Diktat Kuliah s2 Analisis Real

Kfkecscs ]oke

@hkorlfc, ^.^c lffy\;;\13Bykhll.nli

Ecsofsc Al`uiof4 Ecsofsc  Al`uiof4 Nlpyeojt lf Nlpyeojt lf `hkorlfc.fot  ^oeuruh al`uiof ac `hkorlfc.fot  ac  `hkorlfc.fot  akpkt acgufk`kf, akpkt  acgufk`kf, acilacjc`ksc  akf acsomkr`kf sonkrk momks uftu`  tudukf mu`kf `liorscke (flfprljct) `liorscke (flfprljct) ktku k`kaoics (`ofkc`kf  pkfg`kt,  pkfg`kt, sortcjc`ksc, akf somkgkcfyk). Acmleoh`kf ioek`u`kf  pofuecskf  pofuecskf uekfg, aofgkf tkfpk iofakpkt`kf cdcf toreomch akhueu akrc  `hkorlfc.fot. @krofk scjktfyk mu`kf rojorofsc, ik`k acpor`ofkf`kf  dugk uftu` tcak`  uftu`  tcak`  iofyortk`kf ecf` akrc  ecf`  akrc al`uiof al`uiof cfc.

Qoftkfg Al`uiof4 Al`uiof cfc acmukt  cfc  acmukt  uftu`  `opoftcfgkf  prcmkac akf  prcmkac akf glelfgkf, sohcfggk sogkek `oskekhkf akf ‘`otorsosktkf–  ykfg ack`cmkt`kf leoh pofggu leoh  pofggufkkf fkkf al`uiof cfc, pofuecs cfc,  pofuecs tcak`  mortkfggufg  dkwkm. @otorsoackkffyk ac cftorfot  ac  cftorfot  mu`kf morkrtc actudu`kf morkrtc actudu`kf uftu`  pofggufkkf  pofggufkkf uiui, `otorsoackkf torsomut  acik`sua`kf somkgkc  al`uioftksc  lf‒ecfo ykfg acicec`c  pofuecs acicec`c  pofuecs akf acpormleoh`kf acicec`c  leoh sckpk skdk. Quecskf cfc iorupk`kf cfc  iorupk`kf skecfkf uekfg akrc nktktkf akrc  nktktkf  por`ueckhkf  por`ueckhkf Kfkecscs  Kfkecscs ]oke ac  ]oke  ac Cfstctut  Cfstctut  Rortkfckf Mlglr ^o`lekh Mlglr ^o`lekh Rksnkskrdkfk Rrlgrki Ikgcstor  ^kcfs Ikylr Iktoiktc`k Ikylr Iktoiktc`k Qorkpkf ykfg ackipu leoh Mkpk`  Ar. Dkhkrua Ar.  Dkhkruaacf, acf, I^ akf Cmu Moreckf (Mu Kfggc) (Mu  Kfggc)

Lutecfo4 ;. =. >. 1. ?. :.

Qolrc Hcipufkf Qolrc Hcipufkf ^cstoi Mcekfgkf ]oke  _`urkf Eomosguo Cftogrke  Eomosguo Qurufkf akf Cftogrke  ]ukfg Mkfknh

]ojorofsc4 ;. H.E. ]lyaof, ;388, ]oke  Kfkeyscs, ]oke  Kfkeyscs, In. Icekf Rum, Fow Slr`  Fow Slr`  =. Mkrteo, ].G., & A.]. ^hormort, =222, Cftrlauntclf tl ]oke  Kfkeyscs, ]oke  Kfkeyscs, Dlhf  Dlhf Wceoy, Fow Slr`  Fow  Slr`  >. Gleamorg, ]cnhkra  ]., ;37:, Iothlas lj ]oke  lj  ]oke  Kfkeyscs,  Kfkeyscs, Dlhf  Dlhf Wceoy, Fow Slr`  Fow  Slr` 

Qhkf`–s Ql4  Keekh swt ktks swt ktks hcakykh ‒Fyk akf fc`ikt  wk`tu eukfg, Fkmc Iuhkiika  Fkmc  Iuhkiika  skw ktks skw  ktks mcimcfgkf moecku iofgofke`kf `ctk `opkak rkmm `ctk. Mkpk`  Ar. Dkhkruaa Ar.  Dkhkruaacf, cf, I^ akf Cmu Kfggc. Cmu  Kfggc. ]o`kf‒ro`kf sofkscm akf sopofaorctkkf iofghkakpc   Kfkecscs ]oke akf ]oke  akf Kedkmkr   Kedkmkr Ecfokr. Ecfokr. Iy Qcfy  Iy  Qcfy Wom Wom ^cto, Iy Wlfaorjuee  Iy  Wlfaorjuee  ekptlp ykfg skfgkt  sotck. ^ortk Ke  ^ortk  Ke ‒Hkjcaz Iufsykrc  ]ksyca  ykfg sofkftcksk iofoikfc  toecfgk cfc aofgkf cfc aofgkf ekftufkf kykt ‒kykt ‒Fyk ykfg iofggugkh soikfgkt. Akf tk`  eupk, Cstrc ‒`u torncftk akf `uskykfgc  ktks au`ufgkffyk.

Mkgc sotckp ecskf ykfg toekh tor`ufnc, mkh`kf skipkc `opkak hktcfyk…Ik`k czcf`kf akf mckr`kfekh soeoimkr  pofgkaukf cfc iofdkac `ktk ikkj ykfg skfgkt tueus…Fkiuf, dc`k ck torekimkt, ik`k mckr`kfekh eoimkrkf cfc hkfyk iofdkac mkrcs ykfg ioiporcfakh `orkfdkfg skipkh. Gufk`kfekh, dc`k ck mor`ofkf ioikkj`kf…akf mukfgekh, dc`k ioikfg tcak` mor`ofkf…

Mkm ;  Qolrc Hcipufkf Hcipufkf ;.;.

Rofakhueukf ^kekh sktu akrc so`ckf mkfyk` kekt-kekt ykfg poftcfg akeki iktoiktc`k ilaorf kakekh iofgofkc tolrc hcipufkf. Akeki tuecskf cfc, dc`k actuecs`kf hcipufkf  X , ik`k ykfg acik`sua kakekh hcipufkf mcekfgkf roke. Mkgckf portkik, acdoeks`kf momorkpk scimle-scimle akrc tolrc hcipufkf ykfg k`kf skfgkt morgufk fkftcfyk. Aojcfcsc 4 Hcipufkf acaojcfcsc`kf somkgkc `uipuekf lmdo`-lmdo`. Lmdo` akrc suktu hcipufkf acsomut kfggltk/oeoiof akrc hcipufkf torsomut. Aojcfcsc 4  Dc`k x kfggltk hcipufkf X, actuecs x ∈ X.  Dc`k x mu`kf kfggltk hcipufkf X, actuecs actuecs x ∃ X

Hcipufkf X  sonkrk  sonkrk eofg`kp actoftu`kf morakskr`kf kfggltk-kfggltkfyk. Aojcfcsc 4  Dc`k auk hcipufkf X akf S ioicec`c ioicec`c scjkt x ∈ X dc`k akf hkfyk dc`k x ∈ S uftu` soiuk x, ik`k  X 9 S  Aojcfcsc 4  X acsomut hcipufkf mkgckf akrc akrc S, actuecs X

⊍ S, dc`k akf hkfyk dc`k uftu` sotckp x ∈ X ⇝  x ∈  S

Akrc scfc, doeks mkhwk sotckp hcipufkf X  iorupk`kf  iorupk`kf hcipufkf mkgckf akrc  X . Aojcfcsc 4 Auk hcipufkf X akf S ac`ktk`kf skik, ykctu X 9 S, dc`k akf hkfyk dc`k X

⊍ S akf S ⊍ X

@krofk hcipufkf actoftu`kf leoh oeoiof-oeoioffyk, skekh sktu nkrk ykfg sorcfg acek`u`kf uftu` iofaojcsc`kf somukh hcipufkf kakekh aofgkf iofyktk`kf `okfggltkkffyk (oeoiof), somkgkcikfk acaojcfcsc`kf morc`ut cfc4 Aojcfcsc 4  Dc`k K kakekh hcipufkf aofgkf kfggltk-kfgggltk-fyk x ∈ X ykfg ioicec`c scjkt R ik`k K akpkt actuecs4  K 9 {x ∈ X |R ( x )}

^ohcfggk x ∈ K ⇑ x ∈ X akf R ( x ) . @krofk   suakh acdoeks`kf somoeuifyk, ik`k tor`kakfg    X  suakh pofaojcfcsckf K mcsk actuecs4 (x    )}  K 9 {x  4 4 R (  x   Hcipufkf ykfg tcak` ioipufykc kfggltk acfkik`kf aofgkf hcipufkf `lslfg, akf acekimkfg`kf aofgkf Ø.  Qolroik ;.; 4 Hcipufkf `lslfg iorupk`kf hcipufkf mkgckf akrc sotckp hcipufkf.

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

Mu`tc 4 Ac`otkhuc Ø hcipufkf `lslfg. Icske`kf  X  kakekh  kakekh somkrkfg hcipufkf.  K`kf acmu`tc`kf mkhwk Ø ⊍ X , ykctu4 x  ∈  Ø ⇝ x ∈ X Rorfyktkkf ac ktks iorupk`kf porfyktkkf cipec`ksc aofgkf kftosoaoffyk morfcekc skekh, `krofk Ø  tcak` ioipufykc kfggltk. Morakskr`kf tkmoe `omofkrkf uftu` porfyktkkf cipec`ksc, porfyktkkf cipec`ksc ac ktks soekeu morfcekc mofkr. Dkac, tormu`tc mkhwk Ø ⊍ X.

  Dc`k x , y , akf z  kakekh  kakekh oeoiof-oeoiof X , acaojcfcsc`kf hcipufkf-hcipufkf somkgkc morc`ut4 Hcipufkf {x } ykctu hcipufkf ykfg kfggltkfyk hkfyk x  Hcipufkf {x , y } ykctu hcipufkf ykfg kfggltk-kfggltkfyk hkfyk x  akf  akf y  Hcipufkf {x , y , z } ykctu hcipufkf ykfg kfggltk-kfggltkfyk hkfyk x , y , akf z  akf somkgkcfyk. Hcipufkf {x } acsomut aofgkf ufct sot ktku scfgeotlf ktks x . ^ktu hke ykfg somkc`fyk hktc-hktc kftkrk x akf {x }. }. ^omkgkc nlftlh, `ctk ioipufykc x  ∈ {x } totkpc tcak` mofkr x  ∈ x . Akeki {x ,  y } tcak` acmorek`ukf urutkf kftkrk x  ktku  ktku y , ykctu {x ,  y } 9 { y , x }. }. Mogctu dugk aofgkf {x ,  y , z } 9 { x , z ,  y } 9 { y , x , z } 9 { y , z , x } akf somkgkcfyk. @krofk kekskf cfcekh `ctk iofyomut { x ,  y } somkgkc pkskfgkf tk` torurut (uflraoroa pkcr). ^oioftkrk ctu, `ctk iofuecs`kf 5 x ,  y 6 somkgkc pkskfgkf morurut (lraoroa pkcr). Akeki pkskfgkf morurutkf 5 x ,  y 6, 6, x  iorupk`kf  iorupk`kf oeoiof portkik akf  y  oeoiof   oeoiof `oauk. Aojcfcsc 4 5x, y6 9 5k, m6 dc`k akf hkfyk dc`k x 9 k akf y 9 m

Akrc aojcfcsc ac ktks, acscipue`kf mkhwk 4 x ≢ y ⇝ 5x , y 6 ≢ 5 y , x 6. 6. Icske`kf  X   akf S   kakekh auk hcipufkf. Acaojcfcsc`kf Ror`keckf @krtosckf ktku Ror`keckf Ekfgsufg (Nkrtosckf, lr Acront Rrlaunt), X × S somkgkc morc`ut4 Aojcfcsc 4  X × S 9 {5x, y6 | x

∈ X akf y ∈ S }

 Dc`k  X   mcekfgkf roke, ik`k  X  ×  X   iorupk`kf hcipufkf pkskfgkf morurutkf-pkskfgkf morurutkf ktks mcekfgkf roke akf o`ucvkeof aofgkf hcipufkf tctc`-tctc` pkak mcakfg. Qor`kakfg, `ctk sorcfg iofuecs X = uftu` X × X , X > uftu` X × X × X , akf somkgkcfyk. Ektchkf 4  Qufdu``kf mkhwk {x  |  | x ≢ x } 9 Ø Mu`tc 4 Icske K 9 {x  |  | x ≢ x }. }.  K`kf acmu`tc`kf mkhwk K 9 Ø.  Kfakc`kf  K ≢ Ø, ik`k kak x  ∈  K. @krofk x  ∈  K  ik`k x  ≢ x . Qcimue `lftrkac`sc. Dkac, pofgkfakckf skekh, ykfg mofkr  K 9 Ø.

 ;.=.

Jufgsc Icske`kf  X   akf S   kakekh auk hcipufkf. Jufgsc  j   akrc (ktku pkak)  X   `o (ktku `opkak) S  ackrtc`kf somkgkc kturkf ykfg iofgkct`kf sotckp x  ac  X  aofgkf   aofgkf topkt sktu kfggltk  y  ac S  sohcfggk   sohcfggk  y  9 ( x   j (  x ).   Grkjc` jufgsc  j , actuecs G, kakekh `uipuekf pkskfgkf akeki moftu` 5 x ,  j (  (x    )6 x   ac akeki  X  × S .  Dkac, G ⊍  X  × S  acsomut   acsomut grkjc` akrc somukh jufgsc  j   pkak  X   dc`k akf hkfyk dc`k uftu` sotckp x  ∈  X  torakpkt aofgkf tufggke pkskfgkf akeki G ykfg oeoiof portkikfyk kakekh x  ktku  ktku actuecs4 =

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

 9 j (  ( x  G 9 {5x , y 6 | y  9 x )}   ⊍ X × S  Qorechkt, jufgsc akpkt acaojcfcsc`kf ioekeuc grkjc`fyk `krofk jufgsc akpkt actoftu`kf aofgkf grkjc`fyk. Leoh `krofk ctu, acaojcfcsc`kf jufgsc akrc  X  `o S  kakekh   kakekh hcipufkf  j  akeki   akeki  X  × S  acikfk   acikfk sotckp x ∈ X ioipufykc topkt sktu  y ∈ S  sohcfggk  sohcfggk 5x , y 6 ∈ j . Aofgkf `ktk ekcf,  Dc`k 5x , y 6 ∈ j  akf  akf 5x , y– 6 ∈ j  ik`k  ik`k y  9  9 y – @ktk ‘poiotkkf– sorcfg `kec acgufk`kf somkgkc scflfci uftu` `ktk ‘jufgsc–. Jufgsc  j  akrc   akrc  X  `o  `o S  acekimkfg`kf aofgkf4  j  4 4 X ↝ S  mcekikfk 5x , y 6 ∈ j  aofgkf  aofgkf j  jufgsc,  jufgsc, ik`k jufgsc akpkt actuecs y  9  9 j (  (x    )  ktku x   y . x Hcipufkf X  acsomut  acsomut alikcf (ktku akorkh aojcfcsc) akrc  j . Hcipufkf fcekc  j , kakekh ( j   )   9 { y ∈ S  |  | y  9  9 j (  (x    ), ] (  x   x ∈ X } acsomut rkfgo akrc  j . ]kfgo akrc jufgsc  j  sonkrk  sonkrk uiui k`kf eomch `once akrc S , ktku ] (  ( j   )   ⊍ S . Dc`k ] (  ( j   )   9  `opkak S   ), ktku j  surdo`tcj.  surdo`tcj. S , ik`k j acsomut jufgsc lftl (jufgsc  j  `opkak Aojcfcsc 4 Acmorc`kf jufgsc j 4 X ↝  S,  S, j surdo`tcj dc`k akf hkfyk dc`k uftu` sotckp y ∈  S  S torakpkt x ∈  X  X sohcfggk  y 9 j(x). K`cmktfyk, dc`k j surdo`tcj, surdo`tcj, ik`k ](j) 9 S.

 Dc`k K ⊍ X , acaojcfcsc`kf potk  (cikgo)  (cikgo) akrc  K torhkakp j  iorupk`kf  iorupk`kf hcipufkf oeoiof-oeoiof ac akeki S  sohcfggk  sohcfggk y  9  9 j (  (x    )  uftu` soiuk x  ac  ac K, actuecs4 x ( K    ) 9 { y ∈ S  |  | y  9  9 j (  ( x   j (  x ),   x ∈ K} sohcfggk rkfgo akrc j  kakekh  kakekh j (  ( X     ), akf j  surdo`tcj  surdo`tcj dc`k akf hkfyk dc`k S  9  9 j (  ( X     ).  Dc`k M ⊍ S , acaojcfcsc`kf  prkpotk  (cfvors  (cfvors cikgo) akrc M torhkakp  j  iorupk`kf  iorupk`kf hcipufkf oeoiofoeoiof ac akeki x  sohcfggk  sohcfggk y  9  9 j (  (x    )  ∈ M, actuecs4 x  | y  9  9 j (  (x    ),  j  „;( M ) 9 {x ∈ X  | x    y ∈ M}.  Qolroik 4 Jufgsc j 4 X ↝  S  S surdo`tcj ⇑  j   j  „;(M) ≢Ø, ∂M ⊍ S akf M ≢ Ø (   j   surdo`tcj dc`k akf hkfyk dc`k prkpotk akrc hcipufkf mkgckf tk` `lslfg akrc S   iorupk`kf hcipufkf tk` `lslfg). Mu`tc 4 ( ⇝ ) Ac`otkhuc j  surdo`tcj,  surdo`tcj, sohcfggk S  9  9 j (  ( X     ).  K`kf acmu`tc`kf mkhwk ∂M ⊍ S akf M ≢ Ø, j  „;(M) ≢Ø @krofk M ≢ Ø ik`k torakpkt y ∈ M. @krofk y ∈ M akf M ⊍ S  ik`k  ik`k y ∈ S . @krofk y ∈ S  akf  akf S  9  9 j (  ( X     ) ik`k y ∈ j (  ( X     ). @krofk y ∈ j (  ( X     ) ik`k torakpkt x ∈ X  sohcfggk  sohcfggk y  9  9 j (  ( x  x )   „; @krofk y  9  9 j (  ( x  x )  akf y ∈ M, ik`k x ∈ j  ( M ) uftu` suktu x ∈ X.  „;  Dkac, j  (M) ≢Ø. ( ⇒ ) Ac`otkhuc, ∂M ⊍ S akf M ≢ Ø, j  „;(M) ≢Ø  K`kf acmu`tc`kf acmu`tc`kf mkhwk j  surdo`tcj,  surdo`tcj, o`ucvkeof aofgkf ioimu`tc`kf S  9  9 j (  ( X     ) (c)  K`kf acmu`tc`kf acmu`tc`kf mkhwk S ⊍ j (  ( X     )  „; @krofk  j  ( M ) ≢ Ø, ik`k torakpkt x  ∈  j  „;( M ) sohcfggk  y  9  j (  (x    )  uftu` suktu  y  ∈ M. x @krofk M ⊍ S , ik`k y ∈ S . @krofk y  9  9 j (  ( x  ( X     ). x )  ik`k y ∈ j (  >

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

(cc)  K`kf acmu`tc`kf acmu`tc`kf mkhwk j (  ( X     ) ⊍ S  „; @krofk j  ( M ) ≢ Ø, ik`k torakpkt x ∈ j  „;( M ) sohcfggk y  9  9 j (  (x    )  uftu` suktu y ∈ M. x @krofk y  9  9 j (  ( x  ( X     ). x )  ik`k y ∈ j (  @krofk M ⊍ S , ik`k y ∈ S . Zorsc Rk Dkhkruaacf4 Icske y ∈ S . K`kf acmu`tc`kf y ∈ j (  ( X     ).  Kfakc`kf y ∃ j (  ( X     ), ik`k tcak` kak x ∈ X  sohcfggk  sohcfggk y  9  9 j (  (x    ).  sohcfggk x   Dkac, tcak` kak y ∈ S  sohcfggk  9 j (  (x    ).  9 j (  ( x   somkrkfg, ik`k  y  9 x   @krofk M ⊍ S , ik`k tcak` kak  y ∈ M sohcfggk y  9 x ).   @krofk y  somkrkfg,  „; ( X     ).  j  ( M ) 9 Ø. @lftrkac`sc aofgkf hcpltoscs, dkac hkrusekh  y ∈ j (  ( Qho Qho jleelwcfg prllj cs icfo  ) Icske y ∈ j (  ( X     ). K`kf acmu`tc`kf y ∈ S @krofk y ∈ j (  ( X     ) ik`k y  9  9 j (  ( x  x )  uftu` suktu x ∈ X . @krofk  j  „;( M ) ≢Ø akf y  9  9 j (  (x    )  uftu` suktu x ∈ X , ik`k x ∈ j  „;( M ) x @krofk x ∈ j  „;( M ) akf y  9  9 j (  (x    ), x   ik`k y ∈ M @krofk y ∈ M akf M ⊍ S , ik`k y ∈ S .

 Aojcfcsc 4 Jufgsc j 4 X ↝  S  S acsomut sktu-sktu (cfdo`tcj), dc`k ioiofuhc4  j(x ; ) 9 j(x =  ) ⇝ x ; 9 x =  _ftu` sotckp x ; , x =  ∈ X. Aojcfcsc 4 Jufgsc ykfg sktu-sktu akrc X `opkak S (cfdo`tcj akf surdo`tcj) acsomut `lrosplfaofsc sktu-sktu kftkrk X akf S (mcdo`tcj).

Mcek j  mcdo`tcj,  mcdo`tcj, ik`k torakpkt jufgsc  g  4 4 S  ↝ X  sohcfggk  sohcfggk uftu` sotckp x  akf   akf  y  morek`u  morek`u  g (  ( j  (  (x    )) x   9 x   „; akf j (  ( g    (  ( y    )) 9 y . Jufgsc g  acsomut  acsomut cfvors akrc  j  akf  akf actuecs j  . Icske`kf j  4 4  X ↝ S  akf  akf g  4 4 S ↝ T , acaojcfcsc`kf jufgsc h  4 4  X ↝ T , ykctu h (  (x    )  9  g (  ( j  (  (x    )). x x   Jufgsc h  acsomut `liplscsc akrc  g  aofgkf  aofgkf j  akf  akf actuecs  g    j .  Dc`k  j  4  X  ↝ S  akf   akf K ⊍ X acaojcfcsc`kf jufgsc  g  4  K ↝ S  aofgkf   aofgkf ruius  g (  ( x  (x    ), x )  9  j (  x   x  ∈  K. Jufgsc  g  acsomut   acsomut mktkskf ( rostrcntclf   )  j  torhkakp   torhkakp  K akf actuecs  j | K. Jufgsc  j  akf   akf  g  ioicec`c   ioicec`c akorkh hksce rostrcntclf  (rkfgo) akf prkpotk ykfg mormoak. Mkrcskf hcfggk ( f  -tupeo) kakekh suktu jufgsc acikfk alikcffyk iorupk`kf f   mcekfgkf ksec f -tupeo) portkik, ykctu hcipufkf {c ∈ F  |  | c ≡ f }. }. Mkrcskf tk`-hcfggk kakekh jufgsc acikfk alikcffyk iorupk`kf mcekfgkf ksec. @ctk iofggufk`kf cstcekh ―mkrcskf’ uftu` ioikhkic mkrcskf morhcfggk ktku tk`-morhcfggk. Dc`k akorkh hksce akrc mkrcskf cfc morkak akeki hcipufkf  X , `ctk somut mkrcskf akrc ktku ac akeki  X   ktku mkrcskf aofgkf oeoiof-oeoiof akrc  X . Fcekc jufgsc pkak c , actuecs x c  akf iofyomut fcekc torsomut aofgkf oeoiof `o-c  akrc   akrc mkrcskffyk. @ctk dugk iofggufk`mkf fltksc x c  akf mkrcskf tk`-hcfggk aofgkf

x c 

∖ . c 9;

f  c 9;

 uftu` iofuecs`kf f-tupeo torurut,

Fkiuf, tor`kakfg `ctk dugk iofyktk`kf mkrcskf sonkrk

soaorhkfk aofgkf x c  . Akorkh hksce akrc mkrcskf x c   k`kf acfltksc`kf aofgkf { x c }. ^ohcfggk, akorkh f 

f 

hksce akrc f-tupeo torurut x c  c 9;  iorupk`kf hcipufkf tk`-torurut f-tupeo {x c }c 9; . Hcipufkf  K  ac`ktk`kf torhctufg ( nluftkmeo   ) dc`k  K  skik aofgkf akorkh hksce suktu mkrcskf nluftkmeo  (hcfggk ktku tk`-hcfggk). Qotkpc mcekikfk  K  skik aofgkf akorkh hksce akrc mkrcskf hcfggk, ik`k  K acsomut hcipufkf hcfggk (  jcfcto   ). Hcipufkf ykfg mu`kf mu`kf hcipufkf hcfggk hcfggk acsomut hcipufkf tk`-hcfggk tk`-hcfggk ( cfjcfcto   ). cfjcfcto  ^kekh sktu nkrk uftu` iofakpkt`kf mkrcskf tk`-hcfggk (cfjcfcto soquofno) kakekh somkgkc 1

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

morc`ut4 Rrcfscp ]o`urscj 4

Icske`kf  j  4 4  X  ↝  X  akf   akf k  ∈  X , ik`k torakpkt sktu (tufggke) mkrcskf tk`-hcfggk x c  sohcfggk (x     )c   uftu` sotckp c  x ; 9 k , akf x c  + x  + ; 9 j ( 

∖  akrc  X  c 9;

_ftu` sotckp mcekfgkf ksec f , icske`kf  j f  4  X f  ↝  X  akf   akf k  ∈  X . Ik`k torakpkt mkrcskf tufggke (ufcqo) x c   akrc X  sohcfggk  sohcfggk x ; 9 k  akf  akf x c  + x; ,…,x  ). c   + ; 9 j c ( x  Jufgsc  g  4  F  ↝  F  ac`ktk`kf   ac`ktk`kf ilfltlf, dc`k  g (  ( )  c c  6  g (  ( d   )   uftu` c  6  d . Jufgsc h  ac`ktk`kf   ac`ktk`kf mkrcskf mkgckf tk`-hcfggk akrc  j , dc`k torakpkt poiotkkf ilfltlf  g  4 4  F  ↝  F  sohcfggk   sohcfggk h  9  9  j    g . Dc`k  j  9 5j c 6 akf g  9  9 5 g c 6, ik`k j    g  actuecs  actuecs 5 j  gc 6. ;.>.

