshj acò

acosĿcĿđĿ ehdlr1



(i) (f)“dhjl poro`htrh dhđhrchrlele oyeĿsĿeĿ juccoeoroj, yotĿrĿ`Ŀe fođĿ`sĿz ljl

d÷eh`lel saeuedo shrvhtlelzle ??8.>>> dacordoe d obo yójshj acò acosĿcĿđĿ

ehdlr 1

Fobor 6>>; YĿeovĿ, ^rafch` 4

Flr ľsvhë hjaea`lstl acoe @ljohc ^rljs flr sórhdlr baclkoe ojtlvlthchrl, jovkocorĿ,yorocoòcorĿ, vs., ózhrleh ľsvhë paclsl torogĿedoe tapcoeoe dhtoycĿ vhrl lch –Glròe Fays‐ëhthslele óyhchrledhe flrlslele jhedl roparueu juccoeoroj (fjz.

www.crz-ùheibhe.dh/ihs/`ljohc.bt`) gutfac baclkoecorĿ lch lcklcl ëhőltcl hjaea`lj jaeucorĿëocĿőòjtodĿr. Flr òjochsledh dóőòe baclkoe krupcorĿ orosĿedo acosĿ vh shrtjovkocorĿe ehdhechrlel oeoclz hdhr. Fueue lële, jovkocor vh yorocoeòcor ózhrleh flràdhc khclőtlrlr. @adhcdh flr shzaedo dóőòe krupcorĿeĿe acosĿ jorőĿcoőò soyĿsĿ flr

^(8) dođĿcĿ`ĿdĿr (âlssae 0 8). DobosĿ, bhr jovkodo he oz flr yorocoeòeĿe acoiođĿeĿvh khrëhjth, ?>“o jodor bhr yorocoeòeĿe hőlt acosĿcĿjcĿ acduđueu vorsoy`ĿőtĿr.

(o) s÷z jaeusu vorsoyĿ`cor vhrl ljhe, flr yĿc lëhrlsledh ljl dóőòe krufue flrflrlelyorocoò soyĿsĿeĿe fhjchehe dhđhrl ehdlr1 Y÷z jaeusu soyĿeĿe voryoesĿ ehdlr1

(f) WorsoyocĿ` jl fhclrtlchechrle yhrleh, ljl dóőòe krup jorőĿcoőtĿđĿedo jovko acòacosĿcĿđĿ sodhih ?/6“dlr (acosĿ jorőĿcoőòcorĿ fođĿ`sĿz vorsoyoflclrslelz). (o)“yovhrhihđlelz ihvop eosĿc dhđlőlr.



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ ?;

Jaerod @hezhc

?7 Elsoe 6>><

?. @hrjhzl Cl`lt Tharh`l

Khëhe bogto flea` roskhch dhđlőjhechr lële Dh@alvrh-Copcoih tharh`lel k÷r dóđó`ózó botĿrcoyĿeĿz. Fu tharh` hsos ltlforlych, fóyój e dhđhrchrl lële stoedordlzh hdlc`lő \ ~

F(e, p) dođĿcĿ`cĿ roskhch dhđlőjhele ] 0

‑ele yojcoőĿj acoroj stoedort earòc

dođĿcĿ`cĿ acduđueu s÷ychr. Flr flea`, l.l.d. sĿgĿr/flr roskhch dhđlőjhe Rl‑chrle (foőorĿ‐ lch

saeuëcoeoe –dheh`hchr‐le soyĿ Ŀ) tapco`Ŀ acduđu lële, ‑yl R?, …Re ÷rehjch`lele

artocoòsĿ acoroj dóőóehflclrlz.

DacoysĿyco, Dh@alvrh-Copcoih tharh`l khrëhjth l.l.d. sĿgĿr/flr roskhch dhđlőjhelele artocoòsĿeĿe flr saeuiudur. @hrjhzl Cl`lt Tharh`l fueu saecu voryoesĿ acoe dlđhrbhrboekl flr dođĿcĿ`doe hcdh hdlche l.l.d shjoescorĿeĿe ÷rehjch` artocoòcorĿeo khehcchőtlrlr .

Tharh` ? (@hrjhzl Cl`lt Tharh`l). WorsoyocĿ` jl R?, …, R

e artocoòsĿ vh voryoesĿ

3 acoe flr dođĿcĿ`doe hcdh hdlche e fóyójcóđóedh flr roskhch ÷rehjch`dlr.Bhrboekl flr soflt x dhđhrl lële oőođĿdojl yozĿcoflclr=

Furodo ∝ e‑ele artocoòsĿ γ vh voryoesĿ ω6 acoe earòc dođĿcĿò yojĿesoyoiođĿeĿs÷ychyhflclrlz (fozĿcorĿ fueo –yojĿesoò joeueu‐ dhr). Yh`facchr lch lgodh hdhrshj

A bocdh flr artocoò eosĿc acurdo oyeĿ oedo bh` soflt flr dhđhr γ‑yo (Fóyój YoyĿcorJoeueueo k÷rh) vh bh` dh voryoesĿ ? acoe (`hrjhzl cl`lt tharh`leh k÷rh) roskhch flrdhđlőjheh yojĿesor1 Furodojl ÷eh`cl dhtoy `hrjhzl cl`lt tharh`l lch lcklcldlr yoeldođĿcĿ`Ŀe flr eajtoyo dóő`hslel (Fóyój YoyĿcor Joeueuedo fu acur) vhyo saesuzo



jodor potcoòsĿeĿ hekhcchyhihj he uykue aroeĿ vhrhihj őhjlcdh ÷rehjch` artocoòsĿeĿ

∝ ―ele ëorpĿ`Ŀ jodor yujorĿ ëhjhrlz.

Flr dođĿcĿ`Ŀe artocoòsĿeĿ sĿeĿrcoedĿròj lële elyh earòc dođĿcĿ` foőcoòj lële heuykue odoydĿr1 ľjl fođĿ`sĿz earòc roskhch dhđlőjhele tapco`ĿeĿe yleh earòc flr

dođĿcĿ`Ŀ acduđueu (bhr eh jodor voryoescorĿ gorjcĿ acso do, sodhih stoedordlzh hdlc`lőartocoòyo fojtĿđĿ`Ŀz lële fu a jodor do ÷eh`cl dhđlcdlr) tortĿőtĿđĿ`ĿzĿ thjrorbotĿrcoyĿeĿz. \oel fójóc`h (yoel dođĿcĿ`doe fođĿ`sĿz roskhch dhđlőjhe hjch`h)oëĿsĿedoe earòc dođĿcĿ` olchsl lstljrorcĿdĿr. Fueue dlđhr dođĿcĿ`corĿe ëađu lële dađruacòdĿđĿeĿ fhclrthcl` (uelgar` vhyo ósthc klfl).

×rehjch` artocoòsĿ thj thj k÷zch`chrle ođĿrcĿjcoedĿrĿc`Ŀő artocoòsĿ acduđu lële,dlyhcl` jl ÷rehjch`l e‑dhe 6e‑h ëĿjorĿrsoj, Re+?, …, R6e shjoesĿeĿe artocoòsĿ flrleil

artocoòyo hjchelr vh 6‑yh f÷cóeór. DacoyĿsĿyco, hđhr fóyój e‑chr lële flch, e‑ledođĿcĿ`ĿeĿe hcdh hdlche ljl fođĿ`sĿz dhđlőjhele tapco`ĿeĿe oyeĿ dođĿcĿ` olchsledh

acòyoe tór dhe acduđueu vorsoy`Ŀő acsoydĿj, ÷rehjch` artocoòsĿeĿe dođĿcĿ`Ŀ yleh dhlbtlyorl fóyój e dhđhr chrl lële ëaj dhđlőlrdl vh dacoyĿsĿyco lstljrorcĿ flr cl`lth dađruklt`hyhflclrdl. Fu duru`, artocoòeĿe dođĿcĿ`ĿeĿe cl`ltth earòc dođĿcĿò yojcoőòsĿeĿeelyh jofuc hdlchflclr acduđu jaeusuedo yhthrleih flckl vhr lyar acòsĿ khrhjlr .

ŕhjlc ?. e poro otĿőĿeĿe turo soyĿsĿ= ÷rehjch` artocoòsĿ e(sacdo) vh stoedordlzh

hdlc`lő ÷rehjch`le artocoòsĿ ∝ e

(sođdo)

×rehj ?. WorsoyocĿ` jl R ?, …, R e l.l.d. acoe roskhch dhđlőjhechrdlr vh R l ~ XZ>, ?P

dođĿcĿ`Ŀ uelgar`dur, dacoysĿyco p.d.g ő÷ych acur=



ŕl`dl jĿs`Ŀ tapco`corĿe p.d.g.slel bhsopcoòj lële ?> ‑ eueiu dhrsthjl fójóc`h gar`ócóeójuccoeoflclrlz,

j 0 6 lële, oőođĿdojl el hcdh hdhrlz (lethkrocle cl`ltchrleh dljjot ht`hj khrhjlr)

Yaerojl bhsopcoòcor ëaj joròőĿj acoflclr ëóejó lethkrocle cl`ltchrlel vh yađuecujtojl

jĿrĿcò eajtocorĿeĿ sórhjcl jaetrac ht`hj khrhjlr. FozĿ bhsopcoòcordoe saero, j 0 9 lëleőueu hcdh httlj ,

Voskhch dhđlőjhechrle tapco`corĿeĿe fhjchehe dhđhrl lch lcklcl juroco k÷rh

OyrĿio, R ?, R 6 , …, R j fođĿ`sĿz acdujcorĿ lële, tapco`Ŀe voryoesĿ lch lcklcl jurocĿ juccoeoflclrlz

Fuedoe ÷tóró, YJ ‑eĿe stoedordlzh hdlc`lő bocl acoe ] j oőođĿdojl lch vhrlclr=



Furodoe borhjhtch, dhđlőjhe dhđlőtlr`h gar`ócóeó juccoeoroj stoedordlzh hdlc`lő ] ?,

] 6 , ] 9‑óe yađuecujcorĿeĿ bhsopcoyoflclrlz (tórhvle sodhih ‑ yo hőlt acduđueo dljjot

hdlelz)=

Fueue k rogljshc acoroj eosĿc k÷róedóđóeh fojocĿ`=

Xelgar` roskhch dhđlőjhechrle stoedordlzh hdlc`lő tapco`corĿeĿe p.d.g.sl sodhih 9 jodorfođĿ`sĿz ëhjlclőle tapco`corĿeĿe stoedort earòc p.d.g.sleh ëaj fhezh`hjthdlr. Fu flr



sórprlzdlr ëóejó uelgar` yađuecuđue jhedlsl blëflr őhjlcdh earòc roskhchdhđlőjhelejleh fhezh`hz.

Yae ÷rehj, 9 jodor oz (he ozĿedoe aptlj acoroj) soyĿdojl e lële earòc tob`lele acdujëolyl k÷róe`hsl jaeusuedo flroz ocdotĿiĿ acso do, e duruùedo, khehccljch e 4>

vhyo dobo yójshj dhđhrchr lële tob`lele dađru acduđueu lgodh hdhrlz.

×zhtchyhihj acursoj, ÷rehjch`le khrëhj artocoòsĿeĿe p.d.g.slel bhsopcoòjlsth`hdlđl`lz zoòe @hrjhzl Cl`lt Tharh`l ÷zhccljch yororcĿdĿr. Fueuekhrëhjchőhflchihđl ljl duru` vordĿr

Khrëhj p.d.g.‑yl bhsopcoòyoflclrlz ëóejó R‑chrle to` dođĿcĿ`ĿeĿ flc`lyaruzdur,

Khrëhj p.d.g.yl bhsopcoòj lsth`hyhflclrlz ëóejó bhsopcoòcor ëaj joròőĿjacoflclr- Fu khehc fójóc`h gar`ócó (÷rehđh fojĿeĿz), vh dlđhr flr ëaj jhsljcl ÷rehj(sae dhrsthjl Flea` ÷rehj klfl) lële ehrdhysh tortĿőòsĿz dađrudur.

6. Tob`le

Fu dhrsth ől`dlyh jodor , flr roskhch dhđlőjhele dođĿcĿ`ĿeĿe poro`htrhchrlel flcdlđl`lzlvorsoyoroj foőcodĿj. ×rehđle R ~ ^Zκ P acduđueu flclyarduj vh dobo saero a dođĿcĿ`doeacosĿcĿjcorĿ vh roskhch dhđlőjhele dlđhr ÷zhcljchrle bhsopcĿyarduj. ŕl`dl lsh prafch`hthrsthe fojoiođĿz=

poro`htrhchrl flcle`hyhe flr dođĿcĿ`doe hcdh hdlche flr l.l.d. ÷rehjch`h soblp

acduđuùzu vorsoyoioj acursoj, lőchdlđl`lz dođĿcĿ` olchchrledh yhr ocoe boekl dhđhrlele vhrlyl órht`lő acoflchihđl jaeusuedo eosĿc – ojuc‐ flr ihvop hcdh hdhrlz1

×rehj 6. Hđhr flr òdhel poro vhrlc`lőjhe thj flr otĿőĿe turo khc h acosĿcĿđĿeĿ

flc`lyarsoj , ëaj soyĿdo otĿő yopoflclrlz. A zoòe turocorĿe aroeĿeĿ, 0

, dobo

saero oëĿjcoyoiođĿ`Ŀz òetĿjco ^(Turo) acosĿcĿđĿ lële – lyl flr tob`le‐ acoflchihđleldóőóehflclr lz.

g(x|) p.d.g.chrl vhrl ljhe, flr poro`htrh flr dođĿcĿ`cor olchslele soflt hedhjsldlr. Furodo

poro`htrhchrl khehccljch ? …, j acoroj lgodh hdhrlz.

×rehj 9.

Flea` dođĿcĿ` lële poro`htrhchr e dheh`h soyĿsĿ vh p foőorĿ aroeĿdĿr.



Earòc dođĿcĿ` lële poro`htrhchr artocoò

γ

vh stoedort sopò ω ‑dĿr .

âlssae dođĿcĿ`Ŀ flr thj poro`htrhyh soblptlr, κ

ľstotlstlđle ëađu, flclehe dođĿcĿ` olchchrlele boekl óyhslele k÷zch`che`lő flr sórhile vhyoacoyĿe dađru acosĿcĿj dođĿcĿ`ĿeĿ vhrhihđlel fhclrch`hjch lch lcklcldlr. Yh`faclj acoroj, R ~ gZx|>P ljhe flz poro`htrh dhđhrl > ‑Ŀ fucòj lsthrlz. Fu duru`, –dođĿcĿ`Ŀ jorojthrlzhhdhe poro`htrhchrl tob`le ht`h‐ prafch`ldlr.

Bhr zoòe flr roskhch ÷rehjch` R?, …, Re lch foőcoyoiođĿz vh bhr zoòe oőođĿdojlvorsoyĿ`do fucueoiođĿz=

flcle`hyhe > Β lële R ~ gZx|> P

ToeĿ` ?. ‑ele tob`le hdlilsl flr lstotlstljtlr ( yoel R ?, …, R e‑le flr gaejslyaeudur),

Flr ÷rehjch`dh flr tob`le hdlilele khrëhjchő`hsl , (x?, …, xe), ―eĿe tob`lel acoroj

odcoedĿrĿcĿr.

Voskhch flr ÷rehjch`le gaejslyaeu acoroj, tob`le hdlil uykue flr roskhch dhđlőjhedlr,fu ehdhech khehc acoroj p.d.g.sl ilesledhe dođĿcĿ`ĿeĿ oëĿjcoòj vh dođĿcĿ`ĿeĿe à`hetllch lcklchehihđl`lzh dljjot hdlelz.

×rehj 4. WorsoyocĿ` jl R l ~ Fhreauccl( > )‑dlr, yoel R sĿgĿr/flr roskhch dhđlőjhedlr vh ?dhđhrlel acosĿcĿjco ocĿr vh p.d.g.sl ő÷ychdlr=



> ‑ĿeĿ eosĿc tob`le hdhrlz1

×rehjch` artocoòsĿeĿ juccoeoflclrlz,

×rehđle 8 Fhreauccl dheh`hsl ?, >, >, ?, ? lële őueu fucuruz,

8 k÷zch`cl flr ÷rehjch`dh tob`le hdlilsl flr roskhch dhđlőjhe acduđu lële, p.d.g.slel

hcdh hdhflclrlz= botĿrcoyĿeĿz Y8 ( ∟ ) lële Y8 ~ F(8, > )‑dĿr . Jhsljcl roskhch

dhđlőjhechrle p.d.g.slel fucòeĿe y÷eth`chrlel 0 (Y8 )/8 ‑h uykucorsoj, oőođĿdojlel hcdh

hdhrlz=

×zhcljch, tob`le hdlilele dođĿcĿ`Ŀ khrëhj > acosĿcĿđĿeo fođcĿdĿr ’ a acosĿcĿj Z>,?P

orocĿđĿedo bhrboekl flr yhrdh acoflclr- gojot sodhih 7 gorjcĿ jhsljcl dhđhr ocoflclr.

×rehj 8. Hđhr R ~ XZ>, P dođĿcĿ`Ŀ poro`htrhyh fođcĿ acoroj flr orocĿjto uelgar` lsh,

p.d.g.‑sl ő÷ychdlr=

‑yl eosĿc tob`le hdhflclrlz1 OőođĿdojlchrl juccoeoflclrlz,

Dlyhcl` jl, dođĿcĿ`doe óë ÷rehj shëtlj, >.6, >.7, >.4. A zoòe ? 0>.7 vh 6 0>.2 acurduvh f÷ychih oyeĿ poro`htrh lële ljl tob`le hdlil gorjcĿ ihvop vhrlrdl. Fu gorjcĿ tob`lehdlilchr orosĿedoe eosĿc shël` yopoiođĿz1 Fueo flroz saero khrl d÷ehihđlz.

