Carregando Imagens em JavaScript

7/17/2019 Carregando Imagens em JavaScript

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tg` hu quj jigs sj fhrtg` egqujij pheth.

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sjpgrgchs pjih pheth F.

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Ldkurg 7.6

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Gxdh`g DD7. Cgchs trjs pheths cj u`g rjtg, u` j gpjegs u` cjijsihfgidzg-sj jetrj hs hutrhs chds.

7.1 Cjed¿fgh. H fhenueth fhestdtu³ķch phr chds pheths G j A j phr

thchs hs pheths quj sj jefhetrg` jetrj G j A ³j fog`gch sjk`jeth GA. Hspheths G j A sgh cjeh degchs jxtrj`hs hu jxtrj`dcgcjs ch sjk`jeth.@udtgs kurgs pigegs sgh fhestru³ķcgs usgech-sj sjk`jeths. G `gds sd`-

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Ldkurg 7.:

Hasjrvj quj chds pheths G j A cjtjr`deg` cugs sj`d-rjtgs VGA j VAGgs qugds fhetj` h sjk`jeth GA.

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Ldkurg 7.<

7.3 ]rhphsd¿fgh.

g) VGA ∮ VAG ³j g rjtg cjtjr`degcg phr G j A2

a) VGA ∤ VAG 0 GA.

]rhvg. (g) Vjng ` g rjtg cjtjr`degcg phr G j A. Fh`h VGA j VAG sgh

fhestdtu³ķcgs cj pheths cg rjtg ` , jetgh VGA ∮ VAG ⊁ ` . ]hr hutrh igch, sjF ³j u` pheth cg rjtg ` jetgh, cj gfhrch fh` h gxdh`g DD7, u`g cgs trjsphssdadidcgcjs jxfiusdvgs hfhrrj>

7) F jst³g jetrj G j A

1) G jst³g jetrj A j F

6) A jst³g jetrj G j F

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Eh fgsh (7), F pjrtjefj gh sjk`jeth GA. eh fgsh (1), F pjrtjefj g VAG2

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G prhvg cj (a) ³j cjdxgcg fh`h jxjrf³ķfdh pgrg h ijdthr.

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gi³j` cgqujijs fhetdchs eh sjk`jeth GA.

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7.: Cjed¿fgh. Vjng` ` u`g rjtg j G u` pheth quj egh pjrtjefj` g

` . H fhenueth fhestdtu³ķch pjihs pheths cj ` j phr thchs hs pheths A tgdsquj G j A jstgh j` u`g `js`h igch cg rjtg ` ³j fog`gch sj sj`d-pigeh

cjtjr degch phr ` fhetjech G, j sjr³g rjprjsjetgch phr ]`G

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7. Vharj u`g rjtg `grquj qugtrh pheths G, A, F j C, j` hrcj`, cgjsqujrcg pgrg g cdrjdtg. Cjtjr`dej>

g) GA ∮ AFa) GA ∤ AFf) GF ∤ ACc) GA ∤ FC

j) V GA ∤ V AF l) V GA ∤ V GCk) V FA ∤ V AF j) V GA ∮ V AF

1. Wugeths pheths fhùes g pjih `jehs cugs rjtgs phcj tjr u` fhenuethcj 6 rjtgs eh pigeh8 J u` fhenueth cj 3 rjtgs ch pigeh8

6. ]rhvj h dtj` (a) cg prhphsd¿fgh (7.3).

3. ]rhvj g gr`g¿fgh ljdtg, eh tjxth, cj quj jxdstj` deedths pheths jù` sjk`jeth.

:. Vjng` ] 0 {g,a,f}, `7 0 {g, a}, `1 0 {g, f}, `6 0 {a, f}. Fog`j

] cj pigeh j `7, `1 j `6 cj rjtgs. \jrdquj quj ejstg ’kjh`jtrdg‐ vgij hgxdh`g D1.

Cjed¿fgh>Z` suafhenueth ch pigeh ³j fhevjxh sj h sjk`jethidkgech qugdsqujr chds cj sjus pheths jst³g thtgi`jetjfhetdch ejij.

<. Hs jxj`pihs `gds sd`pijs cj fhenueths fhevjxhs sgh h pr³hprdh pigehj qugiqujr sj d-pigeh. @hstrj quj g detjrsj¿fgh cj chds sj`d pigehs ³j u`fhevjxh.

;. `hstrj quj g detjrfjssgh cj e sj`d-pigehs ³j gdecg u` fhevjxh.

?. @hstrj, jxdadech u` fhetrg jxj`pih, quj g uedgh cj fhevjxhs phcjegh sjr u` fhevjxh.

4. Srjs pheths egh fhidejgrjs cjtjr`deg` trjs rjtgs. Wugetgs rjtgssgh cjtjr`degcgs phr qugtrh pheths sjech quj qugdsqujr trjs cjijs sgh eghfhidejgrjs8

7=. Qjpdtg h jxjrf³ķfdh getjrdhr pgrg h fgsh cj < pheths.

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1. ]rhvj quj, sj u`g rjtg detjrfjptg u` igch cj u` trdgekuih j eghpgssg phr ejeou` cj sjus v³jrtdfjs, jetgh jig detjrfjptg tgà³j` u` chshutrhs chds igchs.

6. Qjpdtg h jxjrf³ķfdh 1 pgrg h fgsh cj : j < rjtgs. Lg¿fg u`g fhenjfturgcj qugi sjr³g g rjsphstg eh fgsh cj e rjtgs.

3. @hstrj quj egh jxdstj u` jxj`pih cj u`g ’kjh`jtrdg‐ fh` < pheths,j` quj sjng` vgidchs hs gxdh`gs D7 j D1 j j` quj thcgs gs rjtgs tjeog`jxgtg`jetj 6 pheths.

:. Vj F pjrtjefj g V GA j F 0 G, `hstrj quj> V GA 0 V GF , quj AF ⊁V GA j quj G /∏ AF .

<. Cj`hestrj quj g detjrsj¿fgh cj fhevjxhs gdecg ³j u` fhevjxh.

;. @hstrj quj u` trdgekuih sjpgrg h pigeh j` cugs rjkdhjs, u`g cgsqugds ³j fhevjxg.

?. Kjejrgidzj hs jxjrf³ķfdhs 77 j 71 pgrg h fgsh cj e pheths.

4. ]hcj` jxdstdr chds sjk`jeths cdstdeths tjech chds pheths j` fhù`8J tjech jxgtg`jetj chds pheths j` fhù`8

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]grg sj gprjecjr g nhkgr giku` nhkh, tgi fh`h cg`gs, rh, xgcrjz, jtf.,tj`hs quj, dedfdgi`jetj, gprjecjr gs sugs rjkrgs. Z` pgd tjetgech jesdegrsju ioh g nhkgr cg`gs cdr³g gikh fh`h> ’Jstj ³j h tgauijdrh cj cg`gs j jstgssgh gs pjcrgs fh` quj sj nhkg‐, ’Vgh 71 pgrg fgcg nhkgchr‐, ’Gs pjcrgs sghgrru`gcgs eh tgauijdrh gssd .‐, grru`gr³g gs pjcrgs pgrg h ioh. G³ķ n³g tjr³grjfjadch u`g jexurrgcg cj pjrkuetgs ch tdph> ’]hr quj gs pjcrgs s³h fgègs fgsgs prjtgs8‐, ’]hr quj s³h sgh chzj pjcrgs8‐, ’Ju gfoh `gds ahedtgsgs pjcrgs argefgs egs fgsg prjtgs j gs prjtgs egs fgsgs argefgs, phrquj e gh

³j gssd`8‐, j jtf.Shcgs jstgs pjrkuetgs tj` u`g ³uedfg rjsphstg> ]hrquj jstg ³j u`g cgs

rjkrgs ch nhkh. Vj giku`g cjigs lhr gitjrgcg h nhkh rjsuitgetj, jàhrg phssgsjr tgà³j` ùdth detjrjssgetj, egh sjr³g `gds h nhkh cj cg`gs.

Hasjrvj quj, gh jesdegr u` tgi nhkh, vhfj cdfdi`jetj cjtjr-sj-dg j`cjsfrjvjr h quj sgh gs pjcrgs. H d`phrtgetj sgh gs rjkrgs ch nhkh, dsth ³j,g `gejdrg cj grru`gr gs pjcrgs eh tgauijdrh, g lhr`g cj `hvj-igs, g lhr`gcj ’fh`jr‐ u`g pjcrg ch gcvjrs³grdh, jtf. Wugiqujr frdge¿fg, gphs ch`degrh nhkh, d`prhvdsgr³g tgauijdrhs fh` rdsfhs eh fogh j utdidzgr³g tg`pdeogs cjkgrrglg, ahthjs, fgrthjs, jtf., fh`h pjcrgs.

Gh frdgr-sj u` cjtjr degch nhkh ³j d`phrtgetj quj sugs rjkrgs sjng`

sufdjetjs j fhestgetjs. ]hr sufdjetj qujrj`hs cdzjr quj gs rjkrgs cjvj`jstgajijfjr h quj ³j pjr`dtdch lgzjr j` qugiqujr sdtug¿fgh quj phssg vdr ghfhrrjr eh cjsjerhigr cj u`g pgrtdcg ch nhkh. ]hr fhesdstjetj qujrj`hscdzjr quj gs rjkrgs egh cjvj` fhetrgcdzjr-sj, hu sug gpidfg¿fgh ijvgr³g gsdtug¿fhjs fhetrgcdt³hrdgs.

Kjh`jtrdg, fh`h qugiqujr sdstj`g cjcutdvh, ³j ùdth pgrjfdcg fh` u` nhkh> pgrtd`hs fh` u` fjrth fhenueth cj jij`jeths (pheths, rjtgs, pigehs) j³j ejfjss³grdh gfjdtgr giku`gs rjkrgs a³gsdfgs quj cdzj` rjspjdth ³gs rjig¿fhjs qujsgtdslgzj` jstjs jij`jeths, gs qugds sgh fog`gcgs cj gxdh`gs. H hanjtdvhegi cjstj nhkh ³j h cj cjtjr degr gs prhprdjcgcjs cgs kurgs pigegs j chss³hidchs eh jspg¿fh. Sgds prhprdjcgcjs, fog`gcgs Sjhrj`gs hu ]rhphsd¿fhjs,

cjvj` sjr cjcuzdcgs sh`jetj gtrgv³js ch rgfdhf³ķedh i³hkdfh g pgrtdr chs gx-dh`gs xgchs hu g pgrtdr cj hutrgs prhprdjcgcjs n³g jstgajijfdcgs.

Cj lgth, jxdstj` v³grdgs kjh`jtrdgs cdstdetgs cjpjecjech ch fhenueth cjgxdh`gs xgch. G kjh`jtrdg quj drj`hs jstucgr ejstgs ehtgs ³j fog`gcg cjKjh`jtrdg Jufidcdgeg, j` oh`jegkj` g Jufidcjs quj g cjsfrjvju eh sjuidvrh, cjeh`degch ’Jij`jeths‐.

;



KJH@ JSQDG JZF IDCDGEG ]IGEG G_DH@ GV VH AQJ @ JCDF¿ GH CJ VJK@JESHV

FG]³DSZIH 1

G_DH@GV VHAQJ @JCDF¿ GH CJ VJK@JESHV

H destru`jeth utdidzgch pgrg `jcdr fh`prd`jeth cj sjk`jeths ³j g r³jkugkrgcugcg. Eg kurg gagdxh h sjk`jeth GA `jcj 6f`, h sjk`jeth GF `jcj?f`, j h sjk`jeth AF `jcj :f`.

G A F

= 7 1 6 3 : < ; ?

Hasjrvj quj gh pheth A, fhrrjsphecj (eg r³jkug) h e³u`jrh 6 j gh phethF, h eu`jrh ?. G `jcdcg ch sjk`jeth AF ³j hatdcg pjig cdljrje¿fg ?∑6 0 :.³J figrh quj g r³jkug phcjrdg tjr sdch fhihfgcg j` ùdtgs hutrgs phsd¿fhjs je³u`jrhs cdljrjetjs fhrrjsphecjrdg` ghs pheths A j F. Eh jetgeth, j` fgcgfgsh, g cdljrje¿fg jetrj jijs sjrdg sj`prj : quj ³j g `jcdcg ch sjk`jeth AF.

Jstjs lgths sgh detrhcuzdchs j` ehssg kjh`jtrdg gtrgv³js cj gxdh`gs.

Gxdh`g DDD7. G thch pgr cj pheths ch pigeh fhrrjsphecj u` e³u`jrh`gdhr hu dkugi g zjrh. Jstj e³u`jrh ³j zjrh sj j s³h sj hs pheths sgh fhdefd-cjetjs.

H e³u`jrh g quj sj rjljrj jstj gxdh`g ³j fog`gch cj cdstgefdg jetrj hspheths hu ³j rjljrdch fh`h h fh`prd`jeth ch sjk`jeth cjtjr`degch pjihschds pheths. Jst³g d`pi³ķfdth eh jeuefdgch ch gxdh`g, g jsfhiog cj u`guedcgcj cj `jcdcg quj sjr³g xgcg cj gkhrg j` cdgetj gh ihekh cjstgs ehtgs.

Gxdh`g DDD1. Hs pheths cj u`g rjtg phcj` sjr sj`prj fhihfgchs j`fhrrjsphecjefdg adue³ķvhfg fh` hs e³u`jrhs rjgds, cj `hch quj g cdljrje¿fgjetrj jstjs e³u`jrhs `j¿fg g cdstgefdg jetrj hs pheths fhrrjsphecjetjs.

Jstj gxdh`g aj` phcjrdg rjfjajr h gpjidch cj gxdh`g cg ’r³jkug deedtg‐phds, gh jstgajijfjr g fhrrjsphecjefdg adue³ķvhfg jetrj hs e³u`jrhs rjgds jhs pheths cg rjtg, g pr³hprdg rjtg threg-sj fh`h quj u`g r³jkug deedtg qujphcj sjr usgcg pgrg `jcdr h fh`prd`jeth cj sjk`jeths ejig fhetdchs.

Gh gpidfgr`hs jstj gxdh`g, h eu`jrh quj fhrrjsphecj g u` pheth cgrjtg ³j cjeh`degch fhhrcjegcg cgqujij pheth.

?




Cj gfhrch fh` h gxdh`g DDD7 h fh`prd`jeth cj u` sjk`jeth GA ³j sj`-

prj `gdhr ch quj zjrh. Gssd`, sj g j a sgh gs fhhrcjegcgs cgs jxtrj`dcgcjscjstj sjk`jeth, h sju fh`prd`jeth sjr³g g `gdhr cdljrje¿fg jetrj h `gdhr jh `jehr cjstjs e³u`jrhs. Dsth ³j jqudvgijetj g th`gr-sj g cdljrje¿fg jetrj gj a j` qugiqujr hrcj` j, j` sjkudcg, fhesdcjrgr h sju vgihr gashiuth. E³hsdecdfgrj`hs h fh`prd`jeth ch sjk`jeth GA pjih s³ķàhih GA . ]hrtgeth

GA 0 |a ∑ g|

Gxdh`g DDD6 Vj h pheth F jefhetrg-sj jetrj G j A jetgh>

GF + FA 0 GA

Fh` g detrhcu¿fgh chs gxdh`gs DDD7, DDD1 j DDD6, phcj`hs rjigfdhegr ghrcjeg¿fgh chs pheths cj u`g rjtg, detrhcuzdcg gtrgv³js chs gxdh`gs DD7 jDD1, fh` g hrcj` chs e³u`jrhs rjgds. Hs e³u`jrhs rjgds sgh g hrcjeg¿fgh pjigrjig¿fgh ’`jehr ch quj‐ (hu pjig rjig¿fgh ’`gdhr ch quj‐), j lgz sjetdch cdzjr-sj quj u` e³u`jrh f jst³g jetrj chds hutrhs g j a, qugech hfhrrj g 9 f 9 ahu a 9 f 9 g.

1.7 ]rhphsd¿fgh. Vj, j` u`g sj`d-rjtg VGA, fhesdcjrg`hs u` sjk`jethGF fh` GF 9 GA, jetgh h pheth F jstgr³g jetrj G j A.

]rhvg. Fjrtg`jetj h pheth G egh phcj jstgr jetrj A j F, n³g quj A jF jstgh eg `js`g sj`d-rjtg cj hrdkj` G. Vj h pheth A jstdvjssj jetrj G jF jetgh, pjih gxdh`g DDD6, tjr³ķg`hs GA + AF 0 GF j, fh`h fhesjqujefdg,GA 9 GF . @gs jstg cjsdkugicgcj ³j fhetr³grdg g odp³htjsj GF 9 GA. ]hr-tgeth, ³j h pheth F quj jst³g jetrj G j A.

Sjhrj`g> Vjng` G, A j F pheths cj u`g `js`g rjtg fungs fhhrcjegcgssgh, rjspjftdvg`jetj g, a j f. H pheth F jst³g jetrj G j A sj j s³h sj h eu`jrhf jst³g jetrj g j a.

]rhvg. Vj F jst³g jetrj G j A jetgh, pjih gxdh`g DDD6, tj`-sj quj

GF + FA 0 GA , hu sjng

|f ∑ g| + |a ∑ f| 0 |g ∑ a|

7◬ fgsh> \g`hs suphr dedfdgi`jetj quj g 9 a. Ejstj fgsh, cg dkugicgcjgfd`g, hatj`-sj

|f ∑ g| 9 a ∑ g j |a ∑ f| 9 a ∑ g

4




Fh`h fhesjqujefdg, f ∑ g 9 a ∑ g j a ∑ f 9 a ∑ g. ]hrtgeth, f 9 a j

g 9 f. Gssd`, rjsuitg quj f jst³g jetrj g j a.

1◬ fgsh> H fgsh j` quj a 9 g phcj sjr gegidsgch cj `gejdrg ge³gihkg j³j cjdxgch g fgrkh ch ijdthr.

Qjfdprhfg`jetj, sj h e³u`jrh f jst³g jetrj hs e³u`jrhs g j a jetgh>

|f ∑ g| + |a ∑ f| 0 |g ∑ a|

Vjkuj-sj cg³ķ quj GF + FA 0 GA . J` pgrtdfuigr

GF 9 GA j FA 9 GA

Fhesdcjrgech gs sj`d-rjtgs cjtjr`degcgs pjih pheth G. Vj F j A pjr-tjefj` mg `js`g sj`d-rjtg, jetgh ³j u`g fhesjqujefdg cg prhphsd¿fgh getjrdhrquj F jst³g jetrj G j A. Gr`h quj F j A egh phcj` pjrtjefjr gs sj`d-rjtgscdstdetgs, dsth ³j, egh phcj` sjr sjpgrgchs pjih pheth G. Vj jstj lhssj h fgsh,sjrdg h pheth G quj jstgrdg jetrj A j F j tjr³ķg`hs AG + GF 0 A F . @gscg³ķ rjsuitg quj AG 9 AF h quj jst³g j` fhetrgcd¿fgh fh` g cjsdkugicgcj n³ghatdcg gfd`g. Dsth prhvg g gr`g¿fgh j fhefiud g cj`hestrg¿fgh ch tjhrj`g.

1.6 Cjed¿fgh. Fog`g`hs cj pheth `³jcdh ch sjk`jeth GA g u` phethF cjstj sjk`jeth tgi quj GF 0 F A.

1.3 Sjhrj`g. Z` sjk`jeth tj` jxgtg`jetj u` pheth `³jcdh.

]rhvg. (Jxdstjefdg ) Vjng` g j a gs fhhrcjegcgs cgs jxtrj`dcgcjs chsjk`jeth. Fhesdcjrj h eu`jrh f 0 (g + a)/1. Cj gfhrch fh` h gxdh`g DDD1jxdstj u` pheth F cg rjtg quj tj` f phr fhhrcjegcg. Fh`h

GF 0 |g ∑ f| 0

g ∑

g + a

1

0

g

1

∑ a

1

FA 0 |f ∑ a| 0

g + a

1 ∑ a

0

g

1 ∑

a

1

Fhefiu³ķ`hs quj GF 0 FA. Fh`h h eu`jrh (g + a)/1 jst³g jetrj hse³u`jrhs g j a, sjkuj-sj cg prhphsd¿fgh getjrdhr quj F jst³g jetrj G j A. IhkhF ³j h pheth `³jcdh cj GA.

7=




(Zedfdcgcj ) Vjng F fh`h hatdch eg prhvg cg jxdstjefdg j sjng F— u` hutrh

pheth ch sjk`jeth GA tgi quj GF 0 AF . Vjng` g, a j f

gs fhhrcjegcgschs pheths G, A j F— rjspjftdvg`jetj. Jetgh tjrj`hs,

(d) eh fgsh j` quj g 9 f 9 a,

f ∑ g 0 a ∑ f

,

(dd) eh fgsh j` quj a 9 f 9 g,

g ∑ f 0 f ∑ a

J` gàhs hs fgshs g fhefiusgh ³j quj

f 0 g + a

1

Gssd`, j` qugiqujr fdrfuestgefdg, f 0 f j, phrtgeth, pjih gxdh`g DDD1,F 0 F—. Ldfg gssd` prhvgcg g uedfdcgcj ch pheth `³jcdh.

1.: Hasjrvg¿fgh. G eh¿fgh cj cdstgefdg ³j u`g cgs eh¿fhjs `gds a³gsdfgscg kjh`jtrdg. ]jih quj n³g vd`hs jig sgtdslgz ³gs sjkudetjs prhprdjcgcjs>

7) ]grg qugdsqujr chds pheths G j A ch pigeh, tj`-sj GA ≪ =

Gi³j cdssh, GA 0 = sj j sh`jetj sj G 0 A.

1) ]gr qugdsqujr chds pheths G j A tj`-sj quj GA 0 AG. Z`ghutrg d`phrtgetj prhprdjcgcj cg cdstgefdg ³j g cjsdkugicgcj trd-

gekuigr.6) ]grg qugdsqujr trjs pheths ch pigeh G, A j F, tj`-sj GF ≣GA + AF . Dkugicgcj hfhrrj sj j sh`jetj sj F pjrtjefj gh de-tjrvgih GA.

Jstg cjsdkugicgcj sjr³g cj`hestrgcg eh fgp³ķtuih : (vjng h tjhrj`g (:.7=))fh`h fhesjqujefdg chs 3 prd`jdrhs kruphs cj gxdh`gs.

77




1.< Cjed¿fgh. Vjng G u` pheth ch pigeh j r u` eu`jrh rjgi phsdtdvh.

H f³ķrfuih cj fjetrh G j rgdh r ³j h fhenueth fhestdtu³ķch phr thchs hs phethsA ch pigeh tgds quj GA 0 r.

G

A

H

G F A

Ldkurg 1.1

³J u`g fhesjqujefdg ch gxdh`g DDD1 quj phcj`hs trg¿fgr u` f³ķrfuih fh`qugiqujr fjetrh j qugiqujr rgdh.Shch pheth F quj sgtdslgz g cjsdkugicgcj GF 9 r ³j cdth jstgr cjetrh

ch f³ķrfuih. Vj, gh dev³js, GF 5 r, jetgh F ³j cdth jstgr lhrg ch f³ķrfuih. Hfhenueth chs pheths quj jstgh cjetrh ch f³ķrfuih ³j fog`gch cj cdsfh cj rgdhr j fjetrh G.

³J tgà³j` u`g fhesjqujefdg ch gxdh`g DDD1 quj h sjk`jeth cj rjtgidkgech u` pheth cj cjetrh ch f³ķrfuih fh` u` pheth lhrg ch `js`h tjù` pheth j` fhù` fh` h f³ķrfuih.

71




J_JQF³DFDHV

7. Vjng` G, A, F pheths cj u`g rjtg. Lg¿fg u` cjsjeoh rjprjsjetgech-hs, sgajech quj GA 0 6, GF 0 1 j A F 0 :.

1. Qjpdtg h jxjrf³ķfdh getjrdhr, sgajech quj F jst³g jetrj G j A j qujGA 0 ; j GF 0 :.

6. Cjsjeoj u`g rjtg j sharj jig `jrquj chds pheths G j A. Vupheogquj g fhhrcjegcg ch pheth G sjng zjrh j g ch pheth A sjng u`. @grquj

gkhrg chds pheths fungs fhhrcjegcgs sgh 6, :, :/1, 7/6, 6/1, 1, -7, -1, -:,-7/6, -:/6.

3. Vjng` G7 j G1 pheths cj fhhrcjegcgs 7 j 1. Cj g fhhrcjegcg chpheth `³jcdh G6 ch sjk`jeth G7G1. Cj g fhhrcjegcg ch pigeh `³jcdh G3 chsjk`jeth G1G6. Cj g fhhrcjegcg G: ch pheth `³jcdh ch sjk`jeth G6G3.

:.]rhvj quj, sj g

a 0 f

c jetgh

g) g

f 0 a

c j c

a 0 f

g

a)g+ag 0 f+c

c j g∑ag 0 f∑c

f

f) g+a

a 0 f+c

c j g∑a

a 0 f∑c

c

<. Vj p ³j pheth cj detjrfjssgh cj f³ķrfuihs cj rgdh r j fjetrhs j` G j A,`hstrj quj ] G 0 ] A.

;. Zsgech u`g r³jkug j u` fh`pgssh, cjsfrjvg u` `³jthch pgrg fhe-stru¿fgh cj u` trdgekuih fh` chds igchs cj `js`h fh`prd`jeth. (Z` tgitrdgekuih ³j fog`gch cj trdgekuih ds³hsfjijs).

?. Cjsfrjvg u` `³jthch pgrg fhestru¿fgh cj u` trdgekuih fh` hs trjsigchs cj `js`h fh`prd`jeth.

4.@hstrj quj, sj g 9 a jetgh g 9 (g + a)/1 j a 5 (g + a)/1.

7=. ³J phss³ķvji cjsjeogr sj u` trdgekuih fh` igchs `jcdech 6, ? j :8

77. H f³ķrfuih cj rgdh r7 fjetrgch j` G detjrfjptg h f³ķrfuih cj rgdh r1fjetrgch j` A j` jxgtg`jetj chds pheths. H quj sj phcj gr`gr sharj

76




GA8

71. Fhesdcjrj u` fdrfuih cj rgdh r j fjetrh G. Vjng` A j F pheths cjstjf³ķrfuih. H quj sj phcj gr`gr sharj h trdgekuih GAF8

76. Fhesdcjrj u` f³ķrfuih cj rgdh r j fjetrh H. Vjng G u` pheth cjstjf³ķrfuih j sjng A u` pheth tgi quj h trdgekuih HGA ³j jqudi³gtjrh. Wugi ³j gphsd¿fgh ch pheth A rjigtdvg`jetj gh f³ķrfuih8

73. Chds f³ķrfuihs cj `js`h rgdh j fjetrhs G j A sj detjrfjptg` j` chdspheths F j C. H quj phcj sjr gr`gch sharj hs trdgekuihs GAF j GFC8 Jsharj h qugcrdi³gtjrh GFAC8

73




]QHAIJ@GV

7. Cgch u` sjk`jeth GA `hstrj quj jxdstj j ³j ³uedfh, u` pheth F jetrj

G j A tgi quj `(GF )

`(AF ) 0 g hecj g ³j qugiqujr rjgi phsdtdvh.

1.Cjsfrjvg u` `³jthch pgrg hatjr u`g ahg gprhxd`g¿fgh ch fh`prd`jethcj u` f³ķrfuih.

6. ]rhvj g sjkudetj gr`g¿fgh ljdtg eh tjxth> h sjk`jeth cj rjtg idkgech

u` pheth lhrg cj u` fdrfuih fh` u` pheth cjetrh ch `js`h, tj` u` phethj` fhù` fh` h fdrfuih.

3. Cgch chds pheths G j A j u` eu`jrh rjgi r `gdhr ch quj `(GA), hfhenueth chs pheths F sgtdslgzjech g `(FG) + `(FA) 0 r ³j fog`gch cjjidpsj. Jstgajij¿fg hs fhefjdths cj rjkdgh detjrdhr j cj rjkdgh jxtjrdhr g u`gjidpsj.

:. Z` fhenueth @ cj pheths ch pigeh ³j id`dtgch sj jxdstj u` fdrfuihF tgi quj thchs hs pheths cj @ jstgh cjetrh cj F. ]rhvj quj qugiqujrfhenueth edth cj pheths ³j id`dtgch. ]rhvj tgà³j` quj sjk`jeths sgh

id`dtgchs. fhefiug h `js`h pgrg trdgekuihs.

<. ]rhvj quj g uedgh cj u`g qugetdcgcj edtg cj fhenueths id`dtgchs ³jgdecg u` fhenueth id`dtgch.

;. @hstrj quj cgch u` pheth ] j u` fhenueth id`dtgch @, jetgh jxdstju` cdsfh fh` fjetrh j` ] quj fhet³j` @. (Hasjrvg¿fgh> jstg`hs gc`dtdechg vgidcgcj cg cjsdkugicgcj trdgekuigr).

?. ]rhvj quj gs rjtgs sgh fhenueths did`dtgchs. (Vukjstgh> usj h prha-ij`g ;.)

7:




FH@JES ³GQDH

Gs prd`jdrgs eh¿fhjs kjh`³jtrdfgs surkdrg` qugech h oh`j` vdu-sj fh`-pjidch g jljtugr `jcdcgs, dsth ³j, g fh`pgrgr cdstgefdgs j g cjtjr`degr gscd`jeshjs chs fhrphs quj h rhcjgvg`. Jk³ķpfdhs, Gssdrdhs j Agadihedfhs n³gfheojfdg` gs prdefdpgds kurgs kjh`³jtrdfgs j g eh¿fgh cj gekuih quj usgvgègs `jcdcgs cj ³grjg j eg Gstrheh`dg.

G `gdhr pgrtj ch cjsjevhivd`jeth cg Kjh`jtrdg rjsuithu chs jslhr¿fhsljdths, gtrgv³js cj ùdth s³jfuihs, pgrg fhestrudr-sj u` fhrph cj chutrdegi³hkdfg quj fhrrjigfdheg-sj hs cgchs kjh`³jtrdfhs hatdchs cg hasjrvg¿fgh j `j-

cdcg. ]jih tj`ph cj Jufidcjs (fjrfg cj 6== g.F) g fdjefdg cg Kjh`jtrdgtdeog gifge¿fgch u` jst³gkdh aj` gvge¿fgch. Ch `gtjrdgi gfuùigch Jufidcjsfh`pdihu hs sjus ’Jij`jeths‐, u` chs `gds eht³gvjds idvrhs n³g jsfrdths.

G kjh`jtrdg, fh`h gprjsjetgcg phr Jufidcjs, lhd h prd`jdrh sdstj`g cjdcjdgs cjsjevhivdch pjih oh`j`, eh qugi u`gs phufgs gr`g¿fhjs sd`pijssgh gc`dtdcgs sj` cj`hestrg¿fgh j jetgh utdidzgcgs pgrg prhvgr hutrgs `gdsfh`pijxgs. Z` tgi sdstj`g ³j fog`gch cjcutdvh. G ajijzg cg Kjh`jtrdg,fh`h u` sdstj`g cjcutdvh, despdrhu oh`jes, cgs `gds cdvjrsgs ³grjgs, ghrkgedzgrj` sug dcjdgs cg `js`g lhr`g. Vgh jxj`pihs cdsth h ’]rdefdpdg‐cj Vdr Dssgf Ejwthe, eh qugi jij tjetg gprjsjetgr g L³ķsdfg fh`h u` sdstj`gcjcutdvh, j g ’³Jtdfg‐ ch ihshlh Vpdehzg.

7<



KJH@JSQDG JZFIDCDGEG ]IGEG G_DH@GV VHAQJ @JCDF¿ GH CJ GEKZIHV

FG]³DSZIH 6

G_DH@GV VHAQJ @JCDF¿ GH CJ GEKZIHV

6.7 Cjed¿fgh. Fog`g`hs cj gekuih g kurg lhr`gcg phr cugs sj`d-rjtgs fh` g `js`g hrdkj`.

Ldkurg 6.7

Gs sj`d rjgs sgh fog`gcgs cj igchs ch gekuih j g hrdkj` fhù`, cjv³jrtdfj ch gekuih. Z` gekuih lhr`gch phr cugs sj`d rjtgs cj u` `js`g

rjtg ³j fog`gch cj ˌ gekuih rgsh.

G

Ldkurg 6.1

Jxdstj` v³grdgs `gejdrgs cdstdetgs cj rjprjsjetgr u` gekuih. ]hr jx-j`pih, h gekuih cg kurg (6.6) phcj sjr cjsdkegch phr A ˌ GF hu F AG. Ghutdidzgr`hs jstg ehtg¿fgh, g ijtrg decdfgtdvg ch v³jrtdfj cjvj sj`prj gpgrjfjr

jetrj gs hutrg cugs, gs qugds rjprjsjetg` pheths cgs sj`d-rjtgs quj lhr`g`h gekuih.

G

A

F

Ldkurg 6.6

7;






]grg lgfdidtgr h jeuefdgch ch pr³hxd h gxdh`g, vg`hs cgr g sjkudetj

cjed¿fgh>

6.1 Cjed¿fgh. Cdrj`hs quj u`g sj`d-rjtg cdvdcj u` sj`d-pigeh sjjig jstdvjr fhetdcg eh sj`d-pigeh j sug hrdkj` lhr u` pheth cg rjtg quj hcjtjr`deg.

Gxdh`g D DD:. ³J phss³ķvji fhihfgr, j` fhrrjsphecjefdg adue³ķvhfg, hse³u`jrhs rjgds jetrj zjrh j 7?= j gs sj`d-rjtgs cj `js`g hrdkj` quj cdvdcjù` cgch sj d-pigeh, cj `hch quj g cdljrje¿fg jetrj jstj eu`jrh sjng g`jcdcg ch gekuih lhr`gch pjigs sj`d-rjtgs fhrrjsphecjetjs.

<=71:

7?= =

H

A

G

Ldkurg 6.<

Gh lgzjr`hs tgi fhrrjsphecjefdg fog`g`hs h eu`jrh quj fhrrjsphecjg u`g cgcg sj`d-rjtg cj fhhrcjegcg cg sj`d-rjtg. Eg kurg (6.<) gfd`g, gsj`d-rjtg VHG tj` fhhrcjegcg <=, g sj`d-rjtg V HA tj` fhhrcjegcg 71:. Cjgfhrch fh` h gxdh`g DDD: g `jcdcg ch gekuih G ˌ HA ³j 71: ∑ <= 0 <:. Jstj ³ju` lgth kjrgi. Vj g j a lhrj` fhhrcjegcgs chs igchs ch gekuih G ˌ HA , jetgh|g∑a| ³j g `jcdcg cjstj gekuih. Decdfgrj`hs u` gekuih j g sug `jcdcg pjih`js`h sdàhih. Gssd` jsfrjvjrj`hs cj u`g `gejdrg kjrgi

G ˌ HA 0 |g ∑ a|

]grg sdkedfgr quj |g ∑ a| ³j g `jcdcg ch gekuih Gˌ

HA . Hasjrvj quj gssj`d-rjtgs quj lhr`g` u` gekuih rgsh sjrgh sj`prj eu`jrgcgs phr = j 7?=sjech g `jcdcg cj tgds gekuihs sj`prj 7?=◬.