Gkmufgkf, Crcskf, akf @lipeoiof Acmorc`kf hcipufkf  X  akf   akf ℘(   ) hcipufkf sumsot akrc  X . Icske`kf  K akf M sumsot akrc  X ,  X  acaojcfcsc`kf lporksc crcskf, gkmufgkf, akf `lipeoiof somkgkc morc`ut4 Aojcfcsc 4 Crcskf hcipufkf K aofgkf M, actuecs K ∣ M, acaojcfcsc`kf somkgkc4  K ∣ M 9 {x ∈ X | x ∈ K akf x ∈ M} Gkmufgkf hcipufkf K aofgkf M, actuecs K ∤ M, acaojcfcsc`kf somkgkc4  K ∤ M 9 {x ∈ X | x ∈ K ktku x ∈ M} @lipeoiof hcipufkf K, actuecs „K ktku K ,n  acaojcfcsc`kf somkgkc4  „K 9 Kn  9 {x ∈ X |x ∃ K}

Akrc, aojcfcsc-aojcfcsc ac ktks, acturuf`kf scjkt-scjkt lporksc pkak hcipufkf akeki tolroik morc`ut4  Qolroik 4 Acmorc`kf hcipufkf  X  akf  akf K akf M kakekh sumsot akrc  X . Ik`k morek`u4 ;.  K ∣ M 9 M ∣ K =.  K ∤ M 9 M ∤ K >.  K ∣ M ⊍ K 1.  K ⊍ K ∤ M ?.  K ∣ M 9 K ⇑ K ⊍ M :.  K ∤ M 9 K ⇑ M ⊍ K 7. (   9 K ∣ ( M ∣ N   ) 9 K ∣ M ∣ N   K ∣ M ) ∣ N  9 8. (   9 K ∤ ( M ∤ N   ) 9 K ∤ M ∤ N   K ∤ M ) ∤ N  9 3.  K ∣ ( M ∤ N   ) 9 (   )  K ∣ M ) ∤ (   K ∣ N  ;2.  K ∤ ( M ∣ N   ) 9 (   )  K ∤ M ) ∣ (   K ∤ N  Mu`tc 4 ;. x ∈ K ∣ M ⇑ x ∈ K akf x ∈ M ⇑ x ∈ M akf x ∈ K ⇑ M ∣ K =. x ∈ K ∤ M ⇑ x ∈ K ktku x ∈ M ⇑ x ∈ M ktku x ∈ K ⇑ M ∤ K >. x ∈ K ∣ M ⇑ x ∈ K akf x ∈ M ?

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

⇝ x ∈ K  Dkac K ∣ M ⊍ K 1. x ∈ K ⇝ x ∈ K ktku x ∈ M  Dkac K ⊍ K ∤ M ?.  K ∣ M 9 K ⇑ K ⊍ M (c) Ac`otkhuc K ∣ M 9 K, k`kf acmu`tc`kf mkhwk  K ⊍ M Ackimce somkrkfg x ∈ K. @krofk K ∣ M 9 K akf x ∈ K, ik`k x ∈ K ∣ M @krofk x ∈ K ∣ M, ik`k x ∈ K akf x ∈ M.  Dkac, tormu`tc mkhwk uftu` sotckp x ∈ K ik`k x ∈ M. (cc) Ac`otkhuc K ⊍ M, k`kf acmu`tc`kf mkhwk  K ∣ M 9 K Rortkik, acmu`tc`kf  K ∣ M ⊍ K. Aofgkf iofggufk`kf (>) tormu`tc. @oauk, acmu`tc`kf K ⊍ K ∣ M. Ackimce somkrkfg x ∈ K. @krofk K ⊍ M, ik`k x ∈ M. Dkac, x ∈ K akf x ∈ M, ktku x ∈ K ∣ M  Dkac, tormu`tc mkhwk mkhwk K ∣ M 9 K. :.  K ∤ M 9 K ⇑ M ⊍ K (c) Ac`otkhuc K ∤ M 9 K, k`kf acmu`tc`kf mkhwk M ⊍ K Ackimce somkrkfg x ∈ M.  Kfakc`kf x ∃ K, ik`k x ∃ K ∤ M. @krofk x ∃ K ∤ M ik`k x ∃ K akf x ∃ M. @lftrkac`sc aofgkf ac`otkhuc x ∈ M. Dkac pofgkfakckf skekh, ykfg mofkr x ∈ K. (cc) Ac`otkhuc M ⊍ K, k`kf acmu`tc`kf mkhwk  K ∤ M 9 K Rortkik, acmu`tc`kf  K ∤ M ⊍ K Ackimce somkrkfg x ∈ K ∤ M, ik`k x ∈ K ktku x ∈ M. @krofk M ⊍ K akf x ∈ M, ik`k x ∈ K. Dkac, uftu` sotckp x ∈ K ∤ M ik`k x ∈ K. @oauk, acmu`tc`kf K ⊍ K ∤ M. Aofgkf iofggufk`kf (1) tormu`tc.  Dkac tormu`tc K ∤ M 9 K  K ∣ M ) ∣ N ⇑ x ∈ (   K ∣ M ) akf x ∈ N  7. x ∈ (  ⇑ x ∈ K akf x ∈ M akf x ∈ N  ⇑ x ∈ (  K ∣ M ∣ N  )  ) x ∈ M akf x ∈ N  ⇑ x ∈ K akf ( x  ⇑ x ∈ K akf x ∈ ( M ∣ N  ) ⇑ x ∈ K ∣ ( M ∣ N  ) 8. x ∈ (   K ∤ M ) ∤ N ⇑ x ∈ (   K ∤ M ) ktku x ∈ N  ⇑ x ∈ K ktku x ∈ M ktku x ∈ N  ⇑ x ∈ (  K ∤ M ∤ N  )  ) x ∈ M ktku x ∈ N  ⇑ x ∈ K ktku ( x  ⇑ x ∈ K ktku x ∈ ( M ∤ N  ) ⇑ x ∈ K ∤ ( M ∤ N  ) 3. x ∈ K ∣ ( M ∤ N   ) ⇑ x ∈ K akf x ∈ ( M ∤ N   ) x ∈ M ktku x ∈ N) ⇑ x ∈ K akf ( x   ) x ∈ K akf x ∈ M ) ktku ( x  x ∈ K akf x ∈ N  ⇑ ( x  ⇑ x ∈ (  K ∣ M ) ktku x ∈ (  K ∣ N  ) ⇑ x ∈ (  K ∣ M ) ∤ (  K ∣ N  ) ;2. x ∈ K ∤ ( M ∣ N   ) ⇑ x ∈ K ktku x ∈ ( M ∣ N   )  ) x ∈ M akf x ∈ N  ⇑ x ∈ K ktku ( x  :

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

 ) x ∈ K ktku x ∈ M ) akf ( x  x ∈ K ktku x ∈ N  ⇑ ( x  ⇑ x ∈ (  K ∤ M ) akf x ∈ (  K ∤ N  ) ⇑ x ∈ (  K ∤ M ) ∣ (  K ∤ N  )

 Hcipufkf `lslfg Ø  ioikcf`kf porkfkf ykfg poftcfg ac akeki rukfg  X , akf acsomut`kf ioekeuc scjkt morc`ut4  Qolroik 4  Dc`k Ø hcipufkf `lslfg, X  somkrkfg  somkrkfg hcipufkf akf  K ⊍ X , ik`k morek`u4 ;.  K ∤ Ø 9 K =.  K ∣ Ø 9 Ø >.  K ∤ X  9  9 X  1.  K ∣ X  9  9 K Mu`tc 4 ;. Ackimce somkrkfg x ∈ K ∤ Ø, ik`k x ∈ K ktku x ∈ Ø. @krofk x ∈ Ø skekh, ik`k hkrusekh x ∈ K. ^omkec`fyk, ackimce somkrkfg x ∈ K. Ik`k x ∈ K ktku x ∈ Ø. Dkac, x ∈ K ∤ Ø =.  Kfakc`kf K ∣ Ø ≢ Ø ik`k torakpkt x ∈ K ∣ Ø. @krofk x ∈ K ∣ Ø ik`k x ∈ K akf x ∈ Ø. Akrc scfc tcimue `lftrkac`sc, `krofk x  ∈ Ø  skekh. Dkac, pofgkfakckf skekh, ykfg mofkr kakekh K ∤ Ø 9 Ø. >. Ackimce somkrkfg x  ∈  K ∤  X , ik`k x  ∈  K ktku x  ∈  X . Dkac, x  ∈  X . ^omkec`fyk, ackimce somkrkfg x ∈ X . Ik`k x ∈ K ktku x ∈ X . Dkac, x ∈ K ∤ X . 1. Ackimce somkrkfg x  ∈  K ∣  X , ik`k x  ∈  K akf x  ∈  X . Dkac, x  ∈  K. ^omkec`fyk, ackimce somkrkfg x ∈ K. @krofk K ⊍ X   akf x ∈ K, ik`k x ∈ X . Dkac, x ∈ K ∣ X .

 Akrc aojcfcsc `lipeoiof  K, acporleoh tolroik somkgkc morc`ut4  Qolroik 4  Dc`k Ø hcipufkf `lslfg, X  somkrkfg  somkrkfg hcipufkf akf  K ⊍ X , ik`k morek`u4 n  ;. Ø  9 X  =.  X n 9 Ø >. (   K )n  n  9 K 1.  K ∤ Kn  9 X  ?.  K ∣ Kn  9 Ø :.  K ⊍ M ⇑ Mn  ⊍ Kn  Mu`tc 4 ;. @krofk Ø 9 {x ∈ X  |  | x ∃ X }, }, ik`k Øn  9 {x ∈ X  |  | x ∈ X } 9 X  n  =. @krofk X  9  9 {x ∈ X  |  | x ∈ X }, }, ik`k X  9 {x ∈ X  |  | x ∃ X } 9 Ø >. @krofk Kn  9 {x ∈ X  |  | x ∃ K}, ik`k (   | x ∈ K} 9 K  K )n  n  9 {x ∈ X  | n  1. Ackimce somkrkfg x  ∈  K ∤  K , ik`k x  ∈  K  ktku x  ∈  Kn . @krofk  K ⊍  X   ik`k x  ∈  X . ^omkec`fyk, ackimce somkrkfg x  ∈  X . @krofk  K ⊍  X , ik`k x  ∈  K ktku x  ∃  K. Dkac, x  ∈ K ktku x ∈ Kn . Ik`k, x ∈ K ∤ Kn . ?.  Kfakc`kf K ∣ Kn  ≢ Ø, ik`k torakpkt x ∈ K ∣ Kn . @krofk x ∈ K ∣ Kn  ik`k x ∈ K akf x  ∈ Kn . K`cmktfyk, x ∈ K akf x ∃ K. Qordkac `lftrkac`sc. Dkac, pofgkfakckf skekh, ykfg mofkr  K ∣ Kn  9 Ø. :.  K ⊍ M ⇑ Mn  ⊍ Kn  (c) Ac`otkhuc K ⊍ M, k`kf acmu`tc`kf mkhwk Mn  ⊍ Kn . Ackimce somkrkfg x ∈ Mn . @krofk x ∈ Mn , ik`k x ∃ M. @krofk K ⊍ M, ik`k x ∃ K. Dkac, x ∈ Kn . 7

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

(cc) Ac`otkhuc Mn  ⊍ Kn , k`kf acmu`tc`kf mkhwk  K ⊍ M Ackimce somkrkfg x  ∈  K. @krofk x  ∈  K, ik`k x  ∃  Kn . @krofk Mn  ⊍  Kn , ik`k x  ∃ Mn . @krofk x ∃ Mn , ik`k x ∈ M.

 Auk hu`ui ykfg iofgkct`kf `lipeoiof suktu hcipufkf aofgkf gkmufgkf akf crcskffyk acmorc`kf akeki Hu`ui Ao Ilrgkf somkgkc morc`ut4  Qolroik (Hu`ui Ao Ilrgkf) Ilrgkf) 4  Dc`k X  somkrkfg  somkrkfg hcipufkf akf  K akf M hcipufkf mkgckf akrc  X , ik`k morek`u4 ;. (   K ∤ M )n  9 Kn  ∣ Mn  =. (   K ∣ M )n  9 Kn  ∤ Mn  Mu`tc 4 ;. x ∈ (   K ∤ M )n  ⇑ x ∃ (   K ∤ M ) ⇑ x ∃ K akf x ∃ M ⇑ x ∈ Kn  akf x ∈ Mn  ⇑ x ∈ (  Kn  ∣ M )n  =. x ∈ (   K ∣ M )n  ⇑ x ∃ (   K ∣ M ) ⇑ x ∃ K ktku x ∃ M ⇑ x ∈ Kn  ktku x ∈ Mn  ⇑ x ∈ (  Kn  ∤ M )n 

  Dc`k  K akf M iorupk`kf hcipufkf mkgckf akrc  X , acaojcfcsc`kf moak ( acjjorofno   ) K akf M, actuecs acjjorofno  M ~  K ktku `lipeoiof roektcj akrc  K ac M somkgkc hcipufkf ykfg oeoiof-oeoioffyk ac  K totkpc tcak` ac M. Dkac, M ~ K 9 {x ∈ X | x ∈ K akf x ∃ M} Akrc scfc torechkt mkhwk, M ~ K 9 K ∣ Mn . Moak sciotrc ( syiiotrcn hcipufkf K akf M acaojcfcsc`kf somkgkc4 syiiotrcn acjjorofno  ) akrc auk hcipufkf  K ∊ M 9 (   K ~ M ) ∤ ( M ~ K ) Moak sciotrc akrc auk hcipufkf morcsc soiuk oeoiof ykfg iofdkac kfggltk akrc hcipufkf ykfg sktu ktku ykfg ekcffyk totkpc mu`kf kfggltk `oaukfyk.  Dc`k crcskf akrc `oauk hcipufkf kakekh `lslfg, ac`ktk`kf `oauk hcipufkf torsomut skecfg eopks ktku acsdlcft. @leo`sc hcipufkf-hcipufkf, ύ ac`ktk`kf `leo`sc acsdlcft ktks hcipufkf-hcipufkf dc`k sotckp auk hcipufkf ac ύ kakekh acsdlcft. Rrlsos iofgkimce crcskf ktku gkmufgkf akrc auk hcipufkf akpkt acporeuks aofgkf ioek`u`kf poruekfgkf uftu` ioimorc`kf crcskf ktku gkmufgkf akrc somkrkfg `leo`sc morhcfggk ktks hcipufkf. @ctk mcsk ioimorc`kf aojcfcsc akrc crcskf uftu` somkrkfg `leo`sc ύ ktks hcipufkf-hcipufkf. hcipufkf-hcipufkf. Crcskf akrc `leo`sc ύ kakekh hcipufkf ykfg oeoiof-oeoiof akrc  X  iorupk`kf  iorupk`kf kfggltk uftu` sotckp kfggltk akrc ύ. @ctk iofuecs`kf crcskf cfc aofgkf ∣  K  ktku ∣{ K | K ∈ ύ } . Dkac,  K∈ύ 

∣  K 9 ∣{ K| K ∈ ύ } 9 {x ∈ X | x ∈ K,

uftu` sotckp K ∈ ύ }

 K∈ύ 

^onkrk skik, aojcfcsc akrc gkmufgkf somkgkc morc`ut4 ∤  K 9 ∤{ K| K ∈ ύ } 9 {x ∈ X | x ∈ K, uftu` suktu K ∈ ύ }  K∈ύ 

 Qolroik (Hu`ui Ao Ilrgkf) Ilrgkf) 4 n 

⎫ ⎡ ;. ⎭ ∣  K ⎯ 9 ∤ K n  ⎮ K∈ύ ⎪ K∈ύ  8

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c n 

⎫ ⎡ =. ⎭ ∤  K ⎯ 9 ∣ K n  ⎮ K∈ύ ⎪ K∈ύ  Mu`tc 4 n 

⎫ ⎡ ;. Ackimce somkrkfg x ∈ ⎭ ∣ K ⎯ ⎮ K∈ύ  ⎪ n  ⎫ ⎡ x ∈ ⎭ ∣ K⎯ . ⇑ x ∃ ∣ K K∈ύ  ⎮ K∈ύ ⎪ . ⇑ x ∃ K , uftu` uftu` suktu K ∈ ύ  . ⇑ x ∈ K n , uftu` uftu` sukt suktu K ∈ ύ  . ⇑ x ∈ ∤ K n   K∈ύ  n 

⎫ ⎡ =. Ackimce somkrkfg x ∈ ⎭ ∤ K ⎯ ⎮ K∈ύ  ⎪ n  ⎫ ⎡ x ∈ ⎭ ∤ K⎯ . ⇑ x ∃ ∤ K K∈ύ  ⎮ K∈ύ ⎪ . ⇑ x ∃ K , ∂K ∈ ύ  . ⇑ x ∈ K n , ∂K ∈ ύ  . ⇑ x ∈ ∣ K n   K∈ύ 

  Qolroik (Hu`ui Acstrcmutcj) 4

⎫

⎡

;. M ∣ ⎭ ∤ K ⎯ 9

∤ ( M ∣ K)

⎮ K∈ύ ⎪ K∈ύ  ⎫ ⎡ =. M ∤ ⎭ ∣ K ⎯ 9 ∣ ( M ∤ K ) ⎮ K∈ύ ⎪ K∈ύ  Mu`tc 4

⎫

⎡

;. Ackimce somkrkfg x ∈ M ∣ ⎭ ∤ K ⎯ , acporleoh4

⎮ K∈ύ  ⎪

⎫

⎡

x ∈ M ∣ ⎭ ∤ K ⎯ . ⇑ x ∈ M akf x ∈ ∤ K

⎮ K∈ύ ⎪

K∈ύ 

. ⇑ x ∈ M akf x ∈ K uftu` suktu K ∈ ύ  . ⇑ x ∈ ( M ∣ K ) uftu` suktu K ∈ ύ  . ⇑ x ∈ ∤ (M ∣ K)  K∈ύ 

⎫

⎡

=. Ackimce somkrkfg x ∈ M ∤ ⎭ ∣ K ⎯ , acporleoh4

⎮ K∈ύ  ⎪

3

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

⎫

⎡

x ∈ M ∤ ⎭ ∣ K ⎯ . ⇑ x ∈ M ktku x ∈ ∣ K

⎮ K∈ύ ⎪

K∈ύ 

. ⇑ x ∈ M ktku x ∈ K uftu` sotckp K ∈ ύ  . ⇑ x ∈ ( M ∤ K ) uftu` sotckp K ∈ ύ  . ⇑ x ∈ ∣ (M ∤ K)  K∈ύ 

 Mkrcskf pkak hcipufkf mkgckf akrc  X  `ctk   `ctk krtc`kf mkrcskf akrc ℘(   ), ykctu somukh poiotkkf  X  akrc  F  `o ℘(   ). Dc`k 5 Kc 6 kakekh somukh mkrcskf tk`-hcfggk pkak hcipufkf mkgckf akrc  X , `ctk  X  ∖

iofltksc`kf

∤ Kc   uftu` gkmufgkf akrc akorkh hksce (rkfgo) mkrcskffyk. ^ohcfggk, c 9;

∖

∤ Kc 9 {x ∈ X | x ∈ Kc ,

uftu` suktu c } 0 akf

c 9;

∖

∣ Kc 9 {x ∈ X |x ∈ Kc ,

uftu` sotckp c }

c 9;

^onkrk skik, dc`k

Mc 

f  c 9;

  iorupk`kf mkrcskf hcfggk pkak hcipufkf mkgckf akrc  X , `ctk

f 

iofuecs`kf

∣ Mc   somkgkc crcskf akrc akorkh hksce mkrcskffyk, leoh `krofk ctu4 c 9; f 

∣ Mc 9 M; ∣ M= ∣  ∣ Mf  c 9;

Hcipufkf mkgckf akrc  X  ykfg  ykfg morcfaox kakekh suktu jufgsc pkak hcipufkf cfao`s ΐ `o  X  ktku  ktku `o hcipufkf mkgckffyk. Dc`k ΐ hcipufkf mcekfgkf ksec, ik`k fltksc hcipufkf morcfao`s skik aofgkf fltksc mcekfgkf ksec. Mckskfyk iofggufk`kf fltksc x ΰ akrc pkak x (  (ΰ    ) akf iofuecs`kf cfao`s-fyk aofgkf {x ΰ} ktku {x ΰ 4 ΰ ∈ΐ}. Morc`ut cfc aojcfcsc crcskf akf gkmufgkf akrc hcipufkf morcfao`s4 ∣ Kΰ 9 {x ∈ X |x ∈ Kΰ  , uftu` sotckp ΰ ∈ ΐ} ΰ ∈ΐ

∤ Kΰ 9 {x ∈ X | x ∈ Kΰ  ,

uftu` suktu ΰ ∈ ΐ}

ΰ ∈ΐ

Mcekikfk ΐ 9 F  ik`k  ik`k acporleoh

∖

∣ Kc 9 ∣ Kc   (sorupk dugk uftu` gkmufgkf) c ∈F

c 9;

 Dc`k j  ioiotk`kf  ioiotk`kf X  `opkak  `opkak S  akf  akf { Kΰ} `leo`sc hcipufkf mkgckf akrc  X , ik`k4

⎫ ⎮ΰ

⎡ ⎪

⎫ ⎮ΰ

⎡ ⎪

 j ⎭∤ Kΰ ⎯ 9 ∤ j P Kΰ  Y akf  j ⎭∣ Kΰ ⎯ ⊍ ∣ j ( Kΰ  ) ΰ 

ΰ 

Mu`tc 4 Zorsc Rk Dkhkruaacf4

⎫ ⎡  9 j (  (x    ). x   ∈  j ⎭∤ Kΰ  ⎯ , ik`k torakpkt x ∈ ∤ Kΰ   sohcfggk y  9 ΰ  ⎮ ΰ  ⎪ @krofk x ∈ ∤ Kΰ  , ik`k x  ∈  Kΰ uftu` suktu ΰ ∈ ΐ. @krofk x  ∈  Kΰ  sohcfggk  y  9  j (  ( x  x ),  

(c) Ackimce somkrkfg

ΰ 

ik`k  y ∈ j ( Kΰ  )  uftu` suktu ΰ. Dkac,

∈ ∤  j P Kΰ  Y ΰ 

(cc) Eojt ks kf oxorncso!  Iy Zorsclf4

;2

Mkm ; „ Qolrc Hcipufkf

⎫ ⎮ ΰ 

Nlipceoa my 4 @hkorlfc, ^.^c

⎡ ⎪

ftu` suktu ΰ   y ∈ j ⎭∤ Kΰ ⎯ . ⇝ y 9 j ( x ), ∂x ∈ Kΰ  uftu . ⇝  y ∈ j ( Kΰ  ), uf u ftu` suktu ΰ  . ⇝  y ∈ ∤ j ( Kΰ  ) ΰ 

 y ∈ ∤ j ( Kΰ ) ⇝ y ∈ j ( Kΰ  ), uftu` suktu ΰ  ΰ 

⇝  y 9 j ( x ), ∂x ∈ Kΰ  , uftu` suktu ΰ  ⎫ ⎡ ⇝  y ∈ j ⎭∤ Kΰ  ⎯ ⎮ ΰ  ⎪ ⎫ ⎡ ^oek ^oekfd fdut utfy fyk, k, k`kf k`kf acmu acmu`t `tc` c`kf kf mkhw mkhwkk 4  j ⎭∣ Kΰ ⎯ ⊍ ∣ j ( Kΰ  ) . ⎮ΰ ⎪ ΰ  ⎫ ⎡  y ∈ j ⎭∣ Kΰ ⎯ . ⇝ y 9 j ( x ), ∂x ∈ Kΰ  , ∂ΰ  ⎮ ΰ  ⎪ . ⇝  y ∈ j ( Kΰ  ), ∂ΰ  . ⇝  y ∈ ∣ j ( Kΰ  ) ΰ 

 _ftu` prkpotk, icske`kf { Mΰ} `leo`sc hcipufkf mkgckf akrc S , ik`k  j

∝;

⎫ ⎡ ⎡ ∝; ∝; ⎫ ∝; ⎭∤ Mΰ ⎯ 9 ∤ j ( Mΰ  )  akf  j ⎭∣ Mΰ ⎯ 9 ∣ j ( Mΰ  ) ⎮ ΰ ⎪ ΰ  ⎮ ΰ ⎪ ΰ 

Mu`tc 4

⎫ ⎮ΰ

⎡ ⎪

 K`kf acmu`tc`kf mkhwk  j ∝; ⎭∤ Mΰ ⎯ 9 ∤ j ∝;( Mΰ  ) .

⎫ ⎮ΰ

ΰ 

⎡ ⎪

Rortkik, acmu`tc`kf mkhwk  j ∝; ⎭∤ Mΰ ⎯ ⊍ ∤ j ∝; ( Mΰ  ) x∈ j

∝;

ΰ 

⎫ ⎡ M ∤ ΰ ⎭ ⎯ . ⇝ ∂x ∈ Mΰ  , y 9 j ( x ), uf  tu` suktu ΰ  ⎮ ΰ  ⎪ . ⇝ x ∈ j ∝; ( Mΰ  ), uftu` suktu ΰ  . ⇝ x ∈ ∤ j ∝;( Mΰ  ) ΰ 

@oauk, acmu`tc`kf mkhwk

∝; ∤  j ( Mΰ ) ⊍ j ΰ

∝;

∝;

⎫ ⎡ M ⎭∤ ΰ  ⎯ ⎮ ΰ  ⎪

∝;

x ∈ ∤ j ( Mΰ ). ⇝ x ∈ j ( Mΰ  ), uftu` suktu ΰ  ΰ 

. ⇝ ∂x ∈ M ΰ  , y 9 j ( x ),  uftu` suktu ΰ  .⇝ x∈ j

∝;

⎫ ⎡ ⎭∤ Mΰ  ⎯ ⎮ ΰ  ⎪

^oekfdutfyk k`kf acmu`tc`kf mkhwk,  j

∝;

⎫ ⎡ ∝; M ⎭∣ ΰ ⎯ 9 ∣ j ( Mΰ  ) ⎮ ΰ ⎪ ΰ 

;;

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

⎫ ⎮ΰ

⎡ ⎪

Rortkik, acmu`tc`kf mkhwk  j ∝; ⎭∣ Mΰ ⎯ ⊍ ∣ j ∝; ( Mΰ  ) x∈ j

∝;

ΰ 

⎫ ⎡ M   ⎭∣ ΰ ⎯ . ⇝ ∂x ∈ Mΰ  , y 9 j ( x ), ∂ΰ  ⎮ ΰ  ⎪ . ⇝ x ∈ j ∝; ( Mΰ  ), ∂ΰ  . ⇝ x ∈ ∣ j ∝;( Mΰ  ) ΰ 

@oauk, acmu`tc`kf mkhwk

∣  j

∝;

( Mΰ ) ⊍ j

∝;

ΰ

⎫ ⎡ M ∣ ΰ  ⎭ ⎯ ⎮ ΰ  ⎪

x ∈ ∣ j ∝;( Mΰ ). ⇝ x ∈ j ∝; ( Mΰ  ), ∂ΰ  ΰ 

. ⇝ ∂x ∈ M ΰ  , y 9 j ( x ), ∂  ΰ  .⇝ x∈ j

∝;

⎫ ⎡ M ∣ ΰ  ⎭ ⎯ ⎮ ΰ  ⎪ 

;.1.