(R?, …, Re) gaejslyaecorĿeo eosĿc ucoőĿrsĿeĿz1

Fu tob`le hdlilchrle òjuc acduđueu eosĿc fhclrchyhflclrlz1



OyeĿ poro`htrh lële flr vhyo ljl tob`le hdlil orosĿedoe eosĿc shël` yopòcĿyĿz1

9. Tob`le Hdlilchrle Khehc ×zhcljchrl

R‑le fhjchehe dhđhrlel poro`htrh lch lgodh hdhihđlz, yoel hđhr khrëhj poro`htrh ‑yo

hőltsh R‑le fhjchehe dhđhrl ő÷ychdlr=

OyeĿ őhjlcdh, poro`htrh lch voryoesĿ ő÷ych yozoiođĿ`=

Flr tob`le hdlilele sopòsĿ(yoecĿcĿđĿ Ë.E.) aeue fhjchehe dhđhrl lch khrëhj dhđhrlorosĿedojl gorjtĿr.

Hcfhtth jl, artocoò acoroj poro`htrhyl dađru vhrhe flr tob`le hdlilyl lsthrlz, yoel ldhocacoeĿ sopòeĿe sĿgĿr acòsĿdĿr.

ToeĿ` 6. Flr tob`le hdlil 0 (R ?, …, R e ) lële sopòsĿz dĿr hđhr fótóe > d hđhrchrllële

.

×rehj 7. WorsoyocĿ` jl, R ?, …,R e E( , ) dođĿcĿ`Ŀedoe hcdh hdlche flr l.l.d.÷rehjch`dlr. Khëhe bogto ÷rehjch` artocoòsĿeĿe fhjchehe dhđhrlele bhrboekl flr γ

dhđhrl lële

acduđueu k÷r`óőtój , fu ehdhech e earòc dođĿcĿ`Ŀe artocoòsĿ γ lële sopòsĿz flrtob`le hdlildlr.



×rehj ;. ľ .l.d. acoe R ?, …, R e roskhch ÷rehjch`ledhe γ artocoòsĿ flcle`hyhe R E( ,

) dođĿcĿ`Ŀ lële voryoes poro`htrhsl ‑ y l tob`le ht`hj lsthdlđl`lzl vorsoyocĿ`. 0

Z(R - ZRP)6 P acduđu lële, shzklshc acoroj lckl ëhjlil flr tob`le hdlil oőođĿdojl klfl acoflclr

(furodo ÷rehjch` artocoòsĿeĿ khrëhj fhjchehe dhđhrl lch yhr dhđlőtlrttlj). HđhrdođĿcĿ`Ŀe khrëhj poro`htrhchrl ( γ > , ω > 6 ) lsh, fu tob`le hdlilele fhjchehe dhđhrl ehdlr1

ZR 6 P 0 ZRP)6 + Wor(R) lclőjlslel botĿrcoyĿeĿz, f÷ychih

DacoyĿsĿyco 6 lële sopòsĿz flr tob`le hdlil dhđlcdlr, gojot 6 lële ëaj jacoy flrsopòsĿz tob`le hdlil acuőturoflclrlz.

Fu sopò ehrhdhe khclyar1 Khehc jaeuőòj khrhjlrsh, fueue ehdhel jorh poroethzle

lëledhjl γ ‑yó – kóróctócó‐ flr tob`le acoe 0 e lch yhr dhđlőtlr h`lzdlr. Hđhr γ >

flcleshydl, tob`le hdlil 6 0

∟ ‑ele ω lële sopòsĿz acoiođĿeĿ jhedlelz

jaetrac hdhflclrslelz.

ArtocoòyĿ tob`le ht`hj vhrldhe flr –shrfhstclj dhrhihsl‐ k÷tórór - ÷rehđle hđhr hcl`lzdhsodhih thj k÷zch`cl flr ÷rehjch` acsoydĿ, tob`le hdlche artocoò a k÷zch`h hőlt acurdu



vh voryoesĿe –sog‐ tob`le hdlilsl flzh 6 0 > ‑Ŀ vhrlrdl, oëĿjëosĿ fu do dađru ihvop

dhđlcdlr.

YopòsĿzcĿj flzl` lcklchedlđl`lz thj őhy acòyoflclr. Artocoò acoroj tob`le hdlilele

khrëhj poro`htrhyh hőlt acòsĿ fhccl flr ÷rehjch` lëledh ÷ych acoiođĿ oeco`Ŀeo

khch`hyhihđledhe, khrëhjth tob`le khrëhj poro`htrhyh yojĿe acoiojtĿr.

ToeĿ` 9. Hđhr e‑ y l ortĿrĿrjhe tob`le hdlil acosĿcĿj fojĿ`Ŀedoe > ‑ o yojĿesorso, flr R ?,

…, R e ÷rehjch`l lële, ‑eĿe β lële tutorcĿ flr tob`le hdlil acduđueu s÷ychyhflclrlz. Fótóe θ

5 > dhđhrchrl lële

Jhcl`hchr lch lgodh hdhihj acursoj, yhthrleih fóyój flr ÷rehjch`dh, tutor cĿ flr tob`le

hdlil yójshj flr acosĿcĿjco khrëhj poro`htrhyh ëaj yojĿe flr orocĿđĿe lëledh acoiojtĿr .Dljjot hdlchihj acurso sopòsĿzcĿj vh tutorcĿcĿj ljl ëaj gorjcĿ ÷rtóőhe jovro`dĿr , oeiojflrl dlđhrlel lò ht`hz.

×rehj 2. Xelgar` dođĿcĿ` lële s÷z jaeusu acoe tob`le hdlilchrl`lzdhe flrleh khrld÷ehcl`, R ~ XZ>, > P . Hđhr oőođĿdojl lgodhyh fojoio acursoj,

?‑le β lële sopòsĿz acòdĿđĿeĿ jacoycĿjco k÷rhflclrlz, ëóejó uelgar` dođĿcĿ`Ŀe

dađosĿedoe ÷tóró, R l ‑ele fótóe acosĿ dhđhrchrl > ‑the jóëójtór. Fu ehdhech, e eh jodorfóyój acurso acsue, ^(òx{ R ?, …, R e } 3 > ) 0?‑dlr. F÷ychih Fhjchehe dhđhr β aZ ? P 3

> ‑dĿr . Oeioj, > lële tutorcĿdĿr= Xelgar` dođĿcĿ`Ŀedoe hcdh hdlche flr thj k÷zch` acoe R

lële i.d.g.‑ele G R (x) 0 x/ > acduđueu jacoyio k÷rhflclrlz. \e =0 òx{R ?, …, R e }

÷rehjch`le e‑eil sĿro lstotlstlđl acduđu lële, dobo ÷eihjl tortĿőòcorĿ`Ŀzdoe > y ?

lële G \e(y) 0 (G R (y))e 0 (y/ > )e‑l hcdh hdhrlz. ? acosĿcĿjco ? 3 β > acduđu lële, bhrboekl flr

÷rehjch` soyĿsĿ e vh bhrboekl flr θ 5 > lële oőođĿdojlel bhsopcoyoflclrlz,

furodo θ 5 > acduđu lële p =0

3 ?‑dlr. Fu ehdhech, e‑l ortĿrĿrjhe, > ‑doe θ ―doe dobo

gozco sopòeĿe acosĿcĿđĿ yaj acur, vh dacoysĿyco ? tutorcĿdĿr.

×rehj <. Fóyój YoyĿcor Joeueueo k÷rh, ÷rehjch` artocoòsĿ acosĿcĿj acoroj ZRP 0 γ ‑yo

yojĿesor. DacoyĿsĿyco, E( , )) acoe roskhch dhđlőjhelele flr l.l.d. acoe R ?, …, R e ÷rehjch`l lële, ÷rehjch` artocoòsĿ γ ‑ eóe tutorcĿ flr tob`le hdlilsldlr.



OyrĿ flr shëhehj acoroj, –òjuc acòyoe‐ flr (R ?, …, R e ) tob`le hdlilsleh fojocĿ`. Fuduru`do,

DacoysĿyco fu tob`le hdlil sopòsĿzdĿr. Oeioj, bhrboekl flr e ÷rehjch` fóyójcóđó lële,tob`le hdlilele dođĿcĿ`Ŀ s÷z jaeusu R E( , ω )‑le oyeĿsĿdĿr, fu ehdhech ÷rehđle θ 0 ω >

lële, fótóe e‑chrle acosĿcĿđĿ

furodo stoedordlzh hdlche ] =0 E(>, ?) dođĿcĿ`c ĿdĿr. Fu orkóòecordoe borhjhtch,

sopòsĿzdĿr oeioj tutorcĿ dhđlcdlr.



?4.9> Hjaea`ldh ľstotlstljshc \÷eth`chrh Klrlő Dhrs EatcorĿ ?2

Jaerod @hezhc

69 Elsoe 6>><

?. Tob`le Hdlilele ×zhcljchrl (dhvo`)

?.?. Ytoedort Boto

YĿj sĿj tob`le hdlilele jhsle dađrucuđu bojjĿedo do lgodhchr khclőtlr hj lsthrlz ’

tob`lele dhđhrlel bhr zoòe artoyo jayoflclrlzz, gojot aeue khrëhjthe dh khrëhj

poro`htrhyh yojĿe acduđuedoe eh jodor h`lelz1

ToeĿ` ?. Flr tob`lele stoedort botosĿ ω(), tob`le hdlilele stoedort sopòsĿdĿr (yo dotob`le hdlc`lő stoedort sopòsĿ). ŕ÷ych k÷sthrlclr=

Flr tob`le hdlilele flr roskhch dhđlőjhele gaejslyaeu acduđueu botĿrcoòeĿz khrhjlyarvh fu ehdhech fu roskhch dhđlőjhe lële fhjchehe dhđhrl, voryoesĿ vh dlđhr à`hetchrlbhsopcoyoflclrlz.

×rehj ?. Flr l.l.d. acoe ÷rehjch` R ?, …, R e“le artocoòsĿ e“d lr, furodo Wor(R l ) 0 /e

voryoesĿeo soblptlr. DacoysĿyco stoedort boto

Hđhr

“l flc`lyarsoj , tob`le hdlc`lő stoed ort botoyĿ bhsopcorĿz

y

Ytoedort boto tob`lechrle dađrucuđueu jorőĿcoőtĿròeĿe flr yacudur vh oëĿjëo dobo

dóőój stoedort botocĿ/voryoescĿ tob`le hdlilyl thrilb hdhrlz.



ToeĿ` 6. Hđhr O vh F β lële sopòsĿz tob`le hdlilchr lsh, yoel β aZ O P 0 β aZ F P 0 > , a

zoòe hđhr oőođĿdojl jaőuc sođcoeĿrso O elsphtthe F“yh k÷rh dobo htjledlr dhrlz=

Fozhe tó` tob`le hdlilchrh, Β 0 {?, 6 ,…}, fojorĿz vh tó` Β“cor orosĿedo hđhr O he

dóőój voryoeso soblp lsh htjledlr dhrlz.

×rehj 6. WorsoyocĿ` jl R lch \ ljl gorjcĿ @othòtlj sĿeovĿeĿe eatcorĿdĿr. Ylz flr ëhőlt

–òthòtlj yhthehđl‐ lch lcklchelyarsueuz vh ljl eatto kóróctócódór (ùbth`hche ljl eat

orosĿedo jarhcosyae vordĿr), oyrĿio ZRP 0 Z\P 0 γ , Wor(R) 0 ω R 6 , Wor(\) 0 ω \

6 , vh

Iav(R,\) 0 ω R\ . Flr thj ÷cëó` juccoeòj yhrleh, ljlslel ođĿrcĿjcoedĿrĿc`Ŀő artocoò pR +(? ’ p)\ lch flrchőtlr`hyh joror vhrdlelz. Fu ođĿrcĿjcoedĿrĿc`Ŀő artocoòeĿe fhjchehe

dhđhrl ehdlr1 p“ele boekl dhđhrl ođĿrcĿjcoedĿrĿc`Ŀő artocoòeĿe voryoesĿeĿ `lel`lzh

hdhr1 Fueu sodhih ljl k÷zch`cl flr ÷rehjch` juccoeoroj γ “ yó tob`le ht`hj lsthdlđl`lz

flr tob`le prafch`l acoroj yaru`coyoflclrl z. Fótóe R vh \ ođĿrcĿjcoedĿrĿc`Ŀő artocoòcorĿ

γ acduđu l ële, htjle tob`le hdlilyl fucòyo ëocĿőoiođĿz.

Voskhch dhđlőjhechrle tapco`ĿeĿe voryoesĿeĿe gar`ócóedhe

hcdh hdhrlz. Aptlòc p“yl fucòj lële, flrleil tórhvl sĿgĿro hőltchrlz, yoel

p lële ë÷zóeih, vorsoyocĿ` jl ω R 6 + ω \

6 56 ω R\ vorsoyĿ`Ŀ octĿedo (fueue cajoc flr

`lelù` lële yhthrcl flr jaőuc acduđueo dljjoth dlelz), oőođĿdojlel hcdh hdhrlz

Hđhr R lch \ orosĿedo jarhcosyae yaj so, htjle tob`le hdlil R ózhrleh p* 0

ođrĿcĿđĿeĿ jayor. Fu ođĿrcĿj , elsphthe \“ele voryoesĿeo k÷rh R“le dobo dóőój voryoes

dhđhrchrl lële dobo fóyój dhđhrchr ocĿr.



6. Tob`le Hdlil Acuőturò \÷eth`chrl

6.?. @a`hetchr \÷eth`l

Fu y÷eth`?2<4“th FrltoeyocĿ lstotlstljël Jorc ^horsae torogĿedoe ÷ehrlcdl= vorsoyocĿ` jlflr dođĿcĿ`Ŀe j jodor poro`htrhslel tob`le ht`hj zaruedoyĿz. A zoòe, vhrlele lcj j

÷rehjch` à`hetchrleh fojòcĿyĿz,

vh dođĿcĿò k÷rh bhsopcoeoe flr poro`htrh dhđhrl vhrl ljhe, aecorĿ lcklcl jltch

à`hetchrleh hőltch`hclylz.

A zoòe à`hetchr y÷eth`l (@a@) tob`le hdlilsl oőođĿdojl dhejch` ë÷zóchrhj hcdh

hdlchflclr. Fótóe β“cor lële

×rehj 9. WorsoyocĿ` jl R ?, ….,R e poro`htrhsl, κ , flcle`hyhe flr âlssae dođĿcĿ`Ŀedoe, R ~ ^( κ ) hcdh hdlche flr l.l.d. ÷rehjch`dlr. DođĿcĿ`Ŀe sodhih flr flcle`hyhe poro`htrhsl

vordĿr vh flrleil jltch à`hetl oőođĿdojl lch vhrlclr=

DacoyĿsĿyco @a@ tob`le hdlilsl ő÷ychdlr=



Hđhr khrhjhedhe gozco à`het juccoeoroj poro`htrhc hrl tob`le hdhrshj eh acur1 ’

âlssae dođĿcĿ`Ŀ lële oyrĿio őueu do flclyaruz =

×rehj 4. Flr ëlgt ósthc roskhch dhđlőjhelel p.d.g .sl

a bocdh ljl poro`htrh ( κ , γ )“yó tob`le ht`hj zaruedoyĿz. Flr lstotlstlj jltofĿeo fojĿeio

őueu fucuruz

f÷ychih à`hetchr y÷eth`lele tob`le hdlilsl oőođĿdojlel ë÷zhr.

A bocdh ( , ) lële ë÷zóeih oőođĿdojl, hcdh hdlclr=

6.6. @ojslù` Acoflclrclj Tob`lel

@a`hetchr y÷eth`l sodhih shëlcl soyĿdo jltch à`hetlel ÷rehjch`dhjl jorőĿcĿjcorĿ lch

hőchőtlr`hyh ëocĿőĿrjhe, oyrĿ flr shëhehj acoroj `ó`jóe acduđueio he lyl őhjlcdh÷rehjch` dođĿcĿ`ĿeĿ flr fótóe acoroj jltch dođĿcĿ`Ŀyco hőchőtlrhe flr tob`le hdlilkhclőtlrhflclrlz. Fu, poro`htrh β“eĿe òjslù` acoflclrclj tob`le hdlilslele yoptĿđĿ őhydlr .Y÷z jaeusu poro`htrh, jofoio s÷ych`hj khrhjlrsh, –fóyój lbtlòcch‐ k÷zch`chehe÷rehjch`l artoyo ëĿjoroe dhđhrdlr=



WorsoyocĿ` jl flr l.l.d. acoe flr \?, …, \e ÷rehjch`l`lz vor. \“ele p.d.g.sl poro`htrh β“yojodor flclehe g \(y|β) lch vhrlclyar. @ojslù` acoflclrclj tob`le hdlilsl( @CH) β“eĿe octĿedojl

vhrlele fl chőlj p.d.g.slel òjsl`lzh hdhe “eĿe flr gaejslyaeudur.

Dobo sphslglj acòj khrhjlrsh, ÷rehjch`le acoflclrclđl el oőođĿdojl klfl toeĿ`corĿz=

Khehccljch acoflclrclj gaejslyaeue cakorltòsĿeĿ òjsl`lzh ht`hj ëaj dobo jacoydĿr.

Cakorltò jhsle ortoe flr gaejslyae acduđu lële, №(β) vh C(β)“ele oyeĿ dhđhrchrdhòjsl`lzh acoiođĿeĿ eat hdlelz.

×ehr`h ?. ôro`htrh β > “ do cak- acoflclrclđlele fhjchehe dhđhrl

lch khrëhj poro`htrh β > “do òjsl`lzh acur.

ľYÔT= ózhrledh fhjchehe dhđhrl bhsopcodĿđĿ`Ŀz khrëhj yađuecuj g \(y|β) acduđu lële,Mheshe Hőltslzclđlel juccoeoroj fótóe β dhđhrchrl lële βaZC(\|β) ’ C(\| >)P > vh

cak(.)“ele jaejov acduđueu k÷sthrhflclrlz.

furodo g(y|β) flr yađuecuj acduđu lële, lethkrocl ?“dlr. Fuedoe ÷tóró fótóe β dhđhrchrl lëleβaZC(\| >)P βaZC(\|β)P acduđuedoe, s÷z jaeusu > gaejslyaeu òjsl`lzh hdhr.