6.6 Cjed¿fgh. Vjng` V HG, V HA j V HF sj`d-rjtgs cj `js`g hrdkj`.Vj h sjk`jeth GA detjrfjptgr V HF cdrj`hs quj V HF cdvdcj h gekuih G ˌ HA .

Gxdh`g DDD<. j u`g sj`d rjtg V HF cdvdcj u` gekuih G ˌ HA , jetgh

G ˌ HA 0 G ˌ HF + F ˌ HA

74




6.3 Cjed¿fgh. Chds gekuihs sgh cdths supij`jetgrjs sj g sh`g cj sugs

`jcdcgs ³j 7?=◬

. H supij`jeth cj u` gekuih ³j h gekuih gcngfjetj gh gekuihcgch hatdch pjih prhihekg`jeth cj u` cj sjus igchs.

H

F

A

G

Ldkurg 6.;

³J figrh quj u` gekuih j sju supij`jeth sgh gekuihs supij`jetgrjs. ³Jtgà³j` jvdcjetj quj, sj chds gekuihs tj` g `js`g `jcdcg, jetgh h `js`hhfhrrj fh` sjus supij`jeths.

Wugech cugs rjtgs cdstdetgs sj detjrfjptg`, lhr`g` sj qugtrh gekuihs,fh`h decdfgch eg kurg gagdxh. Hs gekuihs G ˌ HA j C ˌ HF sgh hphsths pjih

v³jrtdfj . Ch `js`h `hch h sgh hs gekuihs G ˌ HC j A ˌ HF .

H

G

A

C

F

Ldkurg 6.?

6.: prhphsd¿fgh. Gekuihs hphsths pjih v³jrtdfj tj` g `js`g `jcdcg.

]rhvg. Cj lgth, sj

G ˌ HA j

C ˌ HF sgh gekuihs hphsths pjih v³jrtdfj, jetghjijs tj` h `js`h supij`jeth> G ˌ HC . Ihkh

G ˌ HA + G ˌ HC 0 7?=◬

C ˌ HF + G ˌ HC 0 7?=◬

]hrtgeth G ˌ HA 0 7?=◬ ∑ G ˌ HC 0 C ˌ HF

1=




6.< Cjed¿fgh. Z` gekuih fung `jcdcg ³j 4=◬ ³j fog`gch cj ˌ gekuih

rjth.³J figrh quj h supij`jeth cj u` gekuih rjth ³j tgà³j u gekuih rjth.

Wugech cugs sj`d rjtgs sj detjrfjptg`, sj u` chs qugtrh gekuihs lhr`gchsphr jigs lhr rjth, jetgh thchs hs hutrhs tgà³j` h sjrgh. Ejstj fgsh cdrj`hsquj gs rjtgs sgh pjrpjecdfuigrjs.

6.; Sjhrj`g. ]hr qugiqujr pheth cj u`g rjtg pgssg u`g ³uedfg rjtgpjrpjecdfuigr g jstg rjtg.

]rhvg. (Jxdstjefdg ). Cgcg u`g rjtg ` j u` pheth G sharj jig, gscugs sj`d-rjtgs cjtjr`degcgs phr G lhr`g` u` gekuih rgsh. Fhesdcjrj u`

chs sj`d pigehs cjtjr`degchs pjig rjtg `. Cj gfhrch fh` gxdh`g DDD:,jetrj thcgs gs sj`d rjtgs fh` hrdkj` G, quj cdvdcj` h sj`d-pigeh xgch,jxdstj u`g fung fhhrcjegcg sjr³g h eu`jrh 4=. Jstg sj`d-rjtg lhr`g, fh`gs cugs sj`d rjtgs cjtjr`degcgs pjih pheth G sharj g rjtg `, gekuihs cj4=◬. ]hrtgeth jig ³j pjrpjecdfuigr g rjtg `

(Zedfdcgcj ). Vupheog quj jxdstdssj` cugs rjtgs e j e pgssgech pjihpheth G j pjrpjecdfuigrjs g `. Ldxj u` chs sj`d-pigehs cjtjr`degchs phr`. Gs detjrfjsshjs cg rjtgs e j e fh` jstj sj`d-pigeh sgh sj`d-rjtgs qujlhr`g` u` gekuih η j, fh`h eg kurg gagdxh, lhr`g` hutrhs chds gekuihsα j ζ fh` gs sj`d rjtgs cjtjr`degcgs pjih pheth G eg rjtg `.

ζ α

η

ee

`

G

Ldkurg 6.4

Fh`h e j e sgh pjrpjecdfuigrjs g ` jetgh α 0 ζ 0 4=◬. ]hr hutrh igch,cjvj`hs tjr η + α + ζ 0 7?=◬. ]hr hutrh igch, cjvj`hs tjr η + α + ζ 0 7?=◬.Ihkh η 0 =◬ j gs rjtgs e j e fhdefdcj`.

17




J_JQF³DFDHV

7. @hstrj quj sj u` gekuih j sju supij`jeth tj` g `js`g `jcdcgjetgh h gekuih ³j rjth.

1. Z` gekuih ³j fog`gch gkuch sj `jcj `jehs cj 4=◬, j ³j hatush sj`jcj `gds cj 4=◬. @hstrj quj h supij`jeth cj u` gekuih gkuch ³j hatush.

3. Chds gekuihs sgh cdths fh`pij`jetgrjs sj sug sh`g ³j u` gekuih rjth.Chds gekuihs sgh fh`pij`jetgrjs j h supij`jeth cj u` cjijs `jcj tgeth

qugeth h supij`jeth ch sjkuech `gds 6=◬

. Wugeth `jcj` hs chds gekuihs8

:.Z`g phidkhegi ³j u`g kurg lhr`gcg phr u`g sjqujefdg cj pheths G7,G1, ..., Ge j pjihs sjk`jeths G7G1, G1G6, G6G3,...,Ge∑7Ge. Hs phethssgh hs v³jrtdfjs cg phidkhegi j hs sjk`jeths sgh hs sjus igchs. Cjsjeoj gphidkhegi GAFC sgajech quj> GA 0 AF 0 FC 0 1.f`, G ˌ AF 0 71=◬ jA ˌ FC 0 7==◬.

<. Z` phi³ķkheh ³j u`g phidkhegi j` quj gs sjkudetjs 6 fhecd¿fhjs sghsgtdsljdtgs> g) Ge 0 G7, a) hs igchs cg phidkhegi sj detjrfjptg` sh`jetj j`sugs jxtrj`dcgcjs j f) chds igchs fh` g `js`g jxtrj`dcgcj egh pjrtjefj`

g u`g `js`g rjtg. Cgs 3 kurgs, sjkudetjs, gpjegs cugs sgh phi³ķkhehs.Cjtjr`dej qugds sgh jigs.

CJ

CF

A

G

J

G

J A

F

C

G

G

A

J C

F

A

F

Z phi³ķkheh cj v³jrtdfj G7, G1, ..., Ge+7 0 G7, sjr³g rjprjsjetgch phr

11




G7GG1G6, ...,Ge. Jij tj` e igchs, e v³jrtdfjs j e gekuihs.

;. Cjsjeoj u` phi³ķkheh cj 3 igchs GAFC tgi quj GA 0 A F 0 F C 0CG 0 1f`, fh` G ˌ AF 0 G ˌ CF 0 7==◬ j fh` A ˌ FC j A ˌ GC 0 ?=◬.

?. H sjk`jeth idkgech hs v³jrtdfjs egh fhesjfutdvhs cj u` phi³ķkheh ³jfog`gch u`g cdgkhegi ch phi³ķkheh. Lg¿fg h cjsjeoh cj u` phi³ķkheh cj sjdsigchs. J` sjkudcg cjsjeoj thcgs gs sugs cdgkhegds. Wugetgs cdgkhegds tjr³gu` phi³ķkheh cj 1= igchs8 J cj e igchs8

4. Z phi³ķkheh ³j fhevjxh sj jst³g sj`prj fhetdch j` u` chs sj`d-pigehscjtjr degchs pjigs rjtgs quj fhetj` hs sjus igchs. Eg kurg sjkudetj h

phi³ķkheh (g) ³j fhevjxh j h (a) ³j egh fhevjxh.

]hi³ķkhehs fhevjxhs rjfjaj` cjsdkeg¿fhjs jspjfdgds. Vgh gs sjkudetjs gscjsdkeg¿fhjs cgcgs g jstjs phi³ķkhehs cj gfhrch fh` sju e³u`jrh cj igchs, gt³j7= igchs.

e◬ cj igchs eh`j ch phi³ķkheh fhevjxh

6 trdgekuih3 qugcrdi³gtjrh: pjet³gkheh< ojx³gkheh

; ojpt³gkheh? hft³gkheh4 ehe³gkheh

7= cjf³gkheh

7=. Cjsfrjvg u` `³jthch, j` quj sj lg¿fg ush gpjegs cj u` fh`pgsshj cj u`g r³jkug egh eu`jrgcg, cj fhestru¿fgh cj u` qugcrdi³gtjrh fh` hsqugtrh igchs cj `js`h fh`prd`jeth. Jstjecg sju `³jthch pgrg h fgsh cj :igchs.

16




]QHAIJ@GV

7. Cgch u` gekuih GHA `hstrj quj jxdstj u`g ³uedfg sj`d-rjtg V HF tgi quj G ˌ HF 0 F ˌ HA . G sj`d rjtg V HF ³j fog`gcg cj adssjtrdz ch gekuihG ˌ HA .

1. @hstrj quj gs adssjtrdzjs cj u` gekuih j ch sju supij`jeth sghpjrpjecdfuigrjs.

6. Cgch u` gekuih G ˌ HA j u` e³u`jrh rjgi phsdtdvh g, = 9 g 9 7, `hstrj

quj jxdstj u`g ³uedfg sj`d-rjtg VHF tgi quj F ˌ HA 0 g»G ˌ HA

3. Cj qugeths krgus `hvj-sj h phetjdrh chs `deuths jequgeth h phetjdrhcgs ohrgs pjrfhrrj u` gekuih rgsh.

:. Cjsfrjvg u` prhfjssh pjih qugi u` cjsjeodstg, sj` usgr u` trgesljrd-chr, phssg ’fhpdgr‐ u` gekuih, dsth ³j, cgch u` gekuih cjsjeogch j` u`glhiog cj pgpji, cjsjng`hs jstgajijfjr u` prhfjcd`jeth pjih qugi phssg`hscjsjeogr u` hutrh gekuih quj sjng fhekrujetj gh prd`jdrh, dsth sj` lgzjrush cj u` trgesljrdchr.

<. Z`g gitjregtdvg pgrg cjedr u` gekuih ³j h cj fhesdcjrgr g detjrfjssghcj sj`d-pigehs. Lhr`gidzj jstg dcjdg. Qjigfdhej fh` ehssg cjed¿fgh.

13




FH@JES ³GQDH

Eh sjkuech j prd`jdrh `dijedhs getjs cj Frdsth, fihrjsfjrg` eg @jshphtg`dg(g rjkdgh jetrj hs rdhs Julrgtjs j Sdkrj, h quj oh nj ³j gprhxd`gcg`jetj hDrgquj) v³grdgs fdvdidzg¿fhjs fheojfdcgs, cj u` `hch kjrgi, fh`h fdvdidzg¿fghagadihedfg. Jetrj jigs, g fdvdidzg¿fgh Vu`³jrdg, quj tjvj sju ³gpdfj eh sjkuech`dijedh g.F., j g fdvdidzg¿fgh quj sj cjsjevhivju j` threh cg fdcgcj fog`gcgAgadihedg eh prd`jdrh `dijedh g.F. Hs agadihedhs gashrvjrg` krgecj pgrtjcg fuiturg `gtj`³gtdfg jk³ķpfdg j g jig gfrjsfjetgrg` sugs pr³hprdgs fhequds-tgs. Jetrj jstgs, kurg h thtgi cjsjevhivd jeth cg ³gikjarg jij`jetgr j g

devje¿fgh cj u` sdstj`g cj eu`jrg¿fgh j` quj hs gikgrds`hs tj` u` vgihrcj phsd¿fgh eg krgg chs e³u`jrhs. Jstj `³jthch cj jsfrjvjr hs e³u`jrhs, de-ljidz`jetj egh lhd gashrvdch pjigs fdvdidzg¿fhjs quj sj sjkudrg` mg fdvdidzg¿fghagadihedfg. J` pgssgch `gds rjfjetj jij lhd cjsfhajrth pjihs odecus cj quj`h d`phrtg`hs, gtrgv³js chs ³grgajs.

Jequgeth g agsj cj eu`jrg¿fgh decu jrg cjfd`gi, jxgtg`jetj fh`h utd-idzg`hs ohnj, g agsj cj eu`jrg¿fgh agadihedfg jrg sjxgkjsd gi. Dsth sdkedfgquj jijs utdidzgvg` <= s³ķàhihs (gikgrds`hs) cdstdeths pgrg jsfrjvjr thchs hse³u`jrhs. Deljidz`jetj h zjrh jrg rjprjsjetgch phr u`g igfueg h quj thregvg,g ijdturg cj gikues e³u`jrhs, fhelusg. Sgivjz jstg tjeog sdch g cdfuicgcjjssjefdgi, quj ijvhu jstj sdstj`g g egh sjr gashrvdch pjigs fdvdidzg¿fhjs quj

h sufjcjrg` g fdvdidzg¿fgh agadihedfg.]grg jstj phvh, quj utdidzgvg u` sdstj`g cj eu`jrg¿fgh cj agsj <=, lhd

ùdth egturgi cdvdcdr h f³ķrfuih j` 6<= pgrtjs (krgu), j fgcg u`g cjstgspgrtjs j` <= pgrtjs (`deuth) j rjpjtdr h prhfjssh pgrg jstgs suapgrtjs.Gssd` h ’krgu‐ ³j u`g devje¿fgh chs agadihedhs, quj jetrg` pgrg g odst³hrdgcg fdjefdg `gtj`³gtdfg, fh` u`g fhetrdaud¿fgh d`phrtgetj quj utdidzg`hs gt³johnj.

1:



KJH@JSQDG JZFIDCDGEG ]IGEG FHEKQZJEFDG

FG]³DSZIH 3

FHEKQZJEFDG

3.7 Cjed¿fgh. Cdrj`hs quj chds sjk`jeths GA 0 F C sgh fhekrujetjs

qugech GA 0 F C2 cdrj`hs quj chds gekuihs ˌ G j ˌ A sgh fhekrujetjs sj jijstj` g `js`g `jcdcg.

Hasjrvj quj, fh` jstg cjed¿fgh, gs prhprdjcgcjs cg dkugicgcj cj e³u`jrhspgssg` g vgijr pgrg g fhekrujefdg cj sjk`jeths j cj gekuihs. Fh`h fhe-sjqujefdg, u` sjk`jeth ³j sj`prj fhekrujetj g jij `js`h j chds sjk`jeths,fhekrujetjs g u` tjrfjdrh, sgh fhekrujetjs jetrj sd. H `js`h vgijech pgrggekuihs.

]gr sd`pidfgr gh `³gxd`h g ehssg ehtg¿fgh, drj`hs utdidzgr h s³ķàhih’0‐ pgrg sdkedfgr fhekrujetj . Gssd` GA 0 FC cjvj sjr idch fh`h GA

³j fhekrujetj g FC j ˌ G 0 ˌ A cjvj sjr idch fh`h gekuih G ³j fhekrujetj

gh ˌ gekuih A. J` kjrgi egh ogvjr³g pjrdkh cj fhelusgh fh` g dkugicgcjcj e³u`jrhs hu cj fhenueths. Wugech ohuvjr, rjlhr¿fgrj`hs fh` pgigvrgs hsdkedfgch ch sdàhih.

3.1 Cjed¿fgh. Chds trdgekuihs sgh fhekrujetjs sj lhr phss³ķvji jstgajij-fjr u`g fhrrjsphecjefdg adue³ķvhfg jetrj sjus v³jrtdfjs cj `hch quj igchs jgekuihs fhrrjsphecjetjs sjng` fhekrujetjs.

G

A

F

J

LK

Ldkurg 3.7

Vj GAF j JL K sgh chds trdgekuihs fhekrujetjs j sj

G ↓ J

A ↓ L

F ↓ K

1<




³j g fhrrjsphecjefdg quj cjej g fhekrujefdg, jetgh vgij`, sdùitgejg-

`jetj, gs sjds rjig¿fhjs sjkudetjs>

GA 0 JL AF 0 LK GF 0 JK

G 0 J A 0 L F 0 F

Vj, ehs trdgekuihs gagdxh, fhesdcjrgr`hs g fhrrjsphecjefdg F ↓ L , A ↓ C

j G ↓ J , vjrdfg`hs quj ˌ F 0 ˌ L , ˌ A 0 ˌ C , ˌ G 0 ˌ J , FA 0 L C, AG 0 CJ jGF 0 J L . ]hrtgeth, hs trdgekuihs FAG j L CJ sgh fhekrujetjs.

7=:

3:

6

7=:

3:

6=

<

3

6

<

6

3

F

A

G

CJ

L

Ldkurg 3.1

Jsfrjvj`hs GA F 0 J L K pgrg sdkedfgr quj hs trdgekuihs GA F j J L K

sgh fhekrujetjs j quj g fhekrujefdg ijvg G j` J , A j` L j F j` K.

Gxdh`g D\. Cgchs chds trdgekuihs GAF j JL K, sj GA 0 J L , GF 0JK j ˌ G 0 ˌ J jetgh GAF 0 J L K.

Hasjrvj quj, cj gfhrch fh` g cjed¿fgh (3.1), pgrg vjrdfgr`hs sj chdstrdgekuihs sgh fhekrujetjs tj`hs quj vjrdfgr sjds rjig¿fhjs> fhekrujefdg chs

trjs pgrjs cj igchs j fhekrujefdg chs trjs pgrjs cj gekuihs fhrrjsphecjetjs.H gxdh`g gfd`g gr`g quj ³j sufdjetj vjrdfgr gpjegs trjs cjigs, hu sjng>

GA 0 J L

GF 0 J K

G 0 J

⇑

.

GA 0 JL, GF 0 JK, AF 0 L K

G 0 J , F 0 K, A 0 L

Jstj gxdh`g ³j fheojfdch fh`h prd`jdrh fgsh cj fhekrujefdg cj trdˌ gekuihs .Hutrh chds fgshs sjrgh gprjsjetgchs g sjkudr.

1;




3.6 Sjhrj`g (1 fgsh cj fhekrujefdg cj trdgekuihs). Cgchs chds

trdgekuihs GAF j J L K, sj GA 0 J L , ˌ G 0 ˌ J j ˌ A 0 ˌ L , jetgh GAF 0 J L K.

]rhvg. Vjng` GAF j JL K chds trdgekuihs tgds quj GA 0 J L , ˌ G 0 ˌ J

j ˌ A 0 ˌ L . Vjng C u` pheth cg sj`d-rjtg V GF tgi quj GC 0 J K.

C

F

G A J

K

L

Ldkurg 3.6

Fhesdcjrj h trdgekuih GAC j h fh`pgrj fh` trdgekuih JL K. Fh`hGC 0 JK, GA 0 JL j ˌ G 0 ˌ J , fhefiu³ķ`hs, pjih gxdh`g D\, quj GAC 0JL K. Fh`h fhesjqujefdg, tj`-sj quj G ˌ AC 0 L. @gs, phr odp³htjsj, ˌ L 0G ˌ AF . Ihkh G ˌ AC 0 G ˌ AF .

Fhesjqujetj`jetj gs sj`d-rjtgs V AC j V AF fhdefdcj . @gs jetgh hpheth C fhdefdcj fh` h pheth F j, phrtgeth, fhdefdcj` hs trdgekuihs GAF

j GAC. Fh`h n³g prhvg`hs quj GAC 0 J L K, jetgh GAF 0 J L K.

3.3 Cjed¿fgh. Z` trdgekuih ³j cdth ds³ hsfjijs sj tj` chds igchs fhekru-jetjs. Jstjs igchs sgh fog`gchs cj igtjrgds , j h tjrfjdrh igch ³j fog`gch cjagsj .

3.: ]rhphsd¿fgh. J` u` trdgekuih ds³hsfjijs hs gekuihs cg agsj sghfhekrujetjs.

]rhvg. Vjng GAF u` trdgekuih j` quj GA 0 GF . ]rjtjecj-sj prhvgrquj ˌ A 0 ˌ F . ]grg dssh fh`pgrj h trdgekuih GAF fh` jij `js`h lgzjechfhrrjsphecjr hs v³jrtdfjs cg sjkudetj `gejdrg>

G ↓ G, A ↓ F j F ↓ A.

]hr odp³htjsj, GA 0 GF j GF 0 GA . Fh`hˌ G 0 ˌ G, sjkuj-sj (pjih gxdh`g D\) quj jstg fhrrj-sphecjefdg cjej u`g fhekrujefdg. Fh`h fhe-sjqujefdg tj`-sj ˌ A 0 ˌ F . A F

G

Ldkurg 3.3

1?




Fgsh h ijdthr tjeog giku`g cdfuicgcj j` sjkudr h rgfdhf³ķedh gfd`g, cjvj

cjsjeogr cugs f³hpdgs ch trdgekuih GAF j rjpjtdr h rgfdhf³ķedh pgrg jstjs chdstrdgekuih.

3.< ]rhphsd¿fgh. Vj, j` u` trdgekuih GAF , tj`-sj chds gekuihs fhe-krujetjs, jetgh h trdgekuih ³j ds³hsfjijs.

]rhvg. Vjng GAF j` trdgekuih j` quj ˌ A 0 ˌ F . \g`hs `hstrgr qujGA 0 GF . Ehvg`jetj fh`pgrj`hs h trdgekuih GAF fh` jij pr³hprdh,lgzjech fhrrjsphecjr hs v³jrtdfjs fh`h eg prhvg cg prhphsd¿fgh getjrdhr, dsth³j> G ↓ G, F ↓ F , j F ↓ A. Fh`h ˌ A 0 ˌ F j ˌ F 0 ˌ A phr odp³htjsj, jAF 0 FA, sjkuj-sj (pjih tjhrj`g(3.6)) quj sjng fhrrjsphecjefdg cjej

u`g fhekrujefdg. Fh`h fhesjqujefdg GA 0 A F .

3.; Cjed¿fgh. Vjng GAF u` trdgekuih j sjng C u` pheth cg rjtgquj fhetj` A j F . H sjk`jeth GC fog`g-sj `jcdgeg ch trdgekuih rjig-tdvg`jetj gh igch AF , sj C lhr h pheth `³jcdh cj AF . H sjk`jeth GC

fog`g-sj adssjtrdz ch gekuih ˌ G sj g sj`d-rjtg V GC cdvdcj h gekuih F ˌ GA j`chds gekuihs dkugds, dsth ³j, sj F ˌ GC 0 C ˌ GA . H sjk`jeth GC fog`g-sj gi-

turg ch trdgekuih rjigtdvg`jetj gh igch AF , sj GC lhr pjrpjecdfuigr`jetjg rjtg quj fhet³j` A j F .

Eg kurg (3.:), j` (g) GC ³j `jcdgeg, j` (a) GC ³j adssjtrdz, j j` (f)GC ³j giturg.

A

G

F

C

F

G

A

C

CF

G

A

(g) (a) (f)

Ldkurg 3.:

3.? ]rhphsd¿fgh. J` trdgekuihs ds³hsfjijs j `jcdgcg rjigtdvg`jetj gagsj ³j tgà³j` adssjtrdz j giturg.

]rhvg. Vjng GAF u` trdgekuih ds³hsfjijs fung agsj ³j GA. Vjng FC

sug `jcdgcg rjigtdvg`jetj mg agsj. Cjvj-sj prhvgr quj G ˌ FC 0 A ˌ FC j quj

14




G ˌ CF ³j u` gekuih rjth. ]grg dsth fhesdcjrj hs trdgekuihs GCF j ACF .

Fh`h GC 0 AC (n³g quj FC ³j `jcdgeg), GF 0 AF (n³g quj h trdgekuih ³jds³hsfjijs fh` agsj GA) j ˌ G 0 ˌ A (cj gfhrch fh` g prhphsd¿fgh getjrdhr), jetghGCF 0 AF C. Vjkuj sj cg³ķ quj G ˌ FC 0 A ˌ FC . G prd`jdrg dkugicgcj ehscdz quj F C ³j adssjtrdz ch gekuih G ˌ FA . Fh`h G ˌ CA ³j u` gekuih rgsh j F ˌ CG

0 A ˌ CF jetgh F ˌ CG + A ˌ CF 0 7?=◬. Fh`h n³g sgaj`hs quj F ˌ CG 0 A ˌ CF

jetgh fhefiu³ķ`hs quj F ˌ CG 0 A ˌ CF 0 4=◬ . ]hrtgeth FC ³j pjrpjecdfuigrg GA. dsth fhefiud g prhvg cg prhphsd¿fgh.

G A

F

C

Ldkurg 3.<

3.4 Sjhrj`g (6◬ fgsh cj fhekrujefdg cj trdgekuihs). Vj chdstrdgekuihs tj` trjs igchs fhrrjsphecjetjs fhekrujetjs jetgh hs trdgekuihssgh fhekrujetjs.

]rhvg. Vjng` GAF j JL K chds trdgekuihs tgds quj GA 0 J L , AF 0L K j GF 0 J K. \g`hs prhvgr quj GAF 0 J L K.

G

F

C

A

K

J L

Ldkurg 3.;

]grg dsth, fhestrug, g pgrtdr cg sj`d-rjtg V GA j eh sj`d-pigeh hphsth ghquj fhet³j` h pheth F , u` gekuih dkugi gh gekuih ˌ J . Eh igch cjstj gekuihquj egh fhet³j` h pheth A, `grquj u` pheth C tgi quj GC 0 J K j idkuj

6=




C g A. Fh`h GA 0 JL (phr odp³htjsj), GC 0 JK (phr fhestru¿fgh) j

C ˌ GA 0 J (phr fhestru¿fgh), jetgh GAC 0 JL K. \g`hs gkhrg `hstrgrquj hs trdgekuihs GAC j GAF sgh fhekrujetjs. ]gr dsth trgfj FC. Fh`hGC 0 J K 0 GF j CA 0 L K 0 A F , jetgh hs trdgekuihs GCF j ACF sghds³hsfjijs. Vjkuj-sj quj G ˌ CF 0 G ˌ FC j F ˌ CA 0 C ˌ FA j ihkh quj G ˌ CA 0G ˌ FA . @gs jetgh, pjih prd`jdrh fgsh cj fhekrujefdg cj trdgekuihs, phcj`hsfhefiudr quj GAC 0 J L K, fhefiu³ķ`hs quj GAF 0 J L K.

67




J_JQF³DFDHV

7. Cjsjeoj u` trdgekuih. Fhestrug gkhrg u` hutrh trdgekuih fhekrujetjgh quj vhfj cjsjeohu. Cjsfrjvg h prhfjcd`jeth.

1. Fhestrug u` trdgekuih GA F sgajech quj GA 0 ;.:f`, A F 0 ?.1f`j G ˌ AF 0 ?=◬. @j¿fg h fh`prd jeth cj A F j hs hutrhs gekuihs ch trdgekuih.

6. Eg kurg gh igch hs gekuihs η j α sghdkugds. @hstrj quj GF 0 A F .

η

α

F

G

A

3. Eg kurg g agdxh tj` sj GA 0 GF j AC 0 F J @hstrj quj>

G

A

F

C

J

g) G ˌ FC 0 G ˌ AJ

a) A ˌ FC 0 F ˌ AJ

:. Srjs sgrrglhs cj `gcjdrg sgh prjkgchs, chds g chds, cj `hch g lhr`gr

u` trdgekuih, fh` sh`jetj u` prjkh j` fgcg v³jrtdfj, fh`h decdfgch egkurg sjkudetj

61




G kurg gssd` hatdcg ³j r³ķkdcg8 ]hrquj8]grg fh`pgrg¿fgh fhestrug u` qugcrdi³gtjrh fh` qugtrh sgrrglhs j u`

prjkh j` fgcg v³jrtdfj. ³J jstg kurg r³ķkdcg8

<. Jxpidquj phrquj ³j usugi rjlhr¿fgr-sj u` phrtgh fh` u`g trgvj egcdgkhegi fh`h decdfgch jsquj`gtdfg`jetj eg kurg sjkudetj.

;. Eg kurg gagdxh, GF 0 GC j GA ³j g adssjtrdz ch gekuih F ˌ GC prhvjquj hs trdgekuihs GFA j GCA sgh fhekrujetjs.

G

F

A

C

?. Eg kurg gagdxh h pheth G ³j pheth `³jcdh chs sjk`jeths FA j CJ .]rhvj quj hs trdgekuihs GAC j GFJ sgh fhekrujetjs.

66




C

A

F

JG

4. Eg kurg gagdxh hs gekuihs ˌ G j ˌ F sgh rjths j h sjk`jeth CJ fhrtgFG eh pheth `³jcdh A cj FG. @hstrj quj CG 0 F J .

F G

C

J

A

7=. Cg kurg gagdxh sgaj sj quj HF 0 HA, HC 0 HG j A ˌ HC 0 F ˌ HG.@hstrj quj FC 0 A G.

C

AF

GH

77. Lg¿fg u`g cj`hestrg¿fgh cdljrjetj cg ]rhphsd¿fgh (3.:) lgzjech ushcg shiu¿fgh ch jxjrf³ķfdh 3.

63




]QHAIJ@GV

7. Eg kurg gagdxh F ˌ @G ³j u gekuih rjth j @ ³j pheth `³jcdh cj GA.@hstrj quj FG 0 F A.

G

A

F @

1. G rjkdgh `grfgcg fh` u` @ rjprjsjetg u` igkh. Cjsfrjvg u`prhfjssh pjih qugi sjr³g phssķvji `jcdr g cdstgefdg jetrj hs pheths G j A.(Wugiqujr `jcd¿fgh lhrg ch igkh ³j phssķvji)

@

G

A

F

6. @hstrj quj, sj u` trdgekuih tj` hs trjs igchs fhekrujetjs, jetgh tj`tgà³j hs trjs gekuihs fhekrujetjs.

3. Eg kurg gagdxh GA C j A FC sgh trdgekuihs ds³hsfjijs fh` agsj CA.]rhvj quj hs gekuihs G ˌ AF j G ˌ CF sgh fhekrujetjs.

F

A

G

C

:. Zsgech g `js`g kurg, (ch jxjrf³ķfdh 3), `hstrj quj tgà³j` g rjtgGF ³j adssjtrdz cj A ˌ GC j pjrpjecdfuigr g CA.

<. Eg kurg gagdxh, GA C j A FC sgh trdgekuihs ds³hsfjijs fh` agsj A C.]rhvj quj G ˌ AF 0 G ˌ CF j quj GF ³j adssjtrdz ch gekuih A FC .

6:




G

A

F

C

;. Nustdquj h sjkudetj prhfjcd`jeth pgrg cjtjr`deg¿fgh ch pheth `³jcdhcj u` sjk`jeth. ’sjng GA u` sjk`jeth. Fh` u` fh`pgssh fjetrgch j`G, cjsjeoj u` fdrfuih cj rgdh GA. Cjsfrjvg hutrh fdrfuih cj `js`h rgdhj fjetrh j` A. Jstjs chds f³ķrfuihs sj detjrfjptg` j` chds pheths. Srgfj g

rjtg idkgech jstjs chds pheths. G detjrfj¿fgh cjstg rjtg fh` h sjk`jeth GAsjr³g h pheth `³jcdh cj GA.‐

?. Eg fhestru¿fgh gfd`g ³j rjgi`jetj ejfjss³grdh quj hs chds f³ķrfuihs tje-og` rgdh GA8

4. @hstrj quj, eg fhestru¿fgh cjsfrdtg eh prhaij`g ?, gfd`g, g rjtg qujcjtjr`deg h pheth `³jcdh cj GA ³j pjrpjecdfuigr g GA.

7=. Ztdidzj g dcjdg cg fhestru¿fgh cjsfrdtg eh prhaij`g ? j prhpheog u``³jthch cj fhestru¿fgh cj u`g pjrpjecdfuigr g u`g rjtg cgcg pgssgech phr

u` pheth cjstg rjtg.

77. Eg kurg gagdxh tj`-sj GC 0 CJ , ˌ G 0 C JF j G ˌ CJ 0 A ˌ CF .@hstrj quj hs trdgekuihs GCA j JCF sgh fhekrujetjs.

C F

AJG

71. Eu` trdgekuih ds³hsfjijs GAF , fh` agsj AF , g adssjtrdz ch gekuihG ³j pjrpjecdfuigr mg agsj j ³j `jcdgeg.

6<




FH@JES ³GQDH

Lhd eg Kr³jfdg quj surkdu, pjig prd`jdrg vjz eg odst³hrdg, g kurg ch fd-jetdstg prhssdhegi. Gqujij oh`j` cjvhtgch mg ausfg cj fheojfd`jeth jrjfjajech u` sgi³grdh pgrg lgzjr dsth. Gikues chs eh`js `gds rjprjsjetg-tdvhs cjstg figssj, curgetj g fdvdidzg¿fgh krjkg, vdvjrg` j` Gijxgecrdg, hecj]thih`ju D ljz jrdkdr u` krgecj fjetrh cj pjsqudsgs cjeh`degch ’@usjh‐,fh` sug lg`hsg adaidhtjfg. Gid, g trgcd¿fgh krjkg j` fdjefdgs j Idtjrgturg lhdprjsjrvgcg j cjsjevhivdcg. H sufjssh cjssj j`prjjecd`jeth lhd fhesdcjr³gvji.