 Kedkmkr Hcipufkf Hcipufkf @leo`sc hcipufkf M acsomut kedkmkr hcipufkf ktku kedkmkr Mlleokf, dc`k uftu` sotckp  K, M ∈ M morek`u  K ∤ M ∈ M  akf  Kn  ∈ M. Akeki, mcekfgkf roke, `leo`sc hcipufkf mkgckf  K   akrc  X   acsomut kedkmkr hcipufkf ktku kedkmkr Mlleokf dc`k ∂ K, M ∈ K  morek`u4  morek`u4 (c)  K ∤ M ∈ K  (cc)  Kn  ∈ K  Akrc hu`ui Ao Ilrgkf , (ccc) (   K ∤ M )n  ∈ K ⇑ K ∣ M ∈ K   Qorechkt, dc`k `leo`sc hcipufkf hcipufkf mkgckf  K  akrc  akrc X  ioiofuhc  ioiofuhc (ccc) akf (cc), ik`k aofgkf hu`ui Ao  Ilrgkf   (c) dugk acpofuhc, sohcfggk iorupk`kf kedkmkr hcipufkf. Aofgkf iofgkimce gkmufgkf hcipufkf-hcipufkf, hcipufkf-hcipufkf, torechkt mkhwk4  K;, … , Kf  hcipufkf-hcipufkf hcipufkf-hcipufkf ac  K  ik`k  ik`k  K; ∤ K= ∤ … ∤ Kf  dugk morkak ac K  .

Nlftlh 4 Hcipufkf M 9 {{;}, {=, >, 1}, {;, =, >, 1},

∏} kakekh kedkmkr hcipufkf. 

Rrlplscsc 4 Icske`kf N `leo`sc hcipufkf mkgckf akrc  X , ik`k torakpkt kedkmkr hcipufkf tor`once  K  ykfg   ykfg ioiukt N0 ykctu, dc`k torakpkt kedkmkr hcipufkf  K  ykfg ioiukt N  sohcfggk dc`k M somkrkfg kedkmkr ykfg ioiukt N ik`k M ioiukt K . Mu`tc 4 Icske`kf J `leo`sc hcipukf mkgckf akrc  X   ykfg morupk kedkmkr hcipufkf ykfg ioiukt N. Acaojcfcsc`kf, ] 9 ∣M M∈J

@krofk ]  ⊍ M ∂M ∈ J  akf M  kedkmkr hcipufkf ykfg ioiukt N  ik`k ]   ioiukt N. ^oekfdutfyk acmu`tc`kf ]  suktu  suktu kedkmkr hcipufkf. Icske`kf  K, M ∈ ]  ik`k   ik`k  K, M ∈ M, ∂M ∈ J. @krofk M kedkmkr hcipufkf, ik`k  K ∤ M ∈ M, ,

;=

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

∂M ∈ J, akf Kn ∈ M, ∂M ∈ J. @krofk K ∤ M ∈ M akf Kn  ∈ M, ∂M ∈ J ik`k  K ∤ M ∈ ∣ M  akf  K n  ∈ ∣ M M∈J

M∈J

@krofk  K ∤ M ∈ ∣ M  akf  K n  ∈ ∣ M  ik`k M∈J

M∈J

 K ∤ M ∈ ∣{ M|M ∈ J } 9 ]  akf  K n  ∈ ∣{ M|M ∈ J } 9 ]  

 Dkac ]  kedkmkr  kedkmkr hcipufkf.

  Kedkmkr tor`once ykfg ioiukt N acsomut kedkmkr ykfg acmkfguf leoh N. Rrlplscsc 4 Icske`kf  Kc    kakekh mkrcskf hcipufkf pkak (ktku ac akeki) kedkmkr hcipufkf

] , ik`k

torakpkt mkrcskf hcipufkf Mc   pkak ]  sohcfggk4  sohcfggk4  akf Mf  ∣ Mi  9 ∏  , f ≢ i  akf

∖

∖

c 9;

c 9;

∤ Mc 9 ∤ Kc 

Mu`tc 4 Icske`kf M; 9 K;, M= 9 K=, akf uftu` sotckp mcekfgkf ksec f  6  6 ;, acaojcfcsc`kf4 Mf 9 Kf ~ ( K; ∤ K= ∤ ... ∤ Kf ∝; )

9  Kf ∣ ( K; ∤ K= ∤ ... ∤ Kf ∝; ) 9  Kf ∣ K;n ∣ K=n ∣ ... ∣ Kf n ∝;

n 

^omkgkc ceustrksc, porhktc`kf ackgrki morc`ut4  K ;

 K=

M=

M;

M>  K>

Mf  ⊍  Kf  akf  Kc   mkrcskf hcipufkf pkak kedkmkr hcipufkf ] , ik`k Mf  ∈ ]  ∂f . Dkac, mkrcskf

 5 f , Mi  ⊍ Ki . Dkac, Mc   pkak ] . @krofk Mf  ⊍ Kf  ∂f , ik`k uftu` i  5 Mi ∣ Mf ⊍ Ki ∣ Mf

 

9  Ki ∣{ Kf ∣ K;n ∣ K=n ∣ ... ∣ Kin ... ∣ Kfn ∝;} 9 ( Ki ∣ Kin ) ∣ { Kf ∣ K;n ∣ K=n ∣ ... ∣ Kfn ∝;} 9 ∏ ∣{ Kf ∣ K;n ∣ K=n ∣ ... ∣ Kf n ∝;} 9∏ ^oekfdutfyk, `krofk Mc  ⊍  Kc  ∂c , ik`k

       

∖

∖

∖

∖

c 9;

c 9;

c 9;

c 9;

∤ Mc ⊍ ∤ Kc  . K`kf acmu`tc`kf ∤ Kc ⊍ ∤ Mc  . Icske`kf

∖

 fcekc tor`once akrc { c  |  | x ∈ Kc }, ik`k x ∈ Mf , uftu` x ∈ ∤ Kc  , ik`k x ∈ Kc  uftu` suktu c . Dc`k f  fcekc c 9;

∖

suktu f . Dkac x ∈ ∤ Mc  . c 9;

  Kedkmkr hcipufkf ]  ac`ktk`kf   ac`ktk`kf kedkmkr- ώ ktku ekpkfgkf Mlroe, dc`k gkmufgkf akrc sotckp `leo`sc ;>

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

hcipufkf torhctufg ac ]  dugk   dugk torakpkt ac ] . Skctu, dc`k  Kc   mkrcskf hcipufkf pkak kedkmkr ] , ik`k ∖

  kedkmkr-ώ. Akrc hu`ui Ao Ilrgkf, acporleoh dugk mkhwk crcskf ∤ Kc   dugk morkak ac ] . ^ohcfggk ]  kedkmkrc 9;

akrc sotckp `leo`sc hcipufkf torhctufg ac ]  dugk   dugk torakpkt ac ] . Aofgkf ioek`u`kf soac`ct ilacjc`ksc pkak Rrlplscsc ac ktks (portkik), acporleoh prlplscsc somkgkc morc`ut. Rrlplscsc 4 Icske`kf N `leo`sc hcipufkf mkgckf akrc  X , ik`k torakpkt kedkmkr- ώ tor`once ]  ykfg  ykfg ioiukt N. Mu`tc 4 Icske`kf J `leo`sc hcipukf mkgckf akrc  X   ykfg morupk kedkmkr- ώ  ykfg ioiukt N. Acaojcfcsc`kf, ] 9 ∣M M∈J

@krofk ]  ⊍ M ∂M ∈ J  akf M  kedkmkr-ώ  ykfg ioiukt N  ik`k ]   ioiukt N. ^oekfdutfyk ,

acmu`tc`kf ]  suktu  suktu kedkmkr-ώ. Icske`kf  Kc   mkrcskf hcipufkf pkak ] . @krofk  Kc   morkak ac ] , ik`k  Kc sotckp M ∈ J. @krofk  Kc

∈ M  uftu`

∖

∈ M uftu` sotckp M ∈ J akf M kedkmkr-ώ, ik`k ∤ Kc ∈ M uftu` c 9;

sotckp M ∈ J. Dkac, ∖

∤ Kc  ∈ ∣ M 9 {M|M ∈ J} 9 ] c 9;

 

M∈J

  Kedkmkr-ώ tor`once ykfg ioiukt N acsomut kedkmkr- ώ ykfg acmkfguf leoh N. ;.?.

 K`sclik Rcechkf Rcechkf akf Ror`keckf Ror`keckf Ekfgsufg  K`sclik Rcechkf Rcechkf 4 Icske N  somkrkfg `leo`sc hcipufkf-hcipufkf tk` `lslfg. Ik`k torakpkt jufgsc J   ykfg acaojcfcsc`kf pkak N ykfg ioiotk`kf sotckp  K ∈ N, sohcfggk suktu oeoiof ac J (  ( K    ) toreotk` ac  K.

Jufgsc J   acsomut jufgsc pcechkf akf morgkftufg pkak poicechkf hcipufkf  K ∈ N, sohcfggk suktu oeoiof ac J (  ( K    ) toreotk` ac K. Icske N 9 { X ΰ} iorupk`kf `leo`sc hcipufkf ykfg accfao`s leoh hcipufkf cfaox ΐ. Acaojcfcsc`kf por`keckf ekfgsufg ( acront acront prlaunt  )4

X ΰ 

ΰ 

iorupk`kf `leo`sc akrc soiuk hcipufkf { x ΰ} ykfg accfao`s leoh ΐ sohcfggk x ΰ∈ X ΰ. ^omkgkc nlftlh, dc`k ΐ 9 {;, =}, ik`k acporleoh aojcfcsc kwke por`keckf ekfgsufg  X ; x X = akrc auk hcipufkf  X ; akf X =.  Dc`k z  9  9 {x ΰ} kakekh oeoiof akrc

X ΰ 

ΰ 

 ik`k x ΰ acsomut `llracfkt `o- ΰ akrc z .

 Dc`k skekh sktu akrc  X ΰ `lslfg ik`k

X ΰ 

ΰ 

 kakekh `lslfg. K`sclik pcechkf o`ucvkeof aofgkf

porfyktkkf `lfvorsfyk, ykctu4 Dc`k tcak` kak  X ΰ ykfg `lslfg ik`k

X ΰ 

ΰ 

 tcak` `lslfg. Ktks akskr cfc

Mortrkfa ]ussoee iofyomut K`sclik Rcechkf aofgkf K`sclik Ror`keckf (iuetcpecnktcvo kxcli).

;1

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

Rrlmeoi 4 Icske j  4 4  X ↝ S  kakekh  kakekh jufgsc lftl (pkak) S . Qufdu``kf mkhwk kak jufgsc  g  4 4 S  ↝  X  sohcfggk  sohcfggk  j  g  iorupk`kf jufgsc caoftctks pkak S . Mu`tc 4 Icske`kf N 9 { K | ∎ y ∈ S  aofgkf  aofgkf K 9 j  „;P{ y }Y} }Y} Rortkik, actufdu``kf K tcak` `lslfg uftu` sotckp K ∈ N. Ackimce somkrkfg K ∈ N, ik`k K 9  j  „;P{ y }Y }Y uftu` suktu y ∈ S .  „; x ),   uftu` suktu y ∈ S } @krofk K 9  j  P{ y }Y }Y ik`k K 9 {x ∈ X  |  | y  9  9 j (  ( x  x   Krtcfyk, K tcak` `lslfg. @krofk j  lftl akf y ∈ S  ik`k  ik`k torakpkt x ∈ X  sohcfggk  sohcfggk y  9  9 j (  (x    ). @oauk , aofgkf kxclik pcechkf, `krofk N  `leo`sc hcipufkf-hcipufkf tk` `lslfg ik`k torakpkt jufgsc g  pkak  pkak N sohcfggk uftu` suktu  y– ∈ g (  ( K    ) ik`k y– ∈ K uftu` sotckp K ∈ N. _ftu` sotckp K ∈ N, acpcech jufgsc  g  4 4 S ↝ X  aofgkf  aofgkf aojcfcsc  g (  x ( y    ) 9 x , aofgkf y  9  9 j (  (x    )   akf x ∈ X . Akrc aojcfcsc torsomut, acporleoh4 ∂ y ∈ S  akf  g ( K ) 9 {x ∈ K | y 9 j ( x )} ⊍ K

^ohcfggk, dc`k  y – ∈  g (  ( K    ) ik`k  y – ∈  K  uftu` sotckp  K ∈ N. Dkac, jufgsc  g  ioiofuhc   ioiofuhc k`sclik pcechkf. Krtcfyk, jufgsc cfc `omorkakkffyk acdkicf leoh k`sclik torsomut.  Qork`hcr, actufdu``kf  j  g  jufgsc caoftctks. Ackimce somkrkfg sotckp  y ∈ S  aofgkf  aofgkf g (  ( y    ) 9 x , ik`k y  9  9 j (  (x    )  akf morek`u4 x ( j  g )( y ) 9 j ( g ( y )) ))

9  j ( x ) 9  y   Qorechkt mkhwk  j  g  iorupk`kf jufgsc caoftctks pkak S .

 ;.:.

Hcipufkf Qorhctufg Rkak mkgckf somoeuifyk toekh acaojcfcsc`kf mkhwk suktu hcipufkf ac`ktk`kf torhctufg (nluftkmeo) dc`k hcipufkf torsomut iorupk`kf akorkh hksce akrc suktu mkrcskf. Dc`k akorkh hksce mkrcskf torsomut morhcfggk (jcfcto), ik`k hcipufkf torsomut morhcfggk (jcfcto). Qotkpc, dc`k akorkh hksce hksce mkrcskf torsomut tk`-morhcfggk (cfjcfcto), ik`k hcipufkf torsomut iufg`cf hcfggk (ktku iufg`cf tk`morhcfggk). @ofyktkkffyk sotckp hcipufkf tk` `lslfg ykfg morhcfggk iorupk`kf akorkh hksce akrc suktu mkrcskf tk` hcfggk. ^omkgkc nlftlh, hcipufkf morhcfggk { x ;, …, x f } iorupk`kf akorkh hksce akrc mkrcskf tk` hcfggk ykfg acaojcfcsc`kf aofgkf x c  9 x f  uftu` c  6  6 f ;. ^omukh hcipufkf ac`ktk`kf torhctufg tk`-morhcfggk dc`k hcipufkf torsomut skik aofgkf akorkh hksce suktu mkrcskf tk` hcfggk totkpc mu`kf iorupk`kf akorkh hksce soiuk mkrcskf morhcfggk. Hcipufkf mcekfgkf ksec  F  kakekh  kakekh skekh sktu nlftlh hcipufkf torhctufg tk`-morhcfggk. tk`-morhcfggk. Hcipufkf `lslfg mu`kf iorupk`kf akorkh hksce akrc soiuk mkrcskf. Hcipufkf hcfggk ykfg torhctufg kakekh hcipufkf `lslfg. Dkac poreu acaojcfcsc`kf iofgofkc hcipufkf morhcfggk akf torhctufg sohcfggk hcipufkf `lslfg iorupk`kf hcipufkf morhcfggk akf torhctufg. Aojcfcsc 4 ^uktu hcipufkf ac`ktk`kf hcfggk (jcfcto) dc`k hcipufkf torsomut `lslfg ktku iorupk`kf akorkh hksce suktu mkrcskf hcfggk. ^uktu hcipufkf ac`ktk`kf torhctufg (nluftkmeo lr aofuiorkmeo) dc`k hcipufkf torsomut `lslfg ktku iorupk`kf akorkh hksce suktu mkrcskf (hcfggk ktku tk` hcfggk).

Akrc aojcfcsc ac ktks acporleoh mkhwk potk akrc somkrkfg hcipufkf torhctufg kakekh torhctufg.  Krtcfyk, akorkh hksce akrc somkrkfg jufgsc aofgkf akorkh kske morupk hcipufkf torhctufg kakekh ;

Akorkh hksce mkrcskffyk kakekh 4 {x  { x ;, x =, …, x f , x f , x f , x f , ….} ;?

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

torhctufg. Rrlmeoi 4 Icske j  4 4 X ↝ S  jufgsc  jufgsc akf K ⊍ X . Dc`k K torhctufg mu`tc`kf mkhwk  j (  ( K    ) torhctufg. Mu`tc 4 @krofk K torhctufg, ik`k  K hcipufkf `lslfg ktku  K skik aofgkf akorkh hksce suktu mkrcskf. (c)  Dc`k K 9 ∏, nu`up acmu`tc`kf  j (  ( K    ) 9 ∏  Kfakc`kf  j (  ( K    ) ≢ ∏, ik`k torakpkt  y  ∈  j (  ( K    ) sohcfggk  y  9  j (  (x    )  uftu` suktu x  ∈  K. x @lftrkac`sc aofgkf ac`otkhuc  K 9 ∏. Dkac, j (  ( K    ) 9 ∏. (cc)  Dc`k K ≢ ∏. @krofk K torhctufg, ik`k  K 9 {x c }. Akrc scfc, ik`k

 j ( K ) 9 j

({ x c } ) 9 { y c | y c 9

j ( x c ), x c ∈ K} 9 { y c }

 

 Dkac,  j (  ( K    ) skik aofgkf akorkh hksce suktu mkrcskf, ykctu { y c } aofgkf  y c  9  j (  ( x  x )c   akf x c  ∈  K. @krofk j (  ( K    ) skik aofgkf akorkh hksce suktu mkrcskf, ik`k ik`k j (  ( K    ) torhctufg.

 Morc`ut cfc `lfsop torhctufg acpor`ofke`kf morakskr`kf kak ktku tcak`fyk suktu `lrosplfaofsc sktu-sktu. Skfg poreu acnktkt kakekh sotckp hcipufkf ykfg mor`lrosplfaofsc sktu-sktu aofgkf hcipufkf morhcfggk kakekh morhcfggk akf sotckp hcipufkf ykfg mor`lrosplfaofsc sktu-sktu aofgkf hcipufkf torhctufg kakekh torhctufg. @krofk hcipufkf mcekfgkf ksec  F  kakekh  kakekh hcipufkf torhctufg totkpc tk` hcfggk, sotckp hcipufkf ykfg mor`lrosplfaofsc sktu-sktu aofgkf  F   hkrusekh torhctufg akf tk` hcfggk. Dkac, hcipufkf tk` hcfggk K torhctufg dc`k akf hkfyk dc`k torakpkt `lrosplfaofsc sktu-sktu kftkrk  K akf F . Dc`k hcipufkf tk` hcfggk  O iorupk`kf akorkh hksce akrc mkrcskf 5 x f 6, ik`k  O mor`lrofsplfaofsc sktu-sktu aofgkf  aofgkf prcfscp ro`ursc morc`ut4  F . Acaojcfcsc`kf jufgsc ϟ  4 4 F ↝ F  aofgkf (;) 9 ; ϟ (;) f + f ).   ϟ (  ( f   + ;) 9 mcekfgkf tor`once i  sohcfggk  sohcfggk x i  ≢ x c  uftu` sotckp c ≡ ϟ (  ( f  @krofk  O tk` hcfggk sohcfggk soekeu torakpkt i  akf   akf aofgkf prcfscp woee -lraorcfg  uftu`   uftu`  F , ik`k soekeu torakpkt mcekfgkf ykfg eomch `once akrc i . @lrosplfaofsc f  xϟ ( f )  kakekh `lrosplfaofsc sktusktu kftkrk  F  akf   akf O. ^ohcfggk, acscipue`kf mkhwk somukh hcipufkf torhctufg akf tk` hcfggk dc`k akf hkfyk dc`k torakpkt `lrosplfaofsc sktu-sktu aofgkf  F . Rrlplscsc 4 ^otckp hcipufkf mkgckf akrc hcipufkf torhctufg kakekh torhctufg. Mu`tc 4 Icske`kf O 9 {x f } torhctufg. Ackimce  K ⊍ O.  Dc`k K 9 ∏, ik`k akrc aojcfcsc  K torhctufg  Dc`k K ≢ ∏, ik`k ∎x ∈ K. @oiuackf acaojcfcsc`kf 5 y f 6 somkgkc morc`ut4 ⎧ x  dc`k x f  ∃ K  y f  9 ⎨ ⎣x f dc`k x f  ∈ K  Doeks mkhwk K iorupk`kf akorkh hksce akrc 5 y f 6. Dkac K torhctufg.



Rrlplscsc 4 Icske`kf K hcipufkf torhctufg, ik`k hcipufkf soiuk mkrcskf hcfggk akrc  K dugk torhctufg. Mu`tc 4 @krofk  K  torhctufg ik`k torakpkt `lrosplfaofsc sktu-sktu aofgkf  F   ktku hcipufkf mkgckffyk. Dkac nu`up acmu`tc`kf mkhwk ^   hcipufkf soiuk mkrcskf hcfggk akrc  F   kakekh torhctufg. Icske`kf 5=, >, ?, 7, ;;, …, R `, …6 mkrcskf mcekfgkf prcik, ik`k ∂f  ∈  F  torakpkt  torakpkt jk`tlrcsksc tufggke akrc f , f 9 = x; .>x = .?x = ...R `x ` ;:

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

aofgkf x c c  ∈ F 2 9 F   9 F ∤ {2} akf x ` 6 2. Acaojcfcsc`kf jufgsc  j  pkak F    pkak  F  ykfg   ykfg ioiotk`kf mcekfgkf ksec f  `o   `o mkrcskf hcfggk 5x  5 x ;, …., x `6 akrc  F 2. Ik`k ^   iorupk`kf hcipufkf mkgckf akrc akorkh hksce akrc  j . Aofgkf iofggufk`kf prlplscsc 1, ^  torhctufg.  torhctufg.

 Rrlplscsc 4 Hcipufkf soiuk mcekfgkf rksclfke kakekh torhctufg Mu`tc 4

⎧ p ⎠ Icske  [ 9 ⎨ 0 p , q ∈ T , q ≢ 2 ⎥  hcipufkf mcekfgkf rksclfke akf N `leo`sc mkrcskf hcfggk akrc ⎣ q  ⎩  F . @krofk  F   torhctufg, ik`k iofurut prlplscsc somoeuifyk sotckp mkrcskf hcfggk akrc  F  kakekh torhctufg. Dkac N kakekh `leo`sc mkrcskf hcfggk akf torhctufg. @krofk, f 

N 9 { K | K 9 {x  dc }

c 9;

acaojcfcsc`kf mkrcskf  K d 

∖  d 9;

, x dc  ∈ F }

 aofgkf f 

 K d 9 {x dc }

c 9;

 Qorechkt N  skik aofgkf akorkh hksce mkrcskf  K d 

∖  d 9;

, dkac N  torhctufg. Icske`kf  X  ⊍ N.

@krofk N torhctufg, iofurut prlplscsc 1, X  1,  X  torhctufg.  torhctufg. Icske`kf  X  9 { K| K 9 {x pq } , x pq , p, q ∈ F } . ^oekfdutfyk acaojcfcsc`kf poiotkkf  j  4  X  ↝  [ somkgkc morc`ut4 , q , ; ↝ p / q   p , q , = ↝ ∝ p / q 

;, ;, > ↝ 2  Qorechkt, poiotkkf torsomut iorupk`kf poiotkkf akrc hcipufkf pkskfgkf morurutkf (akrc mcekfgkf ksec) 5 p, q , c 6, 6, c  9  9 ;, =, > `o mcekfgkf rksclfke  [. @krofk hcipufkf pkskfgkf morurutkf akrc mcekfgkf ksec torhctufg, ik`k  [ torhctufg.

 Rrlplscsc 4 Gkmufgkf `leo`sc torhctufg akrc hcipufkf torhctufg kakekh torhctufg Mu`tc 4 Icske`kf N  `leo`sc torhctufg akrc hcipufkf torhctufg. Dc`k hcipufkf akeki N  soiukfyk `lslfg, ik`k gkmufgkffyk `lslfg akf dugk torhctufg. ^oekfdutfyk, acksuisc`kf hcipufkf ac N tcak` soiuk `lslfg, akf `krofk hcipufkf `lslfg tcak` ioimorc`kf pofgkruh pkak gkmufgkf hcipufkf-hcipufkf ac N  ik`k akpkt acksuisc`kf hcipufkf-hcipufkf ac N  tcak` `lslfg.

^ohcfggk N iorupk`kf akorkh hksce akrc mkrcskf tk` hcfggk  Kf 

∖  akf f 9;

sotckp  Kf   iorupk`kf

∖

akorkh hksce akrc mkrcskf tk` hcfggk x fi  i 9; . Qotkpc, poiotkkf akrc 5 f , i 6 `o x fi   kakekh poiotkkf akrc hcipufkf pkskfgkf morurutkf ktks mcekfgkf ksec `o (pkak) gkmufgkf akrc N. @krofk hcipufkf pkskfgkf akrc mcekfgkf ksec kakekh torhctufg ik`k gkmufgkf akrc `leo`sc N dugk torhctufg.

 ;.7.