Fóyój YoyĿcor Joeueueo k÷rh, l.l.d. acoe flr ÷rehjch` lële cak-acoflclrclj ő÷ychdlr=



DacoysĿyco, l.l.d acoe fóyój ÷rehjch`chrle cak acoflclrclđlel òjsl`lzh ht`hele flzh >“h–yojĿe‐ flr poro`htrh vhrhihđlel dóőóehflclrlz.

×rehj 8. WorsoyocĿ` jl R E( > , ω > 6 )“dlr vh flr l.l.d. ÷rehjch` R ?, …, R e“dhe γ vh ω 6

poro`htrhchrlel tob`le ht`hj lstlyaruz. Acoflclrclj gaejslyaeu ő÷ychdlr=

Cak-acoflclrclđl òjsl`lzh ht`hele dobo jacoy acduđueu artoy o jayoflclrlz,

@ojslùù fucòj lële, γ vh ω 6 “ y h k÷rh tórhvchrl ocĿ p sĿgĿro hőltchrlz=

OyeĿ őhjlcdh,

BæclbozĿrdo, fu tob`le hdlilele ω > 6 lële sopòsĿz acòdĿđĿeĿ k÷sthrdlđl`lzl botĿrcoyĿeĿz,

fu ehdhech khehc acoroj @ojslù` Acoflclrclj Tob`le H dlilchrl sopòsĿz acòj zaruedo

dhđlcdlr.

×rehj 7. Xelgar` dođĿcĿ`cĿ ÷rehđh khrl d÷ehcl`= vorsoyocĿ` jl R l ~ XZ>, β P “dĿr vh β “eĿe

tob`lel lch lcklchelyaruz. @a`hetchr y÷eth`l tob`le hdlilsl lële oőođĿdojlel k÷rhflclrslelz,

f÷ychih fueu ÷rehjch` artocoòsĿeo hőltchyhrhj oőođĿdojlel hcdh hdhrlz=



@ojslù` acoflclrclj tob`le hdlilsl ehdlr1 OëĿjëosĿ, flz bhrboekl flr òx { R ?, …,

R e } ocòyoiođĿz ëóejó “ doe fóyój khrëhjchő`lő flr ÷rehjch`le octĿedo sĿgĿr acosĿcĿđĿ

vordĿr. Flël`shc acoroj, acoflclrclj

β òx { R ?, …, R e }“ele bhrboekl flr dhđhrl òjslùù acoòz ëóejó fótóe a

eajtocordo C( β )“le sĿgĿr acduđueu k÷rhflclrlz. OyeĿ zoòedo, β òx { R ?, …, R e } lële

acoflclrclj gaejslyaeu β “do jhsle ozocoedĿr vh fu ehdhech oőođĿdo lgodh hdlcdlđl klfl

òjslù`dur

? acosĿcĿjco R l 3 > acduđu lële, òjslù` acoflclrclj tob`l e hdlilsl dh ? acosĿcĿjco > “ doe

dóőój acoiojtĿr, f÷ychih sopòsĿz dhđlcdlr. Dobo do oëĿj acòj khrhjlrsh, R (e)“le p.d.g.sl

oőođĿdojl klfl vhrlclr=

F÷ychih,

Ëaj jacoy flr őhjlcdh flr sopòsĿz tob`le hdlil , 0

, acuőturoflclrlz.

6.9. @CH“ele ×zhccljchrl

OőođĿdojlchr sodhih @CH lële hcdh hdlche th`hc tharlj saeuëcorĿe ÷zhtldlr(fu oőoòdo

lspotcorĿ yopòyoiođĿz)=

TutorcĿ tob`le hdlilchr krufuedo htjle flr tob`le hdlil vorso, @CH aeu acuőturur.

Fhccl dózhechylil jaőuccor octĿedo, @CH osl`ptatlj acoroj earòc dođĿcĿ` acoflclr(fu hsos ltlforlych @hrjhzl Cl`lt Tharh`lele flr uykucoòsĿedoe khc`hjthdlr).

@ojslù` acoflclrclj bhr zoòe yopĿcòsĿ khrhjhe he lyl őhy `l1 BoyĿr



sopòcĿ acoflclr

khehccljch bhsopcoeòsĿ zardur

lcklcl dođĿcĿ` lch lcklcl yoecĿő vorsoyĿ`coro jorőĿ ëaj bossos acoflclr.



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ ?<

Jaerod @hezhc

62 Elsoe 6>><

?. @ojslù` Acoflclrclj Tob`le= ľcovh ×rehjchr

×rehj ?. WorsoyocĿ` jl R E( > , ω > 6 )‑dlr vh flr l.l.d. ÷rehjch` R ?, …, R e‑dhe γ vh ω 6

poro`htrhchrlel tob`le ht`hj lstlyaruz. Acoflclrclj gaejslyaeu ő÷ychdlr=

Cak-acoflclrclđl òjsl`lzh ht`hele dobo jacoy acduđueu artoyo jayoflclrlz,

@ojslùù fucòj lële, γ vh ω 6 ‑yh k÷rh tórhvchrl ocĿp sĿgĿro hőltchrlz=

OyeĿ őhjlcdh,



BæclbozĿrdo, fu tob`le hdlilele ω > 6 lële sopòsĿz acòdĿđĿeĿ k÷sthrdlđl`lzl botĿrcoyĿeĿz,

fu ehdhech khehc acoroj @ojslù` Acoflclrclj Tob`le Hdlilchrl sopòsĿz acòj zaruedodhđlcdlr .

×rehj 6. Xelgar` dođĿcĿ`cĿ ÷rehđh khrl d÷ehcl`= vorsoyocĿ` jl R l ~ XZ>, β P‑dĿr vh β ‑eĿetob`lel lch lcklchelyaruz. @a`hetchr y÷eth`l tob`le hdlilsl lële oőođĿdojlel k÷rhflclrslelz,

f÷ychih fueu ÷rehjch` artocoòsĿeo hőltchyhrhj oőođĿdojlel hcdh hdhrlz=

@ojslù` acoflclrclj tob`le hdlilsl ehdlr1 OëĿjëosĿ, flz bhrboekl flr òx { R ?, …, R e

} ocòyoiođĿz ëóejó ‑doe fóyój khrëhjchő`lő flr ÷rehjch`le octĿedo sĿgĿr acosĿcĿđĿ

vordĿr. Flël`shc acoroj, acoflclrclj

β òx { R ?, …, R e }‑ele bhrboekl flr dhđhrl òjslùù acoòz ëóejó fótóe a

eajtocordo C( β )‑le sĿgĿr acduđueu k÷rhflclrlz. OyeĿ zoòedo, β òx { R ?, …, R e } lële

acoflclrclj gaejslyaeu β ‑do jhsle ozocoedĿr vh fu ehdhech oőođĿdo lgodh hdlcdlđl klflòjslù`dur

? acosĿcĿjco R l 3 > acduđu lële, òjslù` acoflclrclj tob`le hdlilsl dh ? acosĿcĿjco > ‑doe

dóőój acoiojtĿr, f÷ychih sopòsĿz dhđlcdlr. Dobo do oëĿj acòj khrhjlrsh, R (e)‑le p.d.g.sloőođĿdojl klfl vhrlclr=

F÷ychih,



Ëaj jacoy flr őhjlcdh flr sopòsĿz tob`le hdlil, 0

, acuőturoflclrlz .

?.?. @CH“ele ×zhccljchrl

OőođĿdojlchr sodhih @CH lële hcdh hdlche th`hc tharlj saeuëcorĿe ÷zhtldlr(fu oőoòdo

lspotcorĿ yopòyoiođĿz)=

TutorcĿ tob`le hdlilchr krufuedo htjle flr tob`le hdlil vorso, @CH aeu acuőturur.

Fhccl dózhechylil jaőuccor octĿedo, @CH osl`ptatlj acoroj earòc dođĿcĿ` acoflclr

(fu hsos ltlforlych @hrjhzl Cl`lt Tharh`lele flr uykucoòsĿedoe khc`hjthdlr).

@ojslù` acoflclrclj bhr zoòe yopĿcòsĿ khrhjhe he lyl őhy `l1 BoyĿr

sopòcĿ acoflclr

khehccljch bhsopcoeòsĿ zardur

lcklcl dođĿcĿ` lch lcklcl yoecĿő vorsoyĿ`coro jorőĿ ëaj bossos acoflclr .

6. Kóvhe OrocĿđĿ

Tob`lele dhđhrl vh aeue dađrucuđu(stoedort botosĿ torogĿedoe vhrlcdlđl klfl) bojjĿedojl

flcklchrl flrchőtlr`hj lële, khehccljch yopĿcoe, flr tob`lele htrogĿedo ùbth`hche khrëhj

dhđhrl lëhrhe flr orocĿj fhclrch`hjtlr.

×rehj 9. WorsoyocĿ` jl dhelz juvvhtchrlele flr tapëhjhrlele(fat) joptoeĿ jĿyĿ őhrldl

fayueio flr soblc jaruò bottĿ acuőturoiojtĿr, gojot aedoe ÷eih dhelzdhe dađrudoek÷róc`hyhe soblcdhjl flr òjlehcl tóghđle yaj hdlc`hsl vhyo he ozĿedoe ođĿr tobrlfotouđrotĿcòsĿ khrhjhihjtlr.

Fato boclbozĿrdo soblcdhe flrjoë jhrh otĿő oëĿcĿr vh `hr`lchrle khcdlđl y÷eh doyoeoroj,

joptoe slcobĿe jaeuù bojjĿedo flr tob`le acuőturur. Tob`le, khrëhj > jaeuùe

htrogĿedo ω ζ 6 voryoescĿ flr earòc dođĿcĿ`dĿr.

Joptoe, soblccle flr orocĿđĿeo gózhchrch yoycĿ` othőledh fucueoroj a ocoedojl bhr őhyl yaj

hdhflclr. Joptoe, soblccle boekl orocĿđĿeo othő hdhihđlel eosĿc fhclrchyhflclr jl %<8

acosĿcĿjco òjlehcl tóghđle arodo acup tobrlp acoiođĿedoe h`le acoflclr vh f÷ychih flrcl jchrlkóvhecl flr őhjlcdh soblch ëĿjoroflclr 1




Earòc dođĿcĿ` lële, acosĿcĿj yĿđĿeĿe %<8‑l ele artocoòeĿe bhr ljl torogĿedojl ?.<7

stoedort sopòcĿj orocĿđĿe lëledh acduđueu flclyaruz. F÷ychih, hđhr joptoe IL 0 Z ’

?.<7 ω , + ?.<7 ω P orocĿđĿ lële othő h`rl vhrlrsh, ‑le > IL acò acosĿcĿđĿ %<8‑tlr.

Dobo ÷eih sodhih khrëhj poro`htrh > dhđhrleh yojĿe dhđhr vhrhe thj gaejslyae (R?,…, Re

) orĿyar ljhe, ől`dl fhccl flr dhđhrh hőlt vhyo dobo yójshj acosĿcĿjco khrëhj

poro`htrh dhđhrlel jopsoyoe (lëhrhe) ljl gaejslyae, O(R?, …, Re) 3 F(R?, …, Re),

acuőturòyo ëocĿőoiođĿz.

ToeĿ`?. ôro`htrh > lële flr ?- ‑cĿj kóvhe orocĿđĿ O(.) vh F(.) klfl vhrlyh-doyocĿ ljlgaejslyaeo fođcĿ flr orocĿjtĿr ZO(R ?, …, R e

), F(R ?, …, R e )P. \oel,

Fu gaejslyaecor fhezhrslz dhđlcdlr gojot tho`ócchrh fođcĿ acoroj, O vh F“yl /6 acosĿcĿđĿ orocĿđĿe bhr ljl torogĿeo hőlt dóőhihj őhjlcdh shëlyaruz.

Flr kóvhe orocĿđĿeĿe, ZO(x?, …, xe), F(x?, …, xe)P, khrëhjchő`hsl lële, ^(O(x?, …, xe) >

F(x?, …, xe) 0 ? - acduđueu s÷ych`hj òetĿjcĿ dhđlc, ëóejó orocĿđĿe cl`ltchrl vh

khrëhj poro`htrh ől`dl rhhc soyĿcordĿr, f÷ychih ÷rehjch`le khrëhjchő`hsl vhrl ljhe,



tob`le hdlche orocĿj yo > jopsor(? acosĿcĿjco) yo do jopsoòz. Khrëhj poro`htrh vhrl

ljhe roskhch acoe kóvhe orocĿđĿdĿr , yajso > dhđlc.

OőođĿdojl jhedlsl lële kóvhe orocĿđĿ acuőturòj lsthdlđl`lz he yoykĿe duru`dur.

×rehj 4. W orsoyocĿ` jl

~E( > , )‑d lr vh flr ?-

‑cĿj flr

kóvhe orocĿđĿ acuőturòjlstlyaruz. Hđhr z ?-ο /6 stoedort earòc dođĿcĿ`Ŀe ?- ( ο /6) aedocĿđĿ lsh yoel Χ (z ?-ο /6 ) 0 ?-

( ο /6) lsh, a zoòe oőođĿdojlel e

> ‑Ŀ oőođĿdojl acosĿcĿjco jopsodĿđĿeĿ jaetrac hdhflclrlz,

furodo

‑eĿe stoedordlzosyaeu acduđu lële stoedort earòc dođĿcĿ`cĿdĿr.

F÷ychih hđhr flr %<8‑clj kóvhe orocĿđĿ lstlyarsoj, z ?-ο /6 0 z >.<;8 0 ?.<7‑dĿr , fu ehdhech

kóvhe orocĿđĿ ?.<7 ω lch vhrlclr.

Fu kóvhe orocĿđĿeĿ hcdh ht`hele he yoykĿe yacudur, fu ehdhech fueue eosĿc lőchdlđlel

oecoòeĿz khrhjlyar.

×rehj 8. Oejht saeuëcorĿ khehccljch flr –boto poyĿ‐ lch ropar hdlclr. ×rehđle Koccup‑ue ?2Elsoe roparueo k÷rh shë`hechrle %47‑sĿeĿe @iIole‑h jorőĿ Icletae‑h, %44‑óeóe@iIole‑h ay vhrhihđlel, %?>‑eue lsh yo bhr ljlsl lële dh ay juccoeòyoiođĿeĿ yo dobhrboekl flr gljrl acòdĿđĿeĿ s÷ych`lőtlr. Fu saeuëcor 4928 jlőlych yopĿcoe k÷róő`hyhdoyoeòjtodĿr vh ropar oyrĿio –ucusoc yhtlőjlechrle tapco` ÷rehjch`leh doyocĿ saeuëcorlële, %<8 kóvhelclrcljch òjslù` ÷rehjch` boto poyĿ yózdh ljl acduđu s÷ychehflclr‐lgodhsleh yhr vhr`lőtlr. Fu eh oecoò khc`hjthdlr1 ’ Hđhr flr odoyĿe khrëhj ay aroeĿ p

lsh, e soyĿdojl shë`he ÷rehjch`ledh artocoò poyĿe voryoesĿ WOV(

e ) 0

―dlr . Fu

voryoesĿe p 0 >.8 lële he yójshj acduđueu jhedlelz dh jaetrac hdhflclrslelz. DacoysĿyco

4928 k÷róő`hcl flr ÷rehjch` lële, òjslù` stoedort sopò ()

‑ tlr.

@hrjhzl Cl`lt Tharh`leh k÷rh, e yojcoőĿj acoroj earòc dođĿcĿ`c ĿdĿr vh flr earòc

dođĿcĿ` lële %<8‑clj flr acosĿcĿj jótchslele artocoòeĿe ?.<7 stoedort sopòcĿj orocĿđĿedo



yhr ocdĿđĿeĿ dobo ÷eih k÷r`óőtój. Fu ehdhech, Z e -?.<7(>.;7) , e + ?.<7(>.;7)P orocĿđĿkhrëhj ay aroeĿeĿ %<8‑the dobo fóyój flr acosĿcĿjco lëhrhihjtlr. Yhë`hele dobo jóëójoct krupcorĿ lële boto poyĿ dobo fóyój acoiojtĿr.

×rehj 7. Flr cofarotuvor flr dovodo joeĿt acoroj juccoeĿcoflchihj flr joe ÷rehđl ózhrledh

jl`yosoc oeoclz yopòjtodĿr. JoeĿt acoroj jofuc hdlchflc`hsl lële, fozĿ òddhchrle`hviudlyhtlele %<>‑cĿj kóvhe orocĿđĿedo % >.>>? k/`c‑dhe dobo oz acòsĿ khrhjlr.

Oeoclzchr lële juccoeĿcoe òjleh khrëhj dhđhr htrogĿedo stoedort sopòsĿ ω 0 >.>8k/`c

acoe earòc dođĿcĿ`cĿ saeuëcor vhr`hjthdlr. %<>‑clj kóvhe orocĿđĿeĿe >.>>? k/`c‑dhe ozacduđuedoe h`le acòj lële joë toeh saeuë ocò`Ŀz khrhjlr1

%<8 ‑clj kóvhe orocĿđĿeĿe khelőclđl ő÷ychdlr=

DacoyĿsĿyco, c >.>>? acòsĿ lële, e ?7.48 6 0 6;>.7>68 ‑h lbtlyoiĿ`Ŀz vor, fu ehdhech

dh he oz 6;?(fođĿ`sĿz) saeuë ocò`Ŀz khrhjlr.

Yaerojl ÷rehj tob`le hdlilele dođĿcĿ`ĿeĿe earòc acòdĿđĿ duru`cordo kóvhe orocĿđĿ

acuőturòeĿe flr yacueu k÷sthr`hjthdlr.

×rehj ;. WorsoyocĿ` jl R ?, …, R e l.l.d.‑dlr, dođĿcĿ`Ŀ R ~ XZ>, β P ‑dlr vh > lële %<> ‑clj kóvhe orocĿđĿ acuőturòj lstlyaruz.