Jetrj hs prd`jdrhs pjsqudsgchrjs gsshfdgchs fh` h @usjh cj Gijxge-

crdg jst³g Jufidcjs, u` chs `gdhrjs `gtj`³gtdfhs `gds defiujetjs cj thchs hstj`phs. Jufidcjs gpgrjetj`jetj rjfjaju sug jcufg¿fgh `gtj`³gtdfg j` Gtj-egs, chs cdsf³ķpuihs cj ]igtgh, j sug prdefdpgi harg detdtuig-sj> ’Jij`jeths‐,(fh`phsth cj 16 vhiu`js). Jstj trgagioh cjvj tjr sj thregch u` fi³gssdfhihkh gp³hs sug puaidfg¿fgh. Fjrtg`jetj, cjscj hs tj`phs cj Grqud`jcjs, jijjrg fhestgetj`jetj rjljrdch j utdidzgch fh`h tjxth a³gsdfh. Gh igch cg a³ķaidg³j sj c³uvdcg h idvrh `gds rjprhcuzdch j jstucgch cj thchs hs quj n³g lhrg`jsfrdths eg odst³hrdg ch ùech hfdcjetgi. @gds cj 7.=== jcd¿fhjs cjij n³g lhrg`prhcuzdcgs cjscj g devje¿fgh cg d`prjesg j, getjs cdsth, f³hpdgs `geusfrdtgsch`degvg` thch h jesdeh cj `gtj`³gtdfg. G kjh`jtrdg jesdegcg eg jsfhigsjfuec³grdg ³j, lrjqujetj`jetj, f³hpdg qugsj idtjrgi cj ? hu 4 chs 76 vhiu`js

quj h fhestdtuj`. H pr³hprdh tjxth quj h ijdthr tj` j` `ghs fhetj` ùdtgscj`hestrg¿fhjs quj sgh, jxfjth pjig idekugkj`, pgrtj chs ’Jij`jeths‐.

Fjrtg`jetj Jufidcjs egh frdhu thcg g kjh`jtrdg fhetdcg ehs sjus ’Jij-`jeths‐.

Vju trgagioh lhd ùdth `gds gqujij cj u` fh`pdigchr, cjsjnhsh cj fhihfgrj` u` ³uedfh tjxth, trjs cgs krgecjs cjsfhajrtgs krjkgs>

g) g tjhrdg cj Juchxdh cgs prhphr¿fhjs (idvrh \).a) g tjhrdg cj Sjtjth chs drrgfdhegds (idvrh _) jf) g tjhrdg chs fdefh fhrphs rjkuigrjs quj hfupgvg iukgr cj cjstgquj eg

fhs`hihkdg cj ]igtgh.

Lhd, eh jetgeth, g gpidfg¿fgh sdstj`³gtdfg ch `³jthch cjcutdvh pgrg cjsje-vhivjr g kjh`jtrdg mg pgrtdr cj gikues lgths a³gsdfhs th`gchs fh`h gxdh`gs,quj, sj` cuvdcg, tjvj h `gdhr d`pgfth j defiujefdg sharj h jesdeh j sharj g`gejdrg cj lgzjr fdjefdg g pgrtdr cj jetgh.

6;



KJH@JSQDG JZFIDCDGEG ]IGEG H SJHQJ@G CH GEKZIH J_SJQEH J VZGV FHEVJWZJEFDGV

FG]³DSZIH :

H SJHQJ@G CH GEKZIH J_SJQEH J VZGVFHEVJWZJEFDGV

:.7 Cjed¿fgh. Vj GAF ³j u trdgekuih, hs sjus gekuihs G ˌ AF , A AG

j F ˌ GA sgh fog`gchs cj ˌ gekuihs detjrehs hu sd`pijs`jetj cj gekuihs chtrdgekuih. Hs supij`jeths cjstjs gekuihs sgh fog`gchs cj ˌ gekuihs jxtjrehs

ch trdgekuih.

FG

C

A

Ldkurg :.7

Eg kurg gfd`g h gekuih A ˌ GC ³j u gekuih jxtjreh ch trdgekuih GAFgcngfjetj gh gekuih detjreh F ˌ GA .

:.1 Sjhrj`g (gekuih jxtjreh). Shch gekuih jxtjreh cj u` trdgekuih`jcj `gds ch quj qugiqujr chs gekuihs detjrehs g jij egh gcngfjetjs.

]rhvg Vjng GAF u` trdgekuih. Eg sj`d rjtg V FG `grquj u` pheth Ctgi quj G jstjng jetrj F j C, fh`h decdfgch eg kurg :.1. Cjvj`hs prhvgquj A ˌ GC 5 ˌ A j A ˌ GC 5 ˌ F . \g`hs dedfdgi`jetj prhvgr quj A ˌ GC 5 ˌ A . ]grgdsth fhesdcjrj h pheth `³jcdh J ch sjk`jeth GA .

6?




CGF

J

A L

Ldkurg :.1

Eg sj`d rjtg V FJ , `grquj u` pheth L tgi quj FJ 0 J L . Srgfj GL .

Fh`pgrj hs trdgekuihs FJA j L GJ . Fh`h AJ 0 GJ (N³g quj J ³j hpheth `³jcdh cj GA), F J 0 JL (phr fhestru¿fgh) j A ˌ JF 0 G ˌ JL (phr sjrj`hphsths pjih v³jrtdfj), sjkuj-sj quj AJ F 0 GJ L . Fhesjqujetj`jetj ˌ A 0J ˌ GL . Fh`h g sj`d-rjtg V GL cdvdcj h gekuih A ˌ GC , jetgh J ˌ GL 9 A ˌ GC .]hrtgeth, ˌ A 9 A ˌ GC . Cjdxg`hs g fgrkh ch ijdthr g prhvg cj quj A ˌ GC 5 ˌ F .

:.6 ]rhphsd¿fgh. G sh`g cgs `jcdcgs cj qugdsqujr chds gekuihs detjr-ehs cj u` trdgekuih ³j `jehr ch quj 7?=◬.

]rhvg. Vjng GAF u` trdgekuih. \g`hs `hstrgr quj ˌ A + ˌ F 9 7?=◬.Vjng ο h gekuih jxtjreh cjstj trdgekuih fh` v³jrtdfj j` F . ]jig prhphsd¿fghgetjrdhr tj`hs quj

ο 5 ˌ A

Fh`h ο j ˌ F sgh supij`jetgrjs, jetgh ο+ ˌ F 0 7?=◬. ]hrtgeth,

ˌ A + ˌ F 0 ο + ˌ F 0 7?=◬

:.3 Fhrhi³grdh. Shchs trdgekuihs phssud pjih `jehs chds gekuihs detjr-ehs gkuchs.

]rhvg. Cj lgth, sj u` trdgekuih phssu³ķssj chds gekuihs egh gkuchs, sugsh`g sjrdg `gdhr hu dkugi g 7?=◬, h quj egh phcj hfhrrjr cj gfhrch fh` gprhphsd¿fgh getjrdhr.

:.: Fhrhi³grdh Vj cugs rjtgs cdstdetgs ` j e sgh pjrpjecdfuigrjs g u`gtjrfjdrg, jetgh ` j e egh sj detjrfjptg`.

64




]rhvg. Vj ` j e sj detjrfjp-tgssj` lhr gr-sj-dg u` trdgekuih fh`chds gekuihs rjths, h quj ³j gasurch pjihfhrhi³grdh getjrdhr.

`

e

Ldkurg :.6

:.< Cjed¿fgh. Cugs rjtgs quj egh sj detjrfjptg` sgh cdtgs pgrgijigs.G prhphsd¿fgh sjkudetj lhrejfj u` `³jthch cj fhestru¿fgh cj rjtgs pjr-

pjecdfuigrjs. Fh`h fhesjqujefdg cg prhphsd¿fgh gfd`g, jstj `³jthch phcjsjr utdidzgch pgrg fhestru¿fgh cj rjtgs pgrgijigs.

:.; ]rhphsd¿fgh. ]hr u` pheth lhrg cj u`g rjtg pgssg u`g j sh`jetju`g rjtg pjrpjecdfuigr g rjtg cgcg.

]rhvg. (Jxdstjefdg ). Vjng ` u`g rjtg j G u` pheth lhrg cjstg rjtg.Sh`j sharj ` chds pheths A j F cdstdeths. Srgfj GA. Vj GA n³g ³j pjr-pjecdfuigr g ` , tjr`deg`hs g fhestru¿fgh. Fgsh fhetr³grdh, fhesdcjrj, ehsj`d-pigeh quj egh fhet³j` G, u`g sj`d-rjtg fh` v³jrtdfj A lhr`gech fh`V AF u`g gekuih fhekrujetj g G ˌ AF . Ejstg sj`d-rjtg th`j u` pheth G

tgi quj AG 0 AG. H sjk`jeth GG ³j pjrpjecdfuigr g ` . Cj lgth, fh`hAG 0 AG, h trdgekuih GAG ³j ds³hsfjijs. Fh`h G ˌ AG— ³j ds³hsfjijs. Fh`hG ˌ AF 0 F AG, jetgh AF ³j adssjtrdz ch gekuih G ˌ AG. Vjkuj-sj, jetgh, qujAF ³j pjrpjecdfuigr g GG.

A

G

G

F

`

Ldkurg :.3

(Zedfdcgcj ) Vj jxdstdssj` cugs rjtgs cdstdetgs pgssgech pjih pheth G jsjech gàgs pjrpjecdfuigrjs g rjtg ` , lhr`gr-sj-dg u` trdgekuih fh` chdsgekuihs, rjths h quj ³j u` gasurch cj gfhrch fh` h fhrhi³grdh (:.3).

3=




G

`

Ldkurg :.:

H pheth G hatdch g pgrtdr cj G j ` eg fhestru¿fgh gfd`g (vdcj kurg(:.3)), ³j fog`gech cj rjfijxh ch pheth G rjigtdvg`jetj mg rjtg ` . H rjfijxh³j fgrgftjrdzgch pjigs sjkudetjs fhecd¿fhjs>

g) GG ³j pjrpjecdfuigr g ` , j

a) ` fhrtg GG eh sju pheth `³jcdh.

G lue¿fgh L quj gsshfdg g fgcg pheth ch pigeh, h sju rjfijxh rjigtd-

vg`jetj g u`g rjtg ` xgcg, ³j fog`gcg rjfijx˙ gh j tj` gs sjkudetjs prh-prdjcgcjs>

d) L (L (G)) 0 G pgrg thch pheth G,

dd) L (G) 0 G sj j sh`jetj sj G ³j pheth cg rjtg ` ,

ddd) L (G)L (A) 0 GA , hu sjng, L prjsjrvg g cdstgefdg jetrjpheths ch pigeh, j

dv) Vj G ∏ `, A /∏ ` j A 0 L (A) jetgh ` ³j g adssjtrdz chgekuih dt AGA.

Ldfg g fgrkh ch ijdthr g cj`hestrg¿fgh cg vgidcgcj cjstgs cugs prh-prdjcgcjs.

Cgch u` pheth G j u`g rjtg ` , g pjrpjecdfuigr g ` pgssgech phrG detjrfjptg ` j` u` pheth ] fog`gch> p³j cg pjrpjecdfuigr agdxgcg chpheth G g rjtg ` . Vj W ³j qugiqujr hutrh pheth cj ` , h sjk`jeth GW ³j cdthsjr hai³ķquh rjigtdvg`jetj g ` .

37




G

W ]

`

Ldkurg :.<

Eg kurg h sjk`jeth W] ³j fog`gch cj prhnj¿f˙ gh ch sjk`jeth WG sharjg rjtg ` . ³J u`g fhesjqujefdg cg prhphsd¿fgh sjkudetj quj WG 5 W] j qujWG 5 G] . H eu`jrh G] ³j fog`gch cj cdstgefdg ch pheth G g rjtg ` .

Cgch u` trdgekuih GAF cdrj`hs quj h igch AF hp˙ hj-sj gh ˌ gekuih ˌ Ghu, cj `gejdrg jqudvgijetj, quj h gekuih ˌ G ³j hphsth gh igch A F .

G

A

F

Ldkurg :.;

:.? ]rhphsd¿fgh Vj chds igchs cj u` trdgekuih egh sgh fhekrujetjs jetghsjus gekuihs hphsths egh sgh dkugds j h `gdhr gekuih ³j hphsth gh `gdhr igch.

]rhvg G prd`jdrg pgrtj cg prhphsd¿fgh ³j u`g fhesjqujefdg d`jcdgtgcg prhphsd¿fhjs (3.:) j (3.<). ]grg prhvgr g sjkuecg pgrtj, fhesdcjrj u`trdgekuih GA F j` quj AF 9 GF j vg`hs `hstrgr quj F ˌ GA 9 F ˌ AG.

A G

F

C

Ldkurg :.?

]grg dsth, `grquj sharj g sj`d rjtg V FG, u` pheth C tgi quj FC 0 AF .Fh`h AF 9 GF jetgh jstj pheth C pjrtjefj gh sjk`jeth GF j, fh`hfhesjqujefdg, g sj`d-rjtg V AC cdvdcj h gekuih F AG. ]hrtgeth tj`-sj

31




F ˌ AG 5 F AC .

Gkhrg hasjrvj quj

F ˌ AC 0 F ˌ CA 5 F ˌ GA

G dkugicgcj gfd`g ³j fhesjqujefdg cj FAC sjr u` trdgekuih ds³hsfjijs,j g cjsdkugicgcj hfhrrj phrquj F ˌ CA ³j gekuih jxtjreh ch trdgekuih ACG.]hrtgeth

F ˌ AG 5 F ˌ GA

fh`h qujr³ķg`hs cj`hestrgr.

:.4 ]rhphsd¿fgh Vj chds gekuihs cj u` trdgekuih egh sgh fhekrujetjsjetgh hs igchs quj sj hphj` g jstjs gekuihs tj` `jcdcgs cdstdetgs j h `gdhrigch hphj-sj gh `gdhr gekuih.

]rhvg. Ehvg`jetj gqud, g prd`jdrg pgrtj cg prhphsd¿fgh ³j u`g fhe-sjqujefdg d`jcdgtg cgs prhphsd¿fhjs (3.:) j (3.<). ]grg prhvgr g sjkuecgpgrtj, fhesdcjrj u` trdgekuih GA F j` quj F ˌ GA 9 F AG j vg`hs `hstrgr

quj AF 9 GF . Hasjrvj quj, jxdstj` trjs phssdadidcgcjs> AF 9 GF , AF 5 GF j A F 0 GF .Vj AF 5 GF jetgh, pjig prhphsd¿fgh getjrdhr, cjvjr³ķg`hs tjr F ˌ GA 5

F ˌ AG, h quj ³j fhetr³grdh g ehssg odp³htjsj. Ch `js`h `hch, sj hfhrrjssjAF 0 GF , h trdgekuih sjrdg ds³hsfjijs j F ˌ GA 0 F ˌ AG, h quj jst³g tgà³j`j` cjsgfhrch fh` ehssg odp³htjsj.

Ihkh cjvj hfhrrjr AF 9 GF , fh`h qujr³ķg`hs cj`hestrgr.

:.7= Sjhrj`g. J` thch trdgekuih, g sh`g chs fu`prd`jeths cj chdsigchs ³j `gdhr ch quj h fh`prd`jeth ch tjrfjdrh igch.

]rhvg. Cgch u` trdgekuih GAF `hstrgrj`hs quj GA + AF 5 GF .]grg dsth, `grquj u` pheth C eg sj`d-rjtg V GA, cj `hch quj GC 0GA + AF . Vjkuj-sj quj AC 0 FA j, phrtgeth, h trdgekuih AF C ³j ds³hsfjijsfh` agsj FC. Ihkh, tjrj`hs A ˌ FC 9 G ˌ FC . Vjkuj-sj quj eh trdgekuihGFC tj` sj A ˌ CF 9 G ˌ FC . Ihkh, pjig prhphsd¿fgh getjrdhr, GF 9 GC. @gsjetgh GC 9 GA + AF .

36




G

F

CA

Ldkurg :.4

:.77 Sjhrj`g (Cjsdkugicgcj trdgekuigr). Cgchs trjs pheths chpigeh, G, A j F , tj` sj quj GF ≣ GA + AF . Dkugicgcj hfhrrj sj jsh`jetj sj A pjrtjefj gh detjrvgih GF .

]rhvg. Vj G, A j F egh jstgh sharj u`g `js`g rjtg, jetgh jijs cjtjr-`deg` u` trdgekuih j g cjsdkugicgcj ³j fhesjqujefdg ch tjhrj`g getjrdhr.Vj jstgh sharj u`g `js`g rjtg, sjng` g, a j f, rjspjftdvg`jetj, gs sugsfhhrcjegcgs. Ejstj fgsh ³j sd`pijs vjrdfgr quj

|g ∑ f| ≣ |g ∑ a| + |a ∑ f|

j quj dkugicgcj hfhrrj sj j sh`jetj sj a jst³g jetrj g j f. H rjsuitgch ³j

gkhrg u`g fhesjqujefdg ch tjhrj`g (1.1).G cjsdkugicgcj trdgekuigr ³j g ³uedfg rjstrd¿fgh pgrg quj sj phssg fhestrudr

u` trdgekuih fh` u` fh`prd`jeth chs igchs pr³j-cjtjr`degchs. ]hr jxj`-pih, cj gfhrch fh` jstg cjsdkugicgcj ³j d`phss³ķvji fhestrudr-sj u` trdgekuihfunhs igchs sjng` :, 6 j 4.

:.71 ]rhphsd¿fgh. Vjng` g, a ≣ f trjs e³u`jrhs phsdtdvhs. Vj f 9 g + ajetgh phcj`hs fhestrudr u` trdgekuih funhs igchs `jcj` g, a j f.

]rhvg. Srgfj u`g rjtg sharj jig j `grquj chds pheths G j A tgds qujGA 0 f. Fh` u` fh`pgssh cjsfrjvg u` fdrfuih cj fjetrh G j rgdh a, j u`fdrfuih cj fjetrh A j rgdh g.

33




G A

F

Ldkurg :.7=

Fh`h f 9 g + a, hs chds f³ķrfuihs sj detjrfjptg`. Fog`j qugdsqujr chspheths cg detjrfjssgh cj F . H trdgekuih GAF tj` igchs `jcdech g, a j f

fh`h cjsjngch.\g`hs gkhrg gpidfgr g cjsdkugicgcj trdgekuigr pgrg rjshivjr h sjkudetj

prhaij`g> V˙ gh cgchs chds pheths G j A lhrg cj u`g rjtg `. Cjtjr`degr

u` pheth ] sharj g rjtg ` tgi quj G] + ] A sjng g `jehr phss³ķvji .Dedfdgi`jetj vg`hs suphr quj G j A jstjng` j` sj`d pigehs cdstdeths

rjigtdvg`jetj g rjtg ` eu` pheth ] . Gr`h quj jstj pheth ³j g shiu¿fghch ehssh prhaij`g. Cj lgth, sj ] j qugiqujr pheth cj `, jetgh, pjigcjsdkugicgcj trdgekuigr, tjrj`hs> G] + ] A ≪ GA, hfhrrjech dkugicgcjsj j sh`jetj sj ] 0 ] .

]

]

G

A

`

Ldkurg :.77

Eh fgsh j` quj G j A jstgh j` u` `js`h sj`d-pigeh, sjng A h rjfijxhch pheth A rjigtdvg`jetj g rjtg `. Vj ] ³j qugiqujr pheth cj `, jetgh] A 0 ] A. Fhesjqujetj`jetj, tjrj`hs quj

G] + ] A 0 G] + ] A.

Gssd`, h prhaij`g rjcuz-sj gh fgsh getjrdhr j g shiu¿fgh ³j h pheth ] hatdch pjig detjrfjssgh cj ` fh` h sjk`jeth GA .

3:




`

]

A

A

G

Ldkurg :.71

³J detjrjssgetj hasjrvgr quj jstj prhaij`g surkj j` L³ķsdfg, qugech sjtjetg cjtjr`degr u` pheth ] , sharj u` jspjioh, hecj cjvj hfhrrjr g rjfijxghcj u` rgdh cj iuz quj vgd ch pheth G gh pheth A, rjfijtdech-sj eh jspjioh.Hu qugech tjetg-sj cjtjr`degr u` pheth hecj u`g ahig cj adiogr cjvjfohfgr-sj fh` g igtjrgi cg `jsg, pgrg dr ch pheth G, thfgr eg igtjrgi jgtdekdr u`g ahig quj sj jefhetrg eh pheth A .

\g`hs gkhrg gpidfgr hs rjsuitgchs n³g hatdchs pgrg jstucgr u`g figssjjspjfdgi cj trdgekuihs.

:.76 Cjed¿fgh> Z` trdgekuih quj phssud u` gekuih rjth ³j fog`gchtrdˌ gekuih rjtˌ gekuih. H igch hphsth gh gekuih rjth ³j fog`gch odphtjeusg , jhs hutrhs chds igchs sgh cjtjr`degchs fgtjths .

Cj gfhrch fh` (:.3) hs gekuihs jxphsths ghs fgtjths sgh gkuchs. ³J u`gfhesjqujefdg cj (:.4) quj g odphtjeusg ³j `gdhr ch quj qugiqujr chs fgtjths.]hr hutrh igch, pjig cjsdkugicgcj trdgekuigr, h fh`prd`jeth cg odphtjeusg³j `jehr ch quj g sh`g chs fh`prd jeths chs fgtjths. Vj chds trdgekuihsrjtgekuihs sgh fhekrujetjs, jetgh, ejfjssgrdg`jetj, hs gekuihs rjths cjvj`sj fhrrjsphecjr.

G

FGSJSH

FGSJSH

OD]HSJEZVG FA

Ldkurg :.76

3<




]hr fgusg cdsth, gi³j` chs trjs fgshs cj fhekrujefdg quj n³g fheojfj`hs,

jxdstj` hutrhs trjs jspjf³ķfhs pgrg trdgekuihs rjtgekuihs. Jstjs sgh gprj-sjetgchs eh tjhrj`g sjkudetj.

:.73 Sjhrj`g (Fhekrujefdg cj trdgekuihs rjtgekuihs). Vjng`GAF j GAF chds trdgekuihs rjtgekuihs funhs gekuihs rjths sgh ˌ F j ˌ F . Vjgiku`g cgs fhecd¿fhjs gagdxh hfhrrjr, jetgh chds trdgekuihs sgh fhekrujetjs>

7)AF 0 AF j ˌ G 0 ˌ G

1)GA 0 GA j A F 0 AF , j

6)GA 0 GA j ˌ G 0 ˌ G

Hs fgshs gfd`g phcj` sjr dcjetdfgchs fh`h dkugicgcj jetrj

7) (f » g) fgtjth j gekuih hphsth,

1) (o » f) odphtjeusg j fgtjth, j

6) (o » g) odphtjeusg j gekuih gkuch.

]rhvg. (Fgsh 7) Ehssg s odp³htjsjs sgh, ejstj fgsh, gs sjkudetjs>

ˌ F 0 ˌ F (rjth), AF 0 AF ˌ G 0 ˌ G.

Hasjrvj quj, gpjsgr cj tjr`hs delhr`g¿fhjs sharj chds gekuihs j u` igch,egh phcj`hs gpidfgr h ’1◬ fgsh cj fhekrujefdg‐. ]grg prhvgr quj GAF jGAF sgh fhekrujetjs `grquj u` pheth C sharj g sjd-rjtg V FG cj shrtj

quj FC 0 F

G

. Hs trdgekuihs F CA j F

G

A

sgh jetgh fhekrujetjs, pjihprd`jdrh fgsh cj fhekrujefdg.

3;




F

AG

C

F

AG

Ldkurg :.73

Fh`h fhesjqujefdg, tj` sj quj F ˌ CA 0 ˌ G. Cjscj quj F ˌ GA 0 ˌ G (phrodp³htjsj), fhefiu³ķ`hs quj

F ˌ CA 0 F ˌ GA .

Gr`h quj hs pheths G j C fhdefdcj . Cj lgth, sj tgi egh hfhrrjr G,C j A lhr`g` u` trdgekuih j` quj hs gekuihs F ˌ CA j F ˌ GA sgh gekuihsjxtjrehs j detjrehs egh gcngfjetj. ]hrtgeth, g dkugicgcj gfd`g egh phcjhfhrrjr cj gfhrch fh` h tjhrj`g ch gekuih jxtjreh. Jetgh G j C fhdefdcj`

j ihkh FGA 0 FCA. Fh`h FCA 0 F GA fhefiud-sj quj FGA 0 F GA,fh`h qujr³ķg`hs cj`hestrgr.

Gs cj`hestrg¿fhjs chs hutrhs chds fgshs sgh cjdxgcgs g fgrkh ch ijdthr.

3?




J_JQF³DFDHV

7. ]rhvj quj, sj u` trdgekuih tj` chds gekuihs jxtjrehs dkugds, jetghjij ³j ds³hsfjijs.

1. G kurg gh igch ³jlhr`gcg pjihs sjk`je-ths GF , GJ , F L j J A.Cjtjr dej hs gekuihsquj sgh>

g) `jehrjs quj h gekuih;.a) `gdhrjs ch quj hgekuih :, jf) `jehrjs ch quj hgekuih 3.

JF

A

G

L

7=

7

1

6:

3

4

?

<

;

C

6. Eg kurg gagdxh hs gekuihs jx-tjrehs G ˌ FJ j G ˌ AC sgtdslgzj` g cj-

sdkugicgcj> G ˌ FJ 9 G ˌ AC . @hstrjquj G ˌ AC 5 G ˌ AF .

G

A F C J

3. ]rhvj quj u` trdgekuih rjtgekuih tj` chds gekuihs jxtjrehs hatushs.

:. Eg kurg gagdxh, A, C j G sgh fhi-dejgrjs. Ch `js`h `hch C, J j F sgh

fhidejgrjs. @hstrj quj Gˌ JF 5 C

ˌ AF

G

A

F

CJ

<. J` u` fgrt³hrdh cj rjkdstrh cj d`³hvjds u` jsfrdvgh rjfushu sj gtrgesfrjvjr h rjkdstrh cj u` tjrrjeh trdgekuigr funhs igchs, sjkuech h sjuprhprdjt³grdh `jcdg` 7==`, <=` j 1=`. \hfj phcj cgr u` grku`jeth quj

34






]QHAIJ@GV

7. G kurg gh i gchlhd fhpdgcg cj u` idvrhphr u`g frdge¿fg. Gs`jcdcgs chs gekuihs de-cdfgcgs sgh gs `jcd-cgs fhrrjtgs ch cj-sjeoh hrdkdegi. Fhàgsj ejstg delhr`g¿fgh,

rjsphecg gs sjkudetjsqujsthjs rjigtdvgs gh cj-sjeoh hrdkdegi.

g) Hs trdgekuihs GAF jCFA sgh fhekrujetjs8

a)Wugi h igch chtrdgekuih GAF ³j `gdsihekh8

f) Wugi h igch ch

trdgekuih CFA quj ³j`gds furth8

G

F

C<=◬

A

<=◬

<:◬

::◬ <:◬

::◬::◬

1. Vj, eh prhaij`ggetjrdhr, hs gekuihstdvjssj` sdch decdfgchsfh`h eg kurg ghigch, qugds sjrdg` gsrjsphstgs mgs pjrkuetgsg, a j f gfd`g8

G

F

C<=◬

A

<:◬

<=◬

::◬

::◬

<:◬:◬

:7




6. Eg kurg gh igch tj`-sjAC 5 A F j ˌ G 5 G ˌ AF . ]rhvjquj AC 5 GF .

:◬

G

A

F

C

3. Eg kurg gh igch O lhd jsfhiodch ehsjk`jeth L K cj shrtj quj JO 0 JK.@hstrj quj 7 5 1.

:◬

L

J K

O7 1

:. Vj u` trdgekuih GA F ³j jqudi³gtjrh j C ³j u` pheth ch sjk`jeth AF `hstrj quj GC 5 CA.

<. Eg kurg gh igch ` j e sgh cugsrjtgs pjrpjecdfuigrjs. Wugi h fg`-deoh `gds furth pgrg sj dr ch phethG gh pheth A thfgech-sj egs cugsrjtgs8

`

]

W e

G

A

;. Eg kurg gh igch 7 0 1.@hstrj quj gs rjtgs ` j e sghpgrgijigs.

7

1

`

e

?. Cjtjr`dej h sjk`jeth `gds furth idkgech u` pheth G g u` pheth cj

:1




u`g rjtg `.

4. Eh trdgekuih GAF cg kurg gh igchtj`-sj F C pjrpjecdfuigr g GA , A J pjr-pjecdfuigr g GF j FC 0 AJ . @hstrj qujGAF ³j u trdgekuih ds³hsfjijs.

A F

G

C J

7=. Eg kurg gh igch GAF ³j u`trdgekuih jqudi³gtjrh j GC 0 AJ + FL .Vj, gi³j` cdssh, C ˌ GA 0 J AF , `hstrjquj J L C ³j tgà³j jqudi³gtjrh.

F

G A

J

L C

77. Zsgech g `js`g kurg j suphech gpjegs quj GAF ³j jqudi³gtjrh j

quj GC 0 AJ 0 FL , `hstrj quj J L C ³j tgà³j jqudi³gtjrh.

71. Eg kurg gh igch GC 0 AF , G ˌ CF

j A ˌ FC sgh gekuihs rjths, j @ j E sghpheths `³jcdhs chs sjk`jeths GA j CF rjspjftdvg`jetj. @hstrj quj @ E ³j pjr-pjecdfuigr g GA j g FC.

E

@G

C

A

F

76. Cj`hestrj hs fgshs (1) j (6) cg prhphsd¿fgh (:.76).

73. Vjng` GAF j GAF chds trdgekuihs egh rjtgekuihs fh` ˌ F 0 ˌ F ,GA 0 GA j AF 0 AF . Cj u` jxj`pih pgrg `hstrgr quj jstgs odp³htjsjsegh gfgrrjtg` quj hs trdgekuihs cjvg` sjr fhekrujetjs.

:6




FH@JES ³GQDH

Wugech sj cjsjng `hstrgr u`g prhphsd¿fgh, rjshivjr u` jxjrf³ķfdh hu sd`-pijs`jetj jetjecjr h jeuefdgch cj u` tjhrj`g, ³j ùdth d`phrtgetj qujsjng`hs fgpgzjs cj sjpgrgr gs odp³htjsjs ch quj sj cjsjng prhvgr (tjsj hufhefiusgh). Jsquj`gtdfg`jetj h jeuefdgch cj u`g prhphsd¿fgh (hu tjhrj`g,hu fhrhi³grdh, hu prhaij`g, hu jxjrf³ķfdh jtf.) phcj sjr sj`prj rjprjsjetgchphr

] ↑ W (ijdg> ] d`pidfg W)

hecj ] j W rjprjsjetg` gqud cugs gr`g¿fhjs. G gr`g¿fgh ] ³j godp³htjsj j g gr`g¿fgh W ³j g tjsj. J` ùdths fgshs, g odp³htjsj vj` prj-fjcdcg cj u` ’sj‐ hu u` ’qugech‐, j g tjsj cj u` ’jetgh‐. Z` jxj`pihcdsth ³j g sjkudetj prhphsd¿fgh>

’Vj cugs rjtgs cdstdetgs phssuj` u`g pjrpjecdfuigr fhù`, jetgh jigsegh sj detjrfjptg`.‐

Ejstj jeuefdgch tj`hs

Odp³htjsj> Cugs rjtgs cdstdetgs phssuj` u`g pjrpjecdfuigr fhù`.Sjsj> Gs cugs rjtgs egh sj detjrfjptg`.

Jvdcjetj`jetj, ej` thchs hs jeuefdgchs cj prhphsd¿fhjs jstgh gprjsje-tgchs eh lhr`gth ’sj..., jetgh..‐. ]hr jxj`pih, g `js`g prhphsd¿fgh phcjrdgtjr sdch jeuefdgcg cg sjkudetj `gejdrg>

’Qjtgs pjrpjecdfuigrjs g u`g tjrfjdrg egh sj jefhetrg`‐.

Gkhrg, g odp³htjsj jst³g cdslgr¿fgcg eh pjcg¿fh cj lrgsj ’Qjtgs pjrpje-cdfuigrjs g u`g tjrfjdrg...‐, j g tjsj, eh rjstgetj. ]jih `jehs qugtrhprhphsd¿fhjs fhestgetjs cjstj fgp³ķtuih ehs cgh jxj`pihs cj jeuefdgchs cjstjtdph. Wugech jefhetrg`hs cdfuicgcj j` rjfheojfjr g odp³htjsj j g tjsj cju` cgch jeuefdgch, ³j sj`prj u`g ahg phidtdfg tjetgr rjsfrjvj-ih eh lhr`gth’sj..., jetgh...‐. ]hr jxj`pih, g prhphsd¿fgh (3.?) tj` h sjkudetj jeuefdgch>

:3




’J` u` trdgekuih ds³hsfjijs g `jcdgeg rjigtdvg`jetj mg agsj ³j tgà³j`

adssjtrdz j giturg.‐

Z`g `gejdrg cj rjsfrjvj-ig eh lhr`gth ’sj..., jetgh...‐ ³j h sjkudetj>

’Vj GA F ³j u trdgekuih ds³hsfjijs fh` agsj AF j C ³j h pheth `³jcdh cjAF , jetgh GC ³j adssjtrdz ch gekuih A ˌ GF j ³j pjrpjecdfuigr gh igch A F .‐

Jàhrg jstjs chds jeuefdgchs sjng` jxtrj`g`jetj cdljrjetjs, jijs ehs

cdzj` jxgtg`jetj g `js`g fhdsg. H prd`jdrh ³j sj` cuvdcg `gds jijkgetj,`gs h sjkuech ³j h jeuefdgch fh` quj rjgi`jetj trgagiog`hs qugech cj`he-strg`hs jstg prhphsd¿fgh.

Fhesdcjrj`hs gkhrg gs cugs prhphsd¿fhjs sjkudetjs>

g) ’ Vj u` trdgekuih ³j ds³hsfjijs, jetgh jij phssud chds gekuihsdkugds‐.

a) ’Vj chds gekuihs cj u` trdgekuih sgh dkugds, jetgh h trdgekuih³j ds³hsfjijs‐.

Hasjrvj quj g odp³htjsj cg prd`jdrg ³j g tjsj cg sjkuecg j quj g tjsj cgprd`jdrg ³j g odp³htjsj cg sjkuecg. J` tjr`hs jsquj`³gtdfhs, sj

] ↑ W

rjprjsjetg g prhphsd¿fgh (g), jetgh

W ↑ ]

rjprjsjetg g prhphsd¿fgh (a). G sjkuecg prhphsd¿fgh ³j cdtg sjr g devjrsg

cg prd jdrg. Fgcg prhphsd¿fgh tj` sj`prj u`g devjrsg g qugi phcj sjrvjrcgcjdrg hu egh. Jxj`pihs>

7 d.) Vj cugs rjtgs phssuj` u`g pjrpjecdfuigr fhù` jetgh jigs sghrjtgs pgrgijigs.

::




7 dd.) Vj cugs rjtgs sgh pgrgijigs, jetgh phssuj` u`g pjrpjecdfuigr

fhù`.