]oeksc akf O`ucvkeofsc Auk oeoiof x  akf  akf y  mcsk  mcsk ‘acroeksc`kf– sktu skik ekcf akeki mkfyk` nkrk soportc x  9  9  y , x ∈  y , x ⊍  y , ktku uftu` mcekfgkf x  5  y . ^onkrk uiui, icske`kf ] iofyktk`kf roeksc dc`k acmorc`kf x   akf  y , x  moroeksc ] aofgkf  y  actuecs  actuecs x  ]  ]  y , ktku x  tcak`  tcak` moroeksc ] aofgkf  y . ] ac`ktk`kf roeksc pkak hcipufkf  X  ;7

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

dc`k ∂x ∈ X  akf  akf ∂ y ∈ S , x  ]  ] y . Dc`k ] roeksc pkak hcipufkf  X , acaojcfcsc`kf grkjc` akrc ] somkgkc4 {5x , y 6 | x  ]  ] y } Auk roeksc ] akf ^ ac`ktk`kf skik dc`k ( x   ] y   ) ⇑ ( x   ^ y   ), `krofk roeksc actoftu`kf sonkrk tufggke x ] x ^ leoh grkjc`fyk, akf somkec`fyk sotckp hcipufkf mkgckf akrc  X  ×  X  iorupk`kf   iorupk`kf grkjc` akrc suktu roeksc pkak X . Dkac, roeksc ] pkak X  akpkt  akpkt acaojcfcsc`kf ] ⊍ X × X  soakfg`kf  soakfg`kf roeksc ] kftkrk hcipufkf  X  akf  akf  kakekh ] ⊍ X × S . S  kakekh ]oeksc ] pkak X  ac`ktk`kf4  ac`ktk`kf4 ;. Qrkfsctcj , dc`k uftu` sotckp x , y ∈ ], x  ]  ] y  akf  akf y  ]  ] z ⇝ x  ]   ]  z   z  =. ^ciotrc , dc`k uftu` sotckp x , y ∈ ], x  ]  ] y ⇝ y  ]  ] x  >. ]ojeo`sc , dc`k uftu` sotckp x ∈ ], x  ]  ] x  1.  O`ucvkeof , dc`k roeksc torsomut trkfsctcj , sciotrc , akf rojeo`sc . ]oeksc ykfg o`ucvkeof pkak X  sorcfg  sorcfg acsomut o`ucvkeof pkak X . Icske`kf ≫ roeksc o`ucvkeof pkak  X . _ftu` sotckp x  ∈  X , icske`kf  Ox  kakekh hcipufkf ykfg kfggltk-kfggltkfyk kfggltk-kfggltkfyk o`ucvkeof aofgkf x  (leoh  (leoh roeksc ≫ ). Dkac,  9 { y  |  | y ≫ x }, }, ∂x ∈ X   Ox  9  Dc`k  y , z  ∈  Ox   ik`k  y  ≫ x   akf z  ≫ x . @krofk ≫  o`ucvkeof, ik`k aofgkf iofggufk`kf scjkt trkfsctcj akf sciotrc akrc ≫ acporleoh z ≫ y . ^ohcfggk, sotckp auk oeoiof akeki Ox  kakekh o`ucvkeof. Dc`k y  ∈  Ox   akf z  ≫  y , ik`k z  ≫  y   akf  y  ≫ x , ykfg mork`cmkt z  ≫ x   akf dugk z  ∈  Ox . Dkac sotckp oeoiof  X  o`ucvkeof aofgkf suktu oeoiof Ox , ykfg mork`cmkt oeoiof torsomut dugk oeoiof Ox , 9 ∤ Ox  x ∈X 

Eomch ekfdut, uftu` sotckp oeoiof x  akf  akf y  ac  ac X , ;.  Dc`k x ≫ y , hcipufkf Ox  akf O y  skik (caoftc`) ktku =.  Dc`k x  ≫/  y , hcipufkf Ox  akf O y  skecfg eopks Hcipufkf { Ox  | x  ∈ X } acsomut hcipufkf o`ucvkeof   ktku `oeks o`ucvkeof   akrc  X  torhkakp  torhkakp roeksc ≫. hcipufkf o`ucvkeof  ktku `oeks o`ucvkeof  akrc ^ohcfggk  X   skecfg kscfg aofgkf gkmufgkf `oeks-`oeks o`ucvkeof torhkakp roeksc ≫. ^omkgkc nktktkf, `krofk x ∈ Ox  ik`k tcak` kak `oeks o`ucvkeof ykfg `lslfg. @leo`sc `oeks-`oeks o`ucvkeof torhkakp roeksc o`ucvkeof ≫  acsomut qultcof akrc  X   torhkakp ≫, akf tor`kakfg actuecs`kf aofgkf  X /≫. Roiotkkf  ) akrc X  pkak  pkak (lftl)  X /≫. fkturke ikppcfg  x  Ox   acsomut poiotkkf kekic ( fkturke Lporksc mcfor pkak hcipufkf  X  kakekh  kakekh poiotkkf akrc  X  ×  X  `o  `o  X . ]oeksc o`ucvkeof ≫ acsomut `lipktcmoe ( nlipktcmeo   ) aofgkf lporksc mcfor + dc`k x  ≫ x – akf  y  ≫  y – ik`k ( x  +  y   ) ≫ ( x  –). Akeki hke nlipktcmeo  x  + x – +  y –). cfc + iofaojcfcsc`kf lporksc pkak qultcoft  [ 9 X /≫ somkgkc morc`ut4  Dc`k O akf J  kfggltk  kfggltk [, acpcech x ∈ O akf y ∈ J  akf  akf acaojcfcsc`kf  O + J  somkgkc  somkgkc O( x x  ++ y  ). @krofk ≫ o`ucvkeof, torechkt  O + J  hkfyk  hkfyk morgkftufg pkak  O akf J  akf  akf tcak` morgkftufg pkak poicechkf x  ktku  ktku y . ;.8.

_rutkf Rkrscke akf Rrcfscp Ik`scike ]oeksc ] pkak hcipufkf X  ac`ktk`kf  ac`ktk`kf ;. Qrkfsctcj , dc`k uftu` sotckp x , y ∈ ], x  ]  ] y  akf  akf y  ]  ] z ⇝ x  ]   ]  z   z  =. ^ciotrc , dc`k uftu` sotckp x , y ∈ ], x  ]  ] y ⇝ y  ]  ] x  >. ]ojeo`sc , dc`k uftu` sotckp x ∈ ], x  ]  ] x  1.  O`ucvkeof , dc`k roeksc torsomut trkfsctcj , sciotrc , akf rojeo`sc . ?.  Kftcsciotrc , dc`k uftu` sotckp x , y ∈ ], x  ]  ] y  akf  akf y ] x ⇝ x  9  9 y  ]oeksc ≴  ac`ktk`kf urutkf pkrscke pkak hcipufkf  X  (ktku   (ktku iofgurut`kf  X  sonkrk   sonkrk pkrscke) dc`k roeksc torsomut trkfsctcj akf kftcsciotrc. ^omkgkc nlftlh, ≡ iorupk`kf urutkf pkrscke pkak mcekfgkf roke, akf ⊍ iorupk`kf urutkf pkrscke pkak `leo`sc hcipufkf ℘.  Dc`k ≴  urutkf pkrscke pkak  X  akf   akf dc`k k  ≴ m , `ctk sorcfg iofgktk`kf mkhwk  k  iofakhueuc  iofakhueuc m  ( k  k   ) ktku m iofgc`utc k (m   ). Qor`kakfg `ctk iofgktk`kf mkhwk k `urkfg akrc m  ktku  ktku m eomch  pronoaos m   pronoaos m  k (m  jleelws k   jleelws k  akrc k . ;8

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

 Dc`k O ⊍ X , oeoiof k ∈ O acsomut ;.  Oeoiof portkik  (jcrst  (jcrst oeoiofs) ktku oeoiof tor`once akrc  O dc`k ∂x ∈ O, x ≢ k , k  ≴ x  =.  Oeoiof tork`hcr  (ekst  (ekst oeoioft) ktku oeoiof tormoskr akrc  O dc`k ∂x ∈ O, x ≢ k , x  ≴  k  >.  Oeoiof icfcik e (icfcike oeoioft) akrc  O dc`k tcak` kak  x ∈ O, x ≢ k , x  ≴ k  1.  Oeoiof ik`scike  (ikxcike  (ikxcike oeoioft) akrc  O dc`k tcak` kak  x ∈ O, x ≢ k , k ≴ x   Dc`k ≴   urutkf ecfokr, oeoiof icfcike pkstc iorupk`kf oeoiof tor`once. Qotkpc sonkrk uiui skfgkt ioiufg`cf`kf uftu` ioicec`c oeoiof icfcike ykfg mu`kf oeoiof tor`once. ]oeksc ≴  pkak hcipufkf  X  ac`ktk`kf,  ac`ktk`kf, ;. _rutkf ecfokr  ktku   ktku urutkf soaorhkfk , dc`k ∂x ,  y  ∈  X , x  ≴  y  ktku   ktku  y  ≴ x  (skekh   (skekh sktu). ^omkgkc nlftlh, ≡ iorupk`kf urutkf ecfokr pkak hcipufkf mcekfgkf roke, soioftkrk ⊍ mu`kf urutkf ecfokr pkak `leo`sc hcipufkf ℘, somkm dc`k x  ∣  y  9 ∏  (skecfg kscfg) ik`k `oauk sykrkt torsomut tcak` acpofuhc, ykctu x ⊁ y  akf  akf y ⊁ z . =. _rutkf pkrscke ykfg rojeo`sc , dc`k x  ≴ x , ∂x  ∈  X . ^omkgkc nlftlh, ≡ iorupk`kf urutkf ecfokr ykfg rojeo`sc pkak hcipufkf mcekfgkf roke. >. _rutkf pkrscke ykfg `ukt , dc`k tcak` porfkh x  ≴ x . ^omkgkc nlftlh, 5 iorupk`kf urutkf pkrscke ykfg `ukt pkak ]. _ftu` sotckp urutkf pkrscke ≴   torakpkt aofgkf tufggke urutkf pkrscke ykfg `ukt akf urutkf pkrscke ykfg rojeo`sc. Rrlmeois 4 Icske`kf I 9 {k ∈ |k ≯ ;} ]oeksc ] acaojcfcsc`kf pkak  I  somkgkc  somkgkc morc`ut4  ] m ⇑ m  akpkt  akpkt acmkgc leoh k  k  ] ;.  Kpk`kh ] roeksc urutkf urutkf ecfokr<  Dkwkm 4 Rortkik actufdu``kf kpk`kh ] roeksc urutkf pkrscke< c) ] kftcsciotrcs  ] y ⇑ y  akpkt  akpkt acmkgc leoh x  x  ]  9 `x  ⇑ ∎` ∈ I  sohcfggk y  9  ] x ⇑ x  akpkt  akpkt acmkgc leoh  y   y  ]  9 ey  ⇑ ∎e ∈ I  sohcfggk x  9  Dkac,  9 `ex  x  9 Acporleoh `e  9  9 ;, ykfg mork`cmkt ` 9 „; akf e  9  9 „; ktku ` 9 ; akf e  9  9 ; @krofk k ≯ ;, ∂k ∈ I  ik`k hkrusekh ` 9 ; akf e  9  9 ;. ^ohcfggk  y  9  9 x  ktku  ktku x  9  9 y . Dkac ] scioftrcs. cc) ] trkfsctcj  ] y ⇑ y  akpkt  akpkt acmkgc leoh x  x  ]  9 `x  ⇑ ∎` ∈ I  sohcfggk y  9  ] z ⇑ z  akpkt  akpkt acmkgc leoh  y   y  ]  9 ey  ⇑ ∎e ∈ I  sohcfggk z  9  Dkac, z  9  9 e`x   ktku z  9  9 px  aofgkf  aofgkf  p 9 e` ∈  Aofgkf `ktk ekcf z  akpkt  akpkt acmkgc leoh x . ^ohcfggk x  ]  ] z . Dkac ] trkfsctcj Akrc (c) akf (cc) ] kakekh roeksc urutkf pkrscke. @oauk ackimce somkrkfg x , y ∈ I  aofgkf  aofgkf x ≢ y , acmu`tc`kf mkhwk x  ]  ] y  ktku  ktku y  ]  ] x   Dc`k x  6  6 y , ik`k akpkt actuecs  9 py  +  + r ;, aofgkf 2 ≡ r ; 5 y  x  9  Dc`k y  6  6 x , ik`k akpkt actuecs  9 qx  +  + r =, aofgkf 2 ≡ r = 5 x   y  9 ;3

Mkm ; „ Qolrc Hcipufkf

Nlipceoa my 4 @hkorlfc, ^.^c

 Dc`k x  akf   akf  y  mcekfgkf-mcekfgkf   mcekfgkf-mcekfgkf prcik, ik`k r ; akf r = tcak` `oaukfyk fle. K`cmktfyk x ] y  akf  y ] x . x  . Dkac, ] mu`kf roeksc urutkf ecfokr. =.  Qoftu`kf oeoiof tor`once  oeoiof tor`once ⇑ ∂x ∈ I , x ≢ k , k  ]  ] x  k  oeoiof  akpkt acmkgc k  (soiuk  (soiuk mcekfgkf akpkt acmkgc leoh k   ) ⇑ x  akpkt  Dkac, k  9  9 ; iorupk`kf oeoiof tor`once. >.  Qoftu`kf oeoiof icfcike  oeoiof icfcike ⇑ Qcak` kak x ∈ I , x ≢ k , x  ]  ] k  k  oeoiof   akpkt acmkgc x  (mcekfgkf-mcekfgkf   (mcekfgkf-mcekfgkf ykfg ⇑ Qcak` kak x  ∈  I   sohcfggk k  akpkt tcak` akpkt acmkgc leoh mcekfgkf ekcf `onukec leoh mcekfgkf ykfg skik)  Dkac k  mcekfgkf  mcekfgkf prcik akf k  9  9 ; iorupk`kf oeoiof icfcike

 Nlftlh 4 ]95  K 9   9 ; iorupk`kf oeoiof tor`once, somkm ∂x ∈ K, x ≢ ; ik`k ; 5 x  • k  9  9 ; iorupk`kf oeoiof icfcike, somkm tcak` kak x ∈ K, x ≢ ; sohcfggk x  5  5 ; • k  9

 Rrcfscp morc`ut cfc o`ucvkeof aofgkf k`sclik pcechkf akf eomch sorcfg acgufk`kf. Rrcfscp Ik`scike Hkusaúrj 4 Icske ≴  urutkf pkrscke pkak  X . Ik`k torakpkt hcipufkf mkgckf ik`scike akrc  X  ykfg   ykfg torurut ecfokr leoh ≴ . Aofgkf `ktk ekcf, hcipufkf mkgckf ^   akrc  X   ykfg torurut ecfokr leoh ≴   akf ioicec`c scjkt dc`k ^ ⊍ Q ⊍ X  akf  akf Q  torurut  torurut ecfokr leoh ≴  ik`k ^  9  9 Q . ;.3.

_rutkf ]kpc akf Lracfke Qorhctufg ^uktu roeksc urutkf ecfokr `ukt 5 pkak hcipufkf  X  acsomut   acsomut urutkf rkpc ( woee woee lraorcfg  ) uftu`  X  ktku ac`ktk`kf iofgurutkf  X  aofgkf  aofgkf rkpc dc`k sotckp hcipufkf mkgckf tk` `lslfg akrc  X  ioiukt  ioiukt suktu oeoiof portkik (oeoiof tor`once).  Dc`k ackimce  X  9  F   akf 5 morkrtc ‘`urkfg akrc–, ik`k  F   torurut rkpc aofgkf 5. Ac scsc ekcf, hcipukf soiuk mcekfgkf roke ] tcak` torurut rkpc aofgkf roeksc ―`urkfg akrc’. Rrcfscp morc`ut cfc iorupk`kf k`cmkt akrc k`sclik pcechkf akf akpkt actufdu``kf dugk o`ucvkeof aofgkffyk (mkgkcikfk yk<) Rrcfscp urutkf rkpc 4 ^otckp hcipufkf  X  akpkt   akpkt torurut rkpc. Aofgkf `ktk ekcf, torakpk roeksc 5 ykfg iofgurut`kf  X  aofgkf rkpc Rrlplscsc 4  Qorakpkt hcipufkf hcipufkf tk` torhctufg X  ykfg  ykfg torurut rkpc leoh roeksc 5 sohcfggk4 (c)  Qorakpkt oeoiof tork`hcr tork`hcr K ac X  (cc)  Dc`k x ∈ X  akf  akf x ≢ K ik`k hcipufkf { y ∈ X  |  | y  5  5 x } torhctufg

Rkak prlplscsc ac ktks, oeoiof tork`hcr K ac  X   ac`ktk`kf lracfke tk` torhctufg portkik (  jcrst   acsomut hcipufkf aofgkf lracfke `urkfg akrc ktku skik aofgkf lracfke tk` ufnluftkmeo lracfke  ) akf  X  acsomut torhctufg portkik. Oeoiof-oeoiof x 5 K acsomut lracfke-lracfke lracfke-lracfke torhctufg ( nluftkmeo lracfkes  ).

=2

Mkm = ^cstoi Mcekfgkf ]oke =.;.

 K`sclik Mcekfgkf Mcekfgkf ]oke Icske`kf   kakekh hcipufkf mcekfgkf roke, R  hcipufkf  hcipufkf mcekfgkf plsctcj akf jufgsc ‘+– akf ‘.– akrc  ×   `o   akf ksuisc`kf ioiofuhc k`sclik-k`sclik morc`ut4  K`sclik Ekpkfgkf Ekpkfgkf _ftu` soiuk mcekfgkf roke x , y , akf z  morek`u4  morek`u4  K;. x  +  + y  9  9 y  +  + x   K=. ( x  + y   ) + z  9  9 x  +  + (   + z   ) x  +  y  +  K>. ∎2 ∈   sohcfggk x  +  + 2 9 x , uftu` sotckp x ∈   K1. ∂x ∈  , ∎! w ∈   sohcfggk x  +  + w  9  9 2  K?. xy  9  9 yx   K:. ( xy   )z  9  9 x (  ( yz     ) xy   K7. ∎; ∈   sohcfggk ; ≢ 2, akf x .; .; 9 x ∂x ∈   K8. ∂x ∈  , x ≢ 2, ∎w ∈   sohcfggk xw  9  9 ;  K3. x (  ( y     +  + z   ) 9 xy  +  + xz 

Hcipufkf ykfg ioiofuhc k`sclik ac ktks acsomut ekpkfgkf (torhkakp lporksc + akf .). Acporleoh akrc K; mkhwk oeoiof 2 kakekh tufggke. Oeoiof w   pkak K1 dugk tufggke akf acfltksc`kf aofgkf ‘„ x  x –.–. Oeoiof ; pkak K7 ufc` akf oeoiof w   pkak K8 dugk ufc` akf acfltksc`kf aofgkf ‘ x  „;– @oiuackf acaojcfcsc`kf pofgurkfgkf akf poimkgckf somkgkc morc`ut4 x  9 xy ∝;  „ y  9  9 x  +  + („   ) akf x  „  y   y   K`sclik _rutkf _rutkf Icske`kf R  kakekh  kakekh hcipufkf mcekfgkf roke plsctcj, R  ioiofuhc  ioiofuhc k`sclik morc`ut4 M;. x , y ∈ R ⇝ x  +  + y ∈  M=. x , y ∈ R ⇝ x.y ∈  M>. x ∈  ⇝ (x 9 2) ktku ( x   ) ktku ( x   ) x ∈ R  x ∈ R 

^uktu scstoi ykfg ioiofuhc k`sclik ekpkfgkf akf k`sclik urutkf acsomut ekpkfgkf torurut (lraoroa jcoea). ^ohcfggk mcekfgkf roke kakekh ekpkfgkf torurut. Mogctu dugk aofgkf hcipufkf mcekfgkf rksclfke iorupk`kf ekpkfgkf torurut. Akeki ekpkfgkf torurut acaojcfcsc`kf x  5  5 y  ykfg  ykfg morkrtc x  „  „ y ∈ R . @ctk iofuecs`kf ‘ x ≡ y – uftu` ‘x  5  y – ktku ‘x  9 y– . Hcipufkf mcekfgkf roke aofgkf roeksc 5 iorupk`kf hcipufkf torurut ecfokr. Morakskr`kf k`sclik urutkf acporleoh4 k. ( x 5 y   ) & ( z   5 w   ) ⇝ x  +  + z  5  5 y  +  + w  z 5 m. (2 5 x  5  5 y   ) & (2 5 z  5  5 w   ) ⇝ xz  5  5 yw  n.  Qcak` kak x  sohcfggk  sohcfggk x  5  5 x . Mu`tc 4 k. _ftu` ioimu`tc`kf x  +  + z  5  5 y  +  + w  nu`up  nu`up acmu`tc`kf (   + w   ) „ ( x  + z   ) ∈ R .  y  + x  + @krofk x  5  5 y  ik`k  ik`k y  „  „ x ∈ R  @krofk z  5  5 w  ik`k  ik`k w  „  „ z ∈ R 

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

@krofk y  „  „ x , w  „  „ z ∈ R  ik`k  ik`k morakskr`kf k`sclik M; acporleoh  9 y + w „ x „ z  9  9 (   ) „ ( x + z   ) ∈ R .  y „ x + w „ z  9  y + w  m. _ftu` ioimu`tc`kf xz  5  5 yw  nu`up  nu`up acmu`tc`kf yw  „  „ xz  @krofk 2 5 x  5  5 y  ik`k  ik`k y  „  „ x ∈ R  akf  akf x  „  „ 2 9 x ∈ R  @krofk 2 5 z  5  5 w  ik`k  ik`k w  „  „ z ∈ R  akf  akf w  „  „ 2 9 w ∈ R  @krofk y  „  „ x , y , w  „  „ z , akf w ∈ R  ik`k  ik`k morakskr`kf M; akf M= acporleoh4 ( y     „  „ x   ) + x (  ( w   „ z   ) 9 yw  „  „ wx  +  + wx  „  „ xz  9  9 yw  „  „ xz ∈ R  w (  w „ n.  Kfakc`kf kak x   sohcfggk x  5 x . @krofk x  5 x   ik`k x  „ x  ∈ R . K`cmktfyk 2 ∈ R. @lftrkac`sc aofgkf ac`otkhuc R   hcipufkf mcekfgkf plsctcj. Dkac pofgkfakckf skekh ykfg mofkr tcak` kak x  sohcfggk  sohcfggk x  5  5 x .  Aojcfcsc (^uproiui akf Cfjciui) 4 Icske`kf ^  ⊍  ⊍  . * ;. k  mktks ktks ^ , dc`k x ≡ k * uftu` sotckp x ∈ ^  =. k  mktks  mktks ktks tor`once akrc ^ , dc`k (c) k  mktks  mktks ktks ^  (cc)  Dc`k m  mktks  mktks ktks ik`k k ≡ m  Fltksc 4 k  9  9 sup( ^   | x ∈ ^ } ^ )  9 sup x  9 sup{x  | x ∈^  *

>. n  mktks mkwkh ^ , dc`k n  ≡ x  uftu`  uftu` sotckp x ∈ ^  1. n  mktks  mktks mkwkh tormoskr akrc ^ , dc`k (c) n  mktks  mktks mkwkh ^  (cc)  Dc`k a  mktks  mktks mkwkh ik`k a ≡ n  Fltksc 4 n  9  9 cfj( ^   | x ∈ ^ } ^ )  9 cfj x  9 cfj{x  | *

x ∈^ 

Rorhktc`kf ceustrksc morc`ut4  „ ο k  „ x 2 k   Dc`k k  mktks  mktks ktks tor`once, ik`k uftu` sotckp ο 6 2 k`kf soekeu kak x 2 sohcfggk x 2 6 k  „  „ ο. Krtcfyk,  „ ο mu`kf mktks ktks `krofk kak x 2 ykfg fcekcfyk eomch moskr (ktks) akrcfyk. k  „ n  x 2 n+ο  Dc`k n  mktks   mktks mkwkh tormoskr, ik`k uftu` sotckp ο  6 2 k`kf soekeu kak x 2 sohcfggk x 2 5 n  + ο.  Krtcfyk n  +  + ο mu`kf mktks mkwkh `krofk kak x 2 ykfg fcekcfyk eomch `once (mkwkh) akrcfyk. Akrc auk ceustrksc ac ktks, ik`k aojcfcsc suproiui akf cfjciui akpkt acfyktk`kf akeki fltksc iktoiktcs somkgkc morc`ut4 ;. k  mktks  mktks ktks tor`once akrc ^ , dc`k (c) ∂x ∈ ^ , x ≡ k  (cc) ∂ο 6 2, ∎x 2 ∈ ^ , ∀ x 2 6 k  „  „ ο =. n  mktks  mktks mkwkh tormoskr akrc ^ , dc`k (c) ∂x ∈ ^ , n ≡ x  (cc) ∂ο 6 2, ∎x 2 ∈ ^ , ∀ x 2 5 n  +  + ο

==

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

Nlftlh ^lke 4 Icske`kf K akf M tormktks. Mu`tc`kf mkhwk sup(  K + M ) 9 sup(   K ) + sup( M ) aofgkf  + m  |  | k ∈ K akf m ∈ M}  K + M 9 {k  +  Dkwkm 4 Icske p 9 sup(   9 sup( M ). @krofk  K ) akf q  9  p 9 sup( K ) ⇑ .( c ) p ≯ k , ∂k ∈ K

.( cc ) ∂ο  6 2, ∎k 2 ∈ K ∀ k 2 6 p ∝ ο = akf q 9 sup( M ) ⇑ .( ccc ) q ≯ m , ∂m ∈ M .( cv ) ∂ο  6 2, ∎m2 ∈ M ∀ m2 6 q ∝ ο = Akrc (  )  ) acporleoh c  c  akf ( ccc  ccc   + m , ∂k ∈ K akf m ∈ M. Dkac, p + q ≯ k  +  + m , ∂k  +  + m ∈ K + M ……………….. (*)  p + q ≯ k  +  Dkac p + q  mktks  mktks ktks akrc  K + M Akrc ( cc   ) acporleoh c ) c  akf ( cv  cv   ) „ ο ……………………………. (**) ∂ο 6 2, ∎k 2 ∈ K akf m 2 ∈ M sohcfggk k 2 + m 2 6 (   p + q  Akrc (*) akf (**) tormu`tc mkhwk p + q  9  9 sup(   K + M )

  K`sclik @oeofg`kpkf @oeofg`kpkf ^otckp hcipufkf mkgckf akrc   ykfg tcak` `lslfg akf tormktks ac ktks ioipufykc mktks ktks tor`once ( suproiui   ). suproiui  ^otckp hcipufkf mkgckf akrc    ykfg tcak` `lslfg akf tormktks ac mkwkh ioipufykc mktks mkwkh tormoskr ( cfjciui   ). cfjciui  =.=.

Mcekfgkf ]oke ykfg Acporeuks _ftu` ioiporeuks scstoi mcekfgkf roke  , ik`k actkimkh`kf oeoiof ∖ akf „ ∖. Hcipufkf mkru cfc acsomut hcipufkf mcekfgkf roke ykfg acporeuks  * . ]oeksc 5 acporeuks aojcfcscfyk pkak  * iofdkac „ ∖ 5 x  5  5 ∖ uftu` sotckp x ∈  . @oiuackf acaojcfcsc`kf ∂x ∈  .  + ∖ 9 ∖,  „ ∖ 9 „ ∖ x  + x  „ dc`k x  6  6 2 x .∖ 9 ∖  .„ ∖ 9 „ ∖  dc`k x  6  6 2 x .„  akf ∖ + ∖ 9 ∖, „ ∖ „ ∖ 9 „ ∖ ∖.( »∖ »∖ ) 9 »∖, „ ∖.( »∖ »∖ ) 9 ∞ ∖ ^oakfg`kf lporksc ∖ „ ∖ tcak` acaojcfcsc`kf. Qotkpc, 2.∖ 9 2. ^kekh sktu `ogufkkf  *   kakekh uftu` iofaojcfcsc`kf sup( ^  ^ )  akf cfj( ^  ^ )  uftu` soiuk ^  hcipufkf hcipufkf mkgckf akrc   ykfg tcak` `lslfg ^ .  Dc`k ^  tcak`  tcak` tormktks ac ktks, ik`k sup( ^  ^ )  9 ∖  Dc`k ^  tcak`  tcak` tormktks ac mkwkh, ik`k cfj( ^  ^ )  9 „ ∖ ∏ )   9 „ ∖.  Dkac, acaojcfcsc`kf acaojcfcsc`kf sup( ∏   =.>.