\ujorĿdojl lgodh e‑eil sĿro lstotlstlđl acsue ( ÷eihjl dhrschrdh k÷sthr lcdlđl ózhrh fu oyeĿ

zoòedo flr òjslù` acoflclrclj tob`le hdlilsldlr). Dobo ÷eih k÷rdóđó`óz klfl, β

lële sopòsĿz acòòsĿeo rođ`he, aeu β lële flr kóvhe orocĿđĿ acuőturòjto

juccoeoflclrlz. YĿro lstotlstlđlele saeuëcorĿedoe k÷rdój jl ‑eĿe i.d.g.slel vhrhe ‑le i.d.g.sloőođĿdojl klfl fhclrche`hjthdlr=

furodo XZ>, β>P acoe flr roskhch dhđlőjhele i.d.g.slel, G(x) 0 x/ >, yhrleh jayduj,

O vh F gaejslyaecorĿeĿ hcdh ht`hj lële, ÷eih o vh f soflt dhđhrchrlel fucocĿ`,



o vh f dhđhrchrlel oőođĿdojlchrl ë÷zóeih fucoflclrlz

f÷ychih o 0 ∝ vh f 0 ∝

“l hcdh hdhrlz. Fu flzh bheóz flr kóvhe orocĿđĿ

vhr`h`hjthdlr, ëóejó kóvhe orocĿđĿeĿe toeĿ`Ŀeo k÷rh flz khrëhj > dhđhrlel hőltslzclđle

artosĿedo lsthrlz. Wh bhr ljl torogĿe gaejslyaecorĿ sodhih vhrlyh vh dlđhr flcle`hyhe

fóyójcójchr fođcĿdĿr .

Oeioj oőođĿdojlel yozoflclrlz,

Fuedoe ÷tóró oőođĿdojl > lële flr %<>“clj kóvhe orocĿđĿdĿr.

Fu duru`do, orocĿđĿe sĿeĿrcorĿ sodhih (R?, …, Re) tob`le hdlilchr oroiĿcĿđĿyco vhrlyh

fođcĿdĿr. Fu khehc acoroj dađru acòj zaruedo dhđlcdlr.

ŕl`dl kóvhe orocĿđĿeo eosĿc ucoőtĿđĿ`ĿzĿ thjrorcoyocĿ`=

?. ÷eih (R?, …, Re) tob`le hdlilchrl vh “le dođĿcĿ`ĿeĿ hcdh ht,

6. oőođĿdojl jaőucu sođcoyoioj acoe o(β) vh f(β)“yl fuc

9. β“yĿ ë÷zhrhj acoyĿ yheldhe yoz

4. O(R), F(R) dhđhrchrlel k÷zch`chehe ÷rehjch` R?, …,Re“l juccoeoroj fuc,

8. ? ’ “cĿj kóvhe orocĿđĿ oőođĿdojl lch vhrlclr=



6.? ×eh`cl Duru`cor

?. earòc dođĿcĿ`cĿdĿr, Wor() flcle`lyar= Kóvhe orocĿđĿ oőođĿdojl klfl

acuőturucoflclr

6. earòc dođĿcĿ`cĿdĿr, Wor() flcle`lyar gojot 6 0 tob`le hdlilsl vor=

Kóvhe orocĿđĿ oőođĿdojl lch vhrlclr

Furodo t e-?(p)dhđhrl e -? shrfhstclj dhrhihcl t-dođĿcĿ`ĿeĿe peil yózdhclđldlr.

9. earòc dhđlc, gojot e 5 9> vhyo dobo gozco= ÷ych oecoőĿcĿyar jl k÷rdóđó`óz

fótóe tob`le hdlilchr(uelga` dođĿcĿ` lële ÷rehjch`le òjslùù borlë) `hrjhzl

cl`lt tharh`leh k÷rh osl`ptatlj acoroj earòcdlr (@hrjhzl Cl`lt Tharlslel eosĿc

uykucoyoiođĿ`Ŀz jaeusu bhr zoòe oëĿj dhđlc dhđlcdlr). Fu duru`do kóvhe

orocĿđĿeĿ 6“dhjl klfl acuőtururuz.

4. earòc dhđlc, e jóëój= hđhr “le p.d.g.sl flclelyar lsh, ?“eil juccoeĿcoroj kóvhe

orocĿđĿ acuőturucoflclr(sae ÷rehjthjl klfl). Hđhr p.d.g. flcle`lyar lsh,

yopoflchihđl`lz flr őhy yaj.

6eil duru`do t dođĿcĿ`ĿeĿ juccoeò`ĿzĿe ehdhel őudur= E(, ) acduđu lële,

Dlđhr torogtoe, őueu jaetrac hdhflclrlz

furodo khehccljch artocoòsĿ sĿgĿr vh voryoesĿ ω6 acoe earòc botocorĿe jorhchrlele

tapco`Ŀ lële yozĿcĿr. DacoysĿyco,



OyrĿio 4“óe khehc duruùedo (vh uelgar` lëhrhe sae ÷rehjth), (R?, …, Re) lstotlstlđlele

bhrboekl flr őhyle sopòsĿz vh tutorcĿ tob`le hdlilsl acòsĿeĿ lsth`hdlj, gojot khrëhj

poro`htrhdh jhsle àeatae acòj zaruedoydĿ. Oeioj, earòc duru`cor(“le voryoesĿ

bojjĿedo flckl soblp acsoj to acòsoj to) vh duru` 9 lële kóvhe orocĿđĿeĿ

acuőturduđuùzdo, tutorcĿ acòj zaruedoydĿj.



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ 6>

Jaerod @hezhc

9> Elsoe 6>><

?. Kóvhe OrocĿđĿ (dhvo`)

Yaerojl ÷rehj tob`le hdlilele dođĿcĿ`ĿeĿe earòc acòdĿđĿ duru`cordo kóvhe orocĿđĿacuőturòeĿe flr yacueu k÷sthr`hjthdlr.

×rehj ?. WorsoyocĿ` jl R ?, …, R e l.l.d.“dlr, dođĿcĿ`Ŀ R ~ XZ>, β P“dlr vh > lële %<>“cljkóvhe orocĿđĿ acuőturòj lstlyaruz.

\ujorĿdojl lgodh e“eil sĿro lstotlstlđl acsue (÷eihjl dhrschrdh k÷sthrlcdlđl ózhrh fu oyeĿ

zoòedo flr òjslù` acoflclrclj tob`le hdlilsldlr). Dobo ÷eih k÷rdóđó`óz klfl, β

lële sopòsĿz acòòsĿeo rođ`he, aeu β lële flr kóvhe orocĿđĿ acuőturòjto

juccoeoflclrlz. YĿro lstotlstlđlele saeuëcorĿedoe k÷rdój jl “eĿe i.d.g.slel vhrhe “le i.d.g.sl oőođĿdojl klfl fhclrche`hjthdlr=

furodo XZ>, β>P acoe flr roskhch dhđlőjhele i.d.g.slel, G(x) 0 x/ >, yhrleh jayduj,

O vh F gaejslyaecorĿeĿ hcdh ht`hj lële, ÷eih o vh f soflt dhđhrchrlel fucocĿ`,

o vh f dhđhrchrlel oőođĿdojlchrl ë÷zóeih fucoflclrlz

f÷ychih o 0 ∝

vh f 0 ∝ “l hcdh hdhrlz. Fu flzh bheóz flr kóvhe orocĿđĿ

vhr`h`hjthdlr, ëóejó kóvhe orocĿđĿeĿe toeĿ`Ŀeo k÷rh flz khrëhj > dhđhrlel hőltslzclđle



artosĿedo lsthrlz. Wh bhr ljl torogĿe gaejslyaecorĿ sodhih vhrlyh vh dlđhr flcle`hyhefóyójcójchr fođcĿdĿr.

Oeioj oőođĿdojlel yozoflclrlz,

Fuedoe ÷tóró oőođĿdojl > lële flr %<>“clj kóvhe orocĿđĿdĿr.

Fu duru`do, orocĿđĿe sĿeĿrcorĿ sodhih (R?, …, Re) tob`le hdlilchr oroiĿcĿđĿyco vhrlyhfođcĿdĿr. Fu khehc acoroj dađru acòj zaruedo dhđlcdlr.

ŕl`dl kóvhe orocĿđĿeo eosĿc ucoőtĿđĿ`ĿzĿ thjrorcoyocĿ`=

?. ÷eih (R?, …, Re) tob`le hdlilchrl vh “le dođĿcĿ`ĿeĿ hcdh ht,

6. oőođĿdojl jaőucu sođcoyoioj acoe o(β) vh f(β)“yl fuc

9. β“yĿ ë÷zhrhj acoyĿ yheldhe yoz

4. O(R), F(R) dhđhrchrlel k÷zch`chehe ÷rehjch` R?, …,Re“l juccoeoroj fuc,

8. ? ’ “cĿj kóvhe orocĿđĿ oőođĿdojl lch vhrlclr=

?.? ×eh`cl Duru`cor?. earòc dođĿcĿ`cĿdĿr, Wor()

flcle`lyar= Kóvhe orocĿđĿ oőođĿdojl klfl

acuőturucoflclr



6. earòc dođĿcĿ`cĿdĿr, Wor() flcle`lyar gojot 6 0 tob`le hdlilsl vor=

Kóvhe orocĿđĿ oőođĿdojl lch vhrlclr

Furodo t e-?(p)dhđhrl e -? shrfhstclj dhrhihcl t-dođĿcĿ`ĿeĿe peil yózdhclđldlr.

9. earòc dhđlc, gojot e 5 9> vhyo dobo gozco= ÷ych oecoőĿcĿyar jl k÷rdóđó`ózfótóe tob`le hdlilchr(uelga` dođĿcĿ` lële ÷rehjch`le òjslùù borlë) `hrjhzlcl`lt tharh`leh k÷rh osl`ptatlj acoroj earòcdlr (@hrjhzl Cl`lt Tharlslel eosĿcuykucoyoiođĿ`Ŀz jaeusu bhr zoòe oëĿj dhđlc dhđlcdlr). Fu duru`do kóvheorocĿđĿeĿ 6“dhjl klfl acuőtururuz.

4. earòc dhđlc, e jóëój= hđhr “le p.d.g.sl flclelyar lsh, ?“eil juccoeĿcoroj kóvhe

orocĿđĿ acuőturucoflclr(sae ÷rehjthjl klfl). Hđhr p.d.g. flcle`lyar lsh,yopoflchihđl`lz flr őhy yaj.

6eil duru`do t dođĿcĿ`ĿeĿ juccoeò`ĿzĿe ehdhel őudur= E(, ) acduđu lële,

Dlđhr torogtoe, őueu jaetrac hdhflclrlz

furodo khehccljch artocoòsĿ sĿgĿr vh voryoesĿ ω6 acoe earòc botocorĿe jorhchrleletapco`Ŀ lële yozĿcĿr. DacoysĿyco,

OyrĿio 4“óe khehc duruùedo (vh uelgar` lëhrhe sae ÷rehjth), (R?, …, Re) lstotlstlđlelebhrboekl flr őhyle sopòsĿz vh tutorcĿ tob`le hdlilsl acòsĿeĿ lsth`hdlj, gojot khrëhj

poro`htrhdh jhsle àeatae acòj zaruedoydĿ. Oeioj, earòc duru`cor(“le voryoesĿbojjĿedo flckl soblp acsoj to acòsoj to) vh duru` 9 lële kóvhe orocĿđĿeĿacuőturduđuùzdo, tutorcĿ acòj zaruedoydĿj.



6 Blpathz Thstl

6.? Oeo Gljlr

Gljlr= flr jltchdhe hcdh hdlche flr roskhch ÷rehjch` vhrlc`lő acsue, jltch bojjĿedojl fozĿlddlocoro jorőĿ ëĿjòj lële yhthrleih joeĿt vor `ĿdĿr1 ×eih fozĿ ÷eh`cl jovro`corĿtoeĿ`coyocĿ`=

flr blpathz flr jltchdhjl (papócosyae) flr roskhch dhđlőjhele dođĿcĿ`Ŀ bojjĿedo flrvorsoyĿ`dĿr

soflt (òletolehd) blpathz thst hdlch`hyhe oeioj eh acurso acsue dađru acduđuvorsoyĿcoe flr blpathzdlr.

thst hdlchflclr blpathz roskhch flr dhđlőjhedhe hcdh hdlche joeĿtcoro k÷rh thst

hdlchflclr vh thst hdlchihj flr blpathzdlr.

faő blpathz thst hdlchihj flr blpathzdlr

octhreotlg blpathz faő blpathz dĿőĿedo jltch bojjĿedojl dlđhr acosĿ vorsoyĿ`dĿr.

Thst prafch`l R?, …, Re ÷rehjch`lel hcdh httlđl`lz yađuecuj g(x|β>) lch lcletlcl acoe >

poro`htrhslele > acosĿ poro`htrh dhđhrchr jó`hsleh olt acup acòòsĿ acoroj lgodhhdlchflclr. Khehccljch faő blpathzl oőođĿdojl klfl yozorĿz=

Fueu oőođĿdojl octhreotlg blpathzh jorőĿ thst hdhrlz.

furodo

Hđhr > 0 {>} sodhih flr thj poro`htrh dhđhrl lëhrlrsh, blpathzle foslt acduđueus÷ychrlz. Flr flchőlj blpathz flrdhe gozco dhđhr vhyo flr soyĿ orocĿđĿeĿe tó`óeó lëhrhe flr jó`hsl torogĿedoe vhrlclr .

×rehj 6. H e yoykĿe jurucuùyco, B > foslttlr vh B O flchőljtlr. ×rehđle, R E( , ),

flcle`lyar vh γ 0 > acup acòdĿđĿeĿ thst ht`hj lstlyaruz. Fu jurucu`do, soflt blpathz

R l “chr l.l.d. earòc vh Wor(R l ) 0 “dlr. Faő blpathz B > = γ 0 > (foslt)“tlr vh fueu octhreotlg

blpathz B > = γ > ( flchőlj)“h jorőĿ thst ht`hj lstlyaruz.



Blpathzl thst ht`hj lële vhrl tapcoòj zaruedoyĿz vh vhrlyh doyoeoroj faő blpathzl rhtvhyo jofuc hdhflclrlz. Oeioj, vhrl`lz bhr zoòe fótóe jltchele flr ÷rehjch`l acduđu lële,vhrdlđl`lz jororcordo boto yopò lbtlòcl`lz vordĿr. Fhclrcl flr thst lële ?. Tlp boto yopò

acosĿcĿđĿ ő÷ych vhrlclr=

ο 0 ^(?.Tlp Boto) 0 ^(Vht|B>)

Fu thstle kóvhelrclj dózhyl (oyeĿ zoòedo fóyójcóđó) acoroj odcoedĿrĿcĿr. Hđhr oőođĿdojllgodhyl yozorsoj,

ζ 0 ^(6.Tlp Boto) 0 ^(Vht ht`h|B O)

a zoòe ? ’ thstle kóiódór .

Khehccljch thstle kóvhelrclj dózhylel acuőtururuz, ÷rehđle % 8 klfl vh dobo saero

kóvhelrclj thstl vhrl ljhe he yójshj kóih soblp flr thst acuőturòyo ëocĿőĿrĿz. DacoysĿyco

soejl faő blpathzl rht ht`h`h botosĿeĿ yopòyĿ thrilb hdlyaruz.

Fueue orjosĿedojl òetĿj lcj foőto shzklshc ockĿcoòyo thrs klfl k÷róeór bocfujl fu

k÷zch`chrdhe, fótóe jltchyh khehcchőtlr lche o`plrlj saruedoe vhyo flcl`shc joeuedoe

joyeojcoeòjtodĿr. Flcl`shc joeue= jltch bojjĿedojl blpathzl`lzl dađrucoyoe flrjoë oeyojoco`Ŀő flch acsoj, uy oyoe flr oe k÷zch`ch`hj ëórót`hj lële yhthrcldlr. Fu ehdhech

o`plrlj joeĿtcorĿ sodhih flr blpathzl rht ht`hj lële juccoeoflclrlz, blëflr zoòe joeĿtcoòj

lële dhđlc. OőođĿdojl Fhrtoed Vusshc“Ŀe `hőbur flr ÷rehđldlr=

–Hvilcchőtlrlc`lő boyvoecor jhedlchrlel earòcdh fhschyhe jlőlchrl k÷rdóđózoòe ylyhihj fhjchrchr. Acdujëo jofo fu dovroeĿőĿe thjdózhclđl,yoeĿckĿcorĿe saru`cusu acduđueu flclrlz. TovujcorĿ ÷`órchrl fayueio bhrkóe fhschyhe odo` saeuedo aeue fayeueu japorĿr, soejl dađoeĿe

thjdózhclđl tovujcor lële yoror coy`Ŀő fojĿő oëĿsĿeo leih flr oyor yopor klfl.

Z..P Khrëhj acoe őu jl, flr őhyle fhccl soyĿdo thjrorcoe`Ŀő acòsĿboyvoecordo vh lesoecordo oyeĿ őhyle thjror acoiođĿ fhjchetlsleleacuőòsĿeo ehdhe acur. ľëkódó`óz jhslecljch yorĿe kóehőle dađoiođĿeo

flzl leoedĿrĿr, gojot fhjche`hyhe flr őhjlcdh fayeu japortĿcoe tovujtoedobo lyl duru`do acòyoflclrlz.‐ (Vushcc, Tbh ^rafch`s ag ^blcasapby) ,



Fu ehdhech, hđhr, ÷rehđle, fhccl flr lcoiĿe bostoeĿe duruùeu fhclrkle flr őhjlcdhlylchőtlrdlđl jaeusuedo flr joeĿt sueòj lsthrshj, faő blpathz B> = –lcoiĿe bostoeĿeduruù ózhrledh blëflr htjlsl yajtur‐ acur. Fu blpathzl rht ht`hj, lcoiĿe htjlsl jaeusuedo

ëaj kóëcó joeĿt fucduđuùz oeco`Ŀeo khclr. \oel bhr zoòe faő blpathzl ëórót`hjlsthdlđl`lz lgodh acoroj fhclrclyaruz.