1. d) Vj u` trdgekuih ³j rjtgekuih, jetgh jij phssud chds gekuihs gku-chs.

1 dd) Vj u` trdgekuih phssud chds gekuihs gkuchs, jetgh jij ³j u`trdgekuih rjtgekuih.

G prhphsd¿fgh (7.d) lhd cj`hestrgcg ejstj fgpdtuih. Vug devjrsg, g prhphsd¿fgh(7.dd), ³j vjrcgcjdrg. `gs sug cj`hestrg¿fgh ejstj e³ķvji ch fursh sjrdg ùdthfh`pidfgcg. G prhphsd¿fgh (1.d) tgà³j` lhd cj`hestrgcg ejstj fgp³ķtuih. Vug

devjrsg, g prhphsd¿fgh (1.dd), ³j havdg`jetj lgisg.Wugech hfhrrj quj gs prhphsd¿fhjs ] ↑ W j W ↑ ] sgh sdùitgejg`jetj

vjrcgcjdrgs, cdzj`hs quj ] j W sgh gr`g¿fhjs jqudvgijetjs, j rjprjsjetg`hsjsquj`gtdfg`jetj dsth phr

] ↓ W (] ³j jqudvgijetj g W)

Eh jeuefdgch cj tjhrj`gs jstgajijfjech quj gs cugs gr`g¿fhjs sghjqudvgijetjs ³j fhù` quj sj usj h lhr`gth ’...sj j sh`jetj sj...‐. ]hrjxj`pih>

’Chds igchs cj u` trdgekuih sgh fhekrujetjs sj j sh`jetj sj chds cj sjusgekuihs sgh fhekrujetjs‐.

Hutrhs lhr`gths fhùes sgh ’... sj j s³h sj...‐, ‐...³j fhecd¿fgh ejfjss³grdgj sufdjetj pgrg...‐ j ’...³j jqudvgijetj g...‐.

Fhesdcjrj gkhrg gs cugs sjkudetjs prhphsd¿fhjs>

D) Vj chds igchs cj u` trdgekuih sgh fhekrujetjs jetgh sjus gekuihshphsths sgh tgà³j` fhekrujetjs.

DD) Vj chds gekuihs cj u` trdgekuih egh sgh fhekrujetjs, jetgh hsigchs quj sj hphj` g jstjs gekuihs tgà³j` egh sgh fhekrujetjs.

Gs prhphsd¿fhjs gfd`g sgh rjigfdhegcgs cg sjkudetj `gejdrg. Vj rjp-rjsjetgr`hs phr ] g ejkg¿fgh cg gr`g¿fgh ], jetgh jsquj`gtdfg`jetj gsprhphsd¿fhjs gfd`g phcj` sjr rjprjsjetgcgs phr

:<




D ) ] ↑ W

DD) W ↑ ]

Fog`g`hs g sjkuecg prhphsd¿fgh cj ejkgtdvg cg prd`jdrg. Z` lgth sd`-pijs cj i³hkdfg, j jxtrj`g`jetj ³utdi, ³j quj u`g prhphsd¿fgh j sug ejkgtdvg sghsj`prj sdùitgejg`jetj vjrcgcjdrgs hu sdùitgejg`jetj lgisgs. ]hr dssh ³jjqudvgijetj cj`hestrgr-sj qugiqujr u`g cgs cugs. Cjvj-sj, eh jetgeth,hasjrvgr quj, h trgagioh pgrg cj`hestrg¿fgh cj u`g cjigs phcj sjr `jehsfh`pidfgcg ch quj h trgagioh pgrg sj cj`hestrgr cdrjtg`jetj g hutrg. Hijdthr cjvj hasjrvgr quj g prhphsd¿fgh D gfd`g ³j jxgtg`jetj g prhphsd¿fgh(3.:) j quj DD ³j g prd`jdrg pgrtj cg prhphsd¿fgh (:.?).

:;



KJH@JSQDG JZFIDCDGEG ]IGEG H G_DH@G CGV ]GQGIJIGV

FG]³DSZIH <

H G_DH@G CGV ]GQGIJIGV

G jxdstjefdg cj rjtgs pgrgijigs ³j u`g fhesjqujefdg chs phstuigchs n³ggprjsjetgchs. H Fhrhi³grdh (:.:), gi³j` cj kgrgetdr tgi jxdstjefdg, lhrejfju` `³jthch cj, jljtdvg`jetj, cjsjeogr-sj rjtgs pgrgijigs. H gxdh`g qujgprjsjetg`hs g sjkudr cdz, jssjefdgi`jetj, quj cugs rjtgs pgrgijigs g u`g

tjrfjdrg j fh` u` pheth j` fhù` sgh fhdefdcjetjs.

Gxdh`g \. ]hr u` pheth lhrg cj u`g rjtg ` phcj sj trg¿fgr u`g ³uedfgrjtg pgrgijig g rjtg `.

Cjvj-sj hasjrvgr quj jstj gxdh`g prjsfrjvj g uedfdcgcj, n³g quj g jx-dstjefdg cj rjtg pgrgijig g `, pgssgech phr u` pheth cgch, n³g jrg kgrgetdchphr (:.:). Fh`h fhesjqujefdg d`jcdgtg cjstj gxdh`g tj`-sj>

<.7 ]rhphsd¿fgh. Vj g rjtg ` ³j pgrgijig mgs rjtgs e7 j e1, jetgh e7 j e1

sgh pgrgijigs hu fhdefdcjetjs.

]rhvg. Vupheog quj e7 j e1 egh fhdefdcg` j sgh pgrgijigs g rjtg `.Vj e7 j e1 egh lhssj` pgrgijigs jetrj sd, jigs tjrdg` u` pheth cj detjrsj¿fgh,cdkg`hs, ] . @gs jetgh e7 j e1 sjrdg` cdstdetgs pgrgijigs mg rjtg ` pgssgechphr ] . Dsth fhetrgcdz h gxdh`g \. Ihkh e7 j e1 sgh pgrgijigs.

<.1 Fhrhi³grdh. Vj u`g rjtg fhrtg u`g cj cugs pgrgijigs, jetgh fhrtgtgà³j hutrg.

]rhvg. Vjng` e7 j e1 rjtgs pgrgijigs. Vj u`g rjtg ` fhrtgssj e7 j eghfhrtgssj e1, jetgh ` j e1 sjrdg` pgrgijigs. Gssd` e1 sjrdg pgrgijig g ` j g

e7. Fh`h ` j e7 egh sgh pgrgijigs jetrj sd ej` fhdefdcjetjs, tj`hs u`gfhetrgcd¿fgh fh` g prhphsd¿fgh getjrdhr. Ihkh ` fhrtg tgà³j` e1.G ehssg cjed¿fgh cj rjtgs pgrgijigs egh ³j tgh sd`pijs cj usgr fh`h

gpgrjetg. Cjscj quj rjtgs sgh deedtgs j` fh`prd`jeth, fh`h phcjrj`hsprhvgr quj cugs rjtgs egh sj detjrfjptg`8 ]hr jxj`pih, gs rjtgs ` j e cgkurg gagdxh pgrjfj` sjr pgrgijigs. Fh`h cjfdcdr sj jigs e gh sj jefhetrg`j` giku` pheth ch pigeh ùdth cdstgetj cj G j A 8

:?




`

e

A

G 7

1

Ldkurg <.7

Z`g `gejdrg ùdth sd`pijs cj rjsphecjr g jstg pjrkuetg ³j gtrgv³js cgfh`pgrg¿fgh chs gekuihs 7 j 1, decdfgchs eg kurg, lhr`gchs pjigs cugspgrgijigs fh` g rjtg quj pgssg phr G j A .

<.6 ]rhphsd¿fgh. Vjng` `, e, 7 j 1 fh`h eg kurg (<.7) Vj 7 0 1, jetghgs rjtgs ` j e sgh pgrgijigs.

]rhvg. Cj lgth, sj ` detjrfjptgssj e j` giku` pheth ] , fh`h decdfgcheg p³gkdeg sjkudetj, lhr`gr-sj-dg u` trdgekuih GA] . Ejstj trdgekuih 7 ³jgekuih jxtjreh j 1 ³j u` gekuih detjreh egh gcngfjetj gh gekuih 7, hu vdfj-vjrsg. Gssd , pjih tjhrj`g ch gekuih jxtjreh tjr³ķg`hs 7 0 1 h quj fhetrgcdz

ehssg odp³htjsj. ]hrtgeth, ` j e egh sj detjrfjptg .

G

A

1

7

] ]

G

7

1

A

Ldkurg <.1

Wugech cugs rjtgs sgh fhrtgcgs phr u`g trgesvjrsgi lhr`g` sj hdthgekuihs fh`h decdfgchs eg kurg gagdxh. Wugtrh cjijs sgh fhrrjsphecjetjsghs hutrhs qugtrh, g sgajr

7 ↓ 16 ↓ 3: ↓ <; ↓ ?

:4




7:6;

< 13?

Ldkurg <.6

Hasjrvj quj 7 0 ;, 1 0 ?, 6 0 : j 3 0 < phr sjrj` hphsths pjih v³jrtdfj.Fh`h fhesjqujefdg, sj 7 0 1 jetgh thchs hs hutrhs pgrjs cj gekuihs fhrrj-sphecjetjs sjrgh dkugds. Gi³j` cdssh, tjrj`hs quj 6 + 1 0 7?=◬. Devjrsg-`jetj, sj 6 + 1 0 7?=◬ jetgh 7 0 1. Jstgs hasjrvg¿fhjs pjr`dtj` rjjsfrjvjrg prhphsd¿fgh (<.6) cj cugs `gejdrgs cdstdetgs.

<.6. ]rhphsd¿fgh G. Vj, gh fhrtgr`hs cugs rjtgs fh` u`g trgesvjrsgi,hatdvjr`hs 6 + 1 0 7?=◬ jetgh gs rjtgs sgh pgrgijigs.

<.6 ]rhphsd¿fgh A. Vj, gh fhrtgr`hs cugs rjtgs fh` u`g trgesvjrsgi,h gekuih fhrrjsphecjetjs sgh dkugds, jetgh gs rjtgs sgh pgrgijigs.

H gxdh`g \ pjr`dtj-ehs `hstrgr quj g devjrsg cjstg prhphsd¿fgh ³j tg a³j`vjrcgcjdrg.

<.3 ]rhphsd¿fgh. Vj cugs rjtgs pgrgijigs sgh fhrtgcgs phr u`g trgesvjr-sgi, jetgh hs gekuihs fhrrjsphecjetjs sgh dkugds.

]rhvg. Vjng` ` j ` cugs rjtgs pgrgijigs j sjng e u`g rjtg quj fhrtg `j ` ehs pheths G j A , rjspjftdvg`jetj. Fhesdcjrj u`g rjtg ` pgssgechpjih pheth G j lhr`gech fh` g trgesvjrsg qugtrh gekuihs fhrrjsphecjetjslhr`gchs pjig rjtg ` fh` g `js`g trgesvjrsgi. Cj gfhrch fh` g prhphsd¿fghgetjrdhr ` j ` sgh pgrgijigs. Cj gfhrch fh` g prhphsd¿fgh (<.7) ` j` sgh fhdefdcjetjs. ]hrtgeth, ` lhr`g gekuihs fh` g rjtg e dkugds ghs

fhrrjsphecjetjs lhr`gchs phr `

fh` g rjtg e.

`G`

`

Ae

Ldkurg <.3

<=




\g`hs gkhrg gprjsjetgr cugs fhesjqujefdgs d`phrtgetjs ch gxdh`g \.

<.: Sjhrj`g. G sh`g chs gekuihs detjrehs cj u` trdgekuih ³j 7?=◬.

]rhvg. Vjng AF u` trdgekuih. ]jih v³jrtdfj F trgfj u`g rjtg pgrgijiggh igch GA . Eu`jrj hs gekuihs lhr`gchs fh` v³jrtdfj F , fh`h decdfgch egkurg sjkudetj

F

71

6

G A

Ldkurg <.:

Sj`-sj 7 + 1 + 6 0 7?=◬. Fh`h GF ³j trgesvjrsgi mgs cugs pgrgijigs,³j u`g fhesjqujefdg cdrjtg cg prhphsd¿fgh getjrdhr quj 7 0 ˌ G. Fh`h AF ³jtgà³j` trgesvjrsgi mgs cugs pgrgijigs, jetgh 6 0 ˌ A . ]hrtgeth

ˌ G + ˌ A +G ˌ FA 0 ˌ 7 + ˌ 6 + ˌ 1 0 7?= ◬

G prhphsd¿fgh sjkudetj rjigfdheg u`g s³jrdj cj fhrhi³grdhs d`jcdgths cjstjtjhrj`g.

<.< Fhrhi³grdh.

g) G sh`g cgs `jcdcgs chs gekuihs gkuchs cj u` trdgekuih rjtgekuih ³j4=◬.

a) Fgcg gekuih cj u` trdgekuih jqudi³gtjrh `jcj <=◬.f) G `jcdcg cj u` gekuih jxtjreh cj u` trdgekuih ³j dkugi g sh`g cgs

`jcdcgs chs gekuihs detjrehs quj egh ioj sgh gcngfjetjs.

c) G sh`g chs gekuihs detjrehs cj u` qugcrdi³gtjrh ³j 6<=◬

.G prhvg cjstj fhrhi³grdh ³j cjdxgcg g fgrkh ch ijdthr. H tjhrj`g sjkudetj

ehs cdz quj rjtgs pgrgijigs sgh jqudcdstgetjs.

<.; Sjhrj`g. Vj ` j e sgh rjtgs pgrgijigs, jetgh thchs hs pheths cj` jstgh mg `js`g cdstgefdg cg rjtg e.

]rhvg. Vjng` ` j e rjtgs pgrgijigs. Vharj ` th`j chds pheths G j G,j cjijs agdxj pjrpjecdfuigrjs mg rjtg e. Vjng` A j A rjspjftdvg`jetj hs p³js

<7




cjstgs pjrpjecdfuigrjs. Cjvj`hs prhvgr quj GA 0 GA. ]grg dssh trgfj

G

A fh`h decdfgch eg kurg sjkudetj.

G G

AA

Ldkurg <.<

Hasjrvj quj G ˌ GA 0 G ˌ AA j quj G— ˌ GA 0 4=◬. Dsth ³j u`g cjfhrrjefdgcj quj ` j e sgh pgrgijigs j cg gpidfg¿fgh cg prhphsd¿fgh (<.3) gh fhesdcjrgr-sj GA j GA fh`h trgesvjrsgds. ]hrtgeth, hs trdgekuihs GGA j A AG sghtrdgekuihs rjtgekuihs fh` u` gekuih gkuch j odphtjeusg (fhù`) dkugds.Vjkuj-sj chs tjhrj`g (:.73) quj jijs sgh fhekrujetjs. G fhekrujefdg ³j gquj ijvg G j` A , G j` A j A j` G. Ihkh GA 0 GA, fh`h qujr³ķg`hscj`hestrgr.

G devjrsg cjstj tjhrj`g ³j tgà³j` vjrcgcjdrg j sug cj`hestrg¿fgh ³j

prhphstg fh`h jxjrf³ķfdh < cjstj fgp³ķtuih.

<.? Cjed¿fgh. Z` pgrgijihkrg`h ³j u qugcrdi³gtjrh funhs igchs hphs-ths sgh pgrgijihs.

G A

FC

Ldkurg <.;

<.4 ]rhphsd¿fgh. J` u` pgrgijihkrg`h igchs j gekuihs hphsths sghfhekrujetjs.

]rhvg. Vjng GAFC u` pgrgijihkrg`h. Srgfj g cdgkhegi GF . Fh`hGA j CF sgh pgrgijihs, jetgh A ˌ GF 0 G ˌ FC . Fh`h, gi³j` cdssh, GF ³j fhù`

<1




ghs trdgekuihs GAF j FCG, jetgh jstjs trdgekuihs sgh fhekrujetjs. Ihkh

ˌ A 0 ˌ C , GA 0 F C j A F 0 CG. ³J gkhrg l³gfdi vjr quj ˌ G 0 ˌ F .

<.7= ]rhphsd¿fgh. Gs cdgkhegds cj u` pgrgijihkrg`h sj detjrfjptg`j` u` pheth quj ³j pheth `³jcdh cgs cugs cdgkhegds.

G prhvg cjstgs prhphsd¿fgh ³j sd`pijs j ³j cjdxgcg g fgrkh ch ijdthr. Gs cugsprhphsd¿fhjs sjkudetjs cgh fhecd¿fhjs sufdjetjs pgrg quj u` qugcrdi³gtjrh sjngu` pgrgijihkrg`h.

<.77 ]rhphsd¿fgh. Vj hs igchs hphsths cj u` qugcrdi³gtjrh sgh fhekru-jetjs jetgh h qugcrdi³gtjrh ³j u` pgrgijihkrg`h.

]rhvg. Vjng GAFC u` qugcrdi³gtjrh j` quj GA 0 F C j AF 0 GC.Srgfj g cdgkhegi AC ch qugcrdi³gtjrh. Hs trdgekuihs GAC j FCA sghfhekrujetjs cj gfhrch fh` h tjrfjdrh fgsh cj fhekrujefdg cj trdgekuihs.Ihkh F ˌ AC 0 A ˌ CG j F ˌ CA 0 C ˌ AG. G prd`jdrg dkugicgcj kgrgetj quj AF jGC sgh pgrgijihs, g sjkuecg kgrgetj quj FC j AG tgà³j sgh pgrgijihs.Ihkh GAFC ³j u` pgrgijihkrg`h.

A F

CG

Ldkurg <.?

<.71 ]rhphsd¿fgh. Vj chds igchs hphsths cj u` qugcrdi³gtjrh sgh fhe-krujetjs j pgrgijihs, jetgh h qugcrdi³gtjrh ³j u` pgrgijihkrg`h.

G prhvg cjstg prhphsd¿fgh ³j cjdxgcg g fgrkh ch ijdthr. Hutrgs prhphsd¿fhjssharj pgrgijihkrg`hs sgh prhphstgs fh`h jxjrf³ķfdhs hu prhaij`gs.

<.76 Sjhrj`g. H sjk`jeth idkgech hs pheths `³jcdhs cj chds igchs cju` trdgekuih ³j pgrgijih gh tjrfjdrh igch j tj` `jtgcj cj sju fh`prd`jeth.

]rhvg. Vjng GA F u` trdgekuih. Cjsdkej phr C h pheth `³jcdh cj GAj phr J h pheth `³jcdh cj GF . Cjvj`hs prhvgr quj CJ ³j pgrgijih g AF

j quj CJ 0 7

1AF . ]grg dsth, `grquj eg sj`d-rjtg V JC u` pheth L tgi

<6




quj L C 0 CJ . Fh`h GC 0 CA (n³g quj C ³j pheth `³jcdh cj GA) G ˌ CJ

0 L ˌ CA phr sjrj` hphsths pjih v³jrtdfj, jetgh hs trdgekuihs GCJ j L CAsgh fhekrujetjs. Fh`h fhesjqujefdg tj`-sj quj C ˌ LA 0 G ˌ JC j L A 0 GJ .Ihkh L A j JF sgh pgrgijihs j tj` h `js`h fh`prd`jeth. Vjkuj-sj jetgh cgprhphsd¿fgh (<.71) quj h qugcrdi³gtjrh LAFJ ³j u` pgrgijihkrg`h. ]hrtgethL J ³j pgrgijih g A F j tj` h `js`h fh`prd`jeth. Fh`h C ³j pheth `³jcdh

cj L J jetgh CJ 0 7

1AF , fh`h qujr³ķg`hs cj`hestrgr.

G

J

FA

L C

Ldkurg <.4

<.73 ]rhphsd¿fgh. Vupheog quj trjs rjtgs pgrgijigs, g, a j f, fhrtg` gsrjtgs ` j e ehs pheths G, A j F j ehs pheths G, A j F , rjspjftdvg`jetj.Vj h pheth A jefhetrg-sj jetrj G j F , jetgh h pheth A tgà³j` jefhetrg-sjjetrj G j F . Vj GA 0 AF , jetgh tgà³j` tj`-sj GA 0 A F .

]rhvg. Vjng` g, a j f rjtgs pgr-gijigs j ` j e rjtgs quj detjr-fjptg` jstgs pgrgijigs ehs phe-ths G, A j F j G, A j F fh`hdecdfgch eg kurg gh igch. VjA jst³g jetrj G j F , jetgh G jF jstgh j` sj`d-pigehs cdstdethsrjigtdvg`jetj g rjtg a. Hasjrvjquj G j G jstgh j` u` `js`hsj`d-pigeh cjtjr`degch phr a, n³g

quj g j a sgh rjtgs pgrgijigs j Gj G pjrtjefj` g rjtg g.

` eC

G

A

J FF

f

a

g

A

G

Ch `js`h `hch F j F jstgh j` u` `js`h sj`d-pigeh cjtjr`degch phra. ]hcj`hs phrtgeth fhefiudr quj G j F jstgh j` sj`d-pigehs cdstdethsrjigtdvg`jetj g rjtg a. Ihkh a detjrfjptg h sjk`jeth GF j` u` ³uedfhpheth. Fh`h A ³j h pheth cj detjrfjssgh cg rjtg e fh` g rjtg a, j G jF pjrtjefj` g e fhefiu³ķ`hs quj h pheth cj detjrfjssgh cj GF fh` a ³jjxgtg`jetj h pheth A. Ihkh A pjrtjefj gh sjk`jeth GF j ihkh A jst³gjetrj G j F . Dsth cj`hestrg g prd`jdrg pgrtj cg prhphsd¿fgh.

<3




]grg cj`hestrgr g sjkuecg pgrtj, trgfj pjih pheth A u`g rjtg pgrgijig

mg rjtg `. Jstg fhrtg gs rjtgs g j f j` pheths C j J , rjspjftdvg`jetj. Gr`hquj hs trdgekuihs ACG j AJF sgh fhekrujetjs. Cj lgth, fh`h CAAGj AJF A sgh pgrgijihkrg`hs, jetgh CA 0 A J . Hasjrvj quj hs gekuihsC ˌ A G j J A F sgh dkugds phr sjrj` hphsths pjih v³jrtdfj j A ˌ CG j A ˌ JF

sgh tgà³j` dkugds phr sjrj` gekuihs fhrrjsphecjetjs cjtjr degchs phru`g trgesvjrsgi fhrtgcg pjigs pgrgijigs g j f. Dsth prhvg g ehssg gr`g¿fgh.Cg fhekrujefdg chs trdgekuihs ACG j AJF cjfhrrj d`jcdgtg`jetj qujGA 0 A F .

Jstg prhphsd¿fgh phcj sjr kjejrgidzgcg cj `gejdrg qugsj d`jcdgtg pgrg hfgsh j` quj cugs trgesvjrsgds fhrtg` u` e³u`jrh qugiqujr (`gdhr hu dkugig trjs) cj rjtgs pgrgijigs.

<.7: Fhrhi³grdh. Vupheog quj b rjtgs pgrgijigs g7, g1,...,gb fhrtg`cugs rjtgs ` j e ehs pheths G7, G1,...,Gb j ehs pheths G

7, G

1,...,G

b, rj-

spjftdvg`jetj. Vj G7G1,...,G1G6 0 » » » 0 Gb∑7Gb jetgh G

7G

1 0 G

1G

6 0» » » 0 G

b∑7G

b.

G prhvg cjstj fhrhi³grdh ³j cjdxgcg g fgrkh ch ijdthr. H tjhrj`g quj drj`hsjeuefdgr g sjkudr fhestdtud-sj eu`g jtgpg jssjefdgi pgrg h jstgajijfd`jethcg tjhrdg cgs kurgs sj`jiogetjs quj sjr³g cjsjevhivdcg eh pr³hxd`h fgp³ķtuih.Eg sug cj`hestrg¿fgh drj`hs utdidzgr, cj `gejdrg jssjefdgi, h lgth cj quj hfhrph chs e³u`jrhs rjgds ³j fh`pijth.

<.7< Sjhrj`g. Vj u`g rjtg, pgrgijig g u` chs igchs cj u` trdgekuih,fhrtg hs hutrhs chds igchs, jetgh jig sj cdvdcj eg `js`g rgzgh.

]rhvg. Vjng GAF u` trdgekuih. Fhesdcjrj u`g rjtg pgrgijig gh igchAF quj fhrtg hs igchs GA j GF , rjspjftdvg`jetj, ehs pheths C j J , fh`hrjprjsjetgch eg kurg (<.77). Cjvjrj`hs prhvgr quj>

(GC/GA) 0 (GJ/GF ).

]grg dsth, th`j u` pjqujeh sjk`jeth G] 7 eg sj`d-rjtg V GA cj `hch

quj gs rgzhjs GA/G] 7 j GC/G] 7 egh sjng` e³u`jrhs detjdrhs. Fhesdcjrj-`hs eg sj`d-rjtg V GA hs pheths ] 1, ] 6,...,] b,... tgds quj

b » G] 7 0 G] b

pgrg thch b ≪ 1. Jxdstj` jetgh chds e³u`jrhs detjdrhs ` j e tgds quj>

C jst³g jetrj ] j ] +7 jA jst³g jetrj ] e j ] e+7

<:




A

C

G

J

F

Ldkurg <.77

Sj`-sj phrtgeth>

` » G] 7 9 GC 9 (` + 7) » G] 7 j

e » G] 7 9 GA 9 (e + 7) » G] 7.

³J jetgh sd`pijs fhefiudr cjstgs cjsdkugicgcjs quj

(g) `

e + 7 9

GC

GA9

` + 7

e .

Srgfj`hs pjihs pheths ] 7, ] 1,...,] e+7 rjtgs pgrgijigs g AF . Jstgs rjtgs,sjkuech (<.7:), fhrtg` g sj`d-rjtg V GF j` pheths W7, W1,...,We+7, hs qugds

tgà³j` sgtdslgzj` g

b » GWi 0 GWb,

pgrg thch b, 1 ≪ b ≪ e + 7. Gi³j` cdssh, h pheth J jefhetrg-sj jetrjW` j W`+7 j h pheth F jetrj We j We + 7. H `js`h rgfdhf³ķedh ljdth gfd`gphcj sjr rjpjtdch gqud hatjech-sj fh`h rjsuitgch g cjsdkugicgcj>

(a) `

e + 7 9

GJ

GF 9

` + 7

e .

Gs cjsdkugicgcjs (g) j (a) pjr`dtj`-ehs fhefiudr quj

(f)

GC

GA∑

GJ

GF

9 ` + 7

e ∑

`

e + 7

Hasjrvj quj, fh`h ` ≪ e, jetgh

` + 7

e ∑

`

e + 7 0

` + e + 7

e(e + 7) ≪

1e + 1

e(e + 7) 0

1

e

<<




hu sjng, gs rgzhjs GC/GA j GJ/GF cdljrj` phr egh `gds ch quj 1/e.

Wugeth `jehr lhr h sjk`jeth G] 7 tgeth `gdhr sjr³g h e³u`jrh e j tgeth`jehr sjr³g h quhfdjetj 1/e. Fh`h h igch jsqujrch cg cjsdkugicgcj (f) eghcjpjecj cj e, s³h phcj`hs fhefiudr quj hs quhfdjetjs GC/GA j GJ/GF sgh dkugds, fh`h qujr³ķg`hs cj`hestrgr.

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J_JQF³DFDHV

7. Eg kurg gh igch H ³j h pheth `³jcdh cjGC j ˌ A 0 ˌ F . Vj A , H j F sgh fhidejgrjs,fhefiug quj hs trdgekuihs GAH j CHF sgh fhekrujetjs.

G A

F C

H

1. ]rhvj quj g sh`g cgs `jcdcgs chs gekuihs gkuchs cj u` trdgekuihrjtgekuih ³j 4=◬.

6. ]rhvj quj fgcg gekuih cj u` trdgekuih jqudi³gtjrh `jcj <=◬.

3. ]rhvj quj g `jcdcg ch gekuih jxtjreh cj u` trdgekuih ³j dkugi g sh`gcgs `jcdgs chs gekuihs detjreh g jij egh gcngfjetjs.

:. Z` sjk`jeth idkgech chds phethscj u` fdrfuih j pgssgch phr sju fjetrhfog`g-sj cdg`jtrh. Eg kurg gh igch H ³jh fjetrh ch fdrfuih, GA ³j u cdg`jtrh j

F ³j hutrh pheth ch fdrfuih. @hstrj quj 10 1»7.

G A

F

H

17

<. ]rhvj quj sj ` j e sgh rjtgs jqudcdstgetjs jetgh ` j e sgh pgrgijigshu fhdefdcjetjs.

;. Vjng GAF u` trdgekuih ds³hsfjijs fh` agsj GA. Vjng` @ j E hspheths `³jcdhs chs igchs FG j FA, rjspjtdvg`jetj. @hstrj quj, h rjfijxhch pheth F rjigtdvg`jetj. @hstrj quj, h rjfijxh ch pheth F rjigtdvg`jetj³g rjtg quj pgssg phr @ j E ³j jxgtg`jetj h pheth `³jcdh ch sjk`jeth GA .

?. Cj`hestrgr g prhphsd¿fgh (<.7=).

4. Cj`hestrj g prhphsd¿fgh (<.71).

7=. Z` rjtˌ gekuih ³j u` qugcrdi³gtjrh quj tj` thchs hs sjus gekuihs,

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rjths. @hstrj quj, thch rjtgekuih ³j u` pgrgijihkrg`h.

77. @hstrj quj, gs cdgkhegds cj u` rjtgekuih sgh fhekrujetjs.

71. Z` ihsgekh ³j u` pgrgijihkrg`h quj tj` thchs hs sjus igchs fhekru-jetjs. @hstrj quj, gs cdgkhegds cj u` ihsgekh fhrtg`-sj j` gekuihs rjth jsgh adssjtrdzjs chs sjus gekuihs.

76. Z` qugcrgch ³j u rjtgekuih quj tgà³j` ³j u` ihsgekh. @hstrjquj, sj gs cdgkhegds cj u` qugcrdi³gtjrh sgh fhekrujetjs j sj fhrtg` j` u`pheth quj ³j pheth `³jcdh cj gàgs, jetgh h qugcrdi³gtjrh ³j u` rjtgekuih.

73. Z` trgp³jzdh ³j u` qugcrdi³gtjrh j` quj chds igchs hphsths sgh pgr-gijihs. Hs igchs pgrgijihs cj u` trgp³jzdh sgh fog`gchs agsjs j hs hutrhschds sgh cjeh`degchs cj igtjrgds . Z trgp³jzdh ³j cdth ds³hsfjijs sj sug igtjrgdssgh fhekrujetjs. Vjng GAFC u` trgp³jzdh j` quj GA ³j u`g agsj. Vj jij ³jds³hsfjijs, `hstrj quj ˌ G 0 ˌ A j ˌ F 0 ˌ C .

7:. @hstrj quj, gs cdgkhegds cj u` trgp³jzdh ds³hsfjijs sgh fhekrujetjs.

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]QHAIJ@GV

7. @hstrj quj, hs fgshs (7) j (6) ch tjhrj`g (:.73) sgh fhesjqujefdgsd`jcdgtgs ch sjkuech fgsh cj fhekrujefdg cj trdgekuihs.

1. Cj`hestrj h fgsh (1) ch tjhrj`g (:.73) utdidzgech g fhestru¿fgh suk-jrdcg pjig kurg gagdxh.

G F

A A

G FC

6. @hstrj quj, sj chds gekuihs j h igch hphsth g u` cjijs, j` u`trdgekuih, sgh dkugds ³gs fhrrjsphecjetjs pgrtjs cj u` hutrh trdgekuih, jetghhs trdgekuihs sgh fhekrujetjs.

3. Eg kurg gh igch G, A j F sgh phethscj u` f³ķrfuih cj fjetrh H. @hstrj qujA ˌ HF 0 1»A ˌ GF .

A

F

GH

:. @hstrj quj, sj ` j e sgh cugs rjtgs qujlhr`g`, fh` u`g trgesvjrsgi, gekuihs 1j 6 (fh`h eg kurg gh igch) tgds quj 1 +

6 0 7?=◬

, jetgh ` j e sj detjrfjptg`.

16

<. @hstrj quj, sj hs gekuihs hphsths cj u` qugcrdi³gtjrh sgh fhekrujetjs,jetgh h qugcrdi³gtjrh ³j u` pgrgijihkrg`h.

;. @hstrj quj, sj gs cdgkhegds cj u` qugcrdi³gtjrh sj detjrfjptg` j` u`pheth `³jcdh gàgs, jetgh h qugcrdi³gtjrh ³j u` pgrgijihkrg`h.

?. @hstrj quj, sj gs cdgkhegds cj u` pgrgijihkrg`h sgh fhekrujetjs,jetgh h pgrgijihkrg`h ³j u` rjtgekuih.

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4. @hstrj quj, u` pgrgijihkrg`h fungs cdgkhegds sgh pjrpjecdfuigrjs ³j

u` ihsgekh.

7=. ]rhvj quj h sjk`jeth idkgech hs pheths `³jcdhs cgs igtjrdgs cj u`trgp³jzdh ³j pgrgijih ³gs agsjs j quj sju fh`prd`jeth ³j g `jcdg grdt`³jtdfg chsfh`prd`jeths cgs agsjs.

77. @hstrj quj, hs pheths `³jcdhs chs igchs cj u` qugcrdi³gtjrh qugiqujrsgh v³jrtdfjs cj u` pgrgijihkrg`h.

71. Zsj g prhphsd¿fgh (<.7:) pgrg jstgajijfjr u` `³jthch cj cdvdsgh cju` sjk`jeth qugiqujr j` b pgrtjs dkugds.

76. Gchtj fh`h gxdh`g \ , j` suastdtud¿fgh gh gxdh`g \, g vgidcgcjcg prhphsd¿fgh fhetdcg eh prhaij`g : gfd`g. ]rhvj gkhrg h gxdh`g \ .Jxpidquj phr quj h prhaij`g : j jstj `hstrg` quj hs gxdh`gs \ j \ sghjqudcdstgetjs. H gxdh`g \ ³j jxgtg`jetj h qudeth gxdh`g cj Jufidcjs (vdcjfh`jet³grdh g sjkudr).