Mcekfgkf Ksec akf Mcekfgkf ]ksclfke somkgkc ^umsot akrc Mcekfgkf ]oke @ctk toekh iofggufk`kf scimle ; mu`kf hkfyk uftu` iofyktk`kf mcekfgkf ksec portkik totkpc dugk mcekfgkf roke ‘sposcke– soportc ykfg actuecs`kf akeki k`sclik K7. Rortkik, acaojcfcsc`kf mcekfgkf roke > somkgkc ; + ; + ;. Aofgkf aoic`ckf `ctk akpkt iofaojcfcsc`kf mcekfgkf roke ykfg mor`lrosplfaofsc mor`lrosplfaofsc aofgkf soimkrkfg mcekfgkf ksec. Morakskr`kf prcfscp ro`urscj ik`k torakpkt somukh jufgsc ϟ 4  ↝   ykfg ioiotk`kf mcekfgkf ksec `o mcekfgkf roke aofgkf aojcfcsc somkgkc morc`ut4 =>

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

ϟ (;) (;) 9 ; ϟ (  ( f   + ;) 9 ϟ (  (f  f + f  )  + ; (Nktktkf4 ; iofyktk`kf mcekfgkf roke pkak scsc `kfkf akf mcekfgkf ksec pkak scsc `crc) @ctk hkrus iofufdu``kf mkhwk jufgsc ϟ  kakekh jufgsc sktu-sktu. _ftu` iofufdu``kffyk nu`up actufdu``kf mkhwk jufgsc ϟ  ilfltlf.  ilfltlf.

Mu`tc 4  K`kf acmu`tc`kf ϟ  ilfltlf  ilfltlf fkc`.  Krtcfyk, dc`k p 5 q  ik`k  ik`k ϟ (  ( p  ) 5 ϟ (  (q  q  )  aofgkf  p , q ∈  . @krofk p 5 q  ik`k  ik`k q  9  9 p + f  uftu`  uftu` sotckp f ∈   K`kf acmu`tc`kf mkhwk ϟ (  ( p  ) 5 ϟ (  ( p  + f   ) Mu`tc aofgkf cfau`sc _ftu` f  9  9 ;, acporleoh ( p  ) 5 ϟ (  ( p   + ; ) 9 ϟ (  ( p  ) + ; ϟ (   Dkac, porfyktkkf mofkr mofkr uftu` f  9  9 ;.  Ksuisc`kf porfyktkkf porfyktkkf mofkr uftu` uftu` f  9  9 `, ykctu morek`u ϟ (  ( p  ) 5 ϟ (  ( p  + ` )  K`kf acmu`tc`kf porfyktkkf porfyktkkf mofkr uftu` uftu` f  9  9 ` + ;, ϟ (  ( p   + ( ` + ;)) 9 ϟ ((  ((   p + ` ) + ;) 9 ϟ (  ( p   + ` ) + ; 6 ϟ (  ( p  ) + ; 6 ϟ (  ( p  )  Dkac, ϟ (  ( p  ) 5 ϟ (  ( p   + ( ` + ;)). Krtcfyk porfyktkkf mofkr uftu` f  9  9 ` + ;. Morakskr`kf cfau`sc ac ktks, tormu`tc ϟ  ilfltlf.  ilfltlf. Aofgkf `ktk ekcf, tormu`tc ϟ  sktu-sktu.  sktu-sktu.

 ^oekfdutfyk, dugk akpkt acmu`tc`kf (aofgkf cfau`sc iktoiktc`k) mkhwk ( p   + q   ) 9 ϟ (  ( p  ) + ϟ (  ( q  ϟ (  q )  akf ϟ (  ( pq     ) 9 ϟ (  ( p  ) ϟ (  ( q  q )  Mu`tc 4 Rortkik4 Icske`kf q  9  9 f  , , ∂f ∈  . Acporleoh ϟ (  ( p   + f   ) 9 ϟ (  ( p  ) + ϟ (  (f  f  )  _ftu` f  9  9 ; ϟ ( p + ;) 9 ϟ( p ) + ; 9 ϟ( p ) + ϟ(;)    Dkac porfyktkkf mofkr mofkr uftu` f  9  9 ;.  Ksuisc`kf porfyktkkf porfyktkkf mofkr uftu` uftu` f  9  9 `, ykctu ϟ (  ( p   + ` ) 9 ϟ (  ( p  ) + ϟ (  (`   )  K`kf acmu`tc`kf porfyktkkf porfyktkkf mofkr uftu` uftu` f  9  9 ` + ;, ykctu4 ;) ϟ ( p + ( ` + ;)) 9 ϟ (( p + ` ) + ;)

9 ϟ ( p + ` ) + ; 9 ϟ ( p ) + ϟ ( ` ) + ; 9 ϟ ( p ) + ϟ ( ` + ;)  Dkac porfyktkkf mofkr mofkr f  9  9 ` + ;. Morakskr`kf prcfscp cfau`sc, tormu`tc tormu`tc mkhwk ( p   + q   ) 9 ϟ (  ( p  ) + ϟ (  (q  ϟ (  q  )  @oauk4 Icske`kf q  9  9 f  , , ∂f ∈  . Acporleoh

ϟ (  ( pf     ) 9 ϟ (  ( p  ) ϟ (  (f  f  )  =1

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

_ftu` f  9  9 ; ( p  ;) 9 ϟ (  ( p  ) 9 ϟ (  ( p  ); 9 ϟ (  ( p  )ϟ (;) (;) ϟ (   Dkac porfyktkkf mofkr mofkr uftu` f  9  9 ;.  Ksuisc`kf porfyktkkf porfyktkkf mofkr uftu` uftu` f  9  9 `, ykctu4 ϟ (  ( p`    ) 9 ϟ (  ( p  )ϟ (  (`   )  K`kf acmu`tc`kf porfykkkf morek`u uftu` uftu` f 9 ` + ;, ykctu4 ϟ ( p( ` + ;)) 9 ϟ ( p` + p )

9 ϟ ( p` ) + ϟ ( p ) 9 ϟ ( )ϟ( ` ) + ϟ ( p ) 9 ϟ ( p ) (ϟ ( ` ) + ;) 9 ϟ ( p)ϟ ( ` + ;)  Dkac porfyktkkf mofkr mofkr f  9  9 ` + ;. Morakskr`kf prcfscp cfau`sc, tormu`tc tormu`tc mkhwk ϟ (  ( pq     ) 9 ϟ (  ( p  ) ϟ (  ( q  q )   ^ohcfggk ϟ  ioimorc`kf   ioimorc`kf `lrosplfaofsc sktu-sktu kftkrk hcipufkf mcekfgkf ksec aofgkf sumsot mcekfgkf roke. Krtcfyk kak `lrosplfaofsc sktu-sktu kftkrk hcipufkf mcekfgkf ksec aofgkf hcipufkf mkgckf akrc mcekfgkf roke ykfg iofgkwot`kf lporksc pofduiekhkf, por`keckf, akf roeksc 5. Dkac   akpkt acpkfakfg somkgkc hcipufkf mkgckf akrc  . Aofgkf iofaojcfcsc`kf soecsch mcekfgkf-mcekfgkf ksec, ik`k acporleoh hcipufkf mcekfgkf muekt    ykfg iorupk`kf sumsot akrc  . @oiuackf iofaojcfcsc`kf poimkgckf mcekfgkf-mcekfgkf muekt acporleoh hcipufkf mcekfgkf rksclfke  . Dkac hcipufkf mcekfgkf roke   cslilrj ; aofgkf  ,  , akf . Rrlplscsc 4 ^otckp hcipufkf torurut cslilrj aofgkf  ,  , akf  .  K`sclik Krnhcioaos Krnhcioaos 4 _ftu` sotckp x  ∈  5 f   ∈  , kak f ∈   sohcfggk x  5 (^otckp mcekfgkf roke ykfg acsomut`kf, pkstc kak mcekfgkf muekt ykfg eomch moskr akrcfyk) Mu`tc 4  Dc`k x  5  5 2, ackimce f  9  9 2.  Dc`k tcak` aoic`ckf, aoic`ckf, acaojcfcsc`kf hcipufkf hcipufkf ^ 9 {` ∈ |` ≡ x } ,  x ≯ 2 ^ohcfggk hcipufkf ^  ioipufykc  ioipufykc mktks ktks ykctu x . Akrc aojcfcsc ac ktks, ^  tcak`  tcak` `lslfg `krofk pkecfg tcak` ^   ioiukt ; oeoiof ykctu x . @krofk ^  tcak`   tcak` `lslfg akf tormktks ac ktks ik`k ^  ioipufykc suproiui ( k`sclik  ). Icske`kf k`sclik `oeofg`kpkf   9 sup( ^   y  9 ^ )   ∝ ;=  mu`kf mktks ktks. Leoh `krofk ctu kak ` ∈ ^  sohcfggk @krofk y  suproiui,  suproiui, ik`k  y  ∝  sohcfggk ` 6 y ∝ ;=

 Dc`k `oauk ruks actkimkh ;, acporleoh acporleoh ` + ; 6 y + =; 6 y  @krofk y  suproiui,   suproiui, ik`k ` + ; ∃ ^ . @krofk ` + ; mcekfgkf muekt ykfg mu`kf oeoiof ^ , ik`k ` + ; 6 x   Dkac acpcech f  9  9 ` + ; ∈  . ( nlle  !!) nlle !!)

  K`cmkt 4  Qorakpkt suktu mcekfgkf mcekfgkf rksclfke ackftkrk ackftkrk auk mcekfgkf mcekfgkf roke soimkrkfg Aofgkf `ktk ekcf, dc`k x  5  5 y  ik`k  ik`k ∎r ∈   sohcfggk x  5  5 r  5  5 y . ; O`ucvkeof,

kak `lrosplfaofsc ;-; =?

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

@lfstru`sc mu`tc 4 Ac`otkhuc x  5  5 r  5  5  y , morkrtc acnkrc mcekfgkf f , q ∈   sohcfggk r  9

⎧ f  Acaojcfcsc`kf= hcipufkf ^ 9 ⎨f ∈  + | ≯ q  ⎣ @krofk ^   tormktks `o mkwkh akf ^   tcak`

Ac ekcf pchk` p ∝;

 p ∝ ; q 

5  y  ktku  y  6

 p ∝ ;

x 9 y ∝( y ∝ x ) 5

q   p q

q 

 aofgkf x  5

f 

f   akf 5  y . q  q 

⎠

  ioicec`c mktks mkwkh, ykctu  y . y ⎥ . Akrc scfc doeks ^  ioicec`c

⎩ `lslfg ik`k ^   ioicec`c mktks mkwkh tormoskr >,

icske`kf  p 9 cfj( ^    ik`k ^ )  akf  p ∈  + . @krofk  p ∈ ^  ik`k Leoh `krofk ctu

f 

 p q 

≯

 ktku  y ≡

 p q 

. ^oekcf ctu  p „ ; ∃ ^ .

.

∝( y ∝ x ) 5

p ∝; q 

5 y .   Dkac

 p q

∝ (  y ∝ x ) 5

p ∝; q 

  ktku

; ⇑ q 6 ( y ∝ x )∝; . Mcekfgkf q   cfcekh ykfg ackimce somkgkc q q q  mcekfgkf muekt ykfg eomch moskr 1 akrc (   „ x   ) „;  y  „ Mu`tc 4  Dc`k x ≯ 2, ik`k uftu` sotckp mcekfgkf roke (  y  „  „ x   ) „; kak q ∈   sohcfggk ; ; q 6 ( y ∝ x )∝;  ktku  y ∝ x 6 ⇑ 5 y ∝ x  q q  Icske`kf ⎧ ⎠ f  ^ 9 ⎨f ∈  + | ≯ y ⎥ q  ⎣ ⎩  torsomut ^  tormktks  tormktks `o mkwkh. @krofk ^ ≢ ∏ akf ^ ≢ ∏ `krofk pkecfg tcak`  y ∈ ^ . Akrc aojcfcsc ^  torsomut tormktks `o mkwkh ik`k ^  ioicec`c  ioicec`c cfjciui, icske`kf  p 9 cfj( ^   ik`k ^ ).   @krofk p ∈ ^  ik`k  p  p ≯  y  ktku  y ≡ q  q   p

∝

5 (  y ∝ x ) ⇑ y ∝ x 6

@krofk p ∈ ^  ik`k  ik`k p „ ;∃ ^ . Leoh `krofk ctu,  p ∝ ;  p ∝ ; 5  y  ktku  y  6 q  q  ^ohcfggk ∝; p p ; p ∝; 5  y ≡  akf x 9 y ∝ ( y ∝ x )  5 ∝ 9 q q  q q q   Dkac, ∝;  p ∝ ;  akf x  5 5  y . q  q   p ∝ ; Akrc scfc acpcech r  9  akf y . ∈   ykfg doeks toreotk` ackftkrk x  akf q   Dc`k x  5  5 2, ackimce f ∈   sohcfggk f  6  6 „ x   ktku f  +  + x  6  6 2. Dkac, iofurut poimu`tckf ac ktks, kak x ktku  + x  5  5 r  5  5 y  5  5 y  +  + f  ktku  ktku x  5  5 r  „  „ f  5  5 y . Doeks r  „  „ f  mcekfgkf  mcekfgkf rksclfke. r  ∈  ∈   aofgkf f  + 

= Rofaojcfcsckf

cfc acakskr`kf pkak hcpltoscs mkhwk y  pkecfg  pkecfg moskr. Dkac, acmoftu` hcipufkf aofgkf kfggltk-kfggltk mcekfgkf rksclfke akf morfcekc eomch moskr akrc  y . Caofyk kakekh kgkr hcipufkf ioipufykc cfjciui, icske`kf  p. > Morakskr`kf K`sclik @oeofg`kpkf 1 Morakskr`kf K`sclik Krnhcioaos =:

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

=.1.

Mkrcskf Mcekfgkf ]oke Mkrcskf mcekfgkf roke 5x f 6 kakekh suktu jufgsc ykfg ioiotk`kf sotckp mcekfgkf ksec f  `o mcekfgkf roke x f . Mcekfgkf roke e  ac`ktk`kf  ac`ktk`kf ecict mkrcskf 5x f 6 dc`k uftu` sotckp ο  plsctcj  plsctcj torakpkt mcekfgkf  sohcfggk uftu` sotckp f ≯ F  morek`u  morek`u |x f  „ e | 5 ο . ^onkrk iktoiktcs,  F  sohcfggk e 9 eci x f ⇑ ( ∂ο 6 2 ) ( ∎F ) ∀ ( f ≯ F ) ( x f  ∝ e 5  ο )

Mkrcskf mcekfgkf roke 5 x f 6 acsomut mkrcskf Nkunhy   dc`k uftu` sotckp ο  plsctcj  plsctcj torakpkt mcekfgkf  F mkrcskf Nkunhy  dc`k sohcfggk uftu` sotckp f , i ≯ F  morek`u  morek`u |x f  „ x i | 5 ο . Dkac x f mkrcskf Nkunhy ⇑ ( ∂ο 6 2 ) ( ∎F ) ∀ ( f , i ≯ F ) ( x f ∝ x i  5 ο ) @rctorck Nkunhy  4 Mkrcskf mcekfgkf roke 5 x f 6 `lfvorgof? dc`k akf hkfyk dc`k 5 x f 6 mkrcskf Nkunhy. Fltksc ecict cfc acporeuks uftu` ioiksu``kf mcekfgkf ∖ (pkak  * )somkgkc morc`ut. morc`ut. eci x f 9 ∖, dc`k ∂∊ 6 2, ∎F ∀ ∂f ≯ F , x f  6  ∊

eci x f 9 ∝∖, dc`k ∂∊ 6 2, ∎F ∀ ∂f ≯ F , x f  5 ∝  ∊ Icske`kf ^ (  ( e e,  ο  ) 9 {x ∈  4 |x  „   „  e   e | 5 ο}, ik`k  9 eci x f , dc`k ∂ο  6  6 2, ∎ F , ∀ x f  ∈ ^ (  ( e e,  ο  ), ∂f ≯ F  e  9 Rkak `ksus cfc e  kakekh   kakekh tctc` ecict ( neustor   ) akrc 5x f 6. Dkac tctc` e  ac`ktk`kf   ac`ktk`kf tctc` ecict ( Neustor  neustor  plcft   plcft  Neustor   ) akrc mkrcskf 5 x f 6 dc`k ∂ο  6   6 2, torakpkt soac`ctfyk sktu tctc` x  F  sohcfggk |x  F  „ e| 5 ο . Mcekikfk Rlcft  * `lfsop cfc acporeuks pkak  , e  9 ∖ tctc` ecict akrc mkrcskf 5 x f 6, dc`k ∂∊ 6 2 torakpkt pkecfg soac`ct sktu tctc` x  F  sohcfggk x  F  ≯ ∊.  Dc`k 5x f 6 kakekh suktu mkrcskf, acaojcfcsc`kf ecict suporclr somkgkc eci x f 9 eci sup x f 9 cfj su sup x ` 9 cfj{sup{ x ;, x = , ....}, sup{ x = , x > , ...}  , ...} : f 

` ≯ f 

^cimle eci akf eci sup `oaukfyk acgufk`kf uftu` ecict suporclr. Mcekfgkf roke e   ac`ktk`kf ecict suporclr akrc mkrcskf 5 x f 6 dc`k akf hkfyk dc`k 4 (c) ∂ο  6  6 2, ∎f ∀ ∂` ≯ f , x ` 5 e  +  + ο  (cc) ∂ο  6  6 2 akf ∂f , ∎` ≯ f , x ` 6 e  „  „ ο  (kak  (kak pkecfg soac`ct sktu tctc` x ` sohcfggk x ` 6 e  „  „ ο  _ftu` mcekfgkf roke ykfg acporeuks ∖ kakekh ecict suporclr 5 x f 6 dc`k akf hkfyk dc`k ∂∊ akf f  torakpkt ` ≯ f  soaoic`ckf  soaoic`ckf sohcfggk x ` ≯ ∊. Mcekfgkf roke „ ∖ kakekh ecict suporclr 5 x f 6 dc`k akf hkfyk dc`k „ ∖ 9 eci x f . Ecict cfjorclr acaojcfcsc`kf somkgkc eci x f 9 ececi cf cfj x f 9 sup cfj x ` 9 sup{cfj{x ;, x = , ....}, cf cfj{ x = , x > , ...}  , ...} f 

` ≯ f 

^cjkt-scjkt4 ;) eci ( ∝x f ) 9 ∝ eci x f  =) eci x f ≡ eci x f  >) eci x f 9 e (pkak  * ) ⇑ e 9 eci x f 9 eci x f 

 

1) eci x f + eci yf ≡ eci ( x f + y f ) ≡ eci x f + eci y f

  

≡ eci ( x f + y f ) ≡ eci x f + eci y f     =.?.

Hcipufkf Qormu`k akeki Mcekfgkf ]oke ^oekfg mu`k ( k   ) 9 {x  | k  5 x  5 m }. }. Fltksc M( x  k,  m  x,  α  ) 9 { y  | |x  „  y | 5 α } 9 ( x  x  „ α , x  + α  ) iofyktk`kf mlek ykfg morpuskt ac x  akf  akf mordkrc-dkrc α . Akeki mcekfgkf roke, M( x   ) kakekh soekfg mu`k. x,  α 

? Ecictfyk : Ackftkrk

kak suproiui-suproiui suproiui-suproiui torsomut, ikfk`kh cfjciuifyk< =7

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

Aojcfcsc 4 Hcipufkf L ac`ktk`kf tormu`k ac   dc`k ∂x ∈ L , ∎α 6 2 ∀ M( x , α ) ⊍ L

Aofgkf `ktk ekcf, ∂x ∈ L soekeu torakpkt soekfg mu`k C  ykfg  ykfg ioiukt x  sohcfggk  sohcfggk C ⊍ L. ^oekfg mu`k kakekh nlftlh akrc hcipufkf tormu`k. Hcipufkf `lslfg akf   dugk nlftlh akrc hcipufkf tormu`k. Rrlplscsc 4  Dc`k L; akf L= tormu`k ik`k L; ∣ L= tormu`k. Mu`tc 4 Ackimce somkrkfg x ∈ L; ∣ L=. K`kf actufdu``kf ∎α  6  6 2 sohcfggk M( x  x,  α  ) ⊍ L; ∣ L=. @krofk x  ∈ L; ∣ L=, ik`k x  ∈ L; akf x  ∈ L=. @krofk L;  akf L= tormu`k ik`k ∎α ;, α = 6 2 sohcfggk M( x  x,  α ; ) ⊍ L; akf M( x  x,  α = ) ⊍ L=. Krtcfyk |t  „  „ x | 5 α ; akf |t  „  „ x | 5 α = α;  , α = ), acporleoh Aofgkf iofgkimce α  9  9 icf( α  |t  „  „ x | 5 α  5  5 α ; akf |t  „  „ x | 5 α  5  5 α = Aofgkf `ktk ekcf, ∂t ∈ M( x   9 icf{α ;, α =}. x , α  ) morek`u t ∈ M( x  x,  α ; ) akf t ∈ M( x  x , α = ), aofgkf α  9  Dkac M( x  x , α  ) ⊍ L; akf M( x  x , α  ) ⊍ L=. ^ohcfggk M( x  x,  α  ) ⊍ L; ∣ L=.



 K`cmkt 4 Crcskf soduiekh morhcfggk hcipufkf tormu`k kakekh tormu`k. Mu`tc 4 f 

Icske Lc  , c  9  9 ;, …, f  hcipufkf  hcipufkf tormu`k. K`kf acmu`tc`kf

∣L

c 

 tormu`k. Ik`k,

c 9;

f 

x ∈ ∣ Lc ⇑ .x ∈ Lc  , c 9 ;,  , f   c 9;

⇑ .∎α c 6 2 ∀ M( x , α c ) ⊍ Lc , c 9 ;,  , f   ;,  , f ⇑ .M( x , α ) ⊍ Lc , α 9 icf{α c }, c 9 ;,

 

f 

 Dkac,

∣L

c 

 tormu`k.

c 9;

 Kflthor vorsclf vorsclf (ketorfkto slef) 4 f 

Ackimce somkrkfg x ∈ ∣ Lc  , ik`k x ∈ Lc  aofgkf Lc  tormu`k ∂c . c 9;

@krofk x ∈ L; akf L; tormu`k, ik`k torakpkt α ; 6 2 sohcfggk M( x  x , α ; ) ⊍ L; @krofk x ∈ L= akf L= tormu`k, ik`k torakpkt α = 6 2 sohcfggk M( x  x,  α = ) ⊍ L= Aoic`ckf sotorusfyk. @krofk x ∈ Lf  akf Lf  tormu`k, ik`k torakpkt α f 6 2 sohcfggk M( x  x,  α f  ) ⊍ Lf  Ackimce α  9  9 icf{α ;, α =, . . ., α f }, doeks mkhwk α  6  6 2. Ik`k M( x   9 ;, =, …, f  x , α  ) ⊍ M( x  x , α  )c  ⊍ Lc , c  9 f 

ykfg mork`cmkt mkhwk M( x , α ) ⊍ ∣ Lc  . Dkac tormu`tc mkhwk c 9;

f 

∣L

c 

 tormu`k

c 9;

 @lfvors akrc prlplscsc ac ktks acmorc`kf pkak prlplscsc somkgkc morc`ut =8

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

Rrlplscscsc 4 ^otckp hcipufkf tormu`k ac   iorupk`kf gkmufgkf torhctufg akrc soekfg-soekfg tormu`k ykfg skecfg kscfg. Mu`tc 4 Icske`kf L somkrkfg hcipufkf tormu`k ac  . @krofk L tormu`k, ik`k uftu` sotckp x ∈ L torakpkt y  6  6 x  soaoic`ckf  soaoic`ckf sohcfggk ( x   ) ⊍ L. x , y  Icske`kf  9 sup { y  |  | ( x   ) ⊍ L}, akf m  9 x,  y   9 cfj {z  |  | ( z   ) ⊍ L} k  9 z,  x  ik`k k 5 x  5  5 m  akf  akf C x  9 ( k   ) kakekh soekfg tormu`k tormu`k ykfg ioiukt x . k,  m  @ekci C x  ⊍ L. Ackimce somkrkfg w  ∈ C x , somut x  5 w  5 m , morakskr`kf aojcfcsc m  ac   ac ktks, ik`k acporleoh mcekfgkf y  6  6 w  sohcfggk  sohcfggk ( x   ) ⊍ L. Dkac w ∈ L. x,  y  @ekci m ∃ L. Kfakc`kf m ∈ L, ik`k kak ο  6  6 2 sohcfggk ( m   „ ο , m  +  + ο  ) ⊍ L ktku ( x   + ο  ) ⊍ L. m „ x , m  + @lftrkac`sc aofgkf aojcfcsc m . ^onkrk skik akpkt acmu`tc`kf mkhwk k ∃ L. Hcipufkf {C x }, x ∈ L iorupk`kf `leo`sc soekfg-soekfg mu`k. @krofk sotckp x  ac  ac L toriukt ac C x  akf sotckp C x  toriukt ac L, acporleoh L 9 ∤ C x  .

Icske`kf ( k   ) akf ( n   ) somkrkfg auk soekfg ac L  aofgkf momorkpk tctc` ykfg skik. Ik`k k,  m  n , a  hkrusekh n  5  5 m  akf  akf k  5  5 a . @krofk n ∃ L, ik`k n ∃ ( k   ). Acporleoh n ≡ k . k,  m  @krofk k ∃ L, ik`k n ∃ ( n   ). Acporleoh k ≡ n . n,  a   Dkac k  9  9 n . ^onkrk skik, acporleoh m  9  9 a . K`cmktfyk ( k   ) 9 ( n   ). k,  m  n,  a  ^ohcfggk sotckp auk soekfg ykfg mormoak ac { C x } pkstc skecfg kscfg. Dkac, L iorupk`kf gkmufgkf soekfg-soekfg mu`k ykfg skecfg kscfg. Qork`hcr tcfggke actufdu``kf L torhctufg. ^otckp soekfg mu`k ioiukt mcekfgkf rksclfke 7. @krofk L  gkmufgkf soekfg-soekfg mu`k ykfg skecfg kscfg akf sotckp cftorvke mu`k ioiukt mcekfgkf rksclfke ik`k torakpkt `lrosplfaofsc `lrosplfaofsc ;-; kftkrk L aofgkf hcipufkf mcekfgkf rksclfke ktku hcipufkf mkgckffyk. Dkac L torhctufg.

 Rrlplscsc 4  Dc`k N  `leo`sc  `leo`sc hcipufkf tormu`k ac  , ik`k

∤ L  hcipukf tormu`k ac

.

L∈N 

Mu`tc 4 uftu` suktu suktu L ∈ N x ∈ ∤ L ⇑ .x ∈ L , uftu`

 

L∈N 

⇑ .∎α 6 2 ∀ M( x , α ) ⊍ L , uftu` suktu L ∈ N

 

⇑ .∎α 6 2 ∀ M( x , α ) ⊍ ∤ L L∈N 

 Dkac,

∤ L  hcipufkf tormu`k ac

.