Foőjo flr ÷rehj acoroj, bujuj slsth`lel dóőóehflclrlz= –flr sórhëth, ljl torogto –suëcu‐vhyo –suësuz‐ őhjcledh flr saeuio ucoőòj lële artoyo vhrl (joeĿtcor) sórhrchr vh mórl yleh

dh ljl boto yopoflclr= òsu` flr jlőlyl suëcu fucoflclr (?. Tlp Boto) yo do flr suëcuyu

suësuz fucoflclr (6. Tlp Boto). @adhre bujuj slsth`chrlele ëađu yorkĿcoòcorĿ jlőlelesuësuz acduđu vorsoyĿ`Ŀeo doyoedĿrĿr, yoel őópbhclele –suëu lspotcoeoe jodor òsu`‐acduđu vorsoyĿcĿr . Foőjo flr lgodh lch, suëu lspotcoòeĿe yójó yorkĿio vhyo mórlyh,

őópbhclele khrëhjthe dh suëcu acduđu jaeusuedo ljeo ht`hj lële yhthrcl joeĿttĿ artoyojayòj zaruedo acoe soviĿyo flehr.

Blpathz thstchrh k÷rh ocĿeoe jororcor, octhreotlgle dađru acòsĿeo jorőĿ faő blpathzledađrucu lële foőto fhclrtlche acosĿcĿđĿ k÷r`hzcljthe khcdlđl`lz òeosĿedo aptlòc acòjzaruedo dhđlcdlr vh ?. Tlp lch 6.Tlp botocorĿ yopòeĿe lcklcl òclyhtchrlel bhsofojotòzcor. Bujuj slsth`l lële, ÷e suëcoòyĿ (prhh`ptlae) sovueoecor flr ëaj jaeudo,

÷rehđle thr÷rlz` klfl, 6. Tlp botoeĿe ëaj pobocĿyo òc acoflchihđlel sĿj sĿj tortĿőĿrcor. Fu

ehdhech bujuj slsth`l fozĿ vojocordo suësuzcuj vorsoyĿ`Ŀ lële lstlseocoro lzlevhr`hcldlr.

Taporcoyoioj acursoj, R?, …, Re ÷rehjch`lele bhr flr acosĿ acoyĿeĿ –rht ht`h‐ lch –rhtht`h`h‐ jororĿeo fođcoyoioj flr juroc gar`óch ht`hj lstlyaruz.



AcosĿcĿj vh ľstotlstlj Dhrs EatcorĿ 6?

Jaerod @hezhc

8 @oyĿs 6>><

Blpathz Thstl Acuőturòj

Hđhr Rl YR dhsthđleh soblpsh, a zoòe ÷rehjch` R 0 (R?, …, Re) “l dhsthjchr. Flr

thstle jrltlj f÷ckhsl jhedlsl lële faő blpathzl rht hdhihđl`lz ÷rehjch`le dhsthđlele

f÷ckhsldlr.

ľzchyhe ÷rehj flr stoedort jurucuùe he ÷eh`cl uesurcorĿeĿ k÷sthr`hjthdlr, fu ehdhechaeo ëaj dljjotcl fojòeĿz vh oyeĿ odĿ`corĿe fhezhr prafch`chrh eosĿc uykucoeoiođĿeĿ÷đrhe`helz khrhjlyar.

×rehj ?. WorsoyocĿ` jl R ?, …, R e flr l.l.d. ÷rehjch`dlr, R l E( ,4)“tór vh B O = γ 0 ?“h

jorőĿ B > = γ 0 >“Ŀ thst ht`hjch lcklchelyaruz. ×eih e 0 6 duruùeo fojocĿ`=


0 > lch uyu`cu acòyoioj jodor – ëaj fóyój‐ R ? + R 6 “ele dhđhrchrlel rht hdhihj flr thst

tosorcoyoflclrlz. Fu rht f÷ckhslel oyeĿ zoòedo flr dađrueue óst torogĿedo dok÷sthrhflclrlz.

Hđhr e fóyój lsh fu k÷sthrl`le juccoeĿ`Ŀ ëaj jacoydĿr, ëóejó rht f÷ckhslel dađrudoe R ?,…, R e lch boyoc ht`hj ëaj zardur. Oeioj, rhs`l e“dhe thj fayuto ledlrkhyhrhj, jrltlj



f÷ckhchrle korlp őhjlcchrlel fhclr`h yhtl`lzl joyfhdhflclrlz, oeioj aecor protlj uykucoò

oëĿsĿedoe ëajto yororcĿ dhđlcdlrchr zothe.

F÷ychih fu ÷rehjth, thst sórhilel flr thst lstotlstlđleh doyoedĿroiođĿz, T e(R ?, …, R e ) 0 e

vh T e“ ele fóyój dhđhrchrl lële rht hdhihđlz.

j“yl eosĿc shëhihđlz1 ljl boto tóró orosĿedo dhđlő-tajuő lch joőĿ jorőĿyo khc`hj zaruedo

jocoiođĿz. WorsoyocĿ` jl ől`dl e 0 68“tlr. R l E( ,4) acduđu lële,


ŕl`dl ?. Tlp vh 6. Tlp botocorĿe acosĿcĿjcorĿeĿ bhsopcoyoflclrlz.

Fu duru`do, , j“dhe bhrboekl flrlel fhclrch`hj dlđhr ljlslel fhclrchr, vh a shël` ?.eil

vh 6.eil T lp botocorĿe acosĿcĿjcorĿ orosĿedo fhclrcl flr dhđlő-tajuő lëhrlr ’ hđhr j“yl

yójshctlrshj, kóvhelrclj dózhyl dóőhr, oyeĿ őhj lcdh ? ’ kóiódh dóőhr. Yphslglj acòj

khrhjlrsh, hđhr j 0 9/8 acoroj shëhrshj, ≂ %7.; vh ≂ % ?8.2; acur.



GorjcĿ ÷rehjch` fóyójcójchrl lële ?.eil vh 6.eil T lp botocorĿe acosĿcĿjcorĿ orosĿedojl dhđlő-

tajuőu ëhőltcl j shël`chrl lële oőođĿdojl k lfl k÷sthrhflclrlz=

Dóőój flr j dhđhrl dobo fóyój kóë vh oyeĿ zoòedo dobo fóyój kóvhelrclj dózhyl vhrlr,

f÷ychih j“yl yójshct`hj flzl kroglj ózhrledh saco dađru joydĿrĿr.

j“yl eosĿc shë`h`lz khrhjlr1 Earòc jurucu`do ÷eihclđle yoecĿő rht ht`h acosĿcĿđĿ “eĿejaetrac hdlc`hsleh vhrlc dlđlel botĿrcoyĿeĿz, fu ehdhech j“yl ?. T lp boto acosĿcĿđĿeĿ jofuchdlchflclr flr dózhydh tutoioj őhjlcdh shëhihđlz, khehccljch fu %8 vhyo %?“dlr.

Hcfhtth, e ljhe, soflt lële thstle kóió, ? ’ ?“h dađru kldhr. Flr tho`óc acoroj,

khehccljch 0 %8 dózhyledhjl flr rht hdlő –oeco`cĿcĿj ‐tĿr , fhezhr őhjlcdh 0 %?“dh rht

ht`h – yójshj oeco`cĿcĿj –tĿr.


×rehj 6. Flr ÷eihjl ÷rehjth, soflt blpathz γ ∂ {>, ?} ldl, gojot fu ëaj yopoy flr

vorsoyĿ`dĿ vh khehccljch fueue f÷ych flr duru` acduđueo leoeòj lële flr ehdhel`lzyajtur.

Dobo ÷eih acduđu klfl, vorsoyocĿ` jl R ?, …, R e flr l.l.d. ÷rehjch`dlr, oeioj ől`dloőođĿdojlel thst hdhihđlz=

B> = γ 0 >



BO = γ ≮ >

ŕl`dl B O ljl y÷ecó flr flchőlj blpathzdlr (yoel octhreotlg oc tĿedo, flrjoë dhđhr ocoflclr,

fozĿcorĿ “Ŀe saccuedo fozĿcor sođĿedodĿr). \leh sodhih ÷rehjch` artocoòsĿeo, 68 ,

doyocĿ flr thsth fojoiođĿz ’ Jrltlj f÷ckh ől`dl eosĿc k÷róeór1

Yhzklshc acoroj, “le bh` fóyój vh bh` jóëój dhđhrchrl lële B > “Ŀ rht ht`hj oecoòcĿdĿr.\oel hđhr faő blpathz dađru lsh, fóyój lbtlòcch bhr ljl juyrujto do dhđhrchr

k÷r`hyhihđlz . Octhreotlg blpathz, γ “ eóe >“doe yo fóyój yo do jóëój acduđueo dolrjoeĿtcor lch lcklchedlđl`lzl lgodh ht`hjthdlr.


Fu ehdhech ljl dhđhr yoel j ? vh j 6 “yl fhclrchyhihđlz jl f÷ychih

ehdlr1 Octhreotlg thj flr acosĿcĿj joeueu fhclrch`hdlđl vh aeue yhrleh aecorĿesórhjclclđlel vhrdlđl lële, ëaj lyl toeĿ`coeò`ĿőtĿr, yoel soflt flr lële őu yozĿcĿr=



Khehccljch orzucoeoe flr kóvhelrclj dózhyl lële, j ?, j 6 “yl faő blpathz l torogĿedoevorsoyĿcoe dhđhr htrogĿedo sl`htrlj acoroj shëhrlz (earòc dođĿcĿ`Ŀe flr thj thph eajtosĿvh sl`htrlj acduđu lële, fu do jrltlj f÷ckhyl acoflcdlđleih fóyój yopor).

Yae ÷rehj voryoesĿ flclehe flr earòc jltchdhe hcdh hdlche lële kóvhe orocĿđĿeĿ

acuőturò yacueu botĿrcotòcĿ= \ujorĿdojl sórhë oscĿedo oőođĿdojleh fhezhrdlr=

?. lële flr ? ’ “cĿj kóvhe orocĿđĿ ZO(R), F(R)P acuőtur (Duru` ?, sae dhrsthjl eatcorofojĿeĿz)

6. Hđhr γ> 0 > ZO(R), F(R)P lsh, B>'Ŀ rht ht

DacoycĿ acoroj faő blpathz octĿedo ^β(O(R) 3 β 3F(R) 0 ? - ∐ acoioj őhjlcdh ZO(R), F(R)PorocĿđĿeĿ acuőturduđuùz lële,


?. Thstchrle Dhđhrchedlrlc`hsl vh AcuőturucòsĿ

Tob`le lch lcklcl tortĿőò`Ŀzdo acduđu klfl, ÷eih khehc gljrl toeĿttĿj saero flrjoë ÷rehjk÷rdój. ŕl`dl lsh thstchr orosĿedo eosĿc shël` yopoiođĿ`ĿzĿ vh aecorĿ foőtoe ltlforheeosĿc acuőturoiođĿ`ĿzĿ k÷rhihđlz.

?.?. Thstchrle ×zhcljchrl

Bhrboekl flr thstle kóvhelrclj dózhyl 0 ^(?. Tlp) vh aeue kóió ? ’ 0 ? ’ ^(6. Tlp) lch

lcklchelrlz. Hđhr bhr ljl B > vh B O foslt blpathz lsh, vh vhrlcl flr ο lële lyl toeĿ`coe`ĿőtĿr

vh he fosltledhe ? ’ “ sĿ he yójshj, yoel he kóëcó, thstl shëhrlz.






Ehyòe-^horsae Ch`ò“do dađrudoe vhrlche he kóëcó thst, ÷rehj uzoydo bhr x

eajtosĿedo (furodo lethkrocchr ëaj fayutcudur, yoel tlplj acoroj x ∂ e ) fóyójcój

vh kóë

orosĿedojl dhđlő-tajuőu ë÷zhr. vh ?- lgodhchrledhe acoflclrclj aroeĿ

“le jrltlj

f÷ckhyh x hjch`hele –glyotĿ‐eĿ IR f÷ckhsledhe flr eajto hjch`hele kóëthjl jozoeĿ`Ŀeo

elsphtthe ilesledhe eh jodor –÷dhyhihđl`lz‐l vhrdlđlel k÷rhflclrlz.

DacoysĿyco, –he uiuz‐ ’yoel jóëój acoflclrclj aroeĿeo soblp- x eajtocorĿeĿ hjchyhrhj jrltljf÷ckhchr acuőturòyo foőcoòcĿyĿz. A zoòe, acoflclrclj aroeĿeo k÷rh sĿrocoe`Ŀő x

dhđhrchrledhe oőođĿ dađru kldhrlz vh “eĿe fóyójcóđó lsthelche dózhyh leleihyh jodor

eajto shë`hyh dhvo` hdhrlz.

×rehj 4. Flr soeĿj (D) jopjoëëĿcĿjtoe òbjh`hdhdlr. @objú` ht`hj lële, mórl %<8acosĿcĿjco jororĿe dađru acduđueo leoeòj zaruedodĿr.



YoviĿeĿe artoyo jayoflchihđl vhyo jayoòyoiođĿ óë porëo patoeslyhc joeĿt vordĿr. Flrduruőòdo mórl jhedlsleh k÷sthrlche óë porëodoe sodhih flrleh doyoeoroj òbjú`lyhtjororĿ vhrlr. OőođĿdojlchr patoeslyhc joeĿt porëocorĿdĿr, jorőĿcĿjcĿ fođĿ`sĿzcĿjvorsoyĿc`ĿőtĿr. Tofcado oyrĿio soeĿđĿe suëcu acduđu vhrl ljhe bhr porëoeĿe leihche`hslacosĿcĿđĿ lch soeĿđĿe suëcu acòdĿđĿ vhrl ljhe bhr porëoeĿe leihche`hsl acosĿcĿđĿ do

vhrlc`lőtlr.

suëcu suësuz acoflclrclj aroeĿ ?. D paclsle khcdlđlel k÷róeih joëor >.7 >.9 ?/66. D suë lőchedlđledh foőjo yhrdh acduđueu

joeĿtcoyoòz (òzhrht) >.< >.9 ?/9

9. D“ele hvlelele yojĿeĿedo faő ëoeto fucueur >.4 >.? ?/4

Ehyòe-^horsae k÷sthrl`leh k÷rh, x joeĿt porëocorĿeĿe 6 9 acosĿ ja`fleosyaecorĿedoe

bhrboekl flrlsl acoflclr. FođĿ`sĿzcĿj vorsoyĿ`ĿeĿ juccoeoroj, fótóe lpuëcorĿeĿe

ja`fleosyaecorĿeĿ, bhr flr lpuiueue bhr blpathz octĿedojl lcklcl acoflclrcljchrlel vh lcklclacoflclrclj aroecorĿeĿ clsthchyhflclrlz. Clsthyl óëóeió jacaedojl acoflclrclj aroeĿeo k÷rhsĿrocodĿ`. Yae jacaeo sĿrocoe`Ŀő x ja`fleosyae clsthsleh k÷rh flrljl`cl tapco`Ŀ hjchdl`=

Yuëcu g O(x) Yuësuz(g >(x)Acoflclrclj AroeĿ

r(x) 0

(j)

?. bhr óë lpuiu 6?7/?>>> </?>>> >.>4?; </?>>>

6. òzhrht, fucueoe ëoeto ?44/?>>> 6?/?>>> >.?482 9>/?>>>

9. joëĿő, òzhrht 964/?>>> 2?/?>>> >.68 ???/?>>>

4. òzhrht 6?7/?>>> ?2</?>>> >.2;8 9>>/?>>>

8. joëĿő, fucueoe ëoeto 64/?>>> 6?/?>>> >.2;8 96?/?>>>

7. fucueoe ëoeto ?7/?>>> 4</?>>> 9.>768 9;>/?>>>

;. joëĿő 97/?>>> ?2</?>>> 8.68 88</?>>>

2. blëflrlsl 64/?>>> 44?/?>>> ?2.9;8 ?

Mórl, %8“the dobo dóőój acosĿcĿjcĿ yoecĿő òbjú`lyhth jorőĿe (yoel soeĿj òsu` lsh),

hđhr dađru lsh he oz %<8 kóvhelclrcljch òbjú`lyht jororĿ vhrlr. Blpathz thstlthr`leacamlsleh k÷rh, òbjú`lyht jororĿ, 0 %8 fóyójcóđóedhjl he kóëcó thstl

juccoeoroj soeĿđĿe òsu` acduđu faő blpathzlele rht hdlc`hsl lch lcletlcldlr.

Yae jacaedojl (j) dhđhrchrleh fojĿeio, lcj ljl joeĿttĿe ja`fleosyaedoe gozco joeĿt

hjch`hj yoecĿő òbjú`lyht acosĿcĿđĿ “yĿ %8“the gozco ortĿrdĿđĿeĿ ajuyoflclyaruz.DacoysĿyco, mórl suëcueue paclsl k÷rdóđóedh joëĿp joëòòsĿeo fojòdoe, foőjo



yhrdh acduđueu lspotcoòòsĿedoe vh hvlele yojĿeĿedo fucueoe ëoetodoe ÷tóró soeĿđĿòbjú` ht`hcldlr. ^rheslp acoroj, mórl, lcovhthe, soeĿđĿe joëĿőĿeĿ, fucueduđu yhrllspotcoòòyĿ vh ëoetoeĿe fucueòòsĿeĿ roskhch fhclrchyhflclr (9. duru`)= Hđhr fu

duru`do mórl soeĿđĿ

≂

acosĿcĿjco òbju` ht`lő acsoydĿ, yoecĿő òbju`lyhtle

acosĿcĿđĿ to` acoroj %8 acurdu, gojot fu ùbth`hche bujuj slsth`l torogĿedo jofuchdlchflclr flr uykucoò acòzdĿ.

×rehj 8. Artocoòyo doyocĿ flr thstle earòc duru`do he kóëcó thst acduđueu ől`dlk÷sthrhflclrlz. WorsoyocĿ` jl R l E( , 4) vh B O = γ 0 ?“h jorőĿ B > = γ 0 >“Ŀ thst hdhihđlz. Furodo 68 k÷zch`cl l.l.d. acoe flr R ?, …, R 68 ÷rehjch`l`lz vor.