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FH@JES ³GQDH

Jufidcjs agsjhu g fhestru¿fgh cg sug kjh`jtrdg j` 7= gxdh`gs sjpgrgchsj` chds kruphs> fdefh lhrg` figssdfgchs fh`h ’eh¿fhjs fhùes‐ j hs hutrhsfh`h ’phstuigchs‐. G cdstde¿fgh jetrj jijs egh ³j cj thcg figrg. Gs ’eh¿fhjsfhùes‐ pgrjfj` tjr sdch fhesdcjrgcgs fh`h odp³htjsjs gfjdt³gvjds g thcgs gsfdjefdgs hu g thcgs gs pjsshgs detjidkjetjs, jequgeth quj hs phstuigchs jrg`fhesdcjrgchs fh`h odp³htjsjs fgrgftjr³ķstdfgs cg kjh`jtrdg. Gs fdefh eh¿fhjsfhùes jrg`>

7. Fhdsgs quj sgh dkugds g u`g `js`g fhdsg sgh tgà³j` dkugds jetrj sd.1. Vj dkugds sgh gcdfdhegchs g dkugds, hs rjsuitgchs sgh dkugds.6. Vj dkugds sgh suatrg³ķchs cj dkugds, hs rjsths sgh dkugds.3. Fhdsgs quj fhdefdcj` fh` hutrgs fhdsgs sgh dkugds u`g g hutrg.:. H thch ³j `gdhr ch quj qugiqujr cj sugs pgrtjs.

Hs phstuigchs jrg`>

7. ]hcj-sj trg¿fgr u`g rjtg phr qugdsqujr chds pheths.1. ]hcj-sj fhetdeugr u`g rjtg deedtg`jetj.6. ]hcj-sj cjsfrjvjr u`g fdrfueljrjefdg fh` qugiqujr fjetrh j qugiqujr

rgdh.3. Shchs hs gekuihs rjths sgh dkugds.:. Vj u`g rjtg fhrtg cugs hutrgs rjtgs lhr`gech gekuihs fhigtjrgds

detjrehs fung sh`g ³j `jehr ch quj chds rjths, jetgh gs cugs rjtgs, sj fhe-tdeugcgs deedtg`jetj, jefhetrg,-sj eh igch eh qugi jstgh hs gekuihs fungsh`g ³j `jehr ch quj chds rjths.

Jàhrg Jufidcjs egh tjeog cdth jspjfdfg`jetj, fg figrh, gtrgv³js cglhr`g fh`h jij h utdidzhu, quj h prd`jdrh phstuigch rjljrj-sj g u`g ³uedfgrjtg idkgech hs chds pheths. Sgà³j`, ch fhetjxth, fg figrh quj, pgrgJufidcjs, ’rjtg‐ sdkedfgvg h quj oh nj fog`g`hs cj ’sjk`jeth‐. Cg³ķ jij

lgigr j` ’fhetdeugr deedtg`jetj u`g rjtg‐ . Jij gssu`du tgfdtg`jetj qujtgi prhihekg`jeth phcj sjr ljdth cj u`g ³uedfg `gejdrg j` fgcg jxtrj`dcgcjcj u`g ’rjtg‐, cj `hch quj cugs rjtgs cdstdetgs egh phcj` tjr u` sjk`jethfhù`. Cj lgth Jufidcjs utdidzhu-sj cj ùdtgs odp³htjsjs quj egh fhestgvg`,sha ejeou`g lhr`g, ej` cgs ’eh¿fhjs fhùes‐, ej` chs ’phstuigchs‐. Jstgh`dssgh ³j fhesdcjrgcg pjihs kjh`jtrgs fh`h u` chs `gds krgvjs cjljdths chs’Jij`jeths‐.

@js`h u` jxg`j gprjssgch ch idvrh D chs ’Jij`jeths‐ rjvjig quj jijfh`phj-sj cj trjs pgrtjs cdstdetgs (jàhrg Jufidcjs egh gs tjeog sjpgrgchlhr`gi`jetj). G prd`jdrg pgrtj, fhestdtu³ķcg pjigs prd`jdrgs 1< prhphsd¿fhjs,

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trgtg qugsj jxfiusdvg`jetj cg tjhrdg jij`jetgr chs trdgekuihs. Jig gargekj

thch h `gtjrdgi quj gprjsjetg`hs gt³j h egi ch fgp³ķtuih : cjstgs ehtgs. Gsjkuecg pgrtj trgtg cg tjhrdg cgs pgrgijigs. Dedfdg-sj fh` g prhphsd¿fgh 1;j prhssjkuj gt³j g prhphsd¿fgh 63. Gid sgh gpjsjetgcgs gs prhphsd¿fhjs qujgargekj` h `gtjrdgi gprjsjetgch eh fgp³ķtuih < cjstgs ehtgs. G pgrtdr cgprhphsd¿fgh 63, gt³j h egi (prhphsd¿fgh 3?), h idvrh D chs ’Jij`jeths‐ trgtg cgsrjig¿fhjs jetrj ³grjgs cj pgrgijihkrg`hs, trdgekuihs j qugcrgchs j fui`deg fh`h lg`hsh tjhrj`g cj ]dt³gkhrgs (]rhphsd¿fgh 3;) j cj sju devjrsh (]rhphsd¿fgh3?). ³J lgth d`phrtgetj g sjr hasjrvgch quj h qudeth phstuigch egh lhdutdidzgch phr Jufidcjs eg prhvg cj qugiqujr cgs 1< prd`jdrgs prhphsd¿fhjsch idvrh D, gs qugds gdecg sgh v³gidcgs fgsh h qudeth phstuigch sjng jxfiu³ķchhu trhfgch phr u` hutrh fh`pgt³ķvji fh` hs rjstgetjs phstuigchs j eh¿fhjs

fhùes.O³g jvdcjefdg cj quj hs phstuigchs, pgrtdfuigr`jetj h qudeth, lhrg` lhr-

ùigchs phr Jufidcjs `js`h. Vgaj-sj quj h qudeth phstuigch threhu-sj,cj d`jcdgth, givh cj fr³ķtdfgs pjihs `gtj`³gtdfhs cg ³jphfg. Jstj lgth egh³j cj jstrgeogr, qugech fhesdcjrg`hs quj, prd`jdrg`jetj, jij ³j agstgetjcdljrjetj, defiusdvj j` tg`geoh chs hutrhs phstuigchs, pgrjfjech `gds u`gprhphsd¿fgh ch quj u` gxdh`g2 cjphds, tjfedfg`jetj, jij ³j g devjrsg cj u`gcgs prhphsd¿fhjs cj`hestrgcgs ehs ’Jij`jeths‐ fh` agsj gpjegs ehs qugtrhprd`jdrhs phstuigchs, g sgajr g prhphsd¿fgh 1;2 phr ³uitd`h, jij egh phssud,j` ejeou` sjetdch, gqujig fgrgftjr³ķstdfg cj ’guth-jvdcjefdg‐ quj fgrgftjr-dzhu dedfdgi`jetj g jsfhiog chs hutrhs gxdh`gs. Gi³j` cdssh, g sug tgrcdg

utdidzg¿fgh, gp³hs tgetgs prhphsd¿fhjs sjrj` prhvgcgs sj` sju guxdidh, ijvge-thu suspjdtgs cj quj jij sjrdg sd`pijs`jetj u`g prhphsd¿fgh cj`hestr³gvjig pgrtdr chs hutrhs gxdh`gs g qugi Jufidcjs egh fhesjkudrdg cj`hestrgr.Fh`h fhesjqujefdg cjssg suspjdtg, deu`jr³gvjds tjetgtdvgs lhrg` ljdtgs pgrgprhv³g-ih hu jid`de³g-ih gtrgv³js cj u`g rjcjed¿fgh ch fhefjdth cj rjtgs pgr-gijigs. Jetrj hs eh`js lg`hshs chs quj tjetgrg` cj`hestrgr h qudeth phs-tuigch phcj`hs idstgr ]rhfius(3?:-37= g.F.), Egsdrgcde (71=7-71;3), Nheo[jiids (7<7<-7;=6), Kjrhig`h Vgfojrd (7<<;-7;66), Nheo O. Igàjrt (7;1?-7;;;), Gcrdje @. Ijkjecrj (7;:1-7?66), Ihuds Ajrtrgec (7;67-7?71) j FgriL. Kguss (7;;;-7?::). Jstjs cjdxgrg` egs sugs hargs rjljrjefdgs rjijvgetjssharj h gssueth. ³J, eh jetgeth, fjrth quj thchs gqujijs detjrjssgchs sj-

rdg`jetj j` `gtj`³gtdfg gt³j h s³jfuih cjzjssjtj tjetgrg` jvjetugi`jetjcj`hestrgr h qudeth phstuigch.

Lhd sh`jetj eg prd`jdrg `jtgcj ch s³jfuih cjzjehvj quj hs `gtj`³gtdfhsfojkgrg` mg fhefiusgh cj quj h qudeth phstuigch egh jrg cj`hestr³gvji g pgr-tdr chs hutrhs qugtrh. Dsth hfhrrju fh` g cjsfhajrtg cgs fog`gcgs kjh`j-trdgs egh-Jufidcdgegs j` quj h qudeth phstuigch cj Jufidcjs ³j suastdtu³ķchphr u`g hutrg gr`g¿fgh quj ioj ³j fhetrgcdt³hrdg. Jstg cjsfhajrtg jst³g gssh-fdgcg fh` h eh`j cj chds `gtj`³gtdfhs quj g hatdvjrg` decjpjecjetj`jetj>Nhogee Ahiygd (7?=1-7?<=) j Edbhigd D. Ihagfojwsby (7;46-7?:<). Hs trg-

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agiohs cjstjs chds `gtj`³gtdfhs lhd jijvgch mgs sugs cjvdcgs prhphr¿fhjs phr

Lrdcjrdfo A. Qdj`gee (7?1<-7?<<) quj cju defdh g u` sjkuech pjr³ķhch ehcjsjevhivd`jeth cgs kjh`jtrdg Jufidcdgegs j egh-Jufidcdgegs, u` pjr³ķhchfgrgftjrdzgch pjigs devjstdkg¿fhjs ch pheth cj vdstg cg kjh`jtrdg cdljrjefdgi,j` fhetrgtj fh` hs `³jthchs sdet³jtdfhs prjvdg`jetj utdidzgchs. Gsshfdgchs gjstj sjkuech pjr³ķhch jstgh hs eh`js cj Idj, Ajitrg`d, Fgyijy, Bijde, Fidhrcj Odiajrt.

Idj lhd rjsphes³gvji pjig detrhcu¿fgh chs kruphs cj trgeslhr`g¿fgh eh js-tuch cg kjh`jtrdg. Ajitrg`d tj` h frjcdth cj tjr prhcuzdch g prd jdrgprhvg cg fhesdstjefdg cgs kjh`jtrdgs egh jufidcdgegs. Jàhrg Ahiygd jIhagfojwsby egh tjeog` jefhetrgch ejeou`g fhetrgcd¿fgh j` sug kjh`j-trdg gh ihekh cj thcgs gs sugs devjstdkg¿fhjs, gdecg pjr`gejfdg g phssdadid-

cgcj cj quj giku`g defhesdstjefdg pucjssj gpgrjfjr eh cjsjevhivd`jeth cjehvhs trgagiohs cj pjsqudsg. Ajitrg`d `hstrhu fh`h g kjh`jtrdg cj Ahiygdj Ihagfojwsby phcdg sjr rjprjsjetgcg sharj u`g supjrl³ķfdj eh jspg¿fh Ju-fidcdgeh g trjs cd`jeshjs, cj lhr`g quj hs sjus phstuigchs lhssj` hatdchsg pgrtdr chs gxdh`gs cg kjh`jtrdg Jufidcdgeg. Fh`h fhesjqujefdg, qugi-qujr defhesdstjefdg quj pucjssj jxdstdr egs kjh`jtrdgs egh-Jufidcdgegs sjrdgtgà³j` u`g defhesdstjefdg cg kjh`jtrdg Jufidcdgeg.

Hs trgagiohs cj Fgijy, Bijde j Fidhrc prhcuzdrg` u`g idecg figssd-fg¿fgh cjstgs kjh`jtrdgs ch pheth cj vdstg prhnjtdvh-`³jtrdfh. Cg³ķ j` cd-getj, g prjhfupg¿fgh fh` g luecg`jetg¿fgh cg kjh`jtrdg j` agsjs s³hidcgsch`dehu g pjsqudsg `gtj`³gtdfg sharj h gssueth fui`degch fh` g rjfhe-

stru¿fgh cg kjh`jtrdg Jufidcdgeg phr Odiajrt h quj, egi`jetj, j cjedtdvg-`jetj, jefjrrhu g ihekg agtgiog fh` h qudeth phstuigch cj Jufidcjs.

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KJH@JSQDG JZFIDCDGEG ]IGEG VJ@JIOGEF¿ G CJ SQDGEKZIHV

FG]³DSZIH ;

VJ@JIOGEF¿ G CJ SQDGEKZIHV

Cdrj`hs quj chds trdgekuihs sgh sj`jiogetjs sj lhr phss³ķvji jstgajijfjru`g fhrrjsphecjefdg adue³ķvhfg jetrj sjus v³jrtdfjs cj `hch quj gekuihsfhrrjsphecjetjs sjng` dkugds j igchs fhrrjsphecjetjs sjng` prhphrfdhegds.

F

G A J

K

L

Ldkurg ;.7

Fh` dsth qujrj`hs cdzjr quj, sj GAF j JL K sgh chds trdgekuihs sj`ji-ogetjs j sj G ↑ J , A ↑ L , F ↑ K ³j g fhrrjsphecjefdg quj jstgajijfj gsj`jioge¿fg, jetgh vgij` sdùitgejg`jetj gs sjkudetjs dkugicgcjs>

ˌ G 0 ˌ J , ˌ A 0 ˌ L , ˌ F 0 ˌ K j

GA

JL 0

AF

L K0

FG

KJ

H quhfdjetj fhù` jetrj gs jcdcgs chs igchs fhrrjsphecjetjs ³j fog`gchcj rgzgh cj prhphrfdhegidcgcj jetrj hs chds trdgekuihs.

Hasjrvj quj chds trdgekuihs fhekrujetjs sgh sj`jiogetjs fh` rgzgh cjprhphrfdhegidcgcj u`2 devjrsg`jetj, chds trdgekuihs sj`jiogetjs fh` rgzghcj prhphrfdhegidcgcj u`, sgh fhekrujetjs.

;:




H tjhrj`g sjkudetj sjr³g rjljrdch fh`h ’sjkuech fgsh cj sj`jioge¿fg cj

trdgekuihs‐ g ` cj quj hs fgshs cj sj`jioge¿fg j hs fgshs cj fhekrujefdgsj fhrrjsphecg` cj u`g lhr`g egturgi.

;.7 Sjhrj`g. Cgch chds trdgekuihs GAF j JL K, sj ˌ G 0 ˌ J j ˌ A 0 ˌ L

jetgh hs trdgekuihs sgh sj`jiogetjs.

]rhvg. Fh`h g sh`g chs gekuihs cj u` trdgekuih ³j 7?=◬, jetgh gdkugicgcj chs gekuihs ˌ G j ˌ J j chs gekuihs ˌ A j ˌ L gfgrrjtg eg dkugicgcj chsgekuihs ˌ F j ˌ K . Qjstg prhvgr quj hs igchs sgh prhphrfdhegs. ]grg dsth th`jeg sj`d-rjtg V JL h pheth O cj `hch quj JO 0 GA . ]jih pheth O trgfju`g rjtg pgrgijig g L K.

G

F

A J

N

K

LO

Ldkurg ;.1

Jstg fhrtg g sj`d rjtg V JK eu` pheth N , lhr`gech u` trdgekuih J ON quj ³j fhekrujetj gh trdgekuih GAF (n³g quj ˌ G 0 ˌ J , GA 0 JO j ˌ A 0 ˌ L 0J ˌ ON . Jstg uitd`g dkugicgcj cjvj-sj gh pgrgijids`h cj N O j KL ). Vjkuj-sjgkhrg ch tjhrj`g (<.7<) quj (JO/JL ) 0 (JN/JK). Fh`h JO 0 GA jJN 0 GF jetgh, cg dkugicgcj gfd`g hat³j`-sj>

(GA/JL ) 0 (GF/JK).

Cj `gejdrg ge³gihkg cj`hestrg-sj quj (GF/JK) 0 (FA/KL ). Ldfgcj`hestrgch h tjhrj`g.

H tjhrj`g (;.7) pjr`dtj fhestrudr, fh` lgfdidcgcj, jxj`pihs chs trge-kuiohs sj`jiogetjs lgzjech-sj ush cg r³jkug j trgesljrdchr. ]hr jxj`pih,pgrg cjsjeogr u` trdgekuih sj`jiogetj gh trdgekuih GAF cg kurg (;.6),dedfdg-sj trg¿fgech u` sjk`jeth JL qugiqujr2

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GA

F

K

L

J

Ldkurg ;.6

g pgrtdr cj sugs jxtrj`dcgcjs fhestrhj`-sj gekuihs ˌ J j ˌ L dkugds ghsgekuihs ˌ G j ˌ A , rjspjftdvg`jetj (j` u` `js`h sj`dpigeh cjtjr`degch pjigrjtg JL )2 prhihekgech-sj hs igchs cjstjs gekuihs cjtjr`deg-sj u` phethK. Cj gfhrch fh` g prhphsd¿fgh getjrdhr hs trdgekuihs GAF j JL K sghsj`jiogetjs.

H sjkudetj tjhrj`g sjr³g rjljrdch fh`h prd`jdrh fgsh cj sj`jioge¿fg cj

trdˌ gekuihs .

;.1 Sjhrj`g. Vj j` chds trdgekuihs GAF j JL K tj`-sj ˌ G 0 ˌ J j(GA/JL ) 0 (GF/JK), jetgh hs trdgekuihs sgh sj`jiogetjs.

]rhvg. Fhestrug u` trdgekuih ODN quj tjeog OD 0 J L , ˌ O 0 ˌ G j ˌ D

0 ˌ A .

G

F

A J

K N

L O D

Ldkurg ;.3

Cj gfhrch fh` h tjhrj`g (;.7), hs trdgekuihs GAF j ODN sgh sj`ji-

;;




ogetjs.

]hr fhesjkudetj>

(GA/OD ) 0 (GF/ON ).

Fh`h O D 0 JL , g odp³htjsj (GA/JL ) 0 (GF/JK) j g dkugicgcj gfd`gd`pidfg quj>

ON 0 JK.

Fh`h, phr fhestru¿fgh O D 0 JL j ˌ O 0 ˌ G 0 ˌ J , phcj`hs fhefiudr, pjih

prd`jdrh fgsh cj fhekrujefdg cj trdgekuihs, quj hs trdgekuihs JL K j ODN sgh fhekrujetjs. Fh`h n³g sga³ķg`hs quj GAF j ODN jrg` sj`jiogetjs,phcj`hs fhefiudr lgfdi`jetj quj GA F j J L K sgh sj`jiogetjs.

H tjrfjdrh fgsh cj sj`jioge¿fg cj trdgekuihs ³j h sjkudetj.

;.6. Sjhrj`g. Vj, j` chds trdgekuihs GA F j J L K, tj sj

GA

JL 0

AF

L K0

FG

KJ ,

jetgh hs chds trdgekuihs sgh sj`jiogetjs.

]rhvg. Fhestrug u` trdgekuih ODN quj tjeog ˌ O 0 ˌ G, OD 0 JL jON 0 JK. Vjkuj-sj jetgh cg odp³htjsj quj

(GA/OD ) 0 (GF/ON ).

]hrtgeth, cj gfhrch fh` h tjhrj`g (;.1), hs trdgekuihs GAF j O DN sghsj`jiogetjs.

G A J L O D

NK

F

Ldkurg ;.:

;?




Cjfhrrj cg³ķ quj, gi³j` cg dkugicgcj gfd`g, tgà³j` hfhrrj.

(GA/OD ) 0 (AF/DN ).

Vjkuj-sj (cg³ķ j cg odp³htjsj ch tjhrj`g) quj DN 0 L K. Fh`h n³g t³ķeog`hsquj O D 0 JL j O N 0 JK (phr fhestru¿fgh) jetgh pjih tjrfjdrh fgsh cj fhe-krujefdg cj trdgekuihs, ODN j JL K sgh fhekrujetjs. Fh`h ODN j GAF sgh sj`jiogetjs, fhefiud-sj quj GAF j JL K sgh tgà³j` sj`jiogetjs. Dsthfhefiud g prhvg ch tjhrj`g.

Vjng GA F u` trdgekuih rjtgekuih fh` gekuih rjth eh v³jrtdfj G. Srgfj

g giturg GC ch v³jrtdfj G gh igch AF . Eh quj sj sjkuj vg`hs lgzjr ush cgsjkudetj ehtg¿fgh g 0 AF , a 0 GF , f 0 GA , o 0 GC, ` 0 AC j e 0 CF .

F

G

AC

a

e

o

g

f

`

Ldkurg ;.<

Fh`h GC ³j pjrpjecdfuigr g A F , jetgh hs trdgekuihs GCA j GCF sghrjtgekuihs. Fh`h ˌ A + ˌ F 0 4=◬ j ˌ A + A ˌ GC 0 4=◬ jetgh

A GC 0 F

Fh`h tgà³j` C

ˌ

GF + ˌ

F 0 4=

◬

jetgh

C ˌ GF 0 ˌ A .

Hs trdgekuihs GCA j FCG sgh phrtgeth gàhs sj`jiogetjs gh trdgekuihGAF j sgh tgà³j` sj`jiogetjs jetrj sd` Cjstgs sj`jioge¿fgs phcj`hscjcuzdr v³grdgs rjig¿fhjs jetrj gs `jcdcgs g, a, f, o, ` j e gfd`g `jefdhegcgs.]hr jxj`pih, g sj`jioge¿fgs jetrj GCA j FCG ³j g quj ijvg G j` F . Aj` G j C j` C. Fh`h fhesjqujefdg cjstg sj`jioge¿fg tj` sj

;4




fa

0 ò

0 oe

Cg ³uitd`g dkugicgcj cjcuz-sj quj

o1 0 è

Gssd` prhvg`hs g sjkudetj prhphsd¿fgh2

;.3 ]rhphsd¿fgh. J` thch trdgekuih rjtgekuih g giturg ch v³jrtdfj chgekuih rjth ³j `jcdg prhphrfdhegi jetrj gs prhnj¿fhjs chs fgtjths sharj godphtjeusg.

H sjkudetj ³j u` chs `gds d`phrtgetjs j `gds utjds tjhrj`gs cg kjh`jtrdgJufidcdgeg pigeg. ³J fheojfdch fh`h ’tjhrj`g cj ]dt³gkhrgs‐ j` oh`hkjejgg u` krgecj kjh`jtrg cg Kr³jfdg getdkg.

;.:. Sjhrj`g (]dt³gkhrgs). J` thch trdgekuih rjtgekuih h qugcrgchch fh`prd jeth cg odphtjeusg ³j dkugi g sh`g chs qugcrgchs chs fh`prd-`jeths chs fgtjths.

J` tjr`hs cg ehtg¿fgh jstgajijfdcg gfd`g h tjhrj`g cj ]dt³gkhrgs gr`g

quj

g1 0 a1 + f1

]rhvg. G prhvg ch tjhrj`g cj ]dt³gkhrgs ³j u`g fhesjqujefdg cg sj`ji-oge¿fg chs trdgekuihs GCA, F CG j GA F . Cg sj`jioge¿fg cj GCA j GAF (G ↑ F, A ↑ A j C ↑ G) fhefiud-sj quj

`

f 0

f

g

Cg sj`jioge¿fg chs trdgekuihs F CG j GAF fhefiud-sj quj

e

a 0

a

g

Ihkh g` 0 f1 j ge 0 a1. ]hrtgeth g(` + e) 0 f1 + a1. Fh`h ` + e 0 g,jetgh g1 0 a1 + f1, fh`h qujr³ķg`hs cj`hestrgr.

G sjkudetj prhphsd¿fgh ³j g devjrsg ch tjhrj`g cj ]dt³gkhrgs.

?=




;.< ]rhphsd¿fgh. Z` trdgekuih phssud igchs `jcdech g, a j f. Vj

g1 0 a1 + f1, jetgh h trdgekuih ³j rjtgekuih j sug odphtjeusg ³j h igch quj`jcj g.

]rhvg. Fhestrug u` rjtgekuih funhs fgtjths `j¿fg` jxgtg`jetj a j f.Ejstj ehvh trdgekuih, cj gfhrch fh` h tjhrj`g cj ]dt³gkhrgs, g odphtjeusg`jcj

∞ a1 + f1 0 g. ]hrtgeth, jstj ehvh trdgekuih (quj ³j rjtgekuih) tj`

igchs `jcdech g, a j f. ]jih tjrfjdrh fgsh cj fhekrujefdg, jij ³j phrtgeth,fhekrujetj gh trdgekuih hrdkdegi. Ihkh h trdgekuih hrdkdegi ³j rjtgekuih j sugodphtjeusg `jcj g.

?7




J_JQF³DFDHV

7. Wugeth `jcj g odphtjeusg cj u` trdgekuih rjtgekuih j` quj hsfgtjths `jcj` u` fjet³ķ`jtrh fgcg8

1. Wugeth `jcj g giturg cj u` trdgekuih jqudi³gtjrh funhs igchs `jcjù` fjet³ķ`jtrh fgcg8

6. Eh trdgekuih GA F , GA 0 :, AF 0 71, FG 0 76. Wugi g `jcdcg chgekuih ˌ A 8

3. Eh trdgekuih CJL , CJ 0 JL 0 < j L C 0 <∞

1. Wugeth `jcj` hsgekuihs ch trdgekuih8

:. Z`g fgdxg `jcj 71 fjet³ķ`jtrhs cj fh`prd`jeth, 3 fjet³ķ`jtrhs cjigrkurg j 6 fjet³ķ`jtrhs cj giturg. Wugeth `jcj` gs cdgkhegds cj fgcg u`gcgs lgfjs cg fgdxg8

<. @hstrj quj chds trdgekuihs jqudi³gtjrhs sgh sj`prj sj`jiogetjs.

;. @hstrj quj sgh sj`jiogetjs chds trdgekuihs ds³hsfjijs quj tj` dkugdshs gekuihs hphsths mg agsj.

?. Eg kurg gh igch C ³j pheth `³jcdh cjGA j J ³j pheth `³jcdh cj GF . @hstrjquj hs trdgekuihs GCJ j GAF sgh sj`ji-ogetjs. G

CA

F

J

4. Eg kurg gh igch tj`-sj quj ACG

j GAF sgh sj`jiogetjs, sjech g sj`ji-oge¿fg g quj ijvg A j` G, C j` A j Gj` F . Fhefiug quj h trdgekuih ACG ³jds³hsfjijs.

G

A

FC

7=. @hstrj quj thch trdgekuih rjtgekuih cj igchs p1∑ q 1, 1 pq j p1 + q 1 ³ju` trdgekuih rjtgekuih. Gqud p j q sgh qugdsqujr e³u`jrhs detjdrhs phsdtdvhsfh` p 5 q .

77. Shchs hs trdgekuihs decdfgchs eg kurg gagdxh sgh rjtgekuihs. Cj-

?1




tjr`dej g, a, f, c j j.

L

HG

A

FC

J

7

7 7

7

7

7

g

afc

j

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]QHAIJ@GV

7. ]rhvj h sjkuech fgsh cj sj`jioge¿fg cj trdgekuihs suphech fheojfdchh tjhrj`g (;.1) j sj` lgzjr ush ch tjhrj`g (<.7<).

1. ]rhvj quj g rjig¿fgh ’³j sj`jiogetj g‐ ³j trgesdtdvg, dsth ³j, prhvj quj,chds trdgekuihs sgh sj`jiogetjs g u` tjrfjdrh, jetgh sgh sj`jiogetjs jetrjsd.

6. ]rhvj quj giturgs fhrrjsphecjetjs j` trdgekuihs sj`jiogetjs jstgh

eg `js`g rgzgh quj hs igchs fhrrjsphecjetjs.

3. ]rhvj quj g adssjtrdz cj u` gekuih cj u` trdgekuih cdvdcj h igchhphsth j` sjk`jeths prhphrfdhegds ghs hutrhs chds igchs. Dsth ³j, sj GAF ³jh trdgekuih j A C ³j g adssjtrdz ch gekuih ˌ A sjech C u` pheth cj igch GF ,jetgh (GC/CF ) 0 (GA/AF ). (Gnucg> trgfj pjih pheth G u`g rjtg pgr-gijig gh igch AC. Jstg detjrfjptg g sj`d-rjtg V FA eu` pheth J lhr`gechtrdgekuihs sj`jiogetjs).

:. Jeuefdj j prhvj g gr`g¿fgh devjrsg ch jxjrf³ķfdh getjrdhr.

<. ]rhvj quj, sj u` trdgekuih rjtgekuih tj` gekuihs gkuchs cj 6=◬ j<=◬, jetgh sju `jehr fgtjth `jcj `jtgcj ch fh`prd`jeth cg odphtjeusg.(gnucg> Lg¿fg ush ch quj lhd hatdch eh jxjrf³ķfdh 1).

;. ]rhvj quj, sj j` u` trdgekuih rjtgekuih h `jehr fgtjth `jcj `jtgcjch fh`prd`jeth cg odphtjeusg, jetgh sjus gekuihs gkuchs sgh cj 6=◬ j <=◬.

?. ]rhvj quj, sj chds trdgekuihs tj` igchs fhrrjsphecjetjs pgrgijihs,jetgh jijs sgh sj`jiogetjs. ]rhvj tgà³j` quj gs rjtgs idkgech hs v³jrtdfjsfhrrjsphecjetjs sgh fhefhrrjetjs hu pgrgijigs. (Vupheog quj hs v³jrtdfjsfhrrjsphecjetjs sgh cdsnueths).

?3




FH@JES ³GQDH

]dt³gkhrgs, quj `hrrju j` 34= g.F., lhd fheojfdch phr sjus fhetj`-phrgejhs fh`h luecgchr cj u` `hvd`jeth cj fueoh rjidkdhsh quj vjdh g sjrfheojfdch fh`h ]dtgkhrds`h. Hs pdtgk³hrdfhs detjrjssgvg`-sj pjig fdjefdgcj u` `hch kjrgi j pgrtdfuigr`jetj pjig Ldihshg j pjig @gtj`³gtdfg. Ehquj fhefjrej mg @gtj`³gtdfg g `gdhr fhetrdaud¿fgh chs pdtgk³hrdfhs lhd h cjsje-vhivd`jeth cg tjhrdg chs e³u`jrhs, j g cjsfhajrtg chs e³u`jrhs drrgfdhegds.Lhrg` jijs quj prhvgrg`, pjig prd`jdrg vjz, quj h eu`jrh

∞ 1 ³j drrgfdhegi.

G prhvg cjstj lgth gprjsjetgcg eh 7=◬ idvrh cj Jufidcjs ³j g sjkudetj>

Vupheog quj∞

1 ³j u` eu`jrh rgfdhegi. Jetgh∞

1 phcj sjr rjprjsjetgcheg lhr`g

∞ 1 0 `/e hecj ` j e sgh chds e³u`jrhs detjdrhs prd`hs jetrj sd.

Ihkh 1e1 0 `1. Fh`h fhesjqujefdg `1 ³j u` eu`jrh pgr. @gs jetgh ` ³jpgr j phcj`hs jsfrjvjr ` fh`h ` 0 1 p. ]hrtgeth 1e1 0 `1 0 3 p1. @gsjetgh e1 0 1 p1. Vjkuj-sj quj e1 ³j u` eu`jrh pgr j, fh`h fhesjqujefdg, e³j u` eu`jrh pgr. @gs sj e ³j pgr j ` ³j pgr jetgh hs chds egh sgh prd`hsjetrj sd. ]hr hutrh igch, eh dedfdh, ogv³ķg`hs jsfhiodchs ` j e prd`hs jetrjsd. Jstg fhetrgcd¿fgh prhvj` cg odp³htjsj cj quj

∞ 1 ³j rgfdhegi. ]hrtgeth,∞

1 egh ³j rgfdhegi.

Jstg cjsfhajrtg lhd, sj` cuvdcg, g krgecj fhetrdaud¿fgh ch ]dtgkhrds`hg Kjh`jtrdg Krjkg. Jig defiujefdhu cj lhr`g cjedtdvg h cjsjevhivd`jethquj tjvj g @gtj`³gtdfg Kr³jfdg g pgrtdr cg³ķ.

G ijecg sharj g hrdkj` ch tjhrj`g cj ]dt³gkhrgs cdz quj jij lhd cj-sfhajrth phr ]dt³gkhrgs h qugi sgfrdfhu 7== ahds ghs Cjusjs fh`h prhvg cjsug krgtdcgh phr tjr fhesjkudch jstg cjsfhajrtg.

Eh jetgeth, g vjrcgcj odst³hrdfg ³j quj h tjhrj`g cj ]dt³gkhrgs n³g jrgfheojfdch, j` fgshs pgrtdfuigrjs eh Jkdth (6.=== g.F.), j j` sug thtgi kje-jrgidcgcj pjihs Vu`³jrdhs j Agadihedhs (1=== g 7=== g.F.). J ³j aj` phss³ķvjiquj sug cj`hestrg¿fgh tjeog sdch hatdch eg Kr³jfdg j` ³jphfg getjrdhr g cj]dt³gkhrgs.

O³g u` krgecj e³u`jrh cj cj`hestrg¿fgh cjstj tjhrj`g. Ejssj fgp³ķtuihgprjstg`hs u`g cjigs, j giku`gs hutrgs sjrgh gprjsjetgcgs eh fgp³ķtuihrjigtdvh g ³grjgs, sha g lhr`g cj jxjrf³ķfdhs j prhaij`gs.

?:



KJH@JSQDG JZFIDCDGEG ]IGEG H F³DQFZIH

FG]³DSZIH ?

H F³DQFZIH

Eh fgp³ķtuih 1 cjed`hs f³ķrfuih cj fjetrh G j rgdh r fh`h h fhenuethchs pheths ch pigeh quj jstgh g u`g cdstgefdg r ch pheth G. Sgà³j`fog`grj`hs cj rgdh gh sjk`jeth quj uej h fjetrh ch f³ķrfuih g qugiqujr cjsjus pheths. H sjk`jeth idkgech chds pheths cj u` f³ķrfuih sjr³g cjeh`degch

cj dt fhrcg. Shcg fhrcg quj pgssg pjih fjetrh ch f³ķrfuih ³j u` cdˌ g`jtrh.Sgà³j` fog`grj`hs cj cdg`jtrh g cdstgefdg 1r. Egh ogvjr³g pjrdkh cjfhelusgh eh ush cg `js`g pgigvrg pgrg sdkedfgr cugs fhdsgs cdljrjetjs. Ehfhetjxth sjr³g sj`prj figrh g quj jstg`hs ehs rjljrdech.