L∈N 

 Kflthor vorsclf vorsclf (wcth nluftkmeo rovcsclf) rovcsclf) 4  Ackimce somkrkfg x ∈ ∤ L , ik`k torakpkt L ∈ N  sohcfggk   sohcfggk x  ∈ L. @krofk L tormu`k ik`k L∈N 

torakpkt α   6 2 sohcfggk M( x  x,  α  ) ⊍ L ⊍

∤ L . Dkac tormu`tc mkhwk uftu` sotckp

L∈N 

torakpkt α  6  6 2 sohcfggk M( x  x,  α  ) ⊍

x ∈ ∤L L∈N 

∤ L  ykfg morkrtc ∤ L  tormu`k.

L∈N 

L∈N 



7 K`sclik

Krnhcioaos =3

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

Roreu acporhktc`kf mkhwk, dc`k N   `leo`sc hcipufkf tormu`k ac    ik`k

∣L

8

  moeui toftu

L∈N 

⎐ ; ;⎖ hcipufkf tormu`k ac  . ^omkgkc nlftlh, Lf  9 ⎕ ∝ , ⎜   soekfg tormu`k, totkpc ⎙ f f ⎢

∖

∣L

f 

{2}   mu`kf 9 {2}

f 9;

hcipufkf hcfggk ac  . Rrlplscsc (Ecfaoeúj) 4 Icske`kf N  `leo`sc   `leo`sc hcipufkf tormu`k ac  , ik`k torakpkt {Lc } sum`leo`sc torhctufg akrc N  soaoic`ckf sohcfggk ∖

∤ L 9 ∤L

c 

L∈N

c 9;

Mu`tc 4 Icske _ 9 ∤{L |L ∈ N } 

Ackimce somkrkfg x ∈ _ . Ik`k torakpkt hcipufkf L ∈ N , aofgkf x ∈ L. @krofk L  tormu`k, ik`k torakpkt soekfg mu`k C x   sohcfggk x  ∈ C x  ⊍ L. Acporleoh3  mkhwk torakpkt soekfg mu`k  D x aofgkf tctc` k`hcr mcekfgkf rksclfke sohcfggk x  ∈ D x  ⊍ C x . @krofk `leo`sc soiuk soekfg mu`k aofgkf tctc` k`hcr mcekfgkf rksclfke kakekh torhctufg, ik`k hcipufkf { D x }, x   torhctufg akf _ 9 ∤ D x  . ∈ _  torhctufg x ∈_ 

_ftu` sotckp soekfg ac { D x } pcech hcipufkf L ac N  ykfg  ykfg ioiukt  D x . Acporleoh sumsot torhctufg ∖

∖

{Lc }c 9;  akrc N , akf _ 9 ∤ L 9 ∤ Lc  L∈N

c 9;

 =.:.

Hcipufkf Qortutup Rofutup hcipufkf  O acfltksc`kf  O Aojcfcsc 4 x ∈ O ⇑ ∂α 6 2, ∎y ∈ O ∀| x ∝ y |5  α 

Aofgkf `ktk ekcf, x ∈ O , dc`k sotckp soekfg mu`k ykfg ioiukt x  dugk  dugk ioiukt suktu tctc` ac  O;2.  Dkac, doeks  O ⊍ O . Nlftlh 4  O 9 (2, ;Y. Qoftu`kf  O .  Kpk`kh x  9  9 2 ∈  O <

∂α 6 2, ∎ y ∈ O 9 ( 2, ;Y ∀ | x ∝ y |5 α 

Rorhktc`kf mkhwk  y f  9

; ↝2 f 

∂α  6  6 2, ∎f 2 ∈ F , ∀ | y f  „ 2| 5 α , ∂f ≯ f 2 ktku, ∂α  6  6 2, | y f  „ 2| 5 α  ; ; Rcech  y f 2 9 ∈ (2,;Y . @krofk ↝ 2 , ik`k ∂α  6  6 2, | y f  „ 2| 5 α . ^ohcfggk x  9  9 2 ∈  O . f 2

f 

 Dkac,  O  9 P2,;Y . 8 Crcskf

tk` morhcfggk hcipufkf-hcipufkf tormu`k prlplscsc 4 Dc`k x  akf  akf y  mcekfgkf  mcekfgkf roke akf x  5  5 y  ik`k  ik`k torakpkt mcekfgkf rksclfke r  sohcfggk  sohcfggk x  5  5 r  5  5 y  ;2 |x  „  „ y | 5 α , morkrtc y ∈ ( x   „ α , x + α  ). ^ohcfggk ∂α  6  6 2, y ∈ ( x   „ α , x + α  ). Dkac, sotckp soekfg tormu`k tormu`k ykfg ioiukt x, dugk ioiukt x „ x „ suktu tctc` (ykctu  y   ) ac O. 3 Echkt

>2

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

Rrlplscsc 4 ;. Dc`k ;.  Dc`k  K ⊍ M  ik`k  K ⊍ M

=.  K ∤ M 9 K ∤ M Mu`tc 4 ;. Ackimce somkrkfg α 6 2 akf x ∈ K . @krofk x ∈ K , ik`k ∎ y ∈ K ∀ | y  „  „ x | 5 α . @krofk K ⊍ M, ik`k y ∈ M ∀ | y  „  „ x | 5 α . Iofurut aojcfcsc, x ∈ M . Dkac tormu`tc  K ⊍ M . =. @krofk K ⊍ K ∤ M, morakskr ;) ac ktks ik`k  K ⊍ K ∤ M . Hke ykfg skik, `krofk M ⊍ K ∤ M ik`k M ⊍ K ∤ M . Dkac,  K ∤ M ⊍ K ∤ M . @oiuackf, k`kf acmu`tc`kf mkhwk  K ∤ M ⊍ K ∤ M . Acscfc acmu`tc`kf `lftrkplscscfyk, ykctu dc`k x ∃ K ∤ M  ik`k x ∃ K ∤ M . @krofk x ∃ K ∤ M , ik`k x ∃ K  akf x ∃ M . x ∃ K ⇝ ∎α ; 6 2 ∀ tcak` kak y ∈ K aofgkf | x ∝ y |5 α  ; x ∃ M ⇝ ∎α = 6 2 ∀ tcak` kak y ∈ M aofgkf | x ∝ y |5 α  =

Ackimce α  9  9 icf{α ;, α =}, ik`k tcak` kak y ∈ K ∤ M aofgkf | y  „  „ x | 5 α .  Dkac, x ∃ K ∤ M . Cfc morkrtc, dc`k x ∃ K ∤ M  ik`k x ∃ K ∤ M . Mu`tc ekcf 4 Ackimce somkrkfg α  6  6 2 akf x ∈ K ∤ M . x ∈ K ∤ M ⇝ ∂α 6 2, ∎y ∈ K ∤ M ∀| y ∝ x |5  α  @krofk y ∈ K ∤ M, ik`k y ∈ K ktku y ∈ M. _ftu` y ∈ K aofgkf | y  „  „ x | 5 α  acporleoh  acporleoh x ∈ K _ftu` y ∈ M aofgkf | y  „  „ x | 5 α  acporleoh  acporleoh x ∈ M  Dkac, x ∈ K ∤ M .

 Aojcfcsc 4 Hcipufkf J  acsomut   acsomut tortutup ( nelsoa   ) dc`k J 9 J  nelsoa 

Iofurut aojcfcsc J ⊍ J  , ik`k hcipufkf J acsomut tortutup dc`k J ⊍ J  , ykctu dc`k J ioiukt soiuk tctc`-tctc` neustorfyk. Nlftlh 4 ;) J  9  9 (2, ;Y mu`kf hcipufkf tortutup, somkm J 9 P2,;Y ≢ J 

=) >) 1) ?)

 9 P2, ;Y hcipufkf tortutup, somkm J 9 P2,;Y 9 J  J  9 ^oekfg P k   Y akf P;, ∖ Y kakekh hcipufkf hcipufkf tortutup k,  m    kakekh hcipufkf tortutup  9 ∏. K`kf acmu`tc`kf mkhwk ∏ 9 ∏ J  9 Mu`tc 4 Akrc aojcfcsc, ∏ ⊍ ∏ . Dkac, tcfggke acmu`tc`kf ∏ ⊍ ∏ . x ∈ ∏ ⇑ .∂α 6 2, ∎y ∈ ∏, ∀| y ∝ x |5 α  

⇑ .∂α 6 2, ∎ y ∈ ∏, ∀ y ∈ M( x , α ) ⇑ .x ∈ ∏  Dkac, ∏ ⊍ ∏ . Leoh `krofk ctu tormu`tc mkhwk ∏ ⊍ ∏ .  Rrlplscsc 4 Rofutup hcipufkf O kakekh tortutup. >;

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

Mu`tc 4  K`kf acmu`tc`kf  O 9 O . Akrc aojcfcsc,  O ⊍ O . Dkac, tcfggke acmu`tc`kf  O ⊍ O . Icske`kf x ∈ O . x ∈ O ⇑ ∂α  6 2, ∎y ∈ O , ∀| y ∝ x |5  α 

= @krofk  y ∈ O  ik`k uftu` α  ac  ac ktks, torakpkt z ∈ O so  sohcfggk |z ∝ y |5 α  = .

 Dkac, uftu` α  ac  ac ktks, torakpkt z ∈ O sohcfggk |z ∝ x |. 9|z ∝ y + y ∝ x |  . 5|z ∝ y | +| y ∝ x |  . 5 α = + α  = 9 α  Cfc morkrtc x ∈ O

 Rrlplscsc 4  Dc`k J ; akf J = tortutup, ik`k J ; ∤ J = tortutup. Mu`tc 4

 K`kf acmu`tc`kf acmu`tc`kf mkhwk J; ∤ J= 9 J; ∤ J= .  Akrc aojcfcsc, doeks mkhwk J; ∤ J= ⊍ J; ∤ J=  sohcfggk nu`up acmu`tc`kf J; ∤ J= ⊍ J; ∤ J=     Ackimce somkrkfg x ∈ J; ∤ J = . K`kf acmu`tc`kf mkhwk x ∈ J; ∤ J = . Iofurut prlplscsc somoeuifyk, x ∈ J; ∤ J= 9 J; ∤ J= .  @krofk J ; akf J = tortutup, ik`k x ∈ J; ∤ J = .

 Rrlplscsc 4 Crcskf `leo`sc hcipufkf tortutup kakekh tortutup Mu`tc 4 Icske`kf N  `leo` leo`ssc hcipu cipufk fkff-hc hci ipufkf ufkf tor tortutu utup. K` K`kf kf acm acmu`tc u`tc``kf mkh mkhwk

∣{J | J ∈ N }

∣{J | J ∈ N } 9 ∣{J | J ∈ N } . Iofurut aojcfcsc, nu`up acmu`tc`kf ∣{J | J ∈ N } ⊍ ∣{J | J ∈ N } . Ackimce somkrkfg x ∈ ∣{J | J ∈ N } . Ik`k uftu` sotckp α  6 2 torakpkt  y ∈ ∣{J | J ∈ N }

tortutup, ykctu

sohcfggk | y  „  „ x | 5 α . @krofk ∈ ∣{J | J ∈ N }  ik`k  y  ∈ J  uftu`   uftu` sotckp J  ∈ N  aofgkf   aofgkf | y  „ x | 5 α . Iofurut aojcfcsc, acporleoh mkhwk x ∈ J  uftu` sotckp J ∈ N . @krofk J  ∈ N  ik`k   ik`k J  tortutup.   tortutup. @krofk J  tortutup   tortutup ik`k J 9 J   k`cmktfyk x  ∈ J , uftu` sotckp J ∈ N . Akrc scfc ik`k, x ∈ ∣{J | J ∈ N } .  Dkac tormu`tc mkhwk

∣{J | J ∈ N } ⊍ ∣{J | J ∈ N } . ^ohcfggk ∣{J | J ∈ N }  tortutup. 

Rrlplscsc 4 ;. @lipeoiof hcipufkf tormu`k kakekh tortutup =. @lipeoiof hcipufkf tortutup kakekh tormu`k Mu`tc 4

;. Icske`kf L hcipufkf tormu`k. K`kf acmu`tc`kf mkhwk L n 9 L n  . Akrc aojcfcsc L n ⊍ L n  .  Dkac nu`up acmu`tc`kf acmu`tc`kf L n ⊍ L n  . K`kf acmu`tc`kf `lftrkplscscfyk. `lftrkplscscfyk. @krofk L tormu`k, ik`k ∂x ∈ L, ∎α  6  6 2 sohcfggk M( x  x , α  ) ⊍ L. @krofk x ∈ L, ik`k x ∃ Ln . Icske`kf y ∈ M( x  x,  α  ). @krofk M( x  x,  α  ) ⊍ L ik`k y ∈ L. >=

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

^ohcfggk, dc`k | y  „  „ x | 5 α  ik`k   ik`k y  ∈ L. Krtcfyk, tcak` kak  y  ∈ Ln  sohcfggk | y  „  „ x | 5 α . ^osukc aojcfcsc pofutup, x ∃ L n  =. Icske`kf J  hcipufkf  hcipufkf tortutup. K`kf acmu`tc`kf mkhwk J n  tormu`k. Ackimce somkrkfg x  ∈ J n , k`kf acmu`tc`kf mkhwk torakpkt α  6   6 2 sohcfggk M( x  x , α  ) ⊍ J n .  Dc`k α  6   6 2 ackimce soimkrkfg, ik`k nu`up acmu`tc`kf M( x  x , α  ) ⊍ J n . K`kf acmu`tc`kf `lftrkplscscfyk. @krofk x  ∈ J n   ik`k x  ∃ J . @krofk J   tortutup, ik`k x ∃ J 9 J  . Krtcfyk, tcak` kak  6 2 ykfg acmorc`kf morek`u | y  „  „ x | 5 α . ^ohcfggk, uftu`  y ∈ J  sohcfggk uftu` sotckp α  6 sotckp y ∈ J  morek`u  morek`u y ∃ M( x  sotckp y ∃ J n  ik`k y ∃ M( x  x,  α  ). Dkac, uftu` sotckp x , α  ).  @leo`sc hcipufkf N  acsomut  acsomut soeciut ( nlvors   ) akrc hcipufkf hcipufkf J  dc`k  dc`k nlvors  J ⊍ ∤{L 4 L ∈ N }  akeki hke cfc `leo`sc hcipufkf N   acsomut iofyoeciutc ( nlvorcfg   ) J . Dc`k sotckp L ∈ N   tormu`k, ik`k nlvorcfg  `leo`sc N   acsomut soeciut tormu`k ( lpof  ) akrc J . Dc`k N   hkfyk ioiukt soduiekh morhcfggk lpof nlvorcfg  hcipufkf-hcipufkf, hcipufkf-hcipufkf, ik`k `leo`sc N  acsomut  acsomut soeciut hcfggk (   ). Akeki hke ‘soeciut tormu`k–,  jcfcto nlvorcfg  `ktk scjkt ‘tormu`k– torsomut iofufdu``kf scjkt hcipufkf-hcipufkf akeki soeciut akf tcak` morik`fk ‘acsoeciutc leoh hcipufkf tormu`k–. Aoic`ckf dugk aofgkf cstcekh ‘soeciut hcfggk– tcak` iofufdu``kf mkhwk soeciutfyk iorupk`kf hcipufkf morhcfggk.  Qolroik (Hocfo-Mlroe) (Hocfo-Mlroe) 4 Icske`kf J  tortutup   tortutup akf tormktks pkak  . Ik`k sotckp soeciut tormu`k akrc J  ioipufykc   ioipufykc soeciut mkgckf ykfg morhcfggk. Aofgkf `ktk ekcf, dc`k N  kakekh `leo`sc hcipufkf tormu`k sohcfggk J ⊍ ∤{L 4 L ∈ N }  f 

ik`k kak `leo`sc morhcfggk { L;, L=, . . ., Lf } pkak N  sohcfggk  sohcfggk J ⊍ ∤ Lc  . c 9;

Mu`tc 4 (soo ‘]oke Kfkeyscs–, >ra oa., H.E. ]lyaof, pkgo 1?) =.7.

Jufgsc @lftcfu Icske`kf j jufgsc morfcekc roke aofgkf alikcf  O iorupk`kf hcipufkf mcekfgkf roke. Morc`ut cfc aojcfcsc-aojcfcsc aojcfcsc-aojcfcsc `lftcfu ac tctc`, `lftcfu pkak  O, akf `lftcfu sorkgki pkak  O;;. Aojcfcsc 4 Jufgsc j  ac`ktk`kf  ac`ktk`kf `lftcfu ac tctc` ( nlftcfulus  ) x ∈ O dc`k ∂ο  6  6 2, ∎α  6  6 2 sohcfggk ∂ y  nlftcfulus kt tho plcft   „ x | 5 α  ik`k  ik`k | j (  (x    )  „ j (  ( y    )| 5 ο . ∈ O, aofgkf | y  „ x Jufgsc j  ac`ktk`kf  ac`ktk`kf `lftcfu pkak ( nlftculus  ) K sumsot akrc  O dc`k j  `lftcfu  `lftcfu ac sotckp tctc` akrc  K. nlftculus lf  Jufgsc j  ac`ktk`kf  ac`ktk`kf `lftcfu sorkgki pkak ( ufcjlriey  ) O, dc`k ∂ο  6  6 2, ∎α  6  6 2 sohcfggk ufcjlriey nlftcfulus lf   „ x | 5 α  ik`k  ik`k | j (  ( x  ( y    )| 5 ο . ∂x , y ∈ O, aofgkf | y  „ x )  „ j ( 

_ftu` soekfdutfyk, dc`k acsomut`kf  j   `lftcfu, ik`k ykfg acik`sua kakekh  j   `lftcfu pkak alikcffyk. Rrlplscsc 4 Icske`kf  j   jufgsc morfcekc roke ykfg `lftcfu akf acaojcfcsc`kf pkak J . Dc`k J   `lftcfu akf tormktks, ik`k j  tormktks  tormktks pkak J  akf  akf ioipufykc tctc` ik`sciui akf icfciui pkak  J .  Krtcfyk kak tctc` x ; akf x = ac akeki J  sohcfggk  sohcfggk j (  (x    ;  ) ≡ j (  ( x  (x     = ), ∂x ∈ J . x x )  ≡ j (  x Mu`tc 4 ;; Roimoakkf

x,  y , ktku α  ) cfc hkfyk torechkt akrc mkgkcikfk `otorgkftufgkf poicechkf α  torhkakp  torhkakp ykfg ekcf ( x  >>

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c

(soo ‘]oke Kfkeyscs–, >ra oa., H.E. ]lyaof, pkgo 17) Rrlplscsc 4 Icske`kf  j   jufgsc morfcekc roke ykfg acaojcfcsc`kf pkak  . Jufgsc  j   `lftcfu pkak  dc`k akf hkfyk dc`k j  „;( L ) tormu`k uftu` sotckp L hcipufkf tormu`k ac  . Mu`tc 4 (soo ‘]oke Kfkeyscs–, >ra oa., H.E. ]lyaof, pkgo 17„18)  Qolroik (Qolroik Fcekc Kftkrk) 4 Icske`kf  j  jufgsc   jufgsc morfcekc roke akf `lftcfu pkak P k   Y. Dc`k  j (  (k  ( y    ) ≡  j (  ( m  (m  ( y    ) ≡  j (  (k  k,  m  k  )  ≡  j (  m )  ktku  j (  m  )  ≡  j (  k  )  ik`k kak n ∈ P k   Y soaoic`ckf sohcfggk sohcfggk j (  (n  k,  m  n  )  9 y . Rrlplscsc 4  Dc`k  j  jufgsc   jufgsc morfcekc roke akf `lftcfu pkak hcipufkf tortutup akf tormktks J  ik`k   ik`k  j   `lftcfu sorkgki pkak J . Mu`tc 4 (soo ‘]oke Kfkeyscs–, >ra oa., H.E. ]lyaof, pkgo 18) Aojcfcsc 4 Icske`kf 5 j f 6 mkrcskf jufgsc pkak  O. Mkrcskf 5 j f 6 ac`ktk`kf `lfvorgof tctc` aoic tctc` ( nlfvorgo  nlfvorgo  ;=  ) pkak  O `o jufgsc  j , dc`k ∂x  ∈  O akf ο  6   6 2, ∎ F   sohcfggk | j (  (x    )  „  j f ( x   plcftwcso  x x )|   5 ο, ∂f  ≯  F . Mkrcskf 5 j f 6 ac`ktk`kf `lfvorgof sorkgki ( nlfvorgo   ) pkak  O `o jufgsc  j , dc`k ∂ο   6 2, nlfvorgo  ufcjlriey  ;> (x    )  „ j f ( x  ∎ F   sohcfggk ∂x ∈ O, | j (  x x )|   5 ο , ∂f ≯ F . =.8.

Hcipufkf Mlroe  Wkekupuf crcskf akrc somkrkfg `leo`sc hcipufkf tortutup kakekh tortutup akf gkmufgkf akrc `leo`sc morhcfggk akrc hcipufkf tortutup dugk tortutup, totkpc gkmufgkf akrc `leo`sc torhctufg hcipufkf-hcipufkf hcipufkf-hcipufkf tortutup tcak` hkrus tortutup. ^omkgkc nlftlh, hcipufkf mcekfgkf rksclfke kakekh gkmufgkf akrc `leo`sc torhctufg hcipufkfhcipufkf tortutup ykfg sotckp hcipufkffyk ioiukt topkt sktu kfggltk. Aojcfcsc 4 @leo`sc hcipufkf Mlroe M kakekh kedkmkr-ώ tor`once ykfg ioiukt soiuk hcipufkf-hcipufkf tormu`k.

O`scstofsc kedkmkr-ώ cfc acdkicf leoh prlplscsc ;1  > ac Mkm C. Eomch ekfdut, kedkmkr-ώ tor`once cfc dugk ioiukt soiuk hcipufkf-hcipufkf tortutup akf ioiukt puek soiuk soekfg-soekfg mu`k. Hcipufkf ykfg iorupk`kf gkmufgkf torhctufg akrc hcipufkf-hcipufkf tortutup acsomut J ώ ktku ac`ktk`kf ioicec`c tcpo J ώ ( J    ac`ktk`kf J uftu`  uftu` tortutup, ώ uftu` duiekh). ^ohcfggk, hcipufkf A  ac`ktk`kf ∖

ioicec`c tcpo J ώ dc`k akpkt actuecs A 9 ∤ J f   uftu` sotckp hcipufkf tortutup J f  ac ] . f 9;

 Dc`k J  hcipufkf  hcipufkf tortutup, ik`k J  ioicec`c  ioicec`c tcpo J ώ somkm J  akpkt  akpkt actuecs iofdkac ∖

J 9 ∤ J f  f 9;

aofgkf J ; 9 J 0 J = 9 J > 9 J 1  9 . . . 9 ∏ ykfg iorupk`kf hcipufkf tutup. Dugk, soekfg mu`k ( k , m   ) ioicec`c tcpo J ώ, somkm

;= Roicechkffyk

morgkftufg pkak x  tcak` morgkftufg pkak x  ;1 Rrlplscsc 4 Icske`kf N `leo`sc hcipufkf mkgckf akrc  X , ik`k torakpkt kedkmkr- ώ tor`once ]  ykfg  ykfg ioiukt N. ;> Roicechkffyk

>1

Mkm = „ ^cstoi Mcekfgkf ]oke

Nlipceoa my 4 @hkorlfc, ^.^c ∖

;⎡ ⎫ ; ( k , m ) 9 ∤ ⎭ k + , m ∝  ⎯ f f ⎪ f 9; ⎮ Akrc scfc acporleoh mkhwk sotckp hcipufkf tormu`k ioicec`c tcpo J ώ. ^omkm, dc`k L mu`k ik`k 4 ∖ ;⎡ ⎫ ; L 9 ∤ ⎭k + , m ∝ ⎯ f f ⎪ f 9; ⎮ Aofgkf k 9 mktks mkwkh L, akf m 9 mktks ktks L. Crcskf torhctufg akrc soiuk hcipufkf tormu`k ac`ktk`kf ioicec`c tcpo Gα. Dkac, suktu hcipufkf ac`ktk`kf ioicec`c tcpo Gα dc`k hcipufkf torsomut iorupk`kf crcskf torhctufg akrc soiuk hcipufkf tormu`k.  Dkac, `lipeoiof akrc hcipufkf ykfg ioicec`c tcpo J ώ kakekh hcipufkf ykfg ioicec`c tcpo Gα akf aoic`ckf dugk somkec`fyk. ^omkm, n

∖

n 

∖ ⎐ ⎖ ⎐∖ ⎖ n  Jώ 9 ∤ Jf ⇝ ⎕ Jώ 9 ∤ Jf ⎜ ⇑ ( Jώ  ) . 9 ⎕ ∤ Jf  ⎜   f 9; f 9; ⎙ ⎢ ⎙ f 9; ⎢ ∖

. 9 ∣ J f n  f 9;

@krofk J f   tortutup uftu` sotckp J f  ac    ik`k iofurut prlplscsc, J f n   tormu`k. Qorechkt ( J  Jώ  )n  iorupk`kf crcskf torhctufg akrc hcipufkf-hcipufkf tormu`k. Dkac tormu`tc mkhwk ( J ώ )n  ioicec`c tcpo Gα. Mu`tc somkec`fyk kfkelg. Hcipufkf ykfg ioicec`c tcpo J ώ akf Gα kakekh nlftlh hcipufkf Mlroe, ykctu kedkmkr-ώ tor`once ykfg ioiukt soiuk hcipufkf tormu`k akf tortutup.

>?

Mkm > _`urkf Eomosguo >.;.