K÷zch`chr l.l.d. earòc acdujcorĿ lële, k÷zch`chehe ÷rehjch`h k÷rh bhsopcoeoe acoflclrcljaroeĿ oőođĿdojl lch vhrlclr=

r(R)“le ÷rehjch` artocoòsĿ 68 oroiĿcĿđĿyco ÷rehjch`h fođcĿ acduđueu vh 68 “th jhsle

ozocoe acduđueu k÷rhflclrlz. Fu ehdhech, he kóëcó thstle jrltlj f÷ckhsl oőođĿdojl garù

ocĿr=




Jaerod @hezhc

; @oyĿs 6>><

×ehr`h ? (Ehyòe-^horsae Ch`ò). g O“yo jorőĿ g > “Ŀe thstledh (bhr ljl B > vh B O foslt

blpathzdlr), jrltlj f÷ckh

bhrboekl flr j > thrilbledh he kóëcódór.

j shël`lele thstle fhclrtlche kóvhelrclj dózhyl “yo fođcĿ acduđueu eat hdlelz. Fu, hđhr R?,

…, Re ÷rehjch`l lële oőođĿdojl acoflclrclj aroeĿ dóőój lsh, he kóëcó thst rht hdhroeco`Ŀeo khclr.

\oel, vhrl fóyój flr acosĿcĿjco B O octĿedo acuőùőtur.




Ehyòe-^horsae Ch`ò“do dađrudoe vhrlche he kóëcó thst, ÷rehj uzoydo bhr x

eajtosĿedo (furodo lethkrocchr ëaj fayutcudur, yoel tlplj acoroj x e ) fóyójcój

vh kóë

orosĿedojl dhđlő-tajuőu ë÷zhr. vh ?- lgodhchrledhe acoflclrclj aroeĿ

“le jrltlj

f÷ckhyh x hjch`hele –glyotĿ‐eĿ IR f÷ckhsledhe flr eajto hjch`hele kóëthjl jozoeĿ`Ŀeo

elsphtthe ilesledhe eh jodor –÷dhyhihđl`lz‐l vhrdlđlel k÷rhflclrlz.

DacoysĿyco, –he uiuz‐ ’yoel jóëój acoflclrclj aroeĿeo soblp- x eajtocorĿeĿ hjchyhrhj jrltljf÷ckhchr acuőturòyo foőcoòcĿyĿz. A zoòe, acoflclrclj aroeĿeo k÷rh sĿrocoe`Ŀő x

dhđhrchrledhe oőođĿ dađru kldhrlz vh “eĿe fóyójcóđó lsthelche dózhyh leleihyh jodor

eajto shë`hyh dhvo` hdhrlz.

.

×rehj ?. Flr soeĿj (D) jopjoëëĿcĿjtoe òbjh`hdhdlr. @objú` ht`hj lële, mórl %<8

acosĿcĿjco jororĿe dađru acduđueo leoeòj zaruedodĿr.



YoviĿeĿe artoyo jayoflchihđl vhyo jayoòyoiođĿ óë porëo patoeslyhc joeĿt vordĿr. Flr

duruőòdo mórl jhedlsleh k÷sthrlche óë porëodoe sodhih flrleh doyoeoroj òbjú`lyht

jororĿ vhrlr. OőođĿdojlchr patoeslyhc joeĿt porëocorĿdĿr, jorőĿcĿjcĿ fođĿ`sĿzcĿj

vorsoyĿc`ĿőtĿr. Tofcado oyrĿio soeĿđĿe suëcu acduđu vhrl ljhe bhr porëoeĿe leihche`hsl

acosĿcĿđĿ lch soeĿđĿe suëcu acòdĿđĿ vhrl ljhe bhr porëoeĿe leihche`hsl acosĿcĿđĿ do

vhrlc`lőtlr.

suëcu suësuz acoflclrclj aroeĿ ?. D paclsle khcdlđlel k÷róeih joëor >.7 >.9 ?/66. D suë lőchedlđledh foőjo yhrdh acduđueu

joeĿtcoyoòz (òzhrht) >.< >.9 ?/9

9. D“ele hvlelele yojĿeĿedo faő ëoeto fucueur >.4 >.? ?/4

Ehyòe-^horsae k÷sthrl`leh k÷rh, x joeĿt porëocorĿeĿe 6 9 acosĿ ja`fleosyaecorĿedoe

bhrboekl flrlsl acoflclr. FođĿ`sĿzcĿj vorsoyĿ`ĿeĿ juccoeoroj, fótóe lpuëcorĿeĿe

ja`fleosyaecorĿeĿ, bhr flr lpuiueue bhr blpathz octĿedojl lcklcl acoflclrcljchrlel vh lcklcl

acoflclrclj aroecorĿeĿ clsthchyhflclrlz. Clsthyl óëóeió jacaedojl acoflclrclj aroeĿeo k÷rh

sĿrocodĿ`. Yae jacaeo sĿrocoe`Ŀő x ja`fleosyae clsthsleh k÷rh flrljl`cl tapco`Ŀ hjchdl`=

Yuëcu g O(x) Yuësuz(g >(x)Acoflclrclj AroeĿ

r(x) 0

(j)

?. bhr óë lpuiu 6?7/?>>> </?>>> >.>4?; </?>>>

6. òzhrht, fucueoe ëoeto ?44/?>>> 6?/?>>> >.?482 9>/?>>>

9. joëĿő, òzhrht 964/?>>> 2?/?>>> >.68 ???/?>>>

4. òzhrht 6?7/?>>> ?2</?>>> >.2;8 9>>/?>>>

8. joëĿő, fucueoe ëoeto 64/?>>> 6?/?>>> >.2;8 96?/?>>>

7. fucueoe ëoeto ?7/?>>> 4</?>>> 9.>768 9;>/?>>>

;. joëĿő 97/?>>> ?2</?>>> 8.68 88</?>>>

2. blëflrlsl 64/?>>> 44?/?>>> ?2.9;8 ?

Mórl, %8“the dobo dóőój acosĿcĿjcĿ yoecĿő òbjú`lyhth jorőĿe (yoel soeĿj òsu` lsh),

hđhr dađru lsh he oz %<8 kóvhelclrcljch òbjú`lyht jororĿ vhrlr. Blpathz thstlthr`leacamlsleh k÷rh, òbjú`lyht jororĿ, 0 %8 fóyójcóđóedhjl he kóëcó thstl

juccoeoroj soeĿđĿe òsu` acduđu faő blpathzlele rht hdlc`hsl lch lcletlcldlr.

Yae jacaedojl (j) dhđhrchrleh fojĿeio, lcj ljl joeĿttĿe ja`fleosyaedoe gozco joeĿt

hjch`hj yoecĿő òbjú`lyht acosĿcĿđĿ “yĿ %8“the gozco ortĿrdĿđĿeĿ ajuyoflclyaruz.

DacoysĿyco, mórl suëcueue paclsl k÷rdóđóedh joëĿp joëòòsĿeo fojòdoe, foőjo



yhrdh acduđueu lspotcoòòsĿedoe vh hvlele yojĿeĿedo fucueoe ëoetodoe ÷tóró soeĿđĿ

òbjú` ht`hcldlr. ^rheslp acoroj, mórl, lcovhthe, soeĿđĿe joëĿőĿeĿ, fucueduđu yhrl

lspotcoòòyĿ vh ëoetoeĿe fucueòòsĿeĿ roskhch fhclrchyhflclr (9. duru`)= Hđhr fu

duru`do mórl soeĿđĿ

acosĿcĿjco òbju` ht`lő acsoydĿ, yoecĿő òbju`lyhtle

acosĿcĿđĿ to` acoroj %8 acurdu, gojot fu ùbth`hche bujuj slsth`l torogĿedo jofuchdlchflclr flr uykucoò acòzdĿ.

?. Thstchr le AcuőturucòsĿ

Khehc acoroj, aptlòc flr thstle eosĿc acuőturucòsĿ khrhjtlđl sarusueue dađrudoe flrihvofĿ yajtur. Ehyòe-^horsae Ch`ò flr foslt blpathzle dlđhrleh jorőĿ thsthdlc`hslele he kóëcó thstl lële foslt flr rhëhth vhr`hjthdlr, gojot khrëhj dóeyo

uykucoòcorĿeĿe ëađuedo octhreotlg blpathz flchőljtlr. OőođĿdojl ÷ehrlchr bhr zoòe

tortĿőòsĿz ëaj kóëcó flr thst vhr`hyhe (fozhe vor flch acòyoe) oò khehccljch jofuc

hdlchflclr saeuëcor dađuroe lőch`chr clsthsldlr.

?. Bhr bhr ljl B> vh B O foslt lsh, Ehyòe ^horsae Ch`ò flzh oőođĿdojl klfl flr

lstotlstlj acuőturò`ĿzĿ

vh uykue őhjlcdh shëlc`lő fozĿ j“chr lële hđhr T(R) 5 j lsh rht ht`h`lzl s÷ychr(khehccljch

j, thstle fóyójcóđó jhsle acoioj őhjlcdh shëlclr). Fu thst oyeĿ zoòedo acoflclrclj aroeĿ

thstl (CVT) acoroj to odcoedĿrĿcĿr.

6. hđhr B> = β 0 β> foslt vh B O = β > flchőlj vh 6- torogcĿ lsh, flr tob`le hdlil

juccoeoroj β lële ? ’ ο“clj kóvhe orocĿđĿ ZO(R), F(R)P (khehccljch sl`htrlj)

acuőturoflclrlz. Yaero hđhr β> ∌ ZO(R), F(R)P lsh rht hdhrlz. Fu flzh B> lële

fóyójcóđóedh flr thst vhrlr. 9. hđhr B> = β 0 β> foslt t vh B O = β > flchőlj vh thj-torogcĿ lsh, β lële sl``htrlj ? ’

6ο“cĿj kóvhe orocĿđĿ acuőturoflclrlz vh sodhih hđhr faő blpathz dhđhrl kóvhe

orocĿđĿeĿe dĿőĿedo lsh vh lcklcl fóyójcóđóeó hcdh ht`hj lële lcklcl juyrujto lsh rht

hdhrlz.

4. yo B> = β > yo do B O = β > (yo do bhr ljlsl) flchőlj lsh, oőođĿdojl lstotlstlđltoeĿ`co

vh uykue őhjlcdh shëlc`lő fozĿ j“chr lële hđhr T(R) 5 j lsh rht ht. Fu tór thstchrkhehcchőtlrlc`lő acoflclrclj aroeĿ thstl (KCVT) acoroj odcoedĿrĿcĿr.



Yae duruù dobo ÷eih tortĿőòdĿđĿ`Ŀz lële, fozĿ oëĿjcoòcor yopòj khrhjhihj=

thst ojco uykuedur ëóejó, hđhr vhrl B>“Ŀ dhsthjch`hzsh T(R) jóëój acò hđlcl`dhacoiojtĿr.

yađuecujcor bhr zoòe pazltlgtlr, fu ehdhech lstotlstlj > lch ? orosĿedo acoiojtĿr

(ëóejó ózhrledh yađuecuđu òjsl`lzh httlđl`lz poydojl jó`h, ózhrledhyađuecuđu òjsl`lzh httlđl`lz poydodojl jó`hyl lëhrdlđl lële)

faő blpathz octĿedojl thst lstotlstlđle jhsle dođĿcĿ`ĿeĿ flc`h`lz khrhjlr, f÷ychihuykue flr jrltlj dhđhr j“yl fucoflclrlz. DođĿcĿ`corĿe ëađu lële, aeu fóyój÷rehjch`chrch hcdh hdhrlz=

-6cakT(R)

furodo p 0 dl`(> O) ’ dl`(>).

KCVT CVT“ele aptlòc ÷zhclđlel poycoőòj zaruedo dhđlcdlr, dađrusu flchőljocthreotlg blpathzcl fu jurucu`do, tortĿőòsĿz ëaj kóëcó thst khehccljch vor

acòyoflclr dh.

6. ×rehjchr

×rehj 6. WorsoyocĿ` jl dađu` sĿrosĿedo fhfhjchrle ođĿrcĿđĿ (paued ilesledhe) R ~ E(;,

?)“h k÷rh dođĿcòjtodĿr. Dlyhcl` jl hđhr flr dađu` uzòeĿ fhfhj fhjchyhe flr oeehyh

zoyĿg flr dlyht ÷ehrlsledh fucuesoydĿ, fu ÷ehrl fhfhđle artocoòdoe ? paued dobo boglg

(gojot oyeĿ voryoeso soblp) dađòsĿeo shfhp acurdu. IoecĿ dađoe ?> jlőlclj flr

÷rehjch` lële, ?> 0 7.6“yl k÷zch`chrlz.

Dađu` uzòeĿ j÷tó ÷ehrldh fucueùyar faő blpathzleh jorőĿ j÷tó ÷ehrldh

fucueuyar octhreotlg blpathzl lële %8“clj flr thstl eosĿc acuőtururuz1 Hcl`lzdh

B > = 0 ;“yh jorőĿ B O = 0 7

vor.

Earòc dođĿcĿ` lële, fu foslt thstl sodhih ÷rehjch` artocoòsĿeo, ?> , doyoedĿròeĿe

dobo aptlòc acduđueu k÷sthr`lőtlj, yoel T(x) 0 ?> . B > octĿedo, ?> ~ E(;, >.?) vh B O

octĿedo ?> ~ E(7, >.?)“d l r. Hđhr ?> 3 j lsh thst rht hdhr. Fu ehdhech thst fóyójcóđó %8

acoioj őhjlcdh j“yl shë`hclylz, yoel



furodo (.) stoedort earòc i.d.g.dlr. Fu ehdhech oőođĿdojl dhejch`le thrslel ocoroj j“yl

hcdh hdhrlz

Fuedoe ÷tóró, ?> 0 7.6 3 7.42 0 j acduđu lële rht hdhrlz.

Fu thstle kóió ehdlr1

WorsoyocĿ` jl kóió he oz %<< acoe flr thst lstlyaruz, k÷zch`ch`hj zaruedo

acduđuùz yhel dađoe fhfhj soyĿsĿ e he oz eh jodor acòcĿdĿr1 e lch dhđlőhihj

thj őhy ÷rehjch`le voryoesĿdĿr, fu ehdhech fu ÷rehđle flrleil f÷có`óedhe, jrltlj

dhđhr j e 0 ; ’

∝ “acduđueu fucuruz, dlđhr torogtoe e“h doyocĿ thstle kóió vh

jrltlj dhđhr j e oőođĿdojl lch vhrlclr=

? ’ >.<< acoroj oyorcoyĿeio őu jaőuccu hcdh hdhrlz=

Fu tór kóë bhsopcoòcorĿ khehccljch flr lstotlstljl dhehy vhyo flr oejht ëocĿőòsĿ

pcoecoedĿđĿedo yopĿcĿr ’ ÷rehđle, fhccl flr lcoiĿe fóyójcóđóeóe htjlslel oroőtĿròj lële flr

lcoë thstledh joë toeh bosto juccoeoiođĿ`ĿzĿ fhclrch`hj klfl. Ëaj soyĿdo jlől ózhrledh

ëocĿőòj vhyo oejht yopòj ëađu zoòe òclyhtcldlr, fu ehdhech yhthrleih fóyój flr

acosĿcĿjco oeco`cĿ dhđlőljcljchrl fucoflc`hj lële flr dhehyle fóyójcóđóeóe eh acòsĿ

khrhj tlđlel ÷eihdhe flc`hj lsthrlz .

×rehj 9. WorsoyocĿ` jl ÷eihjl ÷rehđle jurucuùeo acduđu klfl soblflz, gojot voryoesĿflc`lyaruz. Aeue yhrleh, Y6 0 ?.8 klfl flr tob`lel`lz vor. Thstl eosĿc yopordĿeĿz1 Dobo

÷eih tortĿőtĿđĿ`Ŀz klfl,



lstotlstlđl, hđhr khrëhj artocoò > lsh e -? shrfhstclj dhrhihcl flr ÷đrheil t -dođĿcĿ`ĿdĿr.

DacoysĿyco hđhr oőođĿdojl jaőuc sođcoeĿrso, B > “ Ŀ rht hdhrlz =

^rafch`dh vhrlche rojo`corĿ yhrleh jayorsoj, T 0 -

-6.>77 acur. Fu do t <(>.>8)

0 -?.29“the jóëójtór.

×rehj 4. R l ~ Fhreauccl(p), l 0 ?,6,9, acsue. \oùj flr poroyĿ flrflrledhe fođĿ`sĿz acoroj

óë jhrh gĿrcotĿyaruz vh hđhr turo khclrsh R l 0 ?“dlr, dlđhr duru`do R l 0 >“dĿr. B O = p 0 6/9“h

jorőĿ B > = p 0 ?/9“ó thst ht`hj lstlyaruz. Bhr ljl thst foslt acduđu lële, acoflclrclj aroe

thstlel juccoeoflclrlz,

Fu ehdhech hđhr oőođĿdojl jaőuc khrëhjchőlrsh rht hdhrlz.

furodo 9

“ y h hőlttlr. j“yl fhclrch`hj lële, B > vh B O octĿedo 9 “le acosĿ fótóe

dhđhrchrlel vh acosĿcĿjcorĿeĿ clsthchyhcl`=

F÷ychih hđhr thstle fóyójcóđóeóe ∐ 0 ?/6;“yh hőlt acòsĿeĿ orzucuyarsoj, sodhih vh

sodhih 9 5 6/9 lsh rht hdhflclrdlj. OyeĿ saeuiu dađuroioj őhjlcdh j 0 6/9“ó shëhflclrlz.Fu thstle kóió őueo hőlttlr=



×rehj 8. W orsoyocĿ` jl oőođĿdojl gaejslyae torogĿedoe tórhtlc`lő flr thj k÷zch`l`lz vor,

+ tapco`ĿeĿ `lel`lzh hdhe thst prashdóróeó fucueuz ’ hđhr R 0 >.7 lsh rhthdhr `lylz1 Yodhih flr R k÷zch`l`lz acduđu lële, R ilesledhe jrltlj f÷ckhyl

acuőturòj ëaj joròőĿj dhđlcdlr, fozĿ lchrl dózhydh lstotlstljchrl fucòyo

ëocĿőòj ëaj őhy jozoedĿròyoiojtĿr (oeioj Ehyòe-^horsae furodo lőh

yoroyoflclr). \ađuecuj kroglđleh fojoroj, j jrltlj dhđhrchrledh jóëój R dhđhrchrl

lële thstle rht ht`hsl khrhjtlđl jaeusuedo ljeo acoflclrlz. Tlp L vh T lp LL“ele

acosĿcĿjcorĿ, sĿrosĿyco, > j ? lële ő÷ychdlr,

vh

Fu ehdhech, j ózhrledhe boto acosĿcĿjcorĿeĿ `lel`lzh hdhrlz.