H C

F

G

A

GA j FC sgh fhrcgsFC ³j u cdg`jtrh

Ldkurg ?.7

?.7 ]rhphsd¿fgh> Z` rgdh ³j pjrpjecdfuigr g u`g fhrcg (quj egh ³j u`cdg`jtrh) sj j sh`jetj sj g cdvdcj j` chds sjk`jeths fhekrujetjs.

]rhvg. Vjng H h fjetrh ch f³ķrfuih j HF h rgdh quj ³j pjrpjecdfuigr gfhrcg GA. Vjng @ h pheth cj detjrfjssgh cg fhrcg fh` rgdh. Fh` HG 0 HA

(rgdhs) jetgh h trdgekuih HGA ³j ds³hsfjijs fh` agsj GA.

?<




F H

A

G

Ldkurg ?.1

Ihkh ˌ G 0 ˌ A . Vj g fhrcg ³j pjrpjecdfuigr gh rgdh, jetgh hs gekuihs H ˌ @G

j H ˌ @A sgh rjths. Fh`h fhesjqujefdg G ˌ H@ 0 A H@ . Vjkuj-sj jetgh, pjihprd`jdrh fgsh cj fhekrujefdg cj trdgekuihs, quj GH@ 0 AH@ . Fh`hfhesjqujefdg G@ 0 @ A. Devjrsg`jetj, sj G@ 0 @ A, jetgh, pjih tjrfjdrhfgsh cj fhekrujefdg cj trdgekuihs cjcuz-sj quj> GH@ 0 AH@ . Fh`hfhesjqujefdg, H ˌ @G 0 H ˌ @A . @gs fh`h g sh`g cjstjs chds gekuihs ³j u`gekuih rgsh, fhefiud-sj quj fgcg u` cjijs `jcj 4=◬. ]hrtgeth g fhrcg ³jpjrpjecdfuigr gh rgdh pgssgech phr @ . Dsth fh`pijtg g prhvg cg prhphsd¿fgh.

Wugech u`g rjtg j u` f³ķrfuih tj` gpjegs u` pheth j` fhù`, cdzj`hsquj g rjtg tgekjefdg h f³ķrfuih j fog`g`hs g rjtg cj tgekjetj gh f³ķrfuih.H pheth fhù` jetrj u`g tgekjetj j u` f³ķrfuih ³j fog`gch cj pheth cj

tgekjefdg hu pheth cj fhetgth.

?.1 ]rhphsd¿fgh Vj u`g rjtg ³j tgekjetj g u` f³ķrfuih jetgh jig ³j pjr-pjecdfuigr gh rgdh quj idkg h fjetrh gh pheth cj tgekjefdg.

H

`

S

]

S

Ldkurg ?.6

]rhvg. Fhesdcjrj`hs u` f³ķrfuih cj fjetrh H j u`g rjtg ` quj ioj

?;




sjng tgekjetj. Vjng S h pheth cj tgekjefdg. Cjsdkegrj`hs phr ] h p³j cg

pjrpjecdfuigr agdxgcg ch pheth H mg rjtg `. Khstgr³ķg`hs cj cj fhefiudr quj] j S fhdefdcj`. \g`hs jetgh suphr quj ] j S sgh pheths cdstdeths. JetghHS ³j g odphtjeusg ch trdgekuih rjtgekuih H] S > ]hrtgeth H] 9 HS . Fh`hHS ³j u` rgdh, jetgh ] ³j u` pheth quj jst³g cjetrh ch f³ķrfuih. Sh`j`hsjetgh u` pheth S sharj g rjtg `, tgi quj ] S 0 ] S , fh` S 0 S . ]jihprd`jdrh fgsh cj fhekrujefdg cj trdgekuihs fhefiu³ķ`hs quj H] S 0 H] S .]hrtgeth HS 0 HS . @gs jetgh S ³j hutrh pheth cg rjtg ` quj tgà³j`pjrtjefj gh f³ķrfuih. Ihkh g rjtg ` egh ³j tgekjetj. Fhetrgcd¿fgh! Gssd` ]

j S fhdefdcj` j g prhphsd¿fgh fg cj`hestrgcg.G jxtrj`dcgcj cj u` rgdh quj egh ³j h fjetrh ch f³ķrfuih ³j fog`gcg cj

jxtrj`dcgcj ch rgdh.

?.6 ]rhphsd¿fgh. Vj u`g rjtg ³j pjrpjecdfuigr g u` rgdh j` sug jx-trj`dcgcj, jetgh g rjtg ³j tgekjetj gh f³ķrfuih.

]rhvg. Fhesdcjrj`hs u` f³ķrfuih cj fjetrh H j sjng ` u`g rjtg pjr-pjecdfuigr gh rgdh HS pgssgech pjih pheth S . Cjvj`hs prhvgr quj ` ³jtgekjetj gh f³ķrfuih, hu sjng, quj ` egh tj` hutrh pheth cj detjrfjssgh fh`h f³ķrfuih. Vjng ] qugiqujr hutrh pheth cj `, jetgh h trdgekuih HS ] ³j

rjtgekuih j phrtgeth HS 1

+ S ] 1

0 H] 1

. Vjkuj-sj quj H] 5 HS j phr-tgeth ] jst³g lhrg ch f³ķrfuih. Ihkh S ³j h ³uedfh pheth fhù` mg rjtg j ghf³ķrfuih. Dsth fhefiud g cj`hestrg¿fgh.

Vjng G j A chds pheths cj u` f³ķrfuih. Srgfj`hs g rjtg quj pgssg phrjstjs chds pheths. Jig sjpgrg h pigeh j` chds sj`d-pigehs. Fgcg u` cjstjssj`dpigehs fhetj` u`g pgrtj ch f³ķrfuih. Jstgs pgrtjs sgh cjeh`degcgscj grfhs cjtjr`degchs pjih pheths pheths G j A. Wugech G j A sgh jx-trj`dcgcjs cj u` cdg`jtrh, jstjs grfhs sgh cjeh`degchs cj sj`df³ķrfuihs .Wugech g fhrcg GA egh ³j u` cdg`jtrh, cdstdekud`hs hs chds grfhs cjtjr-`degchs phr G j A ch sjkudetj `hch> fh`h h fjetrh ch f³ķrfuih jefhetrg-sjj` u` sj`dpigeh cjtjr`degchs pjig rjtg quj pgssg phr G j A, h grfh qujfg eh `js`h sj`dpigeh quj h fjetrh ch f³ķrfuih ³j fog`gch cj grfh `gdhr ,h hutrh ³j fog`gch cj grfh `jehr . Hasjrvj quj hs rgdhs quj idkg` h fjetrhch f³ķrfuih ghs pheths ch grfh `jehr thchs fhrtg` g fhrcg GA. N³g hs rgdhs

quj idkg` h fjetrh ch f³ķrfuih ghs pheths ch grfh `gdhr egh detjrfjptg` gfhrcg GA.

??




G

A

H

grfh `jehr

grfh `gdhr

Ldkurg ?.3

Vj H ³j h fjetrh ch f³ķrfuih jetgh G ˌ HA ³j fog`gch cj ˌ gekuih fjetrgi .G `jcdcg j` krgus ch grfh `jehr cjtjr`degch pjihs pheths G j A ³j phrcjed¿fgh g `jcdcg ch gekuih fjetrgi G ˌ HA . G `jcdcg j` krgus ch grfh

`gdhr ³j cjedcg fh`h sjech 6<=◬ ∑ g◬, hecj g◬ ³j g `jcdcg j` krgus chgrfh `jehr. Eh fgsh j` quj GA ³j u cdg`jtrh g `jcdcg chs chds grfhs ³j7?=◬.

?.3 ]rhphsd¿fgh. J` u` `js`h f³ķrfuih, hu j` f³ķrfuihs cj `js`h rgdh,fhrcgs fhekrujetjs cjtjr`deg` gekuihs fjetrgds fhekrujetjs j rjfdprhfg-`jetj.

G prhvg cjstg prhphsd¿fgh ³j sd`pijs j ³j cjdxgcg g fgrkh ch ijdthr. Z`gfhesjqujefdg cjig ³j quj fhrcgs fhekrujetjs cjtjr`deg` grfhs `jehrjs cj`js`g `jcdcg j phrtgeth, tgà³j` grfhs `gdhrjs cj `js`g `jcdcg.

H ijdthr cjvj hasjrvgr quj, g `gejdrg sj sh`gr gekuihs quj tj` h `js`hv³jrtdfj pjr`dtj detrhcuzdr u`g `gejdrg cj sh`gr grfhs quj sj nustgphj`.Jstg sh`g ³j gsshfdgtdvg j fhùtgtdvg fh`h h ³j g sh`g cj gekuihs.

Z` gekuih sj cjeh`deg desfrdth j` u` f³ķrfuih sj sju v³jrtdfj G ³j u`pheth pheth ch f³ķrfuih j sjus igchs fhrtg` h f³ķrfuih j` pheths A j F

cdstdeths ch pheth G. Hs pheths A j F cjtjr`deg` chds grfhs. H grfh qujegh fhetdvjr h pheth G ³j fog`gch cj grfh fhrrjsphecjetj gh gekuih desfrdth

cgch. Cdrj`hs tgà³j` quj h gekuih suatjecj h grfh.

?4




H H HG

F

A

G

F

A

G

F

A

Ldkurg ?.:

?.: ]rhphsd¿fgh. Shch gekuih desfrdth j` u` f³ķrfuih tj` g `jtgcj cg`jcdcg ch grfh fhrrjsphecjetj.

]rhvg. Fhesdcjrj`hs prd`jdrh h fgsh j` quj u` chs igchs ch gekuihdesfrdth ³j u` cdg`jtrh. Vjng G h v³jrtdfj ch gekuih desfrdth j A j F hspheths j` quj sjus igchs fhrtg` h f³ķrfuih. Vupheog quj h fjetrh H chf³ķrfuih pjrtje¿fg gh igch GF . Ejstj fgsh, g `jcdcg ch grfh fhrrjsphecjetjgh gekuih desfrdth ³j g `jcdcg ch gekuih A ˌ HF . Fh`h AH 0 GH jetgh htrdgekuih HGA ³j ds³hsfjijs j phrtgeth H ˌ GA 0 H ˌ AG. @gs jetgh A ˌ HF 0H ˌ GA + H ˌ AG 0 1 » F ˌ GA . ]hrtgeth, ejstj fgsh pgrtdfuigr g prhphsd¿fgh ³jvjrcgcjdrg.

HH HG F

A

G

AF

C

GA

F

C

Ldkurg ?.<

Vupheog`hs gkhrg quj ejeou` chs igchs ch gekuih desfrdth ³j u` cdg`jtrh.Srgfj`hs jetgh h cdg`jtrh quj pgssg pjih v³jrtdfj G ch gekuih desfrdth. VjngC g hutrg jxtrj`dcgcj cjstj cdg`jtrh. ]jih prd`jdrh fgsh fhefiudrj`hs qujA ˌ HC 0 1 » A ˌ GC j quj C ˌ HF 0 1 » C ˌ GF .

Ejstj pheth tj`hs cj cdstdekudr chds fgshs> (g) h cdg`jtrh GC cdvdcjh gekuih A ˌ GF . (a) H cdg`jtrh GC egh cdvdcj h gekuih A ˌ GF . (\jng kurggfd g). Eh fgsh (g), tj`hs quj A ˌ GC + C ˌ GF 0 A ˌ GF . G cj`hestrg¿fgh ³jjetgh fh`pijtgcg sh`gech-sj gs dkugicgcjs n³g jefhetrgcgs>

A ˌ HC + C ˌ HF 0 1 » (A ˌ GC + C ˌ GF ) 0 1 » A ˌ GF .

4=




Hasjrvj quj A ˌ HC + C ˌ HF ³j jxgtg`jetj g `jcdcg ch grfh fhrrjsphecjetj

gh gekuih A ˌ GF . Eh fgsh (a), phcj` gdecg gcvdr cugs sdtug¿fhjs cdstdetgs>(d) GF cdvdcj h gekuih A ˌ GC j (dd) GA cdvdcj h gekuih F ˌ GC . G prhvg ehschds fgshs ³j jssjefdgi`jetj g `js`g. Lgrj`hs h fgsh (d). Ejstj fgsh A ˌ GF 0A ˌ GC - F ˌ GC . Jetgh, utdidzgech-sj gs cugs dkugicgcjs hatdcgs dedfdgi`jetj,tj`-sj.

A ˌ HC ∑ F ˌ HC 01 » (A ˌ GC - F ˌ GC ) 0 1 » A ˌ GF .

Gkhrg hasjrvj quj A ˌ HC ∑ F ˌ CH ³j jxgtg`jetj g `jcdcg ch grfh fhrrj-sphecjetj gh gekuih A ˌ GF . Dsth fh`pijtg g cj`hestrg¿fgh.

?.< Fhrhi³grdh. Shchs hs gekuihs desfrdths quj suatjecj` u` `js`hgrfh tj` g `js`g `jcdcg. J` pgrtdfuigr, thchs hs gekuihs quj suatjecjù` sj`d f³ķrfuih sgh rjths.

GA

F

F F

FH

G

F

A

Ldkurg ?.;

G sjkudetj prhphsd¿fgh ³j tgà³j` cj fjrth `hch u` fhrhi³grdh cg prhphsd¿fgh(?.:).

?.; ]rhphsd¿fgh. Vjng` GA j FC fhrcgs cdstdetgs cj u` `js`h f³ķrfuihquj sj detjrfjptg` eu` pheth ] . Jetgh G] » ] A 0 F] » ] C.

]rhvg. Hasjrvj quj ehs trdgekuihs F]A j CG] tj`-sj> F ˌ ]A 0 G

ˌ ]A (hphsths pjih v³jrtdfj) j F A] 0 G ˌ C] (phr sjrj` gekuihs desfrdths quj sua-

tjecj` h `js`h grfh). Ihkh, hs chds trdgekuihs sgh sj`jiogetjs j g sj`ji-

oge¿fg ³j g quj ijvg F j` G, ] j` ] j A j` C . Ihkh F]

G] 0

] A

] C. @gs

jetgh, G] » ] A 0 F] » ] C.

47




G

C

A

F

]

Ldkurg ?.?

?.? ]rhphsd¿fgh. Vj hs chds igchs cj u` gekuih cj v³jrtdfj ] sgh tge-kjetjs g u` f³ķrfuih ehs pheths G j A , jetgh>

g) g `jcdcg ch gekuih ˌ ] ³j dkugi g 7?=◬ `jehs g `jcdcg ch grfh `jehrcjtjr degch phr G j A 2

a) ]G 0 ]A .

]rhvg. Vjng H h fjetrh ch f³ķrfuih. Eh qugcrdi³gtjrh HG]A tj`hs qujG 0 A 0 4=◬. Ihkh ] + H 0 7?=◬ . Fh`h ˌ H ³j jxgtg`jetj g `jcdcg ch grfh`jehr cjtjr`degch phr G j A , fg prhvgch g pgrtj (g). grg prhvgr g pgrtj

(a), trgfj ]H j fh`pgrj hs trdgekuihs ]GH j ]AH . Fh`h ˌ G 0

ˌ A 0 4=

◬

hs chds trdgekuihs sgh rjtgekuihs. Fh`h GH 0 A H (rgdhs) j ] H ³j fhù ,jetgh hs chds trdgekuihs sgh fhekrujetjs (fhelhr`j h tjhrj`g (:.73)). Ihkh] G 0 ] A. Gssd` h rjsuitgch fg cj`hestrgch.

]G

A

H

Ldkurg ?.4

Cdrj`hs quj u` phi³ķkheh jstg desfrdth eu` f³ķrfuih sj sjus v³jrtdfjs pjr-tjefj` gh f³ķrfuih.

?.4 ]rhphsd¿fgh. Shch trdgekuih jst³g desfrdth j` u` f³ķrfuih.

]rhvg. Vjng GAF u` trdgekuih. ]grg `hstrgr quj jij jst³g desfrdth jù` f³ķrfuih cjvjrj`hs jxdadr u` pheth quj sjng g jqudcdstgetj cj G, A j

41




F . Vjng ` u`g rjtg pjrpjecdfuigr g GA j pgssgech pjih sju pheth `³jcdh

@ j sjng e g rjtg pjrpjecdfuigr g AF j pgssgech pjih sju pheth `³jcdh E .Cjsdkej phr ] h pheth cj detjrfjssgh cjstgs cugs rjtgs. Hasjrvj quj thchpheth cg rjtg ` ³j jqudcdstgetj cj G j A, j quj thch pheth cg rjtg e ³jjqudcdstgetj cj A j F . Ihkh h pheth ] sjr³g jqudcdstgetj cj G, A j F .

F

E

A

@

G

e

]

`

Ldkurg ?.7=

Jstg prhphsd¿fgh phcj sjr rjjeuefdgcg cg sjkudetj `gejdrg>

?.7= ]rhphsd¿fgh. Srjs pheths egh fhidejgrjs cjtjr`deg` u` f³ķrfuih.

Fog`g`hs cj `jcdgtrdz cj u` cgch sjk`jeth mg rjtg pjrpjecdfuigr ghsjk`jeth pgssgech pjih sju pheth `³jcdh. Fh` jstg cjed¿fgh phcj`hsjeuefdgr h sjkudetj fhrhi³grdh cg prhphsd¿fgh (?.4).

?.77 ]rhphsd¿fgh. Gs `jcdgtdzjs chs igchs cj u` trdgekuih jefhetrg`-sj j` u` `js`h pheth.

Cj u` `hch kjrgi gpjegs hs trdgekuihs phssuj` g prhprdjcgcj cj sjrj`desfrdt³ķvjds j` f³ķrfuihs. ]grg hutrhs phi³ķkhehs g fhecd¿fgh cj quj h `js`h

phssg sjr desfrdth j` u` fdrfuih gfgrrjtg lhrtjs rjstrd¿fhjs sharj gs sugs`jcdcgs. G sjkudetj prhphsd¿fgh ³j u` jxj`pih cdsth.

?.71 ]rhphsd¿fgh. Z` qugcrdi³gtjrh phcj sjr desfrdth j` u` f³ķrfuih sjj sh`jetj sj phssud u` pgr cj gekuihs hphsths supij`jetgrjs.

]rhvg. \g`hs suphr dedfdgi`jetj quj h qugcrdi³gtjrh phssg sjr desfrdthj` u` f³ķrfuih. Hasjrvj quj fgcg u` cj sjus gekuihs ³j u` gekuih desfrdtheh f³ķrfuih. Vjng GAFC h qugcrdi³gtjrh. Fhesdcjrj hs gekuihs ˌ G j ˌ F . Jijssuatjecj` jxgtg`jetj hs chds grfhs cjtjr`degchs pjihs pheths A j F . Fh`h

46




jstjs chds grfhs sh`g` 6<=◬, jetgh, cj gfhrch fh` g prhphsd¿fgh (?.:), g sh`g

chs gekuihs ˌ G j ˌ F sjr³g 7?=◬

. ]hrtgeth jijs sgh supij`jetgrjs.

A

G

C

F

A

F

C

J

G

Ldkurg ?.77

\g`hs gkhrg suphr quj u` qugcrdi³gtjrh GAFC tj` u` pgr cj gekuihshphsths supij`jetgrjs. Fh`h g sh`g chs gekuihs detjrehs ch qugcrdi³gtjrh³j 6<=◬, jetgh h hutrh pgr cj gekuihs hphsths tgà³j` ³j supij`jetgr. Srgfju` fdrfuih pjihs pheths G, A j F . Dsth sj`prj phcj sjr ljdth (cj gfhrchfh` (?.7=)). V³h jxdstj` trjs gitjregtdvgs pgrg g ihfgidzg¿fgh ch pheth C >jij phcj jstgr sharj, cjetrh, hu lhrg ch f³ķrfuih. \g`hs suphr quj jij jstjnglhrg ch f³ķrfuih. Ejstj fgsh trgfj h sjk`jeth AC . Vjng J h pheth hecjjstj fhrtg h f³ķrfuih. H qugcrdi³gtjrh GAFJ ³j u` qugcrdi³gtjrh desfrdth eh

f³ķrfuih j, phrtgeth, pjig prd`jdrg pgrtj cg prhphsd¿fgh sjus gekuihs hphsthssgh supij`jetgrjs. J` pgrtdfuigr tj`hs

G ˌ AF + G ˌ JF 0 7?=◬

]hr odp³htjsj tgà³j` tj`hs

G ˌ AF + G ˌ CF 0 7?=◬

Cgs cugs dkugicgcjs fhefiu³ķ`hs quj G ˌ CF 0 G ˌ JF . Gkhrg hasjrvj qujG ˌ JA 5 G ˌ CA j A JF + A ˌ CF (gekuihs jxtjrehs). Ihkh>

G ˌ JF 0 G ˌ JA + A JF 5 G ˌ CA + A ˌ CF 0 G ˌ CF

Jstg fhetrgcd¿fgh `hstrg quj C egh phcj jstgr lhrg ch f³ķrfuih. H rjsth cgprhvg `hstrgech quj C tgà³j egh phcj jstgr cjetrh ch f³ķrfuih ³j cjdxgcgfh`h jxjrf³ķfdh.

Z` f³ķrfuih jstg desfrdth j` u` phi³ķkheh sj thchs hs igchs ch phi³ķkheh sghtgekjetjs gh f³ķrfuih. Wugech tgi hfhrrj cdz-sj quj h phi³ķkheh fdrfuesfrjvj

h f³ķrfuih.

?.76 ]rhphsd¿fgh. Shch trdgekuih phssud u` f³ķrfuih desfrdth.

43




]rhvg. Vjng GAF u` trdgekuih. Srgfj gs adssjtrdzjs chs gekuihs ˌ G j

A. Jstgs sj jefhetrg` j` u` pheth ] . Cjstj pheth, agdxj pjrpjecdfuigrjsghs igchs chs trdgekuihs. Vjng` J , L j K hs p³js cjstgs pjrpjecdfuigrjs ehsigchs GA , AJ j FG, rjspjftdvg`jetj. \g`hs prhvgr quj ] J 0 ] L 0 ] K.Gssd` h pheth ] ³j h fjetrh cj u` f³ķrfuih quj pgssg pjihs pheths J , L j K 2gi³j` cdsth, fh`h hs igchs ch trdgekuih GAF sgh pjrpjecdfuigrjs ghs rgdhs]J , ]L j ]K jijs sgh tgà³j` tgekjetjs gh f³ķrfuih. Ihkh h f³ķrfuih jst³gdesfrdth eh trdgekuih.

F

L

A

J

G

K

]

Ldkurg ?.71

]grg prhvgr quj ] J 0 ] L 0 ] K vg`hs fh`pgrgr hs trdgekuihs ] KG

j ] JG, j hs trdgekuihs ]JA j ]LA . Shchs jijs sgh trdgekuihs rjtgekuihs.Ehs chds prd`jdrhs tj`hs ] ˌ GK 0 ] ˌ GJ (]G ³j adssjtrdz) j ]G fh u`. Ehschds ³uitd`hs tj`hs ] ˌ AJ 0 ] ˌ AL (]A ³j adssjtrdz) j ]A fhù`. ]hrtgeth hschds pgrjs cj trdgekuihs sgh fhekrujetjs. Cg fhekrujefdg chs chds prd`jdrhsfhefiu³ķ`hs quj ] K 0 ] J . Cg fhekrujefdg chs chds ³uitd`hs hatj`hs ] J 0] L . Dsth fh`pijtg g cj`hestrg¿fgh.

?.73. Fhrhi³grdh. Gs adssjtrdzjs cj u` trdgekuih jefhetrg`-sj j` u`

pheth.

]rhvg. Eg cj`hestrg¿fgh getjrdhr prhvg`hs quj h pheth cj jefhetrh cjcugs adssjtrdzjs ch trdgekuih GAF ³j h fjetrh cj u` f³ķrfuih desfrdth egqujijtrdgekuih. ]grg hatjr h Fhrhi³grdh ³j sufdjetj prhvgr quj h sjk`jeth uedechh fjetrh cjstj f³ķrfuih desfrdth fh` h tjrfjdrh v³jrtdfj, ³j tgà³j` u`g adssjtrdzch trdgekuih GAF . H ijdthr egh tjr³g cdfuicgcj j` lgzjr jstg cj`hestrg¿fghquj ³j cjdxgcg fh`h jxjrf³ķfdh.

Z` phi³ķkheh rjkuigr ³j u phi³ķkheh quj ³j jqudi³gtjrh j jqudgekuigr. Fh`dsth qujrj`hs cdzjr quj thchs hs sjus igchs sgh fhekrujetjs (jqudi³gtjrh) j

4:




tgà³j` thchs hs sjus gekuihs sgh fhekrujetjs (jqudgekuigr).

?.7: ]rhphsd¿fgh. Shch phi³ķkheh rjkuigr jst³g desfrdth j` u` f³ķrfuih.

]rhvg. Vjng G7G1,...,Ge u` phi³ķkheh rjkuigr. Srgfj`hs h f³ķrfuih qujpgssg pjihs pheths G7, G1 j G6. Vjng H h fjetrh cjstj f³ķrfuih. Fh`h HG1 0HG6 jetgh h trdgekuih HG1G6 ³j ds³hsfjijs j ihkh H ˌ G1G6 0 H ˌ G6G1. Fh`hh phi³ķkheh ³j rjkuigr thchs hs sjus gekuih detjrehs tj` g `js`g `jcdcg.]hrtgeth G7

ˌ G1G6 0 G1 ˌ G6G3.

H

G7

G<

G:

G3

G6

G1

GeG7

G1

G6

G3

G:

Ldkurg ?.76

@gs jetgh, G7 ˌ G1H 0 H ˌ G6G3. Fh`h gi³j` cdssh G7G1 0 G6G3 (igchs

cj u` phi³ķkheh rjkuigr sgh fhekrujetjs) j HG1 0)G6, jetgh hs trdgekuihsHG7G1 j HG3G6 sgh fhekrujetjs. Cg³ķ hatj`-sj HG3 0 HG7. ]hrtgeth G3

tgà³j` ³j u` pheth ch f³ķrfuih. H `js`h rgfdhf³ķedh phcj gkhrg sjr rjpjtdchpgrg prhvgr quj G: tgà³j` pjrtjefj gh f³ķrfuih, j gssd` sufjssdvg`jetj.Fh`h rjsuitgch egi hatj`-sj quj thchs hs pheths ch phi³ķkheh pjrtjefj`gh f³ķrfuih.

?.7< Fhrhi³grdh. Shch phi³ķkheh rjkuigr phssud u` f³ķrfuih desfrdth.

]rhvg. Srgfj h f³ķrfuih eh qugi h phi³ķkheh rjkuigr G7G1...Ge jst³g de-sfrdth. Vjng H h sju fjetrh. Shch hs trdgekuihs ds³hsfjijs G7HG1, G1HG6, G6HG3,...

sgh fhekrujetjs. Fh`h fhesjqujefdg sugs giturgs rjigtdvg`jetj gs agsjs sghtgà³j` fhekrujetjs. H f³ķrfuih cj fjetrh H j fh` rgdh dkugi gh fh`prd`jethcjstgs giturgs jst³g desfrdth eh phi³ķkheh.

4<




J_JQF³DFDHV

7. Wugech h shi jst³g g 1=◬ gfd`g ch ohrdzhetj, qugi h fh`prd`jeth cgshàrg prh njtgcg phr u` jcdl³ķfdh cj :=`8

1. ]rhvj quj, j` u` `js`h f³ķrfuih hu j` f³ķrfuihs cj `js`h rgdh, fhrcgsjqudcdstgetjs ch fjetrh sgh fhekrujetjs.

6. ]rhvj quj, j` u` `js`h f³ķrfuih hu j` f³ķrfuihs cj `js`h rgdh, sjcugs fhrcgs tj` fh`prd`jeths cdljrjetjs, g `gds furtg ³j g `gds glgstgcg ch

fjetrh.

3. @hstrj quj g `jcdgtrdz cj u`g fhrcg pgssg pjih fjetrh ch f³ķrfuih.

:. Jxpidquj phrquj h rjfijxh cj u` f³ķrfuih rjigtdvg`jetj g u`g rjtg qujpgssg pjih sju fjetrh ³j gdecg h `js`h f³ķrfuih.

<. Eg kurg gh igch jxdstj` trjsrjtgs quj sgh tgekjetjs sdùitgejg-`jetj ghs chds f³ķrfuihs. Jstgs rj-tgs sgh cjeh`degcgs cj tgekjetjs

fhùes ghs f³ķrfuihs. Cjsjeoj chdsf³ķrfuihs quj tjeog`>g) Wugtrh tgekjetjs fhùes8a) jxgtg`jetj cugs tgekjetjs fh-ùes8f) sh`jetj u`g tgekjetj fhù`8c) ejeou`g tgekjetj fhù`8j) `gds cj qugtrh tgekjetjs fh-ùes8

4;




;. Eg kurg gh igch GJ ³j tgekjetjfhù` j NV idkg hs fjetrhs chs chdsf³ķrfuihs. Hs pheths J j G sgh phe-ths cj tgekjefdg j A ³j h pheth cjdetjrfjssgh chs sjk`jeths NV j GJ .]rhvj quj h gekuih ˌ N ³j dkugi ghgekuih ˌ V .

V

J

@G

N

?. Eg kurg sjkudetj, @ ³j h fjetrh chs chds f³ķrfuihs j GB ³j tgekjetj ghf³ķrfuih `jehr eh pheth Q. @hstrj quj GQ 0 QB .

@

BGQ

I

Z B

J

4. Eg gfd g mg cdrjdtg, ZB ³j tgekjetj gh f³ķrfuih eh pheth Z j ZJ 0IZ . @hstrj quj IJ 0 JB .

7=. Eg kurg sjkudetj mg jsqujrcg, @H 0 D_ . ]rhvj quj @D 0 H_ .

@ _

H D

DOF

G

E

Chds pheths j` u` f³ķrfuih cjtjr`deg` chds grfhs. Vj hs chds pheths sgh

G j A e³hs drj`hs rjprjsjetgr phr

GA h grfh `jehr cjtjr`degch phr jstjs

chds pheths. Vj ] tgà³j pjrtjefj gh f³ķrfuih e³hs usgrj`hs g ehtg¿fgh

G]Apgrg rjprjsjetgr h grfh cjtjr`degch phr G j A quj fhet³j` h pheth ] .

4?




77. Eg kurg getjrdhr mg cdrjdtg, O ³j h fjetrh ch f³ķrfuih j FD j u`

cdg`jtrh. Vj FG j OE sgh pgrgijihs, `hstrj quj

GE j

DE tj` g `js`g`jcdcg.

71. Eg kurg g jsqujrcg, H ³j h fjetrh ch f³ķrfuih j SG ³j u cdg`jtrh.Vj ]G 0 G^ , `hstrj quj hs trdgekuihs ]GS j ^GS sgh fhekrujetjs.

]

G

^

S

H

A

I

Q

J

T

76. Eg kurg gfd`g mg cdrjdtg, sgaj-sj quj T ³j h fjetrh ch f³ķrfuih j qujAI 0 JQ. @hstrj quj AJ ³j pgrgijih g IQ.

73. Eg kurg sjkudetj mg jsqujrcg, h qugcrdi³gtjrh CDGE ³j u` pgrgijih-krg`h j D , G j @ sgh fhidejgrjs. @hstrj quj CD 0 C@ .

G @J

S

V

TO

CD

G

@

EH

7:. Eg kurg getjrdhr g cdrjtg, qugi chs chds grfhs

GO hu

@T, tj` g`gdhr `jcdcg j` krgus8 Vgaj sj quj hs chds f³ķrfuihs sgh fhefjetrdfhs.

Z`g rjtg detjrfjptg u` f³ķrfuih j` eh `³gxd`h chds pheths (vjng prha-ij`g 7). Gs quj h detjrfjptg` j` jxgtg`jetj chds pheths sgh fog`gchs cjsjfgetjs . Z` ˌ gekuih sjfgetj ³j u gekuih funhs igchs jstgh fhetdchs j` cugssjfgetjs ch f³ķrfuih j quj fgcg igch detjrfjptg` h f³ķrfuih j` pjih `jehs u`pheth jxfiu³ķch h v³jrtdfj. \g`hs fog`gr cj rjkd˙ gh gekuigr gsshfdgcg g u`

44




gekuih G ˌ AF g detjrfjssgh chs sjkudetjs sj`d-pigehs> h quj fhetj` h pheth

F j ³j cjtjr`degch phr GA , j h quj fhetj` h pheth G j ³j cjtjr`degch phrAF . Cgchs u` gekuih j u` fdrfuih, g pgrtj ch f³ķrfuih fhetdcg eg rjkdghgekuigr gsshfdgcg gh gekuih cgch ³j cjsdkegch grfh (hu grfhs) cjtjr`degch

(cjtjr`degchs) pjih ˌ gekuih.

Ehs jxjrf³ķfdhs sjkudetjs decdfgrj`hs phr ’`jc

GA‐ g `jcdcg j` krgus

ch grfh

GA.

FG

A

]

C

] F

C

G

A

7<. @hstrj quj u` gekuih sjfgetj funh v³jrtdfjs jstg cjetrh ch f³ķrfuihtj` `jcdcg dkugi g `jtgcj cg sh`g ch grfh quj cjtjr`deg fh` h grfh quj³j cjtjr`degch pjih gekuih quj sj ioj hphj pjih v³jrtdfj. (Eg kurg getjrdhr

g jsqujrcg> G]A 0 7

1(`jc

GA + `jc

FC).

7;. Eg kurg getjrdhr G]A ³j u gekuih sjfgetj funh v³jrtdfj jstg lhrg

ch f³ķrfuih `hstrj quj G]A 0 7

1(`jc

GA - `jc

FC).

7?. Vufjssdvhs grfhs sgh `grfgchs j` u` f³ķrfuih cj `hch quj fgcg grfhtjeog u`g fhrcg cj `js`h fh`prd`jeth quj h rgdh. ]rhvj quj h sjxth grfhtjr`deg eh pheth hecj h prd`jdrh grfh fh`j¿fg.

74. ]rhvj quj thch pgrgijihkrg`h desfrdth j` u` f³ķrfuih ³j rjtgekuih.

1=. ]rhvj quj thch trgp³jzdh desfrdth j` u` f³ķrfuih ³j ds³hsfjijs.

17. ]rhvj quj h sjk`jeth idkgech u` v³jrtdfj cj u` phi³ķkheh rjkuigrgh fjetrh ch f³ķrfuih j` quj jij jstg desfrdth ³j adssjtrdz ch gekuih cgqujijv³jrtdfj.

11. Cjsjeoj chds jxj`pihs cj phi³ķkhehs jqudgekuigrjs desfrdths j` u`f³ķrfuih, `gs quj egh sgh rjkuigrjs.

16. Cjsjeoj chds jxj`pihs cj phi³ķkhehs jqudi³gtjrhs quj fdrfuesfrjvj`

7==




u` f³ķrfuih, `gs quj egh sgh rjkuigrjs.