Rofakhueukf Rkfdkfg e (  ( C   acaojcfcsc`kf somkgkc soecsch kftkrk `oauk tctc` udufgfyk ykfg hkscefyk C )  akrc cftorvke C  acaojcfcsc`kf morupk mcekfgkf roke. Rkfdkfg soekfg kakekh skekh sktu nlftlh jufgsc hcipufkf. Jufgsc hcipufkf kakekh suktu jufgsc ykfg ioiotk`kf kftkrk mcekfgkf roke ykfg ac poreuks  *  aofgkf sotckp hcipufkf ac akeki suktu `leo`sc hcipufkf-hcipufkf. Akeki hke cfc, alikcf akrc jufgsc-fyk kakekh `leo`sc soiuk cftorvke-cftorvke (soekfg). @ctk hofak` ioiporeuks `lfsop pkfdkfg cfc iofdkac tcak` hkfyk pkak cftorvke ioekcf`kf hcipufkf ykfg nu`up ruict. ^onkrk soaorhkfk, `ctk akpkt iofaojcfcsc`kf ‘pkfdkfg– akrc suktu hcipufkf tormu`k somkgkc duiekhkf pkfdkfg akrc soekfg-soekfg tormu`k ykfg iofyusuf hcipufkf tormu`k torsomut. @krofk `oeks akrc hcipufkf tormu`k iksch torekeu ruict, `ctk k`kf iofg`lfstru`sc jufgsc hcipufkf i ykfg ioiotk`kf sotckp hcipufkf  O ac akeki suktu `leo`sc hcipufkf mcekfgkf roke I ykfg acporeuks aofgkf suktu mcekfgkf roke acporeuks ykfg tk` fogktcj i (  ( O    ). Jufgsc i cfc acsomut u`urkf akrc hcipufkf  O ( ioksuro lj  achkrkp`kf kakekh kgkr i ioiofuhc scjkt-scjkt scjkt-scjkt morc`ut4 ioksuro lj  O  O ). Qudukf ykfg achkrkp`kf c. i (  ( O    ) toraojcfcsc uftu` uftu` sotckp hcipufkf hcipufkf mcekfgkf roke O, ykctu I 9 ℘(  ). cc. i (  ( C  (C  C )  9 e (  C  ),   uftu` sotckp soekfg C 

ccc.  Dc`k 5 Of 6 mkrcskf hcipufkf skecfg eopks akf i (  ( O    ) f  toraojcfcsc, i ( ∤ Of ) 9

∛ i( O ) f 

cv. i  iorupk`kf  iorupk`kf jufgsc trkfseksc cfvkrckft, ykctu dc`k  O hcipufkf acikfk i  toraojcfcsc  toraojcfcsc akf dc`k  9 {x  +  + y  |  | x ∈ O}, acporleoh aofgkf iofggkftc sotckp tctc` ac  O aofgkf x  +  + y  4 4  O + y  9 ( O    + y   ) 9 i (  ( O    ) i (  ^kykfgfyk, tcak` iufg`cf uftu` ioimukt somukh jufgsc hcipufkf ykfg ioicec`c `ooipkt scjkt ac ktks, akf dugk tcak` ac`otkhuc kpk`kh kak jufgsc hcipufkf ykfg ioiofuhc tcgk scjkt ykfg portkik. ^omkgkc k`cmktfyk, skekh sktu akrc scjkt-scjkt ac ktks hkrus actckak`kf. K`kf eomch iuakh dc`k iofckak`kf scjkt ykfg portkik akf iofgkimce `otcgk scjkt ykfg ekcf. ^ohcfggk i (  ( O    ) tcak` poreu toraojcfcsc uftu` soiuk hcipufkf mcekfgkf roke  O, nu`up hkfyk momorkpk skdk. Qoftu skdk, `ctk cfgcf kgkr i (  ( O    ) toraojcfcsc uftu` somkfyk`-mkfyk`fyk hcipufkf ykfg iufg`cf. @ctk k`kf ioicech I, ykctu `leo`sc (   ) hcipufkf acikfk i   toraojcfcsc, ykfg iorupk`kf kedkmkr- ώ. Ik`k u`urkf i   ykfg  jkicey  acporleoh acsomut u`urkf kactcj ykfg torhctufg ( nluftkmey  ), dc`k i   iorupk`kf jufgsc nluftkmey kaactcvo ioksuro  morfcekc roke tk` fogktcj ykfg aojcfcsc alikcffyk iorupk`kf kedkmkr- ώ I (hcipufkf mcekfgkf roke) akf morek`u4 i ( ∤ Of ) 9 i( Of  )

∛

uftu` sotckp 5 Of 6 mkrcskf hcipufkf skecfg eopks ac I. Qudukf `ctk akeki sum-mkm morc`utfyk kakekh k`kf iofg`lfstru`sc somukh u`urkf kactcj torhctufg ykfg trkfseksc cfvkrckft akf ioicec`c scjkt (C  ( C  i (  C  )  9 e (  C )  uftu` sotckp soekfg C . >.=.

_`urkf Eukr _ftu` sotckp hcipufkf mcekfgkf roke  K, pkfakfg `leo`sc torhctufg soekfg-soekfg mu`k { C f } ykfg iofyoeciutc K. Skctu, `leo`sc ykfg ioiofuhc  K ⊍ ∤ C f  Nlftlh4  K 9 (2,;Y

Mkm > „ _`urkf Eomosguo

;.

Nlipceoa my 4 @hkorlfc, ^.^c

⎐ ; ⎖ ⎐> ⎖ {C ; , C = } , C ; 9 ⎕ ∝ , ;⎜ , C = 9  ⎕ , = ⎜ ⎙ = ⎢ ⎙1 ⎢ {C ; , C = }  soeciut mu`k.  K ⊍ C ; ∤ C =

=.

e ( C ; ) + e ( C = ) 9 =

> 1

⎐ ;⎖ {C ;} , C ; 9 ⎕ 2, ; ⎜ ⎙ =⎢ {C ;}  soeciut mu`k.  K ⊍ C ;

e ( C ; ) 9 ;

; =

 Akf uftu` sotckp `leo`sc, pkfakfg duiekhkf pkfdkfg ikscfg-ikscfg soekfg ac akeki `leo`sc torsomut. @krofk pkfdkfg kakekh mcekfgkf plsctcj, ik`k duiekhkf torsomut acaojcfcsc`kf sonkrk tufggke akf cfaopofaof akrc urutkf mkrcskffyk. Icske`kf J   `leo`sc soeciut mu`k  K. Acaojcfcsc`kf u`urkf eukr ( lutor  ) i *(  *(  lutor ioksuro   K ) akrc  K  somkgkc cfjciui akrc soiuk duiekhkf-duiekhkf duiekhkf-duiekhkf torsomut. Acfltksc`kf somkgkc i * ( K ) 9 cfj e ( C f ) 9 cfj e ( C f )    K ⊍

∤ C f 

∛

{C f }∈J 

∛

 K`cmkt 4 ;. i *(  *( ∏   ∏ )   9 2 =. dc`k K ⊍ M ik`k i *(  *(  *( M )  K ) ≡ i *(  >. i *({ *({k }) }) 9 2, ∂ k ∈ ]  Mu`tc 4 ∏ )   ≯ 2. ;.  Qorechkt ekfgsufg akrc aojcfcsc mkhwk i *(  *( ∏   =ο  ⎐ ο ο  ⎖  Kimce somkrkfg ο  6 2. Rcech C f  9 ⎕ ∝ , ⎜ . Akrc scfc ik`k e ( C f  ) 9 9 ο    akf C f  = ⎙ = =⎢ iorupk`kf soeciut mu`k uftu` ∏  . Dkac, ∏ ⊍ C f ⇑ i * ( ∏ ) ≡ e ( C f )  9 ο  . ^ohcfggk, i * ( ∏ ) ≡ ο , ∂ο  6 2 ⇝ i * ( ∏ ) 9 2

=. Ackimce J K

9 {{C f } 4 {C f } soeciut mu`k uftu` K}

JM

9 {{C f } 4 {C f } soeciut mu`k uftu` M}

akf @krofk  K ⊍ M ⊍ ∤ C f  , ik`k morek`u soeciut M dugk mcsk acpk`kc somkgkc soeciut uftu`  K ).   {C f } ∈ JM ⇝ {C f } ∈ J K  (sotckp  Dkac, JM ⊍ J K . K`cmktfyk, cfj J K ≡ cfj J   M Mogctu dugk aofgkf duiekhkffyk. Iofurut aojcfcsc, i *(  *(  *( M )  K ) ≡ i *(  >. Ackimce somkrkfg k ∈ ] , ο  6  6 2 akf J{k } 9 {{C f } 4 {C f } soeciut mu`k uftu` {k}}   ik`k, i * ({k}) 9 cfj

{ C f }∈J { k }

∛ e ( C ) ≡  ο = 5 ο  f 

@krofk ο  6  6 2 somkrkfg ik`k i *({ *({k }) }) 9 2.

 >7

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

Rrlplscsc 4 _`urkf eukr akrc suktu soekfg skik s kik aofgkf pkfdkfg soekfg torsomut. ^omkgkc nlftlh, dc`k K 9 P2,=Y ik`k i *(  *(   K ) 9 =. Mu`tc 4  Y @ksus ; 4 Icske`kf P k  k,  m  ( Zorsc Zorsc Ir. ]lyaof  ) @krofk P k   Y ⊍ ( k   „ ο , m  +  + ο  ), ∂ο  6  6 2 ik`k k,  m  k „ *(P k   Y) ≡ e ((  (( k   „ ο , m  +  + ο  )) 9 m  „  „ k  +  + =ο  i *(P  k,  m  k „ @krofk i *(P  *(P k   Y) ≡ m  „  „ k  +  + =ο , ∂ο  6  6 2 ik`k k , m  *(P k   Y) ≡ m  „  „ k  i *(P  k,  m  ^oekfdutfyk, k`kf acmu`tc`kf i *(P  *(P k   Y) ≯ m  „  „ k . k,  m  Hke cfc skik skdk aofgkf ioimu`tc`kf mkhwk dc`k { C f } soimkrkfg `leo`sc torhctufg akrc soekfg mu`k ykfg iofyoeciutc P k   Y ik`k k,  m  e ( C f  ) ≯ m ∝ k  

∛ f 

^omkm, cfj  K ⊍

∤

C f 

@krofk fk cfjc cfjciu iui, i, ik`k ik`k ∛ e ( C ) ≯ m ∝ k .  ∛ e ( C ) ≯ m ∝ k . @kro f 

f 

f 

Aofgkf Qolroik Hocfo-Mlroe, sotckp `leo`sc soekfg tormu`k ykfg iofyoeciutc P k , m   Y ioiukt sum`leo`sc morhcfggk ykfg dugk iofyoeciutc P k   Y, akf `krofk duiekhkf pkfdkfg soekfg akrc k,  m  sum`leo`sc morhcfggk tcak` eomch moskr akrc duiekhkf pkfdkfg soekfg akrc `leo`sc ksecfyk, ik`k portcak`skikkf ac ktks tormu`tc uftu` `leo`sc morhcfggk { C f } ykfg iofyoeciutc P k   Y. k,  m  @krofk k   toriukt ac akeki ∤ C f    ik`k kak `  sohcfggk C `  ioiukt k . Icske`kf C ` 9 ( k  k; , m ; ). Acporleoh  5 m ; k ; 5 k  5  Dc`k m ; ≡ m , ik`k m ; ∈ P k   Y akf `krofk m ; ∃ ( k  k,  m  k; , m ; ) ik`k torakpkt cftorvke ( k  k= , m = ) ac akeki { C f } soaoic`ckf sohcfggk m ; ∈ ( k  k= , m = ). Dkac, k = 5 m ; 5 m =. Aoic`ckf sotorusfyk, sohcfggk acporleoh mkrcskf ( k  k; , m ; ), ( k  k= , m = ), . . ., ( k  k`  , m ` ) Akrc `leo`sc {C f } soaoic`ckf sohcfggk k c  5 m c  „ „ ; 5 m c . @krofk {C f } `leo`sc morhcfggk, prlsos ac ktks pkstc morhoftc pkak suktu cftorvke ( k  k`  , m ` ). Qotkpc prlsos cfc hkfyk k`kf morhoftc dc`k m ∈ ( k   5 m `. @krofk k c  5 m c  „ „ ;, ik`k k`  , m ` ), ykctu dc`k k ` 5 m  5 e (Cf ) ≯ e ( k c , m c  ) 

∛

∛

9 ( m` ∝ k ` ) + (m` ∝; ∝ k ` ∝; ) +  + (m; ∝ k ; )   9 m` ∝ ( k ` ∝ m` ∝; ) ∝ ( k ` ∝; ∝ m ` ∝= ) ∝  ∝ ( k = ∝ m; ) ∝ k ; 6 m ` ∝ k ;    Qotkpc m ` 6 m  akf  akf k ; 5 k . K`cmktfyk, m ` „ k ; 6 m  „  „ k . Dkac, ∛ e ( C f  ) 6 m ∝ k .   Qormu`tc mkhwk i *(P  *(P k   Y) ≯ m  „  „ k  k,  m  ( Zorsc  ) Zorsc Irs. Kfggc  @krofk P k   Y ⊍ ( k   „ ο , m  +  + ο  ), ∂ο  6  6 2 ik`k k,  m  k „ *(P k   Y) ≡ e ((  (( k   „ ο , m  +  + ο  )) 9 m  „  „ k  +  + =ο  i *(P  k,  m  k „ @krofk i *(P  *(P k   Y) ≡ m  „  „ k  +  + =ο , ∂ο  6  6 2 ik`k k , m  *(P k   Y) ≡ m  „  „ k  i *(P  k,  m  ^oekfdutfyk, k`kf acmu`tc`kf i *(P  *(P k   Y) ≯ m  „  „ k . k,  m  Hke cfc skik skdk aofgkf ioimu`tc`kf mkhwk dc`k { C f } soimkrkfg `leo`sc torhctufg akrc soekfg mu`k ykfg iofyoeciutc P k   Y ik`k k,  m  e ( C f  ) ≯ m ∝ k  

∛ f 

^omkm, cfj  K ⊍

∤ C f 

@krofk fk cfjc cfjciu iui, i, ik`k ik`k ∛ e ( C ) ≯ m ∝ k .  ∛ e ( C ) ≯ m ∝ k . @kro f 

f 

f 

Aofgkf Qolroik Hocfo-Mlroe, sotckp `leo`sc soekfg tormu`k ykfg iofyoeciutc P k , m   Y ioiukt >8

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

sum`leo`sc morhcfggk ykfg dugk iofyoeciutc P k   Y, akf `krofk duiekhkf pkfdkfg soekfg akrc k,  m  sum`leo`sc morhcfggk tcak` eomch moskr akrc duiekhkf pkfdkfg soekfg akrc `leo`sc ksecfyk, ik`k portcak`skikkf ac ktks tormu`tc uftu` `leo`sc morhcfggk { C f } ykfg iofyoeciutc P k   Y. k,  m  @krofk k   toriukt ac akeki ∤ C f    ik`k kak `  sohcfggk C `  ioiukt k . Icske`kf C ` 9 ( k  k; , m ; ). Acporleoh  5 m ; k ; 5 k  5  Dc`k m ; ≡ m , ik`k m ; ∈ P k   Y akf `krofk m ; ∃ ( k  k,  m  k; , m ; ) ik`k torakpkt cftorvke ( k  k= , m = ) ac akeki { C f } soaoic`ckf sohcfggk m ; ∈ ( k  k= , m = ). Dkac, k = 5 m ; 5 m =. Aoic`ckf sotorusfyk, sohcfggk acporleoh mkrcskf ( k  k; , m ; ), ( k  k= , m = ), . . ., ( k  k`  , m ` ) Akrc `leo`sc {C f } soaoic`ckf sohcfggk k c  5 m c  „ „ ; 5 m c . @krofk {C f } `leo`sc morhcfggk, prlsos ac ktks pkstc morhoftc pkak suktu cftorvke ( k  k`  , m ` ). Qotkpc prlsos cfc hkfyk k`kf morhoftc dc`k m ∈ ( k   5 m `. @krofk k c  5 m c  „ „ ;, ik`k k`  , m ` ), ykctu dc`k k ` 5 m  5 e (Cf ) ≯ e ( k c , m c  ) 

∛

∛

9 ( m` ∝ k ` ) + (m` ∝; ∝ k ` ∝; ) +  + (m; ∝ k ; )   9 m` ∝ ( k ` ∝ m` ∝; ) ∝ ( k ` ∝; ∝ m ` ∝= ) ∝  ∝ ( k = ∝ m; ) ∝ k ; 6 m ` ∝ k ;    Qotkpc m ` 6 m  akf  akf k ; 5 k . K`cmktfyk, m ` „ k ; 6 m  „  „ k . Dkac, ∛ e ( C f  ) 6 m ∝ k .   Qormu`tc mkhwk i *(P  *(P k   Y) ≯ m  „  „ k k,  m  @ksus =  4 Icske`kf C   soekfg morhcfggk somkrkfg, ik`k uftu` ο   6 2 ykfg acmorc`kf, torakpkt soekfg tortutup D ⊍ C  sohcfggk  sohcfggk ( D    ) 6 e (  ( C  e (  C )  „ ο  Acporleoh, (C   5 e (  ( D    ) 9 i *(  *(   ) ≡ i *(  *( C  e (  C  )  „ ο  5  D  C )  ≡ i * ( C ) 9 e ( C ) 9 e ( C ) 

^ohcfggk uftu` sotckp ο  6  6 2, ( C   5 i *(  *( C  ( C  e (  C )  „ ο  5 C )  ≡ e (  C )   Dkac, i *(  *( C  ( C  C )  9 e (  C ).   @ksus > 4 Icske`kf C   cftorvke tk` hcfggk, ik`k uftu` sotckp mcekfgkf roke torakpkt soekfg tortutup  D ⊍ C  sohcfggk  sohcfggk e (  ( D    ) 9 ∊. Acporleoh *( C  *(   ) 9 e (  ( D    ) 9 ∊ i *(  C )  ≯ i *(   D  @krofk i *(  *( C  *( C  ( C  C )  ≯ ∊ uftu` sotckp ∊, ik`k i *(  C )  9 ∖ 9 e (  C ).  

∊  ykfg acmorc`kf

 Rrlplscsc 4 Icske`kf { Kf } `leo`sc torhctufg pkak  , ik`k i * ( ∤ Kf ) ≡

∛i *( K ) f 

Mu`tc 4  Dc`k skekh sktu hcipufkf  Kf  ioicec`c u`urkf eukr tk` hcfggk ( ∖ ), ik`k `o-tk`skikkf ac ktks acpofuhc. Icske`kf i *(  *(   K ) f  morhcfggk, ik`k torakpkt `leo`sc torhctufg akrc soekfg mu`k { C f,c }c  sohcfggk  Kf ⊍ ∤ C f ,c  c 

 Dugk, uftu` sotckp ο  6  6 2 morek`u

∛ e (C

f ,c

) 5 i * ( Kf ) + = ∝f ο 

c 

^oekfdutfyk, `krofk `leo`sc

{C , } , 9 ∤{C  , } f c

f c 

fc

f 

>3

c

  torhctufg, iorupk`kf gkmufgkf torhctufg akrc

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

`leo`sc torhctufg, akf iofyoeciutc i*

(∤ K ) ≡ ∛ e (C f

f, c

f ,c

)9

∤ Kf  , ik`k

∛ ∛e (C f

f,c

)5

c

@krofk ο  6  6 2 somkrkfg, ik`k i * ( ∤ Kf ) ≡

∛ ( i * ( K ) + = ο ) 9 ∛ i * ( K ) + ο  ∝;

f

f

f

 

 

∛i *( K ) f 

  K`cmkt 4  Dc`k K torhctufg, i *(  *(   K ) 9 2 (echkt tutlrcke #:)  K`cmkt 4 Hcipufkf P2, ;Y tcak` torhctufg Mu`tc 4 @krofk i *(P2,;Y) *(P2,;Y) 9 ; ≢ 2 ik`k P2,;Y tcak` torhctufg.

 Rrlplscsc 4  Dc`k  O ⊍   akf  y ∈  . Acaojcfcsc`kf  O + y

9 {x + y 4 x ∈ O}

ik`k, i *(  *(   ) 9 i *(  *(   O+ y   O ) Mu`tc 4 Ackimce J O

9 {{C f } 4 {C f } soeciut mu`k uftu` O}

akf

9 {{C f } 4 {C f } soeciut mu`k uftu` O + y }   Morakskr`kf `ofyktkkf mkhwk (fm4 e ( C f ) 9 e ( k , m ) 9 m ∝ k  akf e ( C f  + y ) 9 e ( k + y , m + y ) 9 m ∝ k )  {C f } ∈ J O ⇝ {C f + y} ∈ JO + y  akf {C f } ∈ JO + y ⇝ {C f ∝ y} ∈ JO     J O + y

 Dkac kak `lrosplfaofsc `lrosplfaofsc ;-; kftkrk J  O akf J  O+y , ktku J O ∵ J O + y  Akrc scfc ik`k, i * ( O ) 9 cfj e ( C f  )   (iofurut aojcfcsc)

∛ cfj ∛ e ( C + y )  (sotckp C  kak `lrosplfaofsc ;-; kftkrk J   akf J   ) cfj ∛ e ( C + y )

{C f }∈J O

9 {C } J  f

9

∈

f 

f 

 O

 O+y 

O

{ C f + y }∈J O + y 

f 

9 i * ( O + y )

 Rrlplscsc 4 Acmorc`kf somkrkfg hcipufkf  K akf ο  6  6 2, ik`k 4 ;.  Qorakpkt hcipufkf hcipufkf tormu`k L soaoic`ckf sohcfggk  K ⊍ L akf *( L ) ≡ i *(  *(  i *(   K ) + ο  =.  Qorakpkt G ∈ Gα  soaoic`ckf sohcfggk K ⊍ G akf *(  *( G ) i *(   K ) 9 i *(  Mu`tc 4 Icske`kf ο  6  6 2 acmorc`kf. ;. _ftu` `ksus portkik, icske`kf i *(  *(   K ) 9 ∖ ackimce L 9  akf morek`u *( L ) 9 ∖ ≡ ∖ + ο  9  9 i *(  *(  i *(   K ) + ο  12

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

_ftu` `ksus `oauk, icske`kf i *(  *(   K ) 5 ∖. Iofurut aojcfcsc u`urkf eukr, kak { C f } `leo`sc torhctufg akrc soekfg-soekfg mu`k aofgkf scjkt  K ⊍ ∤ C f  akf (`krofk i *(  *(  ik`k acgosor soac`ct mu`kf mu`kf ekgc cfjciui).  K ) cfjciui, ik`k e (C f ) ≡ i * ( K) + ο  Morakskr prlplscsc ; akf =, acporleoh i * (∤ Cf ) ≡ i * ( Cf ) 9 e ( C f  ) ≡ i * ( K ) + ο 

∛

∛

∛

 Dc`k acpcech L 9 ∤ C f  , ik`k L ioiofuhc  K ⊍ L

akf

i * ( L ) ≡ i * ( K ) + ο 

@krofk C f  mu`k, ik`k L 9 ∤ C f   mu`k. =. @ksus portkik, dc`k i *(  *(   K ) 9 ∖ acpcech G 9   ik`k G ∈ Gα . @krofk G 9   ik`k K ⊍ G akf i *(  *(  *( G ).  K ) 9 ∖ 9 i *(  @ksus `oauk, i *(  *(   K ) 5 ∖. Iofurut mu`tc mkgckf ;), torechkt mkhwk uftu` sotckp mcekfgkf ksec f  ∈  ∈    kak hcipufkf mu`k Lf  aofgkf scjkt  K ⊍ Lf  akf i * (Lf  ) ≡ i * ( K ) + f ∝;  Acaojcfcsc`kf G 9 ∣ Lf  . @krofk Lf  mu`k uftu` sotckp f  ∈  ∈  , ik`k G mu`k. ^ohcfggk G ∈ Gα  @krofk K ⊍ Lf  akf G 9 ∣ Lf   uftu` sotckp f  ∈  ∈   ik`k  K ⊍ G

@krofk K ⊍ G ik`k *(  *( G ) ……………………………… ……………………………….. (;) i *(   K ) ≡ i *(  Ac ekcf pchk`, `krofk G 9 ∣ Lf   uftu` sotckp f  ∈  ∈   ik`k G ⊍ Lf 

@krofk G ⊍ Lf  uftu` sotckp f  ∈  ∈  , ik`k i * (G) ≡ i * (Lf ) ≡ i * ( K) + f ∝;  Akrc scfc acporleoh *(  *( G ) …………………………… ……………………………… … (=) i *(   K ) ≯ i *(  Akrc (;) akf (=) acporleoh *(  *( G ) i *(   K ) 9 i *( 

 ^kipkc ac scfc, u`urkf eukr ykfg acaojcfcsc`kf ac ktks suakh ioiofuhc ksuisc-ksuisc ykfg accfgcf`kf `onukec ksuisc ykfg `o->. Leoh `krofk ctu poreu acpormkc`c. >.>.

Hcipufkf Qoru`ur _`urkf eukr ykfg acaojcfcsc`kf somoeuifyk toraojcfcsc uftu` somkrkfg hcipufkf, k`kf totkpc mu`kf iorupk`kf u`urkf kactcj ykfg torhctufg. _ftu` iofdkac u`urkf kactcj torhctufg ik`k poreu hcipufkf pkak `leo`sc hcipufkf ykfg iorupk`kf kedkmkr- ώ. Aojcfcsc 4 Hcipufkf O ac`ktk`kf toru`ur ( ioksurkmeo   ) dc`k uftu` sotckp hcipufkf K morek`u ioksurkmeo  *(  *(  *(  i *(   K ) 9 i *(   K ∣ O ) + i *(   K ∣ O )n  1;

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

n  @krofk soekeu morek`u i *(  *(  *(  *(  torechkt mkhwk  O toru`ur dc`k uftu`  K ) ≡ i *(   K ∣  O ) + i *(   K ∣  O ), n  sotckp K morek`u i *(  *(  *(  *(   K ) ≯ i *(   K ∣ O ) + i *(   K ∣ O ). Morakskr`kf aojcfcsc, dc`k  O toru`ur ik`k  On  dugk toru`ur. Nlftlh hcipufkf toru`ur kakekh ∏ akf  .

Eoiik 4  Dc`k i *(  *(   O ) 9 2 ik`k O toru`ur Mu`tc 4 Icske`kf K hcipufkf somkrkfg. @krofk  K ∣ O ⊍ O, ik`k *(  K *(  i *(   K ∣ O ) ≡ i *(   O ) 9 2 ^ohcfggk i*(  K  K ∣ O ) 9 2 @krofk K ⊎ K ∣ On , ik`k *(  *(  K *(  K *(  K *(  K i *(   K ) ≯ i *(   K ∣ O )n  9 2 + i *(   K ∣ O )n  9 i *(   K ∣ O ) + i *(   K ∣ O )n   Dkac O toru`ur.