@lel`lzh hdlc`lő thrl`le tórhvle ocĿp sĿgĿro hőltchrshj,

DacoysĿyco, hđhr R 3 ?/6 lsh rht ht`hclylz vh 0 ?/4“tór. Oeioj, R 0 >.7 lële B > “Ŀ

÷zhcljch rht ht`lyaruz.

Fótóe thstchr orosĿedo ∐ >.? klfl, he jóëój dhđhrcl thstl fuc. ehdlr1 R 0 >.4

acso rht hdhr `lydlelz1 ’ ÷eih j lële (j) 0 >.?“l ë÷zhrlz. \ujorĿdojl gar`ócó

juccoeoroj, j 0 ∝ acur. DacoysĿyco,

j 0 ∝ >.9?7 3 >.4 acduđu lële, R 0 >.4 lële B > “Ŀ rht ht`hylz.



×rehj 7. R khđlőjhel Rl~ XZ>, βP dođĿcĿ`cĿdĿr vh vorsoyocĿ` jl flr l.l.d. ÷rehjch` R?, …,Re“l k÷zch`chdlj vh oőođĿdojlel thst ht`hj lstlyaruz

ľjl shëhehđl`lz vor= β lële flr ?- ο“cĿj kóvhe orocĿđĿeĿ acuőturoflclrlz vh hđhr >“Ŀ jopsoòzso rht hdhrlz. Dlđhr flr shëhehj acoroj, flr KCVT thstl acuőturoflclrlz

Acoflclrclj gaejslyaeu oőođĿdojl lch vhrlclr=

T“ele poyĿ òjsl`lzh hdlil ózhrledhe bhsopcoeoe acoflclrclj lch hcdh hdlclr. Fu òjslù`

acoflclrclj tob`le hdlildlr, @CH 0 R(e) 0 òx{ R?, …, Re }, yoel

DacoysĿyco

Thst fóyójcóđóeó orzucoeoe dózhyh hőltchyhe lstotlstlđle jrltlj dhđhrl j“yl fucòj lële,

0 > faő blpathzl octĿedojl dođĿcĿ`Ŀ flc`hj zaruedoyĿz ’ fueue lële sĿro lstotlstlđlf÷có`óeh fojò`Ŀz khrhjhflclr.

Flr dlp eat acoroj, dobo ÷eih fóyój e“chr lële, faő blpathz octĿedo KCVT“ele flr jl -jorh,

6, dođĿcĿ`Ŀ acduđueu s÷ychdlđl`lz bocdh, fu ÷rehj lële fueue dađru acòdĿđĿ oecoőĿcĿyar

ëóejó khrëhj poro`htrh dhđhrledh yađuecuj sórhjcl dhđlcdlr.




Jaerod @hezhc

?6 @oyĿs 6>><

?. ×rehjchr

×rehj ?. WorsoyocĿ` jl dađu` sĿrosĿedo fhfhjchrle ođĿrcĿđĿ (paued ilesledhe) R ~ E(;,

?)“h k÷rh dođĿcòjtodĿr. Dlyhcl` jl hđhr flr dađu` uzòeĿ fhfhj fhjchyhe flr oeehyh

zoyĿg flr dlyht ÷ehrlsledh fucuesoydĿ, fu ÷ehrl fhfhđle artocoòdoe ? paued dobo boglg

(gojot oyeĿ voryoeso soblp) dađòsĿeo shfhp acurdu. IoecĿ dađoe ?> jlőlclj flr

÷rehjch` lële, ?> 0 7.6“yl k÷zch`chrlz.

Dađu` uzòeĿ j÷tó ÷ehrldh fucueùyar faő blpathzleh jorőĿ j÷tó ÷ehrldh

fucueuyar octhreotlg blpathzl lële %8“clj flr thstl eosĿc acuőtururuz1 Hcl`lzdh

B > = 0 ;“yh jorőĿ B O = 0 7

vor.

Earòc dođĿcĿ` lële, fu foslt thstl sodhih ÷rehjch` artocoòsĿeo, ?> , doyoedĿròeĿe

dobo aptlòc acduđueu k÷sthr`lőtlj, yoel T(x) 0 ?> . B > octĿedo, ?> ~ E(;, >.?) vh B O

octĿedo ?> ~ E(7, >.?)“dlr. Hđhr ?> 3 j lsh thst rht hdhr. Fu ehdhech thst fóyójcóđó %8

acoioj őhjlcdh j“yl shë`hclylz, yoel

furodo (.) stoedort earòc i.d.g.dlr. Fu ehdhech oőođĿdojl dhejch`le thrslel ocoroj j“yl

hcdh hdhrlz

Fuedoe ÷tóró, ?> 0 7.6 3 7.42 0 j acduđu lële rht hdhrlz.



Fu thstle kóió ehdlr1

WorsoyocĿ` jl kóió he oz %<< acoe flr thst lstlyaruz, k÷zch`ch`hj zaruedoacduđuùz yhel dađoe fhfhj soyĿsĿ e he oz eh jodor acòcĿdĿr1 e lch dhđlőhihj

thj őhy ÷rehjch`le voryoesĿdĿr, fu ehdhech fu ÷rehđle flrleil f÷có`óedhe, jrltl j

dhđhr j e 0 ; ’

∝ “acduđueu fucuruz, dlđhr torogtoe e“h doyocĿ thstle kóió vh

jrltlj dhđhr j e oőođĿdojl lch vhrlclr=

? ’ >.<< acoroj oyorcoyĿeio őu jaőuccu hcdh hdhrlz=

Fu tór kóë bhsopcoòcorĿ khehccljch flr lstotlstljl dhehy vhyo flr oejht ëocĿőòsĿ

pcoecoedĿđĿedo yopĿcĿr ’ ÷rehđle, fhccl flr lcoiĿe fóyójcóđóeóe htjlslel oroőtĿròj lële flr

lcoë thstledh joë toeh bosto juccoeoiođĿ`ĿzĿ fhclrch`hj klfl. Ëaj soyĿdo jlől óz hrledh

ëocĿőòj vhyo oejht yopòj ëađu zoòe òclyhtcldlr, fu ehdhech yhthrleih fóyój flr

acosĿcĿjco oeco`cĿ dhđlőljcljchrl fucoflc`hj lële flr dhehyle fóyójcóđóeóe eh acòsĿ

khrhjtlđlel ÷eihdhe flc`hj lsthrlz.

×rehj 6. WorsoyocĿ` jl ÷eihjl ÷rehđle jurucuùeo acduđu klfl soblflz, gojot voryoesĿ

flc`lyaruz. Aeue yhrleh, Y6 0 ?.8 klfl flr tob`lel`lz vor. Thstl eosĿc yopordĿeĿz1 Dobo

÷eih tortĿőtĿđĿ`Ŀz klfl,

lstotlstlđl, hđhr khrëhj artocoò > lsh e -? shrfhstclj dhrhihcl flr ÷đrheil t -dođĿcĿ`ĿdĿr.

DacoysĿyco hđhr oőođĿdojl jaőuc sođcoeĿrso, B > “ Ŀ rht hdhrlz=

^rafch`dh vhrlche rojo`corĿ yhrleh jayorsoj, T 0 -

-6.>77 acur. Fu do t <(>.>8)

0 -?.29“the jóëójtór.

×rehj 9. R l ~ Fhreauccl(p), l 0 ?,6,9, acsue. \oùj flr poroyĿ flrflrledhe fođĿ`sĿz acoroj

óë jhrh gĿrcotĿyaruz vh hđhr turo khclrsh R l 0 ?“dlr, dlđhr duru`do R l 0 >“dĿr. B O = p 0 6/9“h



jorőĿ B > = p 0 ?/9“ó thst ht`hj lstlyaruz. Bhr ljl thst foslt acduđu lële, acoflclrclj aroe

thstlel juccoeoflclrlz,

Fu ehdhech hđhr oőođĿdojl jaőuc khrëhjchőlrsh rht hdhrlz.

furodo 9

“yh hőlttlr. j“yl fhclrch`hj lële, B > vh B O octĿedo 9 “le acosĿ fótóe

dhđhrchrlel vh acosĿcĿjcorĿeĿ clsthchyhcl`=

F÷ychih hđhr thstle fóyójcóđóeóe ∐ 0 ?/6;“yh hőlt acòsĿeĿ orzucuyarsoj, sodhih vh

sodhih 9 5 6/9 lsh rht hdhflclrdlj. OyeĿ saeuiu dađuroioj őhjlcdh j 0 6/9“ó shëhflclrlz.

Fu thstle kóió őueo hőlttlr=

×rehj 4. WorsoyocĿ` jl oőođĿdojl gaejslyae torogĿedoe tórhtlc`lő flr thj k÷zch`l`lz vor,

+ tapco`ĿeĿ `lel`lzh hdhe thst prashdóróeó fucueuz ’ hđhr R 0 >.7 lsh rht

hdhr `lylz1 Yodhih flr R k÷zch`l`lz acduđu lële, R ilesledhe jrltlj f÷ckhyl

acuőturòj ëaj joròőĿj dhđlcdlr, fozĿ lchrl dózhydh lstotlstljchrl fucòyo

ëocĿőòj ëaj őhy jozoedĿròyoiojtĿr (oeioj Ehyòe-^horsae furodo lőh

yoroyoflclr). \ađuecuj kroglđleh fojoroj, j jrltlj dhđhrchrledh jóëój R dhđhrchrl



lële thstle rht ht`hsl khrhjtlđl jaeusuedo ljeo acoflclrlz. Tlp L vh Tlp LL“ele

ac osĿcĿjcorĿ, sĿrosĿyco, > j ? lële ő÷ychdlr,

vh

Fu ehdhech, j ózhrledhe boto acosĿcĿjcorĿeĿ `lel`lzh hdhrlz.

@lel`lzh hdlc`lő thrl`le tórhvle ocĿp sĿgĿro hőltchrshj,

DacoysĿyco, hđhr R 3 ?/6 lsh rht ht`hclylz vh 0 ?/4“tór. Oeioj, R 0 >.7 lële B > “Ŀ

÷zhcljch rht ht`lyaruz.

Fótóe thstchr orosĿedo ∐ >.? klfl, he jóëój dhđhrcl thstl fuc. ehdlr1 R 0 >.4

acso rht hdhr `lydlelz1 ’ ÷eih j lële (j) 0 >.?“l ë÷zhrlz. \ujorĿdojl gar`ócó

juccoeoroj, j 0 ∝ acur. DacoysĿyco,

j 0 ∝ >.9?7 3 >.4 acduđu lële, R 0 >.4 lële B > “Ŀ rht ht`hylz.

×rehj 8. R khđlőjhel Rl~ XZ>, βP dođĿcĿ`cĿdĿr vh vorsoyocĿ` jl flr l.l.d. ÷rehjch` R?, …,Re“l k÷zch`chdlj vh oőođĿdojlel thst ht`hj lstlyaruz

ľjl shëhehđl`lz vor= β lële flr ?- ο“cĿj kóvhe orocĿđĿeĿ acuőturoflclrlz vh hđhr >“Ŀjopsoòzso rht hdhrlz. Dlđhr flr shëhehj acoroj, flr KCVT thstl acuőturoflclrlz



Acoflclrclj gaejslyaeu oőođĿdojl lch vhrlclr=

T“ele poyĿ òjsl`lzh hdlil ózhrledhe bhsopcoeoe acoflclrclj lch hcdh hdlclr. Fu òjslù`

acoflclrclj tob`le hdlildlr, @CH 0 R(e) 0 òx{ R?, …, Re }, yoel

DacoysĿyco

Thst fóyójcóđóeó orzucoeoe dózhyh hőltchyhe lstotlstlđle jrltlj dhđhrl j“yl fucòj lële,

0 > faő blpathzl octĿedojl dođĿcĿ`Ŀ flc`hj zaruedoyĿz ’ fueue lële sĿro lstotlstlđlf÷có`óeh fojò`Ŀz khrhjhflclr.

Flr dlp eat acoroj, dobo ÷eih fóyój e“chr lële, faő blpathz octĿedo KCVT“ele flr jl -jorh,

6, dođĿcĿ`Ŀ acduđueu s÷ychdlđl`lz bocdh, fu ÷rehj lële fueue dađru acòdĿđĿ oecoőĿcĿyarëóejó khrëhj poro`htrh dhđhrledh yađuecuj sórhjcl dhđlcdlr .

6. Dlđhr ×zhc Thstchr

WorsoyocĿ` jl l.l.d. ÷zhclđleh soblp ljl ÷rehjch`l`lz vor, fuecor R?, …, Re vh ]?, …, ]e“dlrvh patoeslyhc acoroj ljlsl gorjcĿ fóyójcójthdlr, e? vh e6 klfl. ľjl gorjcĿ dođĿcĿàcuőturoflclrlz=

\opòyo ëocĿőtĿđĿ`Ŀz ljl gorjcĿ thst őuecordĿr=



Fu blpathzchrl eosĿc thst hdhrlz1

?. Furodo sodhih vh

dhđhrchrl flclehe duruù hch ocoiođĿz (dlđhr duru`cor lchlcklcl tor tĿőò lële jltofo fojĿeĿz). B > = γ R 0 γ ] octĿedo

Yhzklshc acoroj, hđhr faő blpathz dađru dhđlc lsh, T fóyój acòcĿ (ùtcoj dhđhrilesledhe). DacoyĿsĿyco, ∐ fóyójcóđh soblp B O“yo jorőĿ B> thstledh B> rht hdlclr

hđhr oőođĿdojl jaőuc sođcoeĿrso=

6. Woryoes thstl lële, dođĿcĿ`cor lch lcklcl saeuëcorĿ botĿrcoòj khrhjlr=

vh

Fuecor flrflrledhe fođĿ`sĿzdĿr. BotĿrcoyĿeĿz, fođĿ`sĿz jl-jorhchrle shrfhstclj dhrhihsl lch

f÷có`ó G dođĿcĿ`cĿydĿ=

OëĿjëosĿ flz lch

“yl flc`lyaruz, gojot B> = 0

octĿedo fu lgodh fosltchőtlrlchflclr ,

DacoyĿsĿyco, hđhr oőođĿdojl jaőuccor sođcoeĿrso fóyójcójcó flr thst rht hdhr=



6.6 ôro`htrlj Acòyoe ËĿjorĿ`cor

ŕl`dlyh jodor, vhrl órhtl` prashsl g(x|β) garùedo acoe vh saecu poro`htrh β fayutueo

jodor flclehe duru`dojl prafch`chr lch lcklchedlj. A duru`dojl thstchrh poro`htrlj

ëĿjorĿ` dhelclr.

ľstlseo acoroj, tob`le jaeusuedo vurkucodĿđĿ`Ŀz klfl, ÷rehjch` artocoòcorĿ, voryoescor Ŀvh dlđhr à`hetchrl bhrboekl flr dođĿcĿ`Ŀe artocoòcorĿeĿe, voryoescor ĿeĿe vh dlđhryójshj-sĿrocĿ à`hetchrlele bhsopcoeòsĿ lële ovoetomcĿ ÷zhcljchrh soblptlr.

Flr roskhch dhđlőjhele dođĿcĿ`ĿeĿe tó`ó jhedl i.d.g.sl lch jorojthrlzh hdlchflcdlđl lële,bhrboekl flr sĿeĿrcoò khtlr`hdhe(hcfhtth jl khëhrcl flr i.d.g. acòcĿ yoel àeatae vhsođdoe sórhjcl), soejl vhrldhe i.d.g.yl tob`le ht`hj lyl flr gljlr`lő klfl k÷róeóyar.

×rehj dođĿcĿ` gaejslyaeu Ge(x) ő÷ychdlr=

Furodo R(m) m. sĿrocĿ lstotlstljtlr(fueue ÷rehjch`dhjl m he jóëój dhđhr acduđueubotĿrcoyĿeĿz), oyrĿio R(>) ≯ - ∕ vh R(e+?) ≯ -∕“dur.

×rehj 7. flr {-?, 9, ?, ?, >.8, 6, >} ÷rehjch`l lële, sĿrocoe`Ŀő ÷rehjch` { -?, >, >.8, ?, ?,

6, 9}“tór vh ÷rehjch` dođĿcĿ` gaejslyaeu Ge(x)“l kroglj lch k÷sthrhflclrlz=




Flcle`hyhe flr dođĿcĿ` olchsledhe hcdh hdlche flr R?, … Re roskhch ÷rehjch` lch

lcklcl ëĿjorĿ` prafch`l lch lcklchelyaruz vh i.d.g.sl G(x) acoe (÷rehđle flr stoedort earòcdođĿcĿ` lële G(x) 0 (x) klfl) fhclrcl flr dođĿcĿ`doe hcdh hdlclp hdlc`hdlđlel thst ht`hyl

orzucuyaruz. Dobo ÷eihjl tortĿőòcordo octĿ ëlzlche thstchrdhe bhrboekl flrlel uykucoòjlële hcl`lzdh sphslglj poro`htrhchr acòdĿđĿ lële, thst gljrl Ge(R)“le G(x)“the –ëaj gozco‐sopĿp sopòdĿđĿeĿ jaetrac ht`hj acur .


6.9. Jacàkarav-Y`lreav Thstl

K÷zch`chehe flr ÷rehjch`le G(x) dođĿcĿ`Ŀ torogĿedoe tórhtlclp tórhtlc`hdlđle thst ht`hjlële, Jacàkarav-Y`lreav lstotlstlđlele fóyój dhđhrchrl lële thstl rht hdhrlz. ľstotlstlj

oőođĿdojl klfl toeĿ`coeĿr ,

furodo sup x G(x) suprhù`dur, yoel {G(x)= x }“le he jóëój óst sĿeĿrdĿr ’ ugojjó`hchrdh sórhjcl gaejslyaecor lële fu òjslùùe oyeĿsĿdĿr , gojot Jacaàkarav-

Y`lreav lstotlstlđl ?/e fóyójcóđóedhjl sĿëroòcorĿ lëhrhe ÷rehjch` dođĿcĿ` gaejslyaeulëhrdlđl vh suprhù` fótóe rhhc soyĿ ëlzklslel jopsodĿđĿ lële, khrëhjth blëflr fhclrcl x

dhđhrleh ucoőĿcò`Ŀő acoflclr.