13. Cgch u` qugcrgch cj igch :f`, qugi h rgdh ch f³ķrfuih gh qugi jijjst³g desfrdth8 Wugi h rgdh ch f³ķrfuih quj jij fdrfuesfrjvj8

1:. Cgch u` trdgekuih jqudi³gtjrh cj igch 3f`, qugi h rgdh ch f³ķrfuih ehqugi jij jst³g desfrdth. J qugi h rgdh ch f³ķrfuih quj jij fdrfuesfrjvj8

7=7




]QHAIJ@GV

7. ]rhvj quj u`g rjtg phcj fhrtgr u` f³ķrfuih j` eh `³gxd`h chds pheths.

1. Eg kurg gagdxh mg jsqujrcg G ˌ ]F ³j u ³gekuih sjfgetj funh v³jrtdfjjefhetrg-sj lhrg ch f³ķrfuih j quj h detjrfjptg j` qugtrh pheths fh`h de-cdfgch. ]rhvj quj G] » ] A 0 F] » ] C. Fh`pgrj jssj rjsuitgch fh` g]rhphsd¿fgh (?.;).

G A

F

C

]

D

Q[

K

V

SO

6. Eg kurg gfd`g mg cdrjdtg [ V j O D sgh fhrcgs quj sj detjrfjptg` eh

pheth K , j QS ³j adssjtrdz ch gekuih [ ˌ KD . ]rhvj quj [ Q » S V 0 OS » QD .

3. Vjng GAF u` trdgekuih j C u` pheth cj AF tgi quj GC ³j adsstrdzch gekuih ˌ G. ]rhvj quj (GC)1 0 GA » GF ∑ AC » CF . (Gnucg> fhesdcjrjh f³ķrfuih eh qugi h trdgekuih jst³g desfrdth, prhihekuj GC gt³j h pheth J

ch f³ķrfuih, fh`h decdfgch eg kurg sjkudetj mg jsqujrcg.) @hstrj quj hstrdgekuihs GAJ j GCF sgh sj`jiogetjs j usj (?.;).

G

F

JA C

:. Eg kurg sjkudetj h f³ķrfuih jst³g desfrdth eh qugcrdi³gtjrh. ]rhvj qujg sh`g chs fh`prd jeths cj u` pgr cj igchs hphsths ³j dkugi g sh`g chs

7=1




fh`prd`jeths ch hutrh pgr.

<. Vjng GAFCJL u` ojx³gkheh quj fdrfuesfrjvj u` f³ķrfuih. ]rhvj qujGA + FC + JL 0 AF + CJ + L G.

;. Jeuefdj j prhvj u`g prhphsd¿fgh sj`jiogetj g ch jxjrf³ķfdh getjrdhrpgrg u` phi³ķkheh cj ? igchs. Vjr³g quj u`g prhphsd¿fgh sj`jiogetj vgijpgrg u` phi³ķkheh cj 7= igchs8 J pgrg u` cj 77 igchs8 Jeuefdj sugsfhefiushjs eg lhr`g `gds kjrgi phss³ķvji. Z`g rjtg quj fhetj` hs fjetrhscj chds f³ķrfuihs ³j fog`gcg cj rjtg chs fjetrhs chs chds f³ķrfuihs.

?. ]rhvj quj sj chds f³ķrfuihs tj` chds pheths j` fhù`, g rjtg chs

fjetrhs ³j `jcdgtrdz ch sjk`jeth idkgech jstjs chds pheths.

4. ]rhvj quj chds f³ķrfuihs cdstdeths egh phcj` tjr `gds ch quj chdspheths j` fhù`.

Vj chds f³ķrfuihs sgh tgekjetjs g u`g `js`g rjtg j` u` `js`h phethjetgh jijs sgh fog`gchs cj f³ķrfuihs tgekjetjs , j h pheth cj pheth cj fhe-

tgfth.

7=. ]rhvj quj sj chds f³ķrfuihs sgh tgekjetjs, g rjtg chs fjetrhs pgssgpjih pheth cj fhetgfth.

77. Eg kurg sjkudetj gs rjtgs sgh tgekjetjs fhùes ghs chds f³ķrfuihs.]rhvj quj `7 j `1 sj detjrfjptg` eg ideog chs fjetrhs. ]rhvj quj sj hsrgdhs chs chds f³ķrfuihs sgh cdljrjetjs, gs rjtgs e7 j e1 tgà³j` sj detjrfjptgèg rjtg chs fjetrhs.

e7

e1

`7

`1

71. Vjng` G j A pheths cj detjrfjssgh cj chds f³ķrfuihs. Vjng` F j C

gs jxtrj`dcgcjs chs cdg`jtrhs chs chds f³ķrfuihs quj sj dedfdg` eh pheth G.

7=6




]rhvj quj g rjtg quj idkg F g C fhet³j h pheth A .

76. ]rhvj quj g `jcdcg cj u` gekuih lhr`gch phr u` tgekjetj j u`gfhrcg cj u` f³ķrfuih ³j dkugi g `jtgcj cg `jcdcg ch grfh quj jij cjtjr`deg.

73. ]rhvj quj g `jcdcg cj u` gekuih lhr`gch phr u`g tgekjetj j u`gsjfgetj ³j dkugi g `jtgcj cg cdljrje¿fg jetrj gs `jcdcgs chs grfhs quj jijcjtjr`deg.

7:. Vjrdg phss³ķvji rjuedr j` u`g ³uedfg prhphsd¿fgh gs delhr`g¿fhjs pgrfd-gds lhrejfdcgs pjihs jxjrf³ķfdhs 7< j 7; j pjihs prhaij`gs 76 j 738

7<. ]rhvj quj u` phi³ķkheh jqudi³gtjrh desfrdth j` u` f³ķrfuih ³j u`phi³ķkheh rjkuigr.

7;. ]rhvj quj u` phi³ķkheh jqudgekuigr quj fdrfuesfrjvj u` f³ķrfuih ³ju` phi³ķkheh rjkuigr.

7?. ]rhvj quj, sj u` f³ķrfuih lhr cdvdcdch j` u` eu`jrh qugiqujr cjgrfhs dkugds, jetgh gs fhrcgs cjstjs grfhs lhr`g` u` phi³ķkheh rjkuigr de-sfrdth eh fdrfuih, j gs tgekjetjs trg¿fgcgs pjihs pheths cj sjpgrg¿fgh chs grfhslhr`g u` phi³ķkheh rjkuigr quj fdrfuesfrjvj h f³ķrfuih.

74. Cjsfrjvg u` `³jthch cj trg¿fgr u` f³ķrfuih cj rgdh cgch quj sjngtgekjetj ghs igchs cj u` gekuih.

1=. Cjsfrjvg u` `³jthch pgrg trg¿fgr u` f³ķrfuih cj rgdh cgch quj sjngtgekjetj ghs igchs cj u` gekuih.

17. Cjtjr`dej h iukgr kjh`³jtrdfh cjsfrdth pjihs (cjtjr`dej h fhenuethcj thchs hs pheths quj tj` g prhprdjcgcj cj sjrj`) p³js cgs pjrpjecdfuigrjsagdxgcgs cj u` pheth G mgs rjtgs quj pgssg` phr u` pheth A .

11. Cjtjr`dej h iukgr kjh`³jtrdfh cjsfrdth pjihs v³jrtdfjs chs trdgekuihscj agsj GA j fh` ˌ F 0 6=◬ . G `jcdcg xgcg ch gekuih ˌ F lgz giku`gcdljrje¿fg eg sug rjsphstg8

16. Chs f³ķrfuihs sj fhrtg` ehs pheths G j A . ]jih pheth A trg¿fg`hsu`g rjtg quj fhrtg h prd`jdrh f³ķrfuih eu` pheth _ j h sjkuech f³ķrfuiheu` pheth T . @hstrj quj g `jcdcg ch gekuih _ ˌ GT egh cjpjecj cg rjtgtrg¿fgcg.

13. @hstrj quj u` f³ķrfuih ³j h iukgr kjh`³jtrdfh cjsfrdth pjihs pheths F

7=3




pgrg hs qugds g rgzgh cj sugs cdstgefdgs gh pheth G ³j fhestgetj j cdljrjetj

cj u`. (Vukjstgh> eh trdgekuih GAF gs adssjtrdzjs chs gekuihs detjrehs jjxtjrehs cj v³jrtdfjs F sgh pjrpjecdfuigrjs j fhrtg` g rjtg quj pgssg phr G

j A sj`prj ehs `js`hs pheths!).

7=:




FH@JES ³GQDH

Gh ihekh cjstgs ehtgs phstuig`hs g `gejdrg cj `jcdr h fh`prd`jethcj sjk`jeths.

G pgrtdr cg³ķ pucj`hs cjedr h fh`prd jeth cj u`g phidkhegi fh`h gsh`g chs fh`prd`jeths chs sjus igchs. Eh jetgeth egh zj`hs rjljrjefdgg qujstgh cj fh`h `jcdr h fh`prd`jeth cj furvgs. Cj lgth, g ³uedfg furvgquj fhesdcjrj`hs gt³j h `h`jeth lhd h f³ķrfuih. Detudtdvg`jetj h fh`prd-`jeth cj u` f³ķrfuih cjvj sjr h fh`prd`jeth ch sjk`jeth quj hatjr³ķg`hs sjpuc³jssj`hs, fhrtgech h f³ķrfuih, cjsjefurvg-ih, fh`h lgr³ķg`hs g u` pjcg¿fh

cj grg`j fdrfuigr. Eh jetgeth jstg eh¿fgh cj ’cjsfurvg`jeth‐ egh jxdstj egkjh`jtrdg quj fhestru³ķ`hs. ]hcj`hs pgrtdr cj u`g dcjdg aj` `gds sd`pijsquj ³j g cj gprhxd`gr h f³ķrfuih phr u`g phidkhegi. Sh`j`hs pgrg dsth u`phi³ķkheh rjkuigr desfrdth eh f³ķrfuih, fh` e igchs, fgcg u` cjijs `jcdechse. Vj h eu`jrh cj igchs lhr sufdjetj`jetj krgecj, g ehssg destdtud¿fghehs cdz quj h pjr³ķ`jtrh cjstj phi³ķkheh 0, e » se , sjr³g ùdth pr³hxd`h chfh`prd`jeth ch f³ķrfuih.

G

F

C H A

J

`+

Ldkurg ?.73

\g`hs cjtjr`degr s1e j` tjr`hs cj se j ch rgdh ch f³ķrfuih. Eg kurggfd`g mg cdrjdtg, tj`hs u` f³ķrfuih cj fjetrh H j rgdh Q, j ejstj, u`g fhrcg

FJ pjrpjecdfuigr g u` cdg`jtrh GA . H trdgekuih GAF ³j rjtgekuih j FC ³j g giturg ch v³jrtdfj G. Ihkh GF

10 GA »GC. Sg a³j` HFC ³j rjtgekuih j

phrtgeth CH1

0 HF 1∑FC

1. Vj F J 0 se, jetgh GF 0 s1e j, cgs lhrùigs

gfd`g, hatj`-sj

7) (s1e)1 0 1Q » `

1) (Q ∑ `)1 ∑ Q1 ∑se

1

1

7=<




hecj ` 0 GC. Jid`degech-sj ` cjstgs jqug¿fhjs rjsuitg>

s1e 0

1Q1 ∑ Q

3Q1 ∑ s1

e

³J l³gfdi vjrdfgr quj h igch s3 cj u` qugcrgch desfrdth eh fdrfuih ³j Q∞

1j quj sju pjr³ķ`jtrh ³j 3Q

∞ 1. Ztdidzgech-sj gkhrg sufjssdvg`jetj g lhrùig

gfd`g phcj`hs jefhetrgr hs pjr³ķ`jtrhs pe chs phi³ķkhehs rjkuigrjs desfrdthseh fdrfuih fh` ?, 7<, 61, <3,... igchs. H ijdthr egh tjr³g cdfuicgcj j`vjrdfgr quj

s? 0 Q

1 ∑∞ 1

s7< 0 Q

1 ∑ 1 + ∞ 1s61 0 Q

1 ∑

1 +

1 +

∞ 1

J phcjr³g lgfdi`jetj jstgajijfjr u`g jxprjssgh kjrgi pgrg s1e, (e ≣ ?).Fh` agsj egs lhrùigs gfd`g, fgifuig`hs, j` tjr`hs gprhxd`gchs, fhrrjthsgt³j g qugrtg fgsg cjfd`gi, hs vgihrjs cg sjkudetj tgajig.

e se pe

3 7,

37317 »Q

:,

<:<? »Q

? =, ;<:6; » Q <, 7114 » Q

7< =, 64=7? » Q <, 131? » Q

61 =, 74<=6 » Q <, 1;6= » Q

<3 =, =4?73 » Q <, 1?=< » Q

71? =, =34=? » Q <, 1?6= » Q

1:< =, =13:3 » Q <, 1?6= » Q

:71 =, =711; » Q <, 1?67 » Q

Gssd`, sjkuech ehssg detud¿fgh, h fh`prd jeth cj u` f³ķrfuih cj rgdhQ cjvj sjr gprhxd`gcg`jetj <, 1?67

»Q. Deljidz`jetj egh cdsph`hs gdecg

cj u`g cjed¿fgh cj fh`prd jeth ch f³ķrfuih quj ehs pjr`dtg vjrdfgr gvgidcgcj cjstg gr`g¿fgh.

Getjs cj tjetgr jstgajijfjr g cjed¿fgh cj fh`prd`jeth cj u` f³ķrfuihvg`hs lgzjr giku`gs hajrvg¿fhjs. Vjng ] u` phi³ķkheh fhevjxh desfrdth eu`f³ķrfuih, j sjng` G j A chds cj sjus v³jrtdfjs fhesjfutdvhs. Sh`j`hs u`

pheth F ch grfh

GA j decdquj`hs phr ] 7 h phi³ķkheh funhs v³jrtdfjs sgh hsv³jrtdfjs ch phi³ķkheh ] , `gds h pheth F . G pgssgkj` ch phi³ķkheh ] pgrg hphi³ķkheh ] 7 fhestdtud-sj eg suastdtud¿fgh gh igch GA pjihs igchs GF j FA .Cjscj quj GA 9 GF + F A, fhefiu³ķ`hs quj h pjr³ķ`jtrh cj ] 7 ³j `gdhr

7=;




ch quj h pjr³ķ`jtrh cj ] . Gssd`, gcdfdhegech-sj g u` phi³ķkheh fhevjxh de-

sfrdth eh f³ķrfuih ehvhs v³jrtdfjs, gu`jetg`hs h sju pjr³ķ`jtrh. Vj` cuvdcgjstj prhfjcd`jeth egh rjsuitg j` frjsfd`jeth did`dtgch. Cj lgth, sj th`gr-`hs u` phi³ķkheh fdrfuesfrdth gh f³ķrfuih, sju pjr³ķ`jtrh sjr³g `gdhr ch quj hpjr³ķ`jtrh cj qugiqujr phi³ķkheh fhevjxh desfrdth.

Fog`g-sj cj fh`prd`jeth ch f³ķrfuih gh `jehr chs e³u`jrhs `gdhrjs chquj h pjr³ķ`jtrh cj qugiqujr phi³ķkheh fhevjxh ejij desfrdth.

³J u`g fhesjqujefdg d`jcdgtg cjstg cjed¿fgh quj, cgch qugiqujr e³u`jrh 5 =, phcj-sj sj`prj desfrjvjr u` phi³ķkheh fhevjxh eh f³ķrfuih tgi quj gcdljrje¿fg ch sju pjr³ķ`jtrh pgrg h fh`prd jeth ch f³ķrfuih (j` vgihr ga-shiuth) sjng `jehr ch quj . H fh`prd`jeth cg fdrfueljrjefdg ³j trgcd-fdhegi`jetj rjprjsjetgch eg lhr`g 1ύQ hecj ύ ³j h fh`prd`jeth cj u`g

sj`dfdrfueljrjefdg cj rgdh 7. H vgihr gprhxd`gch cj ύ, fhrrjth gt³j g :g fgsgcjfd`gi ³j>

ύ 0 6.737:46

h quj ehs c³g u` pjr³ķ`jtrh cj gprhxd`gcg`jetj <, 1?67?< » Q.

7=?





KJH@JSQDG JZFIDCDGEG ]IGEG LZEF¿ HJV SQDKHEH@³JSQDFGV

]rhvg. Fhesdcjrj`hs u` hutrh f³ķrfuih cj fjetrh H j ejstj u` cdg`jtrh

G

A

. Fhesdcjrj`hs u` pheth F

sharj h f³ķrfuih cj `hch quj h gekuihF ˌ H A sjng dkugi gh gekuih η j phrtgeth dkugi g F ˌ HA . Fhesdcjrj hs trdgekuihsFHC j F HC hecj C j C sgh hs p³js cgs pjrpjecdfuigrjs agdxgcgs mgs rj-tgs GA j GA, rjspjftdvg`jetj, g pgrtdr chs pheths F j F . Fh`h F ˌ CH jF ˌ C H sgh gekuihs rjths j n³g sgaj`hs quj F ˌ HA 0 F ˌ H A, jetgh fhefiu³ķ`hsquj hs trdgekuihs fhesdcjrgchs sgh sj`jiogetjs. ]hrtgeth tjrj`hs

F H

FH0 F C

FC0 HC

HC

Fh`h fhesjqujefdg

sje η 0 FC

FH0 F C

F Hj fhs η 0

HC

FH0 HC

F C

Dsth prhvg g ehssg gr`g¿fgh.

4.: Sjhrj`g. Wugiqujr quj sjng h gekuih η tj`-sj>

sje1η + fhs1η 0 7

]rhvg. ]grg η dkugi g =◬, 4=◬ hu 7?=◬, g gr`g¿fgh gfd`g ³j fh`prhvgcg

pjig suastdtud¿fgh cdrjtg chs vgihrjs ch sjeh j ch fhssjeh fhrrjsphecjetjs.Ehs hutrhs fgshs, fhesdcjrj h trdgekuih H ˌ FA cg kurg (4.7). Sj`-sj jetgh

fhs1η + sje1η 0

HC

HF

1

+

FC

HF

1

0 HC

1+ FC

1

HF 1

0 HF

1

HF 1

0 7

hecj ljz-sj ush ch tjhrj`g cj ]dt³gkhrgs eg pje³uitd`g dkugicgcj.

4.< Sjhrj`g (L³hrùigs cj rjcu¿fgh). Vj η ³j u gekuih gkuch jetgh

g) sje(4=

◬

∑ η) 0 fhs ηa) fhs(4=◬ ∑ η) 0 sje ηf) tk (4=◬ ∑ η) 0 7/tk η

]rhvg. (\jng kurg 4.1) Vjng` F j F pheths cj u` sj`df³ķrfuih cjjxtrj`dcgcjs G j A, tgds quj F ˌ HA 0 η j F ˌ H A 0 4=◬ ∑ η.

77=




ηG A

F

F

CH

4=◬-η

C

Ldkurg 4.1

Vjng` C j C

hs p³js cgs pjrpjecdfuigrjs agdxgcgs mg rjtg GA g pgrtdr cjF j F , rjspjtdvg`jetj. Hasjrvj quj, fh`h F ˌ HA 0 4=◬ ∑ η, jetgh H ˌ F C

0 η. Ihkh hs trdgekuihs FHC j HCF sgh fhekrujetjs (HF 0 HF , F ˌ CH

0 F ˌ C H 0 4=◬ j F ˌ HC 0 H F C 0 η) j phrtgeth

F C

HC0 HC

FC0 HF

HF sjkuj-sj quj

sje(4=◬ ∑ η) 0 F C

HF 0 HC

HF 0 fhs η

fhs(4=◬ ∑ η) 0 HC

HF 0 FC

HF 0 sje η

G prhvg cj (f) ³j cjdxgcg g fgrkh ch ijdthr.

4.; Sjhrj`g. Wugiqujr quj sjng η tj`-sj>

g) sje(7?=◬ ∑ η) 0 sje η

a) fhs(7?=◬ ∑ η) 0 ∑fhs η

]rhvg. Wugech η ³j dkugi g =◬, 4=◬ hu 7?=◬, g gr`g¿fgh gfd`g ³j fh`prh-vgcg phr suastdtud¿fgh cdrjtg chs vgihrjs ch sjeh j fhssjeh fhrrjsphecjetjs.Ehs hutrhs fgshs, fhesdcjrj pheths F j F eh sj`df³ķrfuih cj shrtj quj F ˌ HA

0 η j F ˌ HA 0 7?=◬∑η. Vjng` C j C hs p³js cgs pjrpjecdfuigrjs agdxgcgschs pheths F j F mg rjtg GA .

777




C HC

ηηAG

F F

Ldkurg 4.6

G fhekrujefdg chs trdgekuihs HFC j HF C ehs lhrejfj

FC 0 F C j CH 0 C H

Fh`h fhesjqujefdg d`jcdgtg tj`hs quj

sje(7?=◬ ∑ η) 0 F C

F H0 FC

FH0 sje η

j |fhs(7?=◬ ∑ η)| 0 CH

F C0 CH

FH0 |fhs η|

Fh`h η

0 4=◬ jetghηhu7?=◬

∑η ³j hatush j h hutrh ³j gkuch. ]hr dsth,

fhs η j fhs(7?=◬ ∑ η) tj` sdegds hphsths. Ihkh

fhs(7?=◬ ∑ η) 0 ∑fhs η.

Gs cjed¿fhjs cj sjeh, fhssjeh j tgekjetj cgcgs eh dedfdh ch fgpdtuih pjr-`dtj` fhefiudr d`jcdgtg`jetj gs sjkudetjs lhrùigs rjigfdhegech hs igchscj u` trdgekuih rjtgekuih j hs sjus gekuihs gkuchs.

4.? ]rhphsd¿fgh. J` u` trdgekuih rjtgekuih GAF , cj gekuih rjth ˌ F ,tj`-sj

AF 0 GA sje ˌ G, GF 0 GA fhs ˌ G j AF 0 GF tk ˌ G

Z`g fhesjqujefdg cjstg prhphsd¿fgh ³j quj, sj fheojfjr`hs u` gekuihj u` igch cj u` trdgekuih rjtgekuih ³j phss³ķvji fgifuigr`hs g `jcdcgs cjsjus hutrhs chds igchs, suphech-sj quj fheoj¿fg`hs fh`h fgifuigr gs lue¿fhjssjeh, fhssjeh, j tgekjetj cj u` gekuih. Gtugi jetj, u`g `gqudeg cjfgifuigr rgzhgvji`jetj sd`pijs, phssud fdrfudths quj fgifuig` jstgs lue¿fhjsfh` gprhxd`g¿fgh fhrrjtg gt³j g qugrtg hu qudetg fgsg cjfd`gi, h quj ³j `gdsch quj sufdjetj pgrg g krgecj `gdhrdg chs f³gifuihs. ]hcj-sj, eh jetgeth,lgzjr ush cj u`g tgajig cj lue¿fhjs trdkheh`³jtrdfgs hatdcgs lgfdi`jetj j`qugiqujr fh`pjecdh sharj trdkheh`jtrdg.

771




4.4 ]rhphsd¿fgh.

g) sje 3:◬ 0 7/∞

1, fhs 3:◬ 0 7/∞

1 j tk 3:◬ 0 7.

a) sje 6=◬ 0 7/1, fhs 6=◬ 0∞

6/1 j tk6=◬ 0 7/∞

6.

]rhvg. (g) Fhestrug u` trdgekuih rjtgekuih GAF tjech gekuih rjthˌ F , j tjech GF 0 AF . Sj` sj jetgh quj ˌ G 0 ˌ A 0 3:◬ j, utdidzgech-sj htjhrj`g cj ]dt³gkhrgs, GF 0 AF 0 GA/

∞ 1. Ihkh

sje 3:◬ 0 GA/∞ 1GA

0 7/∞ 1

Cg `js`g lhr`g hat³j`-sj h vgihr cj fhs 3:◬. H vgihr cg tgekjetj ³jhatdch pjig sd`pijs cdvdsgh chs vgihrjs ch sjeh j fhssjeh.

(a) Fhestrug u` trdgekuih jqudi³gtjrh GAF . Shchs hs sjus gekuihs `j-cj` <=◬ j thchs hs sjus igchs tj` h `js`h fh`prd`jeth g.

G

A

F

C6=◬

Ldkurg 4.3

Fhesdcjrj g giturg agdxgcg ch v³jrtdfj A gh igch GF j sjng C h p³j cjstggiturg. Hs chds trdgekuihs lhr`gchs sgh fhekrujetjs j CG 0 CF 0 g/1.Gpidfgech h tjhrj`g cj ]dt³gkhrgs gh trdgekuih GAC fhefiu³ķ`hs quj AC 0g∞

6/1. Hasjrvj quj h gekuih G ˌ AC `jcj 6=◬. Ihkh

sje 6=◬ 0 g/1

g 0 7/1

fhs 6=◬ 0 g∞

6/1

g 0

∞ 6/1

776




tk 6=◬ 0 7/1

∞ 6/10 7/

∞ 6

Fh` jstg prhphsd¿fgh j hs rjsuitgchs (4.<) j (4.;) phcj`hs gkhrg lgfdi-`jetj cjtjr`degr hs vgihrjs ch sjeh, fhssjeh j tgekjetj chs gekuihs cj <=◬,71=◬, 76:◬ j 7:=◬.

4.7= Sjhrj`g (Ijd chs fhssjehs). J u` trdgekuih GAF tj` sj>

GA1

0 GF 1

+ AF 1 ∑ 1 »GF »AF » fhs ˌ F

]rhvg. Vj h gekuih ˌ F lhr rjth jetgh g gr`g¿fgh gfd`g ³j jxgtg`jetjh tjhrj`g cj ]dt³gkhrgs. ]hcj`hs phrtgeth suphr quj ˌ F egh ³j u gekuihrjth. Srgfj`hs g giturg ch v³jrtdfj G. Fh`h F egh ³j u gekuih rjth jetghh p³j cjstg giturg, quj cjsdkegrj`hs phr C , egh fhdefdcj fh` h pheth F .Vj C fhdefdcdr fh` h pheth A jetgh h trdgekuih GAF ³j rjtgekuih tjech ˌ A

fh`h gekuih rjth. Ejstj fgsh, GF » fhs ˌ F 0 AF j h rjsuitgch gfd`g ³j u`gcjfhrrjefdg d`jcdgtg ch tjhrj`g cj ]dt³gkhrgs. Gssd` phcj`hs suphr qujA , F j C sgh pheths cdstdeths. Fh`h GCA j GCF sgh trdgekuihs rjtgekuihstj`-sj

GA1

0 GC1

+ AC1

GF 1

0 GC1

+ CF 1

G G G

AAAF F FCCC

Ldkurg 4.:

Ihkh, suatrgdech-sj jstgs cugs jqug¿fhjs hat³j`-sj

GA1 ∑GF

10 AC

1 ∑CF 1

Gkhrg drj`hs suastdtudr h tjr`h AC1

cjstg jqug¿fgh. ]grg dsth, tjrj`hscj fhesdcjrgr trjs phssdadidcgcjs (vjng kurg 4.:).

773




g) F jst³g jetrj A j C .

Ejstj fgsh tj` sj CF + AF 0 AC. Vuastdtudech-sj ACphr CF + AF , cjsjevhivjech-sj h qugcrgch j sd`pidfgech-sjhs tjr`hs, g jqug¿fgh gfd`g threg-sj>

GA1

0 GF 1

+ AF 1

+ 1 »AF »CF

Hasjrvj quj CF 0 GF » fhs(G ˌ FC ) j quj fhs(G ˌ FC ) 0∑fhs(7?=◬ ∑ G ˌ FC ) 0 ∑fhs(G ˌ FA ). Fh`h G ˌ FA ³j jxgtg`jetj hgekuih ˌ F ch trdgekuih GAF , h rjsuitgch fg cj`hestrgch ejstj

fgsh.

a) C jst³g jetrj F j A .

Ejstj fgsh tj`-sj AC+CF 0 AF j, phrtgeth, AC 0 CF ∑AF . Vuastdtudech-sj fh`h eh fgsh getjrdhr jstj vgihr cj AC ,hat³j`-sj

GA1

0 GF 1

+ AF 1 ∑ 1 »AF »CF

Hasjrvgech-sj quj CF 0 GF

» fhs ˌ F hat³j`-sj h rjsuitgch.

f) A jst³g jetrj F j C .

Jstj fgsh ³j trgtgch cj lhr`g sj`jiogetj j ³j cjdxgch g fgrkhch ijdthr fh`pijtgr g cj`hestrg¿fgh.

4.: Sjhrj`g (Ijd chs sjehs). Wugiqujr quj sjng h trdgekuih GAF ,tj` sj>

sje ˌ G

AF

0 sje ˌ A

GF

0 sje ˌ F

GA

]rhvg. Fhesdcjrj h f³ķrfuih quj fdrfuesfrjvj h trdgekuih AF . Vjng H h sjufjetrh j Q h sju rgdh. Fhesdcjrj h cdg`jtrh quj tj` A fh`h jxtrj`dcgcj.

77:




G

A

FC

H

G

A

F

C

H

Ldkurg 4.<

Vjng C sug hutrg jxtrj dcgcj. Vj hs pheths G j C jstdvjrj` cj u``js`h igch cg rjtg AF , jetgh hs gekuihs A ˌ C F j A ˌ GF sgh dkugds phr sjrj`gekuihs desfrdths fhrrjsphecjetjs g u` `js`h grfh. Vj hs pheth G j C

jstdvjrj` j` igchs cdstdeths cg rjtg AF jetgh hs gekuihs A ˌ CF j A ˌ GF

sgh supij`jetgrjs n³g quj fhrrjsphecj` g grfhs quj sj fh`pij`jetg` pgrglhr`gr h fdrfuih. J` gàhs hs fgshs tj`-sj quj

sje ˌ C 0 sje ˌ G.

Fhesjqujetj`jetj

AF 0 1Q sje ˌ G

(Hasjrvj quj gqud utdidzg`hs h lgth cj quj h trdgekuih AFC ³j rjtgekuih).Cj lhr`g ge³gihkg cj`hestrg-sj quj

GA 0 1Q» sje ˌ F j GF 0 1Q» sje ˌ A .

Fh`pgrgech-sj gs trjs lhrùigs hatdcgs fhefiud-sj quj

sje ˌ G

AF 0 sje ˌ A

GF 0 sjeˌ F

GA0

7

1Q.

Ldfg gssd` cj`hestrgch jstj tjhrj`g.

77<






FH@JES ³GQDH

H jstuch cgs lue¿fhjs trdkheh`³jtrdfgs j cj sugs gpidfg¿fhjs ³j cjeh`degchtrdkheh jtrdg. G trdkheh`jtrdg dedfdhu-sj fh`h h jstuchs cgs gpidfg¿fhjs, gprhaij`gs pr³gtdfhs, cgs rjig¿fhjs jetrj hs igchs cj u` trdgekuih. Eg Kr³jfdg,gsshfdgtdvg-sj g u` cgch gekuih g fhrcg fhrrjsphecjetj gh grfh quj jijcjtjr degvg j` u` fdrfuih fjetrgch eh sju v³jrtdfj. ]hstjrdhr`jetj, eg`gtj`³gtdfg odecu, fh`j¿fhu-sj g gsshfdgr `jtgcj cjstg fhrcg. G fhrrj-sphecjefdg jrg jstgajijfdcg gtrgv³js cj tgajigs.

Z`g krgecj cdfuicgcjs quj jxdstdg tgeth eh tj`ph chs krjkhs fh`h

eh tj`ph chs odecus pgrg jigahrg¿fgh j utdidzg¿fgh cjstgs tgajigs jrg g eghjxdstjefdg cj u` sdstj`g gcjqugch cj eu`jrg¿fgh (detrhcuzdch eg grdt`³jtdfgsh`jetj eh 7<◬ s³jfuih). H prd`jdrh trgagioh sharj trdkheh`jtrdg cj qujsj tj` rjgi fheojfd`jeth jst³g fhetdch eh ’Gi`gkjst‐. (Z` trgagioh sharjgstrheh`dg j` 76 vhiu`js, jsfrdth phr ]thih`ju cj Gijxgecrdg phr vhitg cg`jtgcj ch sjkuech s³jfuih.) Eu` cj sjus fgp³ķtuihs ³j gprjsjetgcg u`g tgajigcj fhrcgs. Jstg tgajig rjljrj-sj g u`g sjqujefdg cj gekuihs, cdstgefdgchs cj`jdh krgu u` ch hutrh, j sjus vgihrjs jst gh fhrrjths gt³j pjih `jehs fdefhfgsgs cjfd`gds. Gid tgà³j` ³j jxpidfgch h `³jthch cj fgifuigr g tgajig.Z` hutrh fgpdtuih ³j cjcdfgch g shiu¿fgh cj trdgekuihs. \³grdhs tjhrj`gsrjigtdvhs g fhrcgs sgh cj`hestrgchs. Jstjs tjhrj`gs fhetj` d`pidfdtg`jetj

h fheojfd`jeth cgs lhrùigs `gds d`phrtgetjs cg trdkheh`jtrdg.Eh jetgeth, hs jsfrdthrjs krjkhs ch qugrth s³jfuih fog`g` g Odpgrfh, quj

vdvju ehs sjkuech s³jfuih g.F., cj hrdkdegchr cg fdjefdg cg trdkheh`jtrdg. Jijtjvj g rjputg¿fgh cj tjr fgifuigch u`g tgajig cj fhrcgs quj fhestdtud uìdvrh fh` chzj vhiu`js.

77?



KJH@JSQDG JZFIDCDGEG ]IGEG ³GQJG

FG]³DSZIH 7=

³GQJG

Z`g rjkd˙ gh trdgekuigr (kurg (g) gagdxh) ³j u` fhenueth cj pheths chpigeh lhr`gch phr thchs hs sjk`jeths fungs jxtrj`dcgcjs jstgh sharj hsigchs cj u` trdgekuih. H trdgekuih ³j fog`gch cj lrhetjdrg cg rjkdgh trdge-kuigr. H fhenueth cj pheths cj u`g rjkdgh trdgekuigr quj egh pjrtjefj` g

sug lrhetjdrg ³j fog`gch cj detjrdhr cg rjkdgh trdgekuigr.

(g) (a)

Ldkurg 7=.7

Z`g rjkd˙ gh phidkhegi ³j g uedgh cj u` e³u`jrh edth cj rjkdhjs trdge-kuigrjs quj cugs g cugs egh tj` pheths detjrdhrjs j` fhù` (kurg (a))gfd`g).