 Eoiik 4  Dc`k O; akf O= toru`ur ik`k  O; ∤ O= toru`ur. Mu`tc 4 Icske`kf K soimkrkfg hcipufkf. @krofk  O= toru`ur ik`k *(  *(  *(  i *(   K ∣ O; )n  9 i *(   K ∣ O;n ∣ O= ) + i *(   K ∣ O;n ∣ O= )n  @krofk n  (   K ∣ O; ) ∤ (   K ∣ O= ∣ O; )n  9 K ∣ (   O; ∤ (   O; ∣ O; )) n  9 K ∣ ((   O; ∤ O= ) ∣ (   O; ∤ O; )) 9 (   K ∣ (   O; ∤ O= )) ∣  9 (   K ∣ (   O; ∤ O= )) ik`k i*(  *(  *(   K ∣ (   O; ∤ O= )) ≡ i *(   K ∣ O; ) + i *(   K ∣ O= ∣ O; )n  ^ohcfggk, *(  *(  *(  i *(   K ∣ (   O; ∤ O= )) + i *(   K ∣ (   O; ∤ O= ) )n  9 i *(   K ∣ (   O; ∤ O= )) + i*(   K ∣ O;n  ∣ O= )n  *(  *(  *(  ≡ i *(   K ∣ O; ) + i *(   K ∣ O= ∣ O; )n  + i *(   K ∣ O;n ∣ O= )n  9 i *(  *(  *(   K ∣ O; ) + i *(   K ∣ O; )n  9 i *(  *(   K )

  K`cmkt 4 @leo`sc hcipufkf-hcipufkf toru`ur I iorupk`kf kedkmkr hcipufkf Eoiik 4 Icske`kf  K hcipufkf soimkrkfg akf  O;, . . .,  Of  mkrcskf hcfggk akrc hcipufkf toru`ur ykfg skecfg eopks ( acsdlcft   ), ik`k acsdlcft 

⎐ ⎫ f  ⎡ ⎖ f  i * ⎕ K ∣ ⎭∤ Oc ⎯ ⎜ 9 ∛ i * ( K ∣ Oc  ) ⎮ c 9; ⎪ ⎢ c 9; ⎙ Mu`tc 4 Mu`tc iofggufk`kf cfau`sc pkak f . Rorfyktkkf doeks mofkr uftu` f  9  9 ;  Ksuisc`kf mkhwk porfyktkkf porfyktkkf mofkr uftu` uftu` f  „  „ ;, ykctu 4

⎐ ⎫ f ∝; ⎡ ⎖ f ∝; i * ⎕ K ∣ ⎭∤ Oc ⎯ ⎜ 9 ∛ i * ( K ∣ Oc  ) ⎮ c 9; ⎪ ⎢ c 9; ⎙ @krofk Oc  skecfg eopks, ik`k 1=

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

⎫ f  ⎮ c 9;

⎡ ⎪

 K ∣ ⎭∤ Oc ⎯ ∣ Of

9 ( K ∣ Of  )

akf

⎫f ⎡ ⎫ f ∝; ⎡ n   K ∣ ⎭∤ Oc ⎯ ∣ Of 9 K ∣ ⎭∤ Oc  ⎯ ⎮ c 9; ⎪ ⎮ c 9; ⎪ @krofk Oc  toru`ur ik`k

⎐ ⎐ ⎖ ⎐ ⎖ ⎫ f ⎡⎖ ⎫f ⎡ ⎫ f  ⎡ i * ⎕ K ∣ ⎭∤ Oc ⎯ ⎜ 9 i * ⎕ K ∣ ⎭∤ Oc ⎯ ∣ Of ⎜ + i * ⎕ K ∣ ⎭ ∤ Oc ⎯ ∣ Ofn  ⎜ ⎮ c 9; ⎪ ⎢ ⎮ c 9; ⎪ ⎮ c 9; ⎪ ⎙ ⎙ ⎢ ⎙ ⎢

 

Akrc scfc, ik`k

⎐ ⎐ ⎫ f ⎡⎖ ⎫ f ∝; ⎡ ⎖ i * ⎕ K ∣ ⎭∤ Oc ⎯ ⎜ 9 i * ( K ∣ Of ) + i * ⎕ K ∣ ⎭∤ Oc  ⎯ ⎜ ⎮ c 9; ⎪ ⎢ ⎮ c 9; ⎪ ⎢ ⎙ ⎙ f ∝;

9 i * ( K ∣ Of ) + ∛ i * ( K ∣ Oc  ) c 9;

f 

9 ∛ i * ( K ∣ Oc  ) c 9;

  K`cmkt 4 @leo`sc hcipufkf-hcipufkf toru`ur I iorupk`kf kedkmkr- ώ Mu`tc 4 (soo ‘]oke Kfkeyscs–, >ra oa., H.E. ]lyaof, pkgo :2)

 Eoiik 4 ^oekfg (k, ∖ ) toru`ur Mu`tc 4 Icske`kf K soimkrkfg hcipufkf akf ο  6  6 2 acmorc`kf.  K; 9 K ∣ ( k  k,  ∖ )  Y  K= 9 K ∣ („ ∖, k   K`kf acmu`tc`kf mkhwk *(  *(  *(  i *(   K; ) + i *(   K= ) ≡ i *(   K )  Dc`k i *(  *(   K ) 9 ∖, ik`k mu`tc soeoskc.  Dc`k i *(  *(   K ) 5 ∖, ik`k torakpkt `leo`sc torhctufg hcipufkf mu`k { C f } ykfg iofyoeciutc  K akf morek`u 4 e ( C f  ) ≡ i * ( K ) + ο 

∛

Icske`kf C f 9 C f  ∣ ( k , ∖ )   akf C f 9 C f  ∣ ( ∝∖, k Y   ik`k C f '   akf C f ''   iorupk`kf soekfg (ktku `lslfg) akf (C  (C–  (C––  *( C–  *( C––  e (  C   )  f  9 e (  C  –  ) C  ––  ) C–  ) C–– f  ) f  + e (  f  9 i *(  f  + i *(  @krofk  K; ⊍ ∤ C f ' , ik`k '

''

i * ( K; ) ≡ i * ( ∤ C f' ) ≡

∛ i * ( C ) 

i * ( K= ) ≡ i * ( ∤ C f'' ) ≡

∛ i * ( C ) 

'

f 

@krofk  K= ⊍ ∤ C f '' , ik`k ^ohcfggk, i * ( K; ) + i * ( K= ) ≡

∛ ( i * ( C ) + i * ( C ) ) ≡ ∛ e ( C ) ≡ i * ( K ) + ο  ' f

'' f 

f 

1>

''

f 

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

@krofk ο  6  6 2 soimkrkfg, ik`k i *(  *(  *(  *(   K; ) + i *(   K= ) ≡ i *(   K )



 Qolroik 4 ^otckp hcipufkf Mlroe toru`ur Mu`tc 4 (soo ‘]oke Kfkeyscs–, >ra oa., H.E. ]lyaof, pkgo :;)

Mor`kctkf aofgkf tolroik ac ktks, I  iorupk`kf kedkmkr- ώ  ykfg ioiukt sotckp soekfg ykfg mormoftu` ( k  k,  ∖ ) akf M  kedkmkr-ώ tor`once ykfg ioiukt soekfgsoekfg soportc ctu. Akeki tolroik dugk acporleoh mkhwk sotckp hcipufkf mu`k akf tutup toru`ur. >.1.

_`urkf Eomosguo  Dc`k  O hcipufkf toru`ur, acaojcfcsc`kf u`urkf Eomosguo ( Eomosguo Ioksuro  ) i (  ( O    ) somkgkc u`urkf eukr akrc  O. sohcfggk i  kakekh  kakekh jufgsc hcipufkf ykfg acporleoh aofgkf ioimktksc jufgsc hcipufkf i * `o `leo`sc hcipufkf toru`ur I. Dkac, dc`k I kakekh `leo`sc hcipufkf toru`ur akf  O ∈ I. _`urkf Eomosguo akrc O, kakekh ( O    ) 9 i *(  *(  i (   O )

Auk scjkt poftcfg akrc u`urkf Eomosguo acrcfg`ks akeki prlplscsc-prlplscsc morc`ut4 Rrlplscsc 4 Icske`kf 5 Oc 6 mkrcskf hcipufkf toru`ur, ik`k

∛ i( O ) i ( ∤ O ) 9 ∛ i( O ) , dc`k 5 O 6 skecfg eopks.

(c) i ( ∤ Oc ) ≡

c 

(cc)

c 

c

c 

Mu`tc 4 (c)  Doeks akrc prlplscsc prlplscsc; pkak sum-mkm >.= (cc)  Dc`k 5 Oc 6 mkrcskf hcfggk ykfg skecfg eopks, ik`k iofurut eoiik = pkak sum-mkm >.> aofgkf ioicech K 9  acporleoh 4

⎐ ⎫ f ⎡⎖ f ⎐ f  ⎖ f  i * ⎕  ∣ ⎭∤ Oc ⎯ ⎜ 9 ∛ i * (  ∣ Oc ) ⇑ i * ⎕ ∤ Oc ⎜ 9 ∛ i * ( Oc ) ⎮ c 9; ⎪ ⎢ c 9; ⎙ c 9; ⎢ c 9; ⎙

 

 Dc`k 5 Oc 6 mkrcskf tk`hcfggk ykfg skecfg eopks, ik`k f 

∖

c 9;

c 9;

∤ Oc ⊍ ∤ Oc  akf

⎐∖ ⎖ ⎐ f  ⎖ f  i ⎕ ∤ Oc ⎜ ≯ i ⎕ ∤ Oc ⎜ 9 ∛ i( Oc  ) ⎙ c 9; ⎢ ⎙ c 9; ⎢ c 9; @krofk ruks `crc pkak `otcak`skikkf ac ktks tcak` morgkftufg pkak f , acporleoh

⎐∖ ⎖ ∖ i ⎕ ∤ Oc ⎜ ≯ ∛ i( Oc  ) ⎙ c 9; ⎢ c 9; @omkec`kf akrc `otk`skikkf ac ktks acporleoh akrc (c).

 Rrlplscsc 4 Icske`kf 5 Of 6 mkrcskf hcipufkf toru`ur ykfg ilfltlf turuf. Dc`k i (  ( O   ; ) hcfggk, ik`k

; Icske`kf

{ Kf } `leo`sc = Icske`kf K hcipufkf

torhctufg pkak  , ik`k… (echkt hke. >8) soimkrkfg akf  O;, . . ., Of  mkrcskf hcfggk akrc hcipufkf toru`ur ykfg skecfg eopks ik`k… (echkt hke. >3) 11

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

⎐∖ ⎙ c 9;

⎖ ⎢

i ⎕ ∣ Oc ⎜ 9 eci i( Of ) f ↝∖

Rofdoekskf 4 Ilfltlf turuf krtcfyk  Oc +; +; ⊍ Oc  _`urkf akrc-fyk skik aofgkf ecictfyk.

J c c 

J =

J >

J ;

Mu`tc 4 ∖

Icske  O 9 ∣ Oc   akf J c  9 Oc  „ Oc +; +;. Ik`k O; „ O 9 c 9;

∖

∤ J c   akf J c  skecfg eopks. Dkac, c 9;

∖

⎐ ⎖ ⎜ 9 ∛ i( Jc  )   ; c  ⎙ 9 ⎢ c 9; ∖

i ⎕ ∤ Jc

i( O; ∝ O ) 9

∖

∛ i( J )  c 

c 9;

∖

9 ∛ i( Oc ∝ Oc +; ) c 9;

@krofk O ⊍ O;, ik`k O; 9 O ∤ (   O; „ O ). Dkac, ( O   ; ) 9 i (  ( O    ) + i (  ( O   ; „ O )  i (  ( O   ; „ O ) 9 i (  ( O   ; ) „ i (  ( O    ) i (  @krofk Oc +;  Oc  „ Oc +; +; ⊍ Oc , ik`k Oc  9 Oc +; +; ∤ (  +; ). Dkac, ( O    )c  9 i (  ( O   c +; ( O   c  „ Oc +; ( O   c  „ Oc +; ( O    )c  „ i (  ( O   c +; i (  +; ) + i (  +; )  i (  +; ) 9 i (  +; ) Akrc scfc ik`k, i (O; ) ∝ i ( O)

∖

9 ∛ i( Oc ) ∝ i( Oc +;) c 9;

f ∝;

9 eci ∛ i ( Oc ) ∝ i( Oc +; ) f ↝∖

c 9;

9 eci i (O; ) ∝ i( Of ) f ↝∖

9 i ( O; ) ∝ eci i( Of ) f ↝∖

i ( O)

9 eci i( Of ) f ↝∖

 Rrlplscsc 4 Icske`kf O suktu hcipufkf. Ecik porfyktkkf morc`ut kakekh o`ucvkeof. ;.  O hcipufkf toru`ur =. ∂ο  6  6 2, kak hcipufkf mu`k L ⊎ O sohcfggk i *(  *( L „ O ) 5 ο  >. ∂ο  6  6 2, kak hcipufkf tutup J ⊍ O sohcfggk i *(  *(   ) 5 ο   O „ J  1.  Qorakpkt G ac Gα  aofgkf O ⊍ G sohcfggk i *(  *( G „ O ) 9 2 ?.  Qorakpkt J  ac  ac J ώ  *(   ) 9 2 J ⊍ O sohcfggk i *(   O „ J  ώ aofgkf    Dc`k i *(  *(  `oecik porfyktkkf ac ktks o`ucvkeof aofgkf  O ) hcfggk, ik`k `oecik :. ∂ο  6   6 2 torakpkt hcipufkf _ , ykctu gkmufgkf hcfggk akrc momorkpk soekfg mu`k sohcfggk *( _  i *(  _ ∊ O ) 5 ο  1?

Mkm > „ _`urkf Eomosguo

>.?.

Nlipceoa my 4 @hkorlfc, ^.^c

Jufgsc Qoru`ur Rrlplscsc 4 Icske`kf  j   jufgsc roke aofgkf akorkh kske hcipufkf toru`ur. @ooipkt porfyktkkf morc`ut o`ucvkeof. ;. ∂λ ∈  , {x | j ( x ) 6 λ }  hcipufkf toru`ur =. ∂λ ∈  , {x | j ( x ) ≯ λ }  hcipufkf toru`ur >. ∂λ ∈  , {x | j ( x ) 5 λ }  hcipufkf toru`ur 1. ∂λ ∈  , {x | j ( x ) ≡ λ }  hcipufkf toru`ur @ooipkt porfyktkkf ac ktks iofgk`cmkt`kf ?. ∂λ ∈  , {x | j ( x ) 9 λ }  hcipufkf toru`ur Mu`tc 4 Mu`tc ccc) ⇑ cv) Ackimce somkrkfg mcekfgkf roke λ . @krofk ∖ ;⎠ ⎧ λ | ( ) {x j x ≡ } 9 ∣ ⎨x | j ( x ) 5  λ  + ⎥ f ⎩ f 9; ⎣ akf crcskf akrc mkrcskf hcipufkf toru`ur kakekh toru`ur ik`k hcipufkf { x  |  | j (  ( x) x) ≡ λ } toru`ur. ^omkec`fyk, `krofk ∖ ;⎠ ⎧ {x | j ( x ) 5 λ } 9 ∤ ⎨x | j ( x ) ≡  λ  ∝ ⎥ f ⎩ f 9; ⎣ akf gkmufgkf akrc mkrcskf hcipufkf toru`ur kakekh toru`ur ik`k hcipufkf { x |  j (  (x)   ) 5 λ } x toru`ur. Mu`tc c) ⇑ cc) Ackimce somkrkfg mcekfgkf roke λ . @krofk ∖ ;⎠ ⎧ {x | j ( x ) ≯ λ } 9 ∣ ⎨x | j ( x ) 6  λ  ∝ ⎥ f ⎩ f 9; ⎣ akf crcskf akrc mkrcskf hcipufkf toru`ur kakekh toru`ur ik`k hcipufkf { x  |  | j (  ( x) x) ≯ λ } toru`ur. ^omkec`fyk, `krofk ∖ ;⎠ ⎧ {x | j ( x ) 6 λ } 9 ∤⎨x | j ( x ) ≯  λ  + ⎥ f ⎩ f 9; ⎣ akf gkmufgkf akrc mkrcskf hcipufkf toru`ur kakekh toru`ur ik`k hcipufkf { x |  j (  (x)   ) 6 λ } x toru`ur. (tho rost, soo ‘]oke Kfkeyscs–, >ra oa., H.E. ]lyaof, pkgo :? „ ::)

 Aojcfcsc 4 Jufgsc  j   ac`ktk`kf toru`ur (Eomosguo), dc`k akorkh kske jufgsc  j   toru`ur akf skekh sktu akrc `ooipkt porfyktkkf akeki prlplscsc ac ktks morek`u. Nlftlh 4 Ac`otkhuc  j 4 A  ↝  ↝   jufgsc toru`ur. Acaojcfcsc`kf  j +  9 ikx( 2, j  ) akf  j ∝  9 ikx( 2, ∝ j  ) Aofgkf iofggufk`kf aojcfcsc jufgsc toru`ur, mu`tc`kf mkhwk  j + akf j  „  toru`ur. Mu`tc 4 @krofk j toru`ur, ik`k A 9 A + 9 A  j ∝  toru`ur.

Icske`kf, {x 4 j + ( x ) 6 λ } 9 K . λ  5 2 ⇝  K 9 A  λ

≯ 2 ⇝ {x 4 j + ( x ) 6 λ } 9 {x 4 j ( x )  6 λ } 1:

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

Icske`kf, {x 4 j ∝ ( x ) 6 λ } 9 M . λ  5 2 ⇝ M 9 A  λ ≯ 2 ⇝ {x 4 j ∝ ( x ) 6 λ ≯ 2} 9 {x 4 ∝ j ( x ) 6 λ } 9 {x 4 j ( x ) 5  ∝λ } Iofurut aojcfcsc, tormu`tc mkhwk  j + akf j  „  toru`ur.

 



Roreu acnktkt mkhwk,  j  jufgsc  jufgsc `lftcfu aofgkf akorkh kske ykfg toru`ur kakekh jufgsc toru`ur. Mu`tc 4 Icske`kf  j 4 A  ↝   hcipufkf toru`ur akf  j jufgsc `lftcfu. @krofk A  hcipufkf   hcipufkf  ↝   aofgkf A  hcipufkf toru`ur, ik`k A  j  9 A   hcipufkf toru`ur. K`kf acmu`tc`kf mkhwk ∂λ  ∈    morek`u {x 4 j ( x ) 6 λ }  toru`ur kakekh hcipufkf toru`ur. Ackimce somkrkfg λ  ∈   `krofk   (λ , ∖ )} {x 4 j ( x ) 6 λ } 9 {x 4 j ( x ) ∈

9  j ∝;((λ , ∖ )) ⊊ A  Cftorvke (λ , ∖ )   kakekh hcipufkf mu`k ac  . Morakskr`kf tolroik, acporleoh  j  ∝;((λ , ∖ )) iorupk`kf hcipufkf mu`k ac A . @krofk  j ∝;((λ , ∖ )) 9 ∤ ( k ` , m ` ) `

akf sotckp cftorvke/soekfg kakekh hcipufkf toru`ur ik`k {x 4 j ( x ) 6 λ }  hcipufkf toru`ur.

 ^otckp jufgsc tkfggk iorupk`kf jufgsc toru`ur. Dc`k  j  jufgsc   jufgsc toru`ur akf  O hcipufkf toru`ur akeki akorkh kske, ik`k mktkskf j  torhkakp  torhkakp O, j | O dugk toru`ur. Rrlplscsc 4 Icske`kf n  suktu   suktu `lfstkftk,  j  akf   akf  g  auk   auk jufgsc roke ykfg toru`ur akeki akorkh kske ykfg skik, ik`k jufgsc j  +  + n , nj , j  +  + g , j  „  „ g , akf jg  dugk  dugk toru`ur Mu`tc 4 (soo ‘]oke Kfkeyscs–, >ra oa., H.E. ]lyaof, pkgo ::)  Qolroik 4 Icske`kf 5 j f 6 mkrcskf jufgsc toru`ur, ik`k

sup{ j ; , j = , , j f } , cfj{ j ; , j = ,  , j f } , sup{ j f } , cfj{ j f } , eci j f  , akf eci j f  f 

f 

kakekh jufgsc-jufgsc toru`ur. Mu`tc 4 Rortkik, acmu`tc`kf mkhwk sup{ j ; , j = ,  , j f }  jufgsc toru`ur. @krofk j c , c  9  9 ;, =, . . ., f  toru`ur  toru`ur ik`k A   toru`ur. Icske c 

 j

9 sup{ j ; , j = , , j f }

akf f 

A

9 ∣ A  j  c 9;

c 

ik`k A  j  toru`ur. Ackimce somkrkfg λ  ∈  . K`kf acmu`tc`kf mkhwk {x 4 j ( x ) 6 λ } .  Dc`k  j ( x ) 9 sup{ j ; ( x ), j = ( x ), , j f ( x )} 6 λ   ik`k kak c , ; ≡ c ≡ f  sohcfggk  sohcfggk j c ( x  x )  6 λ . Dkac, f 

{x 4 j ( x ) 6 λ } ⊊ ∤ {x 4 j c ( x ) 6  λ } c 9;

 Dc`k j c ( x   uftu` suktu c , ; ≡ c ≡ f  ik`k  ik`k sup{ j ; ( x ), ) , j = ( x ), ) , , j f ( x )} ) } 6 λ ⇑ j (x ) 6 λ  . Dkac, x )  6 λ  uftu`

17

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c f 

∤{x 4

j c ( x ) 6 λ } ⊊ {x 4 j ( x )  6 λ }

c 9;

@oscipuekffyk f 

{x 4 j ( x ) 6 λ } 9 ∤{x 4 j c ( x ) 6  λ } c 9;

{x 4 j c ( x ) 6 λ }   toru`ur. @krofk gkmufgkf torhctufg akrc hcipufkfhcipufkf toru`ur kakekh toru`ur ik`k {x 4 j ( x ) 6 λ }  toru`ur. Dkac tormu`tc  j  toru`ur.  toru`ur.

@krofk  j c   toru`ur ik`k

(scskfyk echkt ‘]oke Kfkeyscs–, > ra oa., H.E. ]lyaof, pkgo :7)

 Aojcfcsc 4 (c)  j  9  9 g  k.o  k.o dc`k j  akf  akf g  ioipufykc  ioipufykc akorkh kske ykfg skik akf i {x | j (  (x    )  ≢ g (  (x    )} x x   92 (cc)  j f   `lfvorgof `o  g   hkipcr ac ikfk-ikfk, dc`k torakpkt hcipufkf  O  ykfg moru`urkf fle sohcfggk j f ( x  (x    )  uftu` sotckp x ∃ O. x )  `lfvorgof `o g (  x Rrlplscsc 4  Dc`k j  jufgsc  jufgsc toru`ur akf  j  9  9 g  k.o,  k.o, ik`k g  jufgsc  jufgsc toru`ur. Mu`tc 4 @krofk j  9  9 g  (ko)  (ko) ik`k i ({ ({x 4 j (  (x    )  ≢ g (  ( x  ( k  (k  ({k }) }) 9 2. Dkac, x x )})   9 2. Icske`kf j (  k )  ≢ g (  k  ),   ik`k i ({ ∂x ∈ A j 9 A g  ∝ {k }  j ( x ) 9 g ( x ),

Ackimce somkrkfg λ  ∈  . K`kf acmu`tc`kf { x  4 4 g (  (x    )  6 λ } toru`ur. x c)  Dc`k j (  ( k  ( k  k )  5 g (  k )   5 j (  (k  (k  •  Dc`k λ  5 k  )  5 g (  k  )  {x 4 g ( x ) 6 λ } 9 {x 4 j ( x )  6 λ }  toru`ur  5 g (  (k  •  Dc`k j ( (k  k  )  5 λ  5 k  )    {x 4 g ( x ) 6 λ } 9 {x 4 j ( x ) 6 λ } ∤{k}  toru`ur (k  •  Dc`k j ( (k  k  )  5 g (  k  )  5 λ  {x 4 g ( x ) 6 λ } 9 {x 4 j ( x )  6 λ }  toru`ur cc)  Dc`k j (  ( k  ( k  k )  6 g (  k )  (k  •  Dc`k λ  6 j ( (k  k  )  6 g (  k  )  {x 4 g ( x ) 6 λ } 9 {x 4 j ( x )  6 λ }  toru`ur  6 g (  (k  •  Dc`k j ( (k  k  )  6 λ  6 k  )    {x 4 g ( x ) 6 λ } 9 {x 4 j ( x ) 6 λ } ∝ {k}  toru`ur (k) •  Dc`k j ( (k  k  )  6 g (  k  ) 6 λ 

 Rrlplscsc 4 Icske`kf  j   jufgsc toru`ur pkak P k   Y, akf  j  morfcekc   morfcekc »∖ hkfyk pkak hcipufkf ykfg moru`urkf k, m  fle, ik`k uftu` sotckp ο   6 2, torakpkt jufgsc tkfggk  g   akf suktu jufgsc `lftcfu h   ykfg ioiofuhc | j „ g | 5 ο  akf  akf | j  „  „ h | 5 ο  @onukec pkak hcipufkf ykfg moru`urkf eomch `once akrc ο , ykctu 4 (x    )  „ g (  (x    )  ≯ ο } 5 ο , akf i {x | | j (  (x    )  „ h (  (x    )| i {x | | j (  x x x x   ≯ ο } 5 ο  ^omkgkc nktktkf, dc`k i ≡ j ≡ I , pcech jufgsc g  akf  akf h  ykfg  ykfg ioiofuhc i ≡ g ≡ I  akf  akf i ≡ h ≡ I  Mu`tc 4 Mu`tc acmkgc iofdkac auk mkgckf

18

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

;) Icske`kf  j  jufgsc   jufgsc toru`ur ykfg acaojcfcsc`kf pkak P k   Y akf i ( {x 4 j ( x )   9 »∖} ) 9 2.  K`kf k , m  acmu`tc`kf mkhwk ∂ο  6 2, ∎I ∀ j ≡ I    `onukec pkak hcipufkf ykfg u`urkffyk `urkfg akrc ο  . O`ucvkeof aofgkf iofudu``kf kak I sohcfggk i ({x ∈ P k , m Y 4 j ( x ) 6 I } ) 5 ο  . Icske`kf f ∈   somkrkfg. Acaojcfcsc`kf  Of  9 {x ∈ P k , m Y4 Y4 j (x )

6 f }

ykctu,  O;

9 {x ∈ P k , m Y 4 j ( x ) 6 ;}

 O=

9 {x ∈ P k , m Y 4 j ( x ) 6 =} 

Y4 j ( x )  Of  9 { x ∈ P k , m Y4  Of +;

6 f }

9 {x ∈ P k , m Y 4 j ( x ) 6 f + ;}

@krofk  Of  9 {x ∈ P k , m Y4 Y4 j (x )

6 f } 9 {x ∈ P k , m Y 4 j ( x ) 6 f ∨ j ( x ) 5 ∝f}  9 {x ∈ P k , m Y 4 j ( x ) 6 f} ∤ {x ∈ P k , m Y 4 j ( x ) 5 ∝f } 

akf j  toru`ur,  toru`ur, ik`k  Of  toru`ur. @krofk ∂f ∈   morek`u f  +  + ;6 f sohcfggk dc`k  j ( x ) 6 f + ; ik`k  j ( x ) 6 f . K`cmktfyk  Of +;

9 {x ∈ P k , m Y 4 j ( x ) 6 f + ;} ⊍ {x ∈ P k , m Y 4 j ( x ) 6 f} 9 Of 

Rorhktc`kf mkhwk  O; 9 {x ∈ P k , m Y 4 j ( x ) 6 ;} , ik`k i ( O; ) ≡ i (P k , m Y) ⇑ i ( O; ) ≡ m ∝ k 5  +∖

 Dkac, i( O; )  morhcfggk. @krofk  Of   mkrcskf hcipufkf toru`ur aofgkf  Of +; ⊍ Of  , ∂f ∈   akf i