ľstotlstlđle jrltlj dhđhrchrl osl`ptatlj (yoel fóyój e“chr lële) dođĿcĿ` gaejslyaeuedoe hcdhhdlchflclr

Fu lgodhyl dađuroe orkóòecor ëaj oëĿj dhđlcdlr vh ëaj thjeljtlr ëóejó fu, rhhc soyĿcorĿdhđlc, gaejslyaecorĿe dođĿcĿ`ĿeĿ lëhrlyar (roskhch gaejslyaecor khehccljch stajostlj prashs

acoroj odcoedĿcor ).


I.d.g.ele bhsopcoeòsĿ saesuz flr shrlele bhsopcoeòsĿeĿ khrhjtlrdlđl lële, gar`ócjuccoeòj foslt dhđlcdlr. Oeioj, flrëaj dhrs jltofĿ yoykĿe jrltlj dhđhrchr lële tofcaacuőturur.

×rehj ;. W orsoyocĿ` jl flr òdhel poroyĿ thjror thjror, ÷rehđle ?7> jhrh, gĿrcotĿyaruz vh

÷rehjch`le flr F(4, >.8) dođĿcĿ`Ŀedoe tórhtlclp tórhtlc`hdlđle ο 0 >.6 kóvhelrclj dózhyl lch

thst ht`hj lstlyaruz. Dlyhcl` jl oőođĿdojl ÷rehjch` grhjoescorĿeĿ k÷zch`chdlj=

Turo soyĿsĿ > ? 6 9 4

×rehjch` grhjoesĿ ?> 99 7? 49 ?9Jó`ócotlg ÷rehjch` grhjoesĿ Ge(.) ?> 49 ?>4 ?4; ?7>B> G(.) octĿedo Jó`ócotlg grhjoes ?> 8> ??> ?8> ?7>gorj > ; 7 9 >



A zoòe Jacàkarav-Y`lreav lstotlstlđl őueo hőlttlr=

Jltoptojl osl`ptatlj gar`ócóeó juccoeoroj, I >.6> 0

∝ >.28 “tlr . >.443>.28 acduđu lële,

faő blpathzl %6> dózhy ledh rht hdh`lyaruz.

6.4. 6-×rehjch`cl Jacàkarav-Y`lreav Thstl

WorsoyocĿ` jl dođĿcĿ` olchsl flcle`hyhe ljl fođĿ`sĿz roskhch ÷rehjch`l`lz vor vh fuecor

R?, …, Re lch \?, …, \e acsue. Bhr ljl ÷rehjch`ledh oyeĿ dođĿcĿ` torogĿedoe tórhtlclp

tórhtlc`hdlđlel thst ht`hj lstlyaruz. Furodojl dóőóeih Ge(x) lch Ke(x)“le flrflrledhe –ëajuzojto‐ acup acòdĿđĿeĿ thst ht`hjtlr.

Flr lstotlstlj acuőturuyaruz,

vh fóyój D dhđhrchrl lële rht hdlyaruz. Hđhr oőođĿdojl jaőuc sođcoeĿrso,

fóyójcóđóedhjl flr thst lële jrltlj dhđhrchrle osl`ptatlj acoroj lyl flr tob`le l thstl rht

hdhr=

6.8 ^horsae“eĿe Thstl

WorsoyocĿ` jl e toeh l.l.d. k÷zch`cl flr ÷rehjch`dhjl bhr flr Rl dhđhrl j jodor, O?,…, Oj,

jothkarldhe flrleh yhrchőhihj őhjlcdh sĿeĿgcoedĿrĿcdĿ. p?, …, pj bhr flr jothkarlele

acosĿcĿđĿ, vh g ?, …, g j“dh k÷zch`chehe grhjoescor acsue. OőođĿdojl flchőlj blpathzl

octhreotlgh jorőĿ thst httlđl`lzl dóőóehcl`. Octhreotlg blpathz fu hőltcljchrdhe he oz ljl

vhyo dobo gozcosĿeĿe khrëhjchő`h`hsldlr(acosĿcĿjcorĿe tapco`Ŀ ? acduđu lële, furodo toàcoroj flr hőltclj tutòdĿ dlyh`hylz). OőođĿdojl lstotlstlđl juccoeoflclrlz



vh fóyój T dhđhrchrl lële rht hdhrlz. He uykue jrltlj dhđhrchrl fhclrch`hj lële, T“ele eosĿcdođĿcdĿđĿeĿ flc`hj zaruedoyĿz. @oochshg, fu dođĿcĿ` lcklcl àdhch fođcĿdĿr. Oeioj, B>

octĿedo dođĿcĿ` osl`ptatlj acoroj àdhcdhe fođĿ`sĿzĿdĿr vh fóyój e ÷rehjch`chr lële

osl`ptatlj acoroj T ~ “dlr. ^rotlj flr juroc acoroj, hđhr e 4j lsh jl-jorh tob`lel

ëocĿőoflclr.



?4.9> Hjaea`ldh ľstotlstljshc \÷eth`chrh Klrlő Dhrs EatcorĿ 64

Jaerod @hezhc

?4 @oyĿs 6>><

?. Thjror

Eajto Tob`lel

×rehjch`le tob`le hdlil gaejslyaeu (R?, …, Re)

Tob`le hdlilele sopòsĿ

Tob`le hdlilele stoedort botosĿ

Tob`le hdlilchrl dhđhrchedlr`hj lële ÷eh`cl jrlthrchr

YopòsĿzcĿj Htjleclj

TutorcĿcĿj

Tob`le hdlil acuőturò y÷eth`chrl

?. @a`hetchr \÷eth`l

èil jltch à`hetl

èil ÷rehjch` à`hetl

lcj à`hetl bhsopco, ` 0 ?, …, j lële γ`() hőltclđle acuőtur vh lële ë÷z.6. @ojsl u` Acoflclrclj



R?, …, Re ÷rehjch`l lële acoflclrclj gaejslyaeueu yoz

C(β) lch cak(C(β))‑l òjsl`lzh hdhe β dhđhrchrlel fuc

òjslùù fucòj lële lcj tórhvl sĿgĿro hőltch, hđhr β dhsthjcl roskhch dhđlőjhele tórhvl ocĿeòyoioj duru`do lsh, a zoòe gaejslyaeue eosĿc k÷róedóđóehfojòcĿ vh òjslùùe ehrhdh acòsĿ khrhjtlđlel fhclrch`hclslelz.

Kóvhe OrocĿđĿ

O(R?, …, Re) lch F(R?, …, Re) vhrlslele gaejslyaecorĿeĿ fucueuz, yoel

ZO(R?, …, Re), F(R?, …, Re)P β lële kóvhe orocĿđĿdĿr

Whrlcl ?-ο‑clj kóvhelrclj dózhyl lële ëaj soyĿdo acosĿ kóvhe orocĿđĿ vordĿr

Kóvhe orocĿđĿ acuőturòj lële ëađu zoòe oőođĿdojl odĿ`corĿ tojlp ht=

?. o(β>) lch f(β>)‑yl fuc vh fozĿ T(R?, …, Re) lstotlstljchrl lële oőođĿdojlel acuőtur

(dađoc acoroj furodo flr tob`le hdlil juccoeĿcoiojtĿr)

6. acosĿcĿđĿe lëledhjl acoyĿ oőođĿdojl klfl thjror yoz

9. kóvhe orocĿđĿeĿ acuőturòj lële R?, …, Re lch O(.) vh F(.)‑yl bhsopco

FozĿ ×eh`cl Duru`cor=

sopòsĿzdĿr vh earòc dođĿcĿ`cĿdĿr, Wor() flclelyar=

sopòsĿzdĿr vh earòc dođĿcĿ`cĿdĿr, Wor() flcle`lyar vh flr tob`le hdlil`lz vor,

=



earòc dhđlc, e 5 9> vhyo dobo gozco= ől`dlyh jodor k÷rdóđó`óz tob`lehdlilchrle osl`ptatlj acoroj earòc dođĿcĿ`cĿ acdujcorĿeĿ k÷rdój, dacoysĿyco s÷zjaeusu tob`lel juccoeoiođĿz vh flr ÷eihjl duruù uykucoyoiođĿz. WoryoesĿflcshj th flc`hshj th t-dođĿcĿ`ĿeĿ juccoeoroj flr őhjlcdh kóvhe orocĿđĿeĿe tob`lejuccoeĿ`ĿeĿ ihzocoedĿr`Ŀő acoiođĿz.

earòc dhđlc, e jóëój= hđhr (o) ‑le p.d.g.slel flclyarsoj, lcj duruù juccoeorojkóvhe orocĿđĿ acuőturoflclrlz, hđhr (f) p.d.g.yl flc`lyarsoj, yopoflchihđl`lz flr őhyyajtur

Blpathz Thstl

blpathzchr, B O = β ΒO‑yo jorőĿ B> = β Β>

faő vh octhreotlg octĿedo gorjcĿ dođĿcĿò soblp dotoeĿe gaejslyaecorĿeĿe T(R)lstotlstlđlel thst ht

I jrltlj f÷ckhsl= faő blpathzl rht httlđl`lz, T(R)‑le khrëhjchőtlđl f÷ckh.

thst prashdóró= Hđhr T(R) I lsh B>‑Ŀ rht ht.

I‑ele shël`l oőođĿdojlchrl fhclrchr

ο 0 ^(?. Tlp Boto) 0 ^(rht|B>)

ζ 0 ^(6. Tlp Boto) 0 ^(rht ht`h|B O)

ocgo fóyójcój acoroj, ?- ζ lsh thstle kóió acoroj odcoedĿrĿcĿr.

oyeĿ ο fóyójcójthjl ljl thstthe ?- ζ kóió he fóyój acoeĿ thrilb ht

Hđhr ζ 0 ζ(β) lsh he dóőój ζ(β) dhđhrcl thstl thrilb ht, fu uelgar` acoroj he kóëcóthsttlr.

B> vh B O‑eĿe ljlsl dh foslttlr= Ehyòe-^horsae Ch`ò‑yo k÷rh he kóëcó thst

–hđhr

3 j lsh rht ht‐ garùedo acoeĿdĿr.

fozĿ àeatae gaejslyaecor lële –hđhr k(T(R)) 3 k(j) lsh rht ht‐ garùedojl flr thst–hđhr T(R) 3 j rht ht‐ garùedojl flr thstle fhezhrldlr.

Thstle acuőuù B> vh B O‑eĿe garùeo fođcĿdĿr =

?. bhr ljl B> vh B O foslttlr= acoflclrclj aroeĿ thstl



vh uykue flr őhjlcdh shëlc`lő j dhđhrchrl lële hđhr T(R) 3 j lsh rht ht (Ehyòe -^horsaeCh`ò‑yo k÷rh he kóëcó acoeĿ)

6. B> = β 0 β> foslt, B O= β Β O flchőljtlr vh 6-yoecĿdĿr= ? ’ ο‑cĿj kóvhe orocĿđĿ ZO(R),F(R)P acuőtur vh rht ht hđhr

9. B> = β 0 β> foslt, B O= β 5 β> flchőljtlr vh ?-yoecĿdĿr= ? ’ 6ο‑cĿj kóvhe orocĿđĿ ZO(R),F(R)P acuőtur vh hđhr β> 3 O(R) lsh rht ht.

4. Khehc duru`= Khehcchőtlrlc`lő Acoflclrclj Aroe Thstl lstotlstlđl=

uykue flr őhjlcdh shëlc`lő j dhđhrchrl lële hđhr T(R) 3 j lsh rht ht.

ľjl-×rehjch` Thstl

ľjl fođĿ`sĿz R?, …, Re vh ]?, …, ]e klfl dhđlőjhele l.l.d. ÷rehjch`chrl acsue, Rl

E(R, ) vh ]l E(],

).

OőođĿdojlchrl thst hdhflclrlz=

(l)

vhyo

(ll)

(l) duruùedo oőođĿdojl thstl acuőtur



fu faő blpathz octĿedo E(>, ?)‑dlr.

(ll) duruùedo oőođĿdojl thstl acuőtur

fu faő blpathz octĿedo G(e? ’ ?, e6 ’ ?) dođĿcĿ`cĿdĿr, vh rht ht hđhr yo T 5 G-? (

) yo do

T 5 G-? (?-

) lsh.

×rehj Yarucor ( Fobor 6>>> YĿeovĿ)

?. @a`hetchr \÷eth`l= FođĿ`sĿz dhđlőjhechr acoe R?, …, Re Z>, βP dhsthjcl sórhjcluelgar` dođĿcĿ`doe ëhjlc`lőtlr. ?4=9>‑dojl dhrslelzdhe botĿrcorsoeĿz, R l

÷rehjch`ledhe β tob`le hdlilslel hcdh ht`hj lële yo @a`hetchr \÷eth`lel yo do@ojslù` Acoflclrclj y÷eth`lel juccoeoflclrslelz. Oeioj slz ugoj dhđlőljcljchr lstlyarsueuz vh yhel flr roskhch dhđlőjhe toeĿ`cĿyarsueuz, yoel

furodo j torogĿeĿzdoe fhclrche`lő yo do torogĿeĿzdoe flclehe flr soflt dhđhrdlr. \?, …,\e‑l juccoeoroj sodhih β‑yĿ tob`le hdhflclrslelz.

(o) vorsoyocĿ` jl j (>, β)‑dĿr. \l‑ele flr gaejslyaeu acoroj à`hetle óë y÷eth`lel dhjuccoeoroj β lële flr tob`le hdlil tórhtle. OyrĿio, (>, β) orocĿđĿedo acòj lële ehdhej‑yh khrhjslel` duyduđueuzu oëĿjcoyĿeĿz.

(f) ől`dl vorsoyocĿ` jl j (>, ∕)‑dĿr vh j flcle`hyhe poro`htrh β‑doe dobo fóyój

vhyo dobo jóëój acoflclr. Hđhr e 0 >‑cĿ flr ÷rehjch` k÷zch`chyhihj acursoeĿz, j

lch β orosĿedojl lclőjl lële eh s÷ychyhflclrslelz1

(i) β lële òjslù` acoflclrclj tob`le hdlilsl tórhtlelz (ueutòyĿeĿz, tob`lechr lëlesodhih \?, …, \e ―l juccoeoflclrslelz).

Ihvopcor=

(o) β flr-fayutcu acduđu lële, sodhih \l‑ele flrleil à`hetlel juccoeòj zaruedoyĿz.Jltchele fhjchehe dhđhrl



hđhr j 3 β lsh, à`hetchr y÷eth`lele tob`le hdlilsl oőođĿdojl klfl ë÷zóchrhj

hcdh hdlclr

(f) hđhr j 5 β lsh, βZ\lP 0 ^(Rl j) 0 òx{? ’ (j/β), >} 0 > ortĿj β‑yo fođcĿ dhđlcdlr.Hđhr j‑ele β‑doe dobo fóyój vhyo jóëój acduđueu flc`lyarsoj, poro`htrh β

jurucuùeu sĿeĿrcoedĿròj lële à`hetchr y÷eth`lele tob`le hdlilslele òetĿđĿeĿjuccoeoflclrlz=

Hđhr fóyój ÷rehjch`chr lële e 0 > lsh, j fóyój flr acosĿcĿjco β‑doe fóyój acoiojtĿr.

(i) Acoflclrclj gaejslyaeueu tórht`hj lële oőođĿdojlel eat ht`hj khrhjlr=

DacoysĿyco,

CakcorĿ ocĿeio,

β ilesledh ocĿeoe tórhvchrl sĿgĿro hőltchyleih

β lële ë÷zóeih,



Fu tob`le hdlil j 5 β acso flch ëocĿőĿr.

6. Blpathz Thstl= WorsoyocĿ` jl R?, …, Re artocoòsĿ, γ, flcle`hyhe oeioj voryoesĿ,, flclehe vh ?‑h hőlt acoe earòc dođĿcĿ`doe hcdh hdlche flr ÷rehjch` acuőtursue.

(o) OőođĿdojl jurucu` lële, %8‑clj kóvhelrclj dózhyledh he kóëcó thstl vhrhe f÷ckhylfhclrtlelz. Thstle kóióeó bhsopcoyĿeĿz.

(f) OőođĿdojl jurucu` lële, %8‑clj kóvhelrclj dózhyledh he kóëcó thstl vhrhe f÷ckhyl

fhclrtlelz.

(i) e‑ele vh e‑ele boekl dhđhrchrl lële (o)‑do jl γ 0 > blpathz lch (f)‑dhjl γ 0 ?blpathzlel oyeĿ oedo jofuc hdhrslelz1

(d) OőođĿdojl jurucu` lële, %8‑clj kóvhelrclj dózhyledh uelgar` acoroj he kóëcó thstlvhrhe f÷ckhyl fhclrtlelz.

Fu thst lële, kóë gaejslyaeu ? ’ ζγ lële flr gar`óc acuőturup, kroglđle ëlzlelz.

(h) (o) vh (f)‑dhjl thstchrle jrltlj f÷ckhchrl orosĿedo eosĿc flr lclőjl vordĿr1 6. Tlp botoyopò acosĿcĿjcorĿeĿe lclőjlsl ehdlr1

Ihvopcor=

(o) Ehyò-^horsae Ch`ò‑yo k÷rh, he kóëcó thst acoflclrclj aroeĿeo doyoeĿr



Hđhr acoflclrclj aroeĿ jrltlj dhđhrdhe dóőój lsh he kóëcó thst rht hdhr, yo do fhezhr

őhjlcdh, hđhr uykue flr őhjlcdh shëlc`lő j lële e 5 j lsh, rht hdhr (sĿeovdo fueu bocĿbozĿrdo tórhttlđl`lzl fhclrt`helz yhthrcldlr).

Faő blpathz octĿedo, e E(>, ?/e),

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