Z` pheth ³j detjrdhr g u`g rjkdgh phidkhegi sj jxdstj giku`g rjkdgh trd-gekuigr fhetdcg eg rjkdgh phidkhegi j fhetjech h pheth eh sju detjrdhr. Hdetjrdhr cg rjkdgh phidkhegi ³j h fhenueth chs pheths quj ioj sgh detjrdhrjs.G lrhetjdrg cg rjkdgh phidkhegi ³j fhestdtu³ķcg pjihs pheths cg rjkdgh quj eghpjrtjefj` gh sju detjrdhr.

774




p q

p ³j pheth detjrdhr g rjkdgh

q ³j pheth cg lrhetjdrg

kurg 7=.1

G eh¿fgh cg ³grjg cj rjkdhjs phidkhegds ³j detrhcuzdcg eg kjh`jtrdg gtrgv³jschs sjkudetjs gxdh`gs>

Gxdh`g \D.7. G thcg rjkdgh phidkhegi fhrrjsphecj u` e³u`jrh `gdhrch quj zjrh.

H e³u`jrh g quj sj rjljrj jstj gxdh`g ³j fog`gch cj ³ grjg cg rjkdgh.

Gxdh`g \D.1. Vj u`g rjkdgh phidkhegi ³j g uedgh cj cugs hu `gdsrjkdhjs phidkhegds quj cugs g cugs egh tjeog` pheths detjrdhrjs j` fhù`,jetgh sug ³grjg ³j g sh`g cgs ³grjgs cgqujigs rjkdhjs.

Gxdh`g \D.6. Qjkdhjs trdgekuigrjs id`dtgcgs phr trdgekuihs fhekru-jetjs tj ³grjgs dkugds.

³J figrh quj thch phi³ķkheh fhevjxh cjtjr`deg u`g rjkdgh phidkhegi. E³hs

drj`hs th`gr g idajrcgcj cj usgr jxprjsshjs ch tdph ’g ³grjg cj u` qugcrgch‐qugech qujrj`hs cdzjr rjgijetj ’g ³grjg cg rjkdgh phidkhegi fung lrhetjdrg ³ju` qugcrgch‐. J` kjrgi lgigrj`hs cj ’³grjg cj u` cgch phi³ķkheh‐, qugechqujrj`hs cj lgth ehs rjljrdr g ³grjg cg rjkdgh fung lrhetjdrg ³j gqujij phi³ķkheh.Gssd`, h gxdh`g \D.6 gfd g phcjrdg tjr sdch jeuefdgch fh`h> ’trdgekuihsfhekrujetjs phssuj` ³grjgs dkugds‐.

Gxdh`g \D.3. Vj GAFC ³j u rjtgekuih jetgh sug ³grjg ³j cgcg pjihprhcuth GA » AF .

G pgrtdr cjstjs gxdh`gs vg`hs cjtjr`degr g ³grjg cj giku`gs rjkdhjs

phidkhegds sd`pijs. \g`hs dedfdgr pjih pgrgijihkrg`h.Cgch u` pgrgijihkrg`h GAFC cjsdkej`hs phr a h fh`prd`jeth ch

igch GA j phr o h fh`prd`jeth cj u` sjk`jeth idkgech GA g FC qujsjng pjrpjecdfuigr g gàhs. Z` tgi sjk`jeth ³j fog`gch cj giturg chpgrgijihkrg`h rjigtdvg`jetj gh igch GA .

7=.7 ]rhphsd¿fgh. G ³grjg ch pgrgijihkrg`h ³j h prhcuth ch fh`prd`jethcj u` cj sjus igchs pjih fh`prd`jeth cg giturg rjigtdvg g jstj igch.

]rhvg. J` tjr`hs cg ehtg¿fgh xgcg gfd`g cjvj`hs prhvgr quj g ³grjg

71=




ch pgrgijihkrg`h GAFC j a » o. ]grg dsth trgfj, g pgrtdr chs pheths G j A ,

chds sjk`jeths, GJ j AL , pjrpjecdfuigrjs mg rjtg FC . H qugcrdi³gtjrh GALJ ³j u rjtgekuih fung ³grjg ³j GA » AL g qugi, j` tjr`hs cj ehssg ehtg¿fgh,³j jxgtg`jetj a » o. ]grg fhefiudr g cj`hestrg¿fgh hasjrvj quj hs trdgekuihsGCJ j FAL sgh fhekrujetjs j quj

³Grjg(GAFC ) 0 ³Grjg(GAFJ ) + ³Grjg(GCJ ) 0

0 ³Grjg(GAFJ ) + ³Grjg(FAL ) 0

0 ³Grjg(GALJ ) 0 a » o

Dsth fhefiud g cj`hestrg¿fgh.

C J F L

o

AG a

Ldkurg 7=.6

Fh`h fhrhi³grdh cjstg prhphsd¿fgh cjtjr`deg-sj g ³grjg cj u` trdgekuihqugiqujr.

7=.1 ]rhphsd¿fgh. G ³grjg cj u` trdgekuih ³j g `jtgcj ch prhcuth chfh`prd`jeth cj qugiqujr cj sjus igchs pjig giturg rjigtdvg g jstj igch.

]rhvg. Cgch u` trdgekuih GAF , trgfj pjih v³jrtdfj F u`g rjtg pgr-gijig gh igch GA , j pjih v³jrtdfj A u`g rjtg pgrgijig gh igch GF . Js-tgs cugs rjtgs sj detjrfjtg` j` u` pheth C . H phi³ķkheh GFCA ³j u`pgrgijihkrg`h, j hs chds trdgekuihs GAF j FCA sgh fhekrujetjs. Fh`h

³Grjg(GACF ) 0 ³Grjg(GAF ) + ³Grjg(FCA) j ³Grjg(GAF ) 0 ³Grjg(FCA),

jetgh> ³Grjg(GAF ) 0 7

1³Grjg(GACF ). ]grg fh`pij`jetgr g cj`hestrg¿fgh

hasjrvj quj g giturg ch v³jrtdfj F ch trdgekuih GAF ³j jxgtg`jetj g giturgch pgrgijihkrg`h GACF rjigtdvg`jetj gh igch GA .

717




G A

CF

Ldkurg 7=.3

7=.6 ]rhphsd¿fgh. G ³grjg cj u` trgp³jzdh ³j g `jtgcj ch prhcuth chfh`prd`jeth cj sug giturg pjig sh`g chs fh`prd`jeths cj sugs agsjs.

]rhvg. Vjng GAFC u` trgp³jzdh fungs agsjs sgh hs igchs GA j FC .Srgfj g cdgkhegi GF pgrg cdvdcdr h trgp³jzdh j` chds trdgekuihs.

G A

C F

J

L

Ldkurg 7=.:

Srgfj gs giturgs FJ , ch trdgekuih GFA , j GL , ch trdgekuih GFC . Jetgh

tjrj`hs quj GL 0 FJ , n³g quj hs igchs GA j FC sgh pgrgijihs. Fh`hfhesjqujefdg

³Grjg(GAFC) 0 ³Grjg(GFA) + ³Grjg(GFC) 0

0 7

1GA » FJ +

7

1CF » GL 0

0 7

1

GA + CF

» FJ

711




Ldfg gssd` cj`hestrgcg g prhphsd¿fgh.

7=.3 ]rhphsd¿fgh. G ³grjg cj u` phi³ķkheh rjkuigr cj e igchs, desfrdtheu`g fdrfueljrjefdg cj rgdh Q ³j 7

1Q1e » sje(6<=◬/e).

]rhvg. Vjng H h fjetrh ch f³ķrfuih. Idkgech-sj fgcg u` chs v³jrtdfjs chphi³ķkheh gh pheth H lhr`g-sj e trdgekuihs ds³hsfjijs fungs agsjs sgh hs igchsch phi³ķkheh, funhs igchs dkugds tj` fh`prd`jeth Q j funh gekuih ch thph`jcj 6<=◬/e. Vjng HGA u` tgi trdgekuih. Srgfj g giturg ch v³jrtdfj G. Jstggiturg `jcj Qsje(6<=◬/e) j h igch HA `jcj Q. Ihkh g ³grjg cjstj trdgekuih³j 7

1eQ1sje(6<=◬/e) j g ³grjg thtgi ch phi³ķkheh ³j 7

1eQ1sje(6<=◬/e).

G

AHη

Qo

Ldkurg 7=.<

Eh fh`jet³grdh ljdth gh egi ch Fgpdtuih ? e³hs vd`hs fh`h prhfjcjrpgrg jstgajijfjr g eh¿fgh cj fh`prd`jeth cj u` f³ķrfuih. ]hcj`hs prhfjcjrcj `gejdrg ge³gihkg fh` rjig¿fgh mg ³grjg cg rjkdgh id`dtgcg phr u` f³ķrfuih.Fhesdcjrj`hs phi³ķkhehs desfrdths j hasjrvg`hs quj, gh gu`jetgr`hs u`v³jrtdfj j` u` cj tgds phi³ķkhehs, gu`jetg`hs g sug ³grjg. Gssd`, egh jxdstju` phi³ķkheh desfrdth eh f³ķrfuih fh` ³grjg `gxd`gi. ]hr hutrh igch, g ³grjgcj qugiqujr phi³ķkheh desfrdth ³j `jehr ch quj g ³grjg cj qugiqujr phi³ķkhehfdrfuesfrdth. ³J jetgh egturgi cjedr> g ³ grjg ch f³ķrfuih ³j h `jehr e³ u`jrh

`gdhr ch quj g ³ grjg cj qugiqujr phi³ķkheh ejij desfrdth.\gihrjs gprhxd`gcgs pgrg g pgrg cj u` f³ķrfuih phcj` jetgh sjr hatd-

cgs g pgrtdr cg l³hrùig cjtjr`degcg eg prhphsd¿fgh (7=.3) pgrg g ³grjg cju` phi³ķkheh rjkuigr desfrdth. Eg tgajig gagdxh e ³j h e³u`jrh cj igchs chphi³ķkheh j Ge ³j g sug ³grjg.

716




e Ge

7< 6,=<73;61 6,7173:<3 6,76<::

71? 6,73=661:< 6,7371?:71 6,737:7

7=13 6,737:;1=3? 6,737:?3=4< 6,737:4

7=3?:3< 6,737:4

H ijdthr egh tjr³g cdfuicgcj j` rjlhrùigr g prhvg cg prhphsd¿fgh (7=.3)(fhesdcjrgech g giturg ch v³jrtdfj H ch trdgekuih HGA ) cj `hch g hatjr gsjkudetj jxprjssgh pgrg g ³grjg ch phi³ķkheh rjkuigr desfrdth cj e igchs.

Ge 0 eQ1sje

7?=◬

e

» fhs

7?=◬

e

j g sjkudetj l³hrùig pgrg h pjr³ķ`jtrh ch `js`h phi³ķkheh

pe 0 1eQ sje

7?=◬

e

Vjkuj-sj cg³ķ quj

1Ge

Qpe0 fhs (7?=◬/e)

Vj fhesdcjrg`hs u` phi³ķkheh desfrdth fh` u` krgecj e³u`jrh cj igchs,h vgihr cj fhs(7?=◬/e) sjr³g jxtrj`g`jetj pr³hxd`h ch vgihr fhs=◬, jequgethquj h vgihr cj Ge jstgr³g ùdth pr³hxd`h ch vgihr cg ³grjg cg rjkdgh id`dtgcgpjih f³ķrfuih j h vgihr cj pe sjr³g gprhxd`gcg`jetj h vgihr ch fh`prd`jeth chf³ķrfuih. ³J phrtgeth rgzh³gvji jspjrgr quj, pgrg h f³ķrfuih, cugs vjzjs sug ³grjgsjng dkugi g Q vjzjs sju pjr³ķ`jtrh. G prhphsd¿fgh gagdxh tj` jxgtg`jetjjstj jeuefdgch.

713




7=.: Sjhrj`g. G ³grjg cg rjkdgh id`dtgcg phr u` f³ķrfuih ³j dkugi g

`jtgcj ch prhcuth ch rgdh pjih fh`prd`jeth ch f³ķrfuih.

]rhvg. Qjprjsjetgrj`hs phr p h fh`prd`jeth ch f³ķrfuih j phr G g³grjg cg rjkdgh phr jij id dtgcg. Vj ] ³j u` phi³ķkheh desfrdth eh f³ķrfuih,rjprjsjetgrj`hs phr p(]) h sju pjr³ķ`jtrh, phr G(]) g sug ³grjg j phr I(])h fh`prd`jeth ch `gdhr cj sjus igchs.

Sh`j`hs u` e³u`jrh phsdtdvh g qugiqujr, j sjng ] u` phi³ķkheh desfrdthtgi quj

g) I(] ) 9 g

a) G∑

G(] ) 9 g»

Q

f) p ∑ p(] ) 9 g

]grg lgzjr g jsfhiog cjstj phi³ķkheh phcj`hs dedfdgi`jetj jsfhiojr trjsphi³ķkhehs> ] 7, hecj sj vjrdfg (g), ] 1 hecj sj vjrdfg (a) j ] 6, hecj sjvjrdfg (f). G `gejdrg fh`h sj cjedu ³grjg j pjr³ķ`jtrh ch f³ķrfuih pjr`dtjgr`gr quj gs jsfhiogs cj ] 1 j ] 6 sgh phss³ķvjds. Gkhrg, lhr`j u` ehvhphi³ķkheh quj tjeog fh`h v³jrtdfjs hs v³jrtdfjs chs trjs phi³ķkhehs. Jstj ehvhphi³ķkheh sgtdslgz gs trjs fhecd¿fhjs gfd`g. G jij fog`grj`hs cj phi³ķkheh] . G ³grjg cjstj phi³ķkheh phcj sjr fgifuigcg sh`gech-sj gs ³grjgs cj thchs

hs trdgekuihs fh` v³jrtdfjs eh fjetrh ch f³ķrfuih j tjech fh`h igchs u` chsigchs ch phi³ķkheh ] . Vjng HGA u` cjstjs trdgekuihs. Vug ³grjg sjr³g

³Grjg (HGA) 0 7

1GA » HF

hecj HF ³j g giturg ch v³jrtdfj H cjstj trdgekuih.

A

F

GH

Ldkurg 7=.;

71:




Fh`h HG 5 HF 5 HG ∑ GF , tj` sj quj

7

1GA »

HG ∑ GF

9 ³Grjg(HGA) 9

7

1GA » HG

Hasjrvgech quj HG 0 Q j GF 9 I(] ) 9 g, fhefiu³ķ`hs quj

7

1GA » (Q ∑ g) 9 ³Grjg(HGA) 9

7

1GA » Q

Cjscj quj u`g cjsdkugicgcj fh`h jstg vgij j` fgcg u` chs trd gekuihsj` quj suacdvdcd`hs h phi³ķkheh ] , phcj`hs sh`gr thcgs jigs pgrg hatjr

71

] (] ) » (Q ∑ g) 9 G(] ) 9 71

p(] ) » Q

Fh`h h phi³ķkheh ] sgtdslgz g fhecd¿fgh (f), tj`hs quj p ∑g 9 p(] ). ]hrhutrh igch sgaj`hs, cg cjed¿fgh cj pjr³ķ`jtrh ch f³ķrfuih, quj p(] ) 9 p.Ztdidzgech jstgs cugs delhr`g¿fhjs eg cjsdkugicgcj gfd`g hat³j`-sj

7

1( p ∑ g)(Q ∑ g) 9 G(] ) 9

7

1 p » Q

hu sjng

7

1 p » Q ∑

7

1

Q + gp ∑ g1

9 G(] ) 9

7

1 p » Q

Cjstgs cjsdkugicgcj cjfhrrj quj g ³grjg ch phi³ķkheh G(] ) cdljrj cj p »Q/1 j` `jehs quj (gQ + gp ∑ g1)/1. N³g quj, pjig jsfhiog ch phi³ķkheh ] ,G ∑ G(] ) 9 g » Q, jetgh fhefiu³ķ`hs quj

G(] ) ∑

7

1 p » Q

9 g » Q + 7

1 gQ + gp ∑ g1

Fh`h h vgihr cj g ³j gradtr³grdh, phcjech sjr th`gch tgh pjqujehs qugeth

sj qujdrg, j h igch jsqujrch cjstg cjsdkugicgcj egh cjpjecj cg jsfhiog cj

g , s³h phcj`hs fhefiudr quj g cdljrje¿fg G(] ) ∑ 7

1 p » Q ³j zjrh.

Ldfg gssd` cj`hestrgch h tjhrj`g.

71<




J_JQF³DFDH

7. Cjtjr`dej g ³grjg cj u` trdgekuih jqudi³gtjrh cj igch s .

1. Wuj rjig¿fgh sgtdslgzj` gs ³grjgs cj chds trdgekuihs rjtgekuihs sj`ji-ogetjs.

6. H rgdh ch f³ķrfuih desfrdth j` u` phi³ķkheh rjkuigr ³j fog`gch cjgp³ htj`g ch phi³ķkheh rjkuigr. ]rhvj quj g ³grjg cj u` phi³ķkheh rjkuigr ³jdkugi g `jtgcj ch prhcuth ch sju pjr³ķ`jtrh pjig sju gp³htj`g.

3. Cjtjr`dej g ³grjg cj u` ojx³gkheh rjkuigr desfrdth j` u` f³ķrfuih cjrgdh Q.

:. ]rhvj quj g rgzgh jetrj hs fh`prd`jeths cj chds f³ķrfuihs ³j dkugi grgzgh jetrj sjus rgdhs.

<. ]rhvj quj g rgzgh jetrj gs ³grjgs cj chds f³ķrfuihs ³j dkugi g rgzgh jetrjhs qugcrgchs chs sjus rgdhs.

;. Vj hs cdg`jtrhs cj chds f³ķrfuihs sgh 6 j <, qugi g rjig¿fgh jetrj gs sugs³grjgs8

?. Wugi g ³grjg cj u` qugcrgch desfrdth j` u` f³ķrfuih funh rgdh `jcj:f`8

4. Chds ojx³gkhehs rjkuigrjs tj` igchs `jcdech 1f` j 6f`. Wugi ³j grjig¿fgh jetrj gs sugs ³grjgs.

7=. H fh`prd jeth cj u` f³ķrfuih vgij cugs vjzjs h fh`prd jeth cjhutrh f³ķrfuih. Wuj rjig¿fgh jetrj gs sugs ³grjgs8

77. G ³grjg cj u` f³ķrfuih vgij fdefh vjzjs g ³grjgs cj hutrhs f³ķrfuih. Wujrjig¿fgh sgtdslgzj` sjus rgdhs8

71. Wugeth sjrdg ejfjss³grdh cj pgpji pgrg fhardr thcg g lgfj jxtjreg cju` u`g igtg fdi³ķecrdfg fung giturg ³j 7:f` ³j funh rgdh cg agsj ³j :f`8

71;




]QHAIJ@GV

7. @hstrj quj sj chds trdgekuihs sgh sj`jiogetjs jetgh g rgzgh jetrjsugs ³grjgs ³j dkugi g rgzgh jetrj hs qugcrgchs cj qugdsqujr chds sj sjus pgrjscj igchs fhrrjsphecjetjs.

1. @hstrj quj g rgzgh jetrj gs ³grjgs cj chds phi³ķkhehs sj`jiogetjs ³j dkugig rgzgh jetrj hs qugcrgchs j qugdsqujr chds cj sjus igchs fhrrjsphecjetjs.

6. Srjs phi³ķkhehs sj`jiogetjs sgh fhestru³ķchs tjech fgcg u` cjijs, fh`h

igch, u` chs igchs cj u` trdgekuih rjtgekuih. ]rhvj quj g ³grjg ch `gdhrcjijs ³j dkugi g sh`g cgs ³grjgs chs chds `jehrjs.

3. G rjkdgh id`dtgcg phr chds rgdhs j u` grfh cj u` f³ķrfuih ³j fog`gcg

cj sjthr ch f³ķrfuih. @hstrj quj g ³grjgs cj u` sjthr ³j 7

1QV hecj Q ³j h rgdh

ch f³ķrfuih j V ³j fh`prd`jeth ch grfh.

:. ]rhvj quj, j` chds f³ķrfuihs cj rgdhs Q j bQ, fhrcgs cj fh`prd`jethsf j bf , rjspjtdvg`jetj, suatjecj` grfhs cj fh`prd`jeths V j bV .

<. Cjtjr`dej g ³grjg cg rjkdgh id`dtgcg phr u`g fhrcg j pjih grfh cjf³ķrfuih quj jig suatjecj.

;. Cjtjr`dej g ³grjg cg pgrtj cg rjkdgh id`dtgcg phr u` f³ķrfuih quj fgjetrj cugs cj sugs fhrcgs.

?. G kurg gh igch sukjrj u`g hutrg`gejdrg cj cj`hestrgr h tjhrj`g cj]dt³gkhrgs. ]grg lgzjr g cj`hestrg¿fgh jx-prjssj g ³grjg ch qugcrgch `gdhr cj cugs`gejdrgs cdljrjetjs> fh`h prhcuths chs

igchs j fh`h g sh`g cgs ³grjgs chs 3trdgekuihs j ch qugcrgch `jehr. Fh`-pijtj g cj`hestrg¿fgh.

f a

f

a

fa

f

a gg

gg

71?




4. Z`g hutrg prhvg ch tjhrj`g cj

]dt³gkhrgs ³j sukjrdcg pjig kurg gh igch.Cjtjr`dej g ³grjg ch trgp³jzdh cj cugs`gejdrgs cdljrjetjs, cj lhr`g ge³gihkg ghquj lhd ljdth eg qujstgh getjrdhr. Fh`-pijtj g prhvg. Jstg prhvg lhd devjetgcgphr Kgrjicg j` 7?;<.

g Ng`js Gargoge Kgrjic (7?67 ‖ 7??7) lhdh vdk³jsd`h prjsdcjetj chs Jstgchs Zedchs j jrgu` krgecj jstucdhsh j jetusdgstg cg `gtj`³gtdfg.J` 7?;<, jequgeth jstgvg eg Fg`grg cj Qjp-rjsjetgetjs, rgadsfhu eu` pgpji u`g detjrjs-sgetj cj`hestrg¿fgh ch Sjhrj`g cj ]dt³gkhrgs. H

Ejw Jekigec Nhuregi hl Jcufgtdhe puaidfhu jstgcj`hestrg¿fgh.

G A

JC

G

g

a f

g

a

f

7=. Aogsbgrg, u` `gtj`³gtdfh odecu chs³jfuih chzj, frdhu u`g prhvg ch tjhrj`gcj ]dt³gkhrgs agsjgcg eg kurg gh igch.Lg¿fg jstg cj`hestrg¿fgh.

f

f f

f

a g

77. Wugdsqujr chds qugcrgchs phcj` sjr fhrtgchs j` : pjcg¿fhs cj tgilhr`g quj jstjs fdefh pjcg¿fhs phcj sjr rjgrru`gchs pgrg lhr`gr u` ehvhqugcrgch. G `gejdrg cj lgzjr hs fhrtjs ³j decdfgcg eg kurg sjkudetj, eh fgshpgrtdfuigr j` quj hs qugcrgchs fhesdcjrgchs tj` igchs u` h cupih ch hutrh.Cjphds quj vhfj vjrdfgr fh`h fhestrudr h qugcrgch fh` hs : pjcg¿fhs, tjetjcjtjr`degr fh`h lgzjr hs fhrtjs eh fgsh j` quj hs qugcrgchs tj` 71 j ?.

714




FH@JES ³GQDH

³J phss³ķvji jvdtgr g detrhcu¿fgh chs Gxdh`gs \D.7 g \D.3. H fhefjdthcj ³grjgs tjrdg jetgh cj sjr trgagiogchs g pgrtdr chs gxdh`gs j tjhrj`gscj`hestrgchs ehs fgp³ķtuihs getjrdhrjs j h fhetj³uch chs gxdh`gs \D.7 g \D.3gpgrjfjrdg fh`h u` tjhrj`g cg ehssg kjh`jtrdg. \g`hs cgr u`g dcjdg cjfh`h dsth phcjrdg sjr ljdth.

Fh`j¿fg`hs hasjrvgech quj,j` thch trdgekuih, h vgihr chprhcuth ch fh`prd`jeth cj u`

igch pjig giturg quj ioj ³j fhr-rjsphecjetj, egh cjpjecj cg js-fhiog ch igch. Vj usgr`hsg ehtg¿fgh decdfgcg eg kurggagdxh, h quj jstg`hs cdzjech ³jquj gk 0 ao 0 fb. Cjed`hs,jetg h g ³grjg cj u` trdgekuihfh`h jstj prhcuth, vjzjs u`gfhestgetj uedvjrsgi I. Dsth ³j,cgch u` trdgekuih GAF , cjed-`hs sug ³grjg fh`h

G

A

Fg

a f

b k o

³Grjg(GAF ) 0 I»

GA »

o (7)

hecj o ³j h fh`prd`jeth cg giturg ch v³jrtdfj F rjigtdvg`jetj gh igchGA . J` sjkudcg tj`hs cj cj`hestrgr quj, sj suacdvdcdr`hs h trdgekuihGAF j` u` e³u`jrh edth cjtrdgekuihs, j utdidzgr-`hs pgrgfgcg u` chs trdgekuihs cg sua-cdvdsgh g l³hrùig gfd`g, g sh`gcj sugs ³grjgs sjr³g dkugi g ³grjgch trdgekuih hrdkdegi GAF . Jstgcj`hestrg¿fgh egh sjr³g gprjsje-

tgcg gqud. H ijdthr detjrjs-sgch phcjr³g jefhetrg-ig eh idvrh’Kjh`jtrdg Jij`jetgi‐ sj G.\.]hkhr³jihv, Jcdthrg @dr. @gs sjqudsjr tjetgr phr sd `js`h, cjvjdedfdgr fh` h fgsh `gds sd`pijs j` quj g suacdvds gh lhd ljdtg fh`h eg kurggetjrdhr. Hasjrvj quj, ejstj fgsh, o³g u`g giturg quj ³j fhù` g thchstrdgekuihs! ]grg h fgsh kjrgi, sj ]WQ lhr u` trdgekuih cg suacdvdsgh, tjetjcjtjr`degr u`g lhrùig pgrg sug ³grjg j` tjr`hs cgs ³grjgs chs trdgekuihsG]W , GWQ j GQ] .

76=




H vgihr I quj kurg eg l³hrùig (7) ³j gid fhihfgch pgrg pjr`dtdr g

gcgptg¿fgh cgqujig l³hrùig ³gs cdvjrsgs uedcgcjs cj ³grjg.Cgcg u`g rjkdgh phidkhegi, g fhdsg `gds egturgi g lgzjr ³j cjedr sug

³grjg fh`h g sh`g cgs ³grjgs chs trdgekuihs quj g fh`phj. Hasjrvj`hs quju`g `js`g rjkdgh phidkhegi phcj sjr suacdvdcdcg j` rjkdhjs trdgekuigrjs cjùdtgs `gejdrgs cdljrjetjs. Gssd`, tj`hs u`g qujstgh cj fhesdstjefdg g sjrrjshivdcg. ]grg quj phssg`hs usgr jstg cjed¿fgh cjvj`hs vjrdfgr sj h vgihregi cg ³grjg cj u`g rjkdgh phidkhegi ³j decjpjecjetj cg pgrtdfuigr suacdvdsghcg `js`g j` trdgekuihs. Vupheog`hs quj u`g rjkdgh phidkhegi ] tjeogsdch suacdvdcdg j` trdgekuihs S 7, S 1, ..., S e j cjphds j` trdgekuihs S

7, S

1,

..., S `. ]rjfdsg`hs cj`hestrgr quj g sh`g cgs ³grjgs chs trdgekuihs, j`gàhs hs fgshs, ³j g `js`g. Hs trdgekuihs cgs cugs suacdvdshjs, fhesdcjrgchs

j` fhenueths, rjgidzg` u`g suacdvdsgh cg rjkdgh phidkhegi ] j phi³ķkhehsfhevjxhs, g sgajr> j` trdgekuihs, qugcrdi³gtjrhs, pjet³gkhehs j ojx³gkhehs.Vuacdvdcd`hs, fgcg u` cjijs, j` trdgekuihs. Sjrj`hs gkhrg u`g suacdvdsghS

7 , S

1 , ..., S

p cg rjkdgh phidkhegi ] fh` g prhprdjcgcj cj quj qugiqujrtrdgekuih, cg prd`jdrg suacdvdsgh hu cg sjkuecg suacdvdsgh, ³j fh`phsth cjtrdgekuihs cg ehvg suacdvdsgh. Fh`h n³g vd`hs, g ³grjg cj thch trdgekuih cgsjkuecg suacdvdsgh ³j g sh`g cgs ³grjgs chs trdgekuihs quj fh`phj`. ]hrdsth

³Grjg(S d) 0

³Grjg(S

n )

³Grjg (S

b) 0

³Grjg (S

n )

]hrtgeth g sh`g cgs ³grjgs chs trdgekuihs cj qugiqujr cgs pgrtd¿fhjs ³j g`js`g. Dsth fhefiud h ehssh grku`jeth.

Gkhrg ³j l³gfdi hatjr quj g ³grjg cj u` rjtgekuih GAFC j dkugi g 1I »GA »

FC. ]grg dsth agstg suacdvdcdr h rjtgekuih j` chds trdgekuihs rjtgekuihsgtrgv³js cg cdgkhegi GF .

³J jxgtg`jetj fhevjedjetj jsfhiojr h vgihr cj I cj `hch quj, g ³grjg cju` qugcrgch cj igch u`, sjng jxgtg`jetj u`. Cj gfhrch fh` g l³hrùiggfd`g, agstg th`gr I 0 7/1. Jstg jsfhiog trgeslhr`g¿fgh g jqug¿fgh (7) egl³hrùig usugi cg ³grjg cj u` trdgekuih!

767



F]@ =7 ‖ Ihkgrdt`hs

Jihe Igkjs Id`g

F]@ =1 ‖ Geçidsj Fhàdegtördg j ]rhagadidcgcj

Gukusth F. @hrkgch, Nhæh Ahsfh ]dthàjdrg cj Fgrvgioh, ]guih Fjzgr ]deth Fgrvgioh j ]jcrh

Ljegecjz

F]@ =6 ‖ @jcdcg j Lhr`g j` Kjh`jtrdg

(Fh`prd`jeth, çrjg, vhiu`j j sj`jiogeëg)

Jihe Igkjs Id`g

F]@ =3 ‖ @ju ]rhljsshr cj @gtj`çtdfg j hutrgs Odstördgs

Jihe Igkjs Id`g

F]@ =: ‖ Fhhrcjegcgs eh ]igeh

Jihe Igkjs Id`g fh` g fhigahrgëæh cj ]guih Fjzgr ]deth Fgrvgioh

F]@ =< ‖ Srdkheh`jtrdg j Eþ`jrhs Fh`pijxhs

@gdelrjch ]. ch Fgr`h, Gukusth Fjzgr @hrkgch, Jcugrch [gkejr fh` Ehtgss Odstördfgs cj Nhæh

Ahsfh ]dthàjdrg

F]@ =; ‖ Fhhrcjegcgs eh Jspgëh

Jihe Igkjs Id`g

F]@ =? ‖ ]rhkrjssújs j @gtj`çtdfg Ldegefjdrg

Gukusth F. @hrkgch, Jcugrch [gkejr j Vojdig F. ^ged

F]@ =4 ‖ Fhestruëújs Kjh`ätrdfgs

Cugrch [gkejr fh` g fhigahrgëæh cj Nhsä ]guih W. Fgrejdrh

F]@ 7= ‖ Detrhcuëæh í Kjh`jtrdg Jspgfdgi

]guih Fjzgr ]deth Fgrvgioh

F]@ 77 ‖ Kjh`jtrdg Jufidcdgeg ]igeg

Nhæh @grqujs Iufgs Agrahsg

F]@ 71 ‖ Dsh`jtrdgs

Jihe Igkjs Id`g

F]@ 76 ‖ G @gtj`çtdfg ch Jesdeh @äcdh ‖ \hi. 7

Jihe Igkjs Id`g, ]ugih Fjzgr ]deth Fgrvgioh, Jcugrch [gkejr j Gukusth F. @hrkgrch

]grg hatjr `gdhrjs delhr`gëújs j gcqudrdr

idvrhs cg fhijëæh ]rhljsshr cj @gtj`çtdfg cdrdng-sj g

VA@ • Sji. > (=17)143-4=61 • Lgx.>(=17)1:4-3736

Jstgcg Cheg Fgsthrdeg, 77=113<=-61= • Ngrcd` Ahtáedfh

Qdh cj Ngejdrh • QN

FHIJ ÆH ]QHLJVVHQ CJ @GSJ@ÇSDFG ‖ F]@



Jstj idvrh lhd jsfrdth pgrg sjrvdr cj tjxth g u`g cdsfdpideg cj Kjh`jtrdg pgrg

giuehs cj furshs cj Idfjefdgturg j` @gtj`çtdfg. Jij lgz u`g gprjsjetgëæh

cg Kjh`jtrdg Jufidcdgeg ]igeg cj u` pheth cj vdstg `gds gvgeëgch quj h

quj sj cjvj utdidzgr eh jesdeh cj 7¾ j 1¾ krgus. H hanjtdvh ä cgr gh luturh

prhljsshr u`g vdsæh `gds g`pig cgqudih quj jij vgd jesdegr. J` pgrtdfuigr,

h idvrh lgz u`g gprjsjetgëæh gxdh`çtdfg cg Kjh`jtrdg, phrä` ä fhevjedjetj

cjdxgr aj` figrh ‖ sj` qujrjr fh`h dsth sukjrdr quj h luturh prhljsshr cj 7¾

j 1¾ krgus gchtj h `js`h tdph cj gprjsjetgëæh qugech jstdvjr jesdegech.

Z`g gprjsjetgëæh gxdh`çtdfg cg Kjh`jtrdg, giä` cj sjr ihekg j jxdkdr

tj`ph, sö phcj sjr ljdtg fh` prhvjdth qugech hs giuehs nç tdvjrg` gcqudrdch

agstgetj lg`didgrdcgcj fh` hs lgths kjh`ätrdfhs, fhecdëæh decdspjesçvji

pgrg aj` prjpgrg-ihs g jetjecjr j gprjfdgr h phrquî cg gxdh`gtdzgëæh.



Wujr tjr gfjssh g giku`gs shiuëújs cjssj idvrh8 Jetæh gfjssj>

www.euàjr.?4=`.fh`

]grg jetrgr j` fhetgth fh` h jcdthr

cg vjrsæh jijtróedfg cjssj idvrh jsfrjvg pgrg

edaaijcdjkhMk`gdi.fh`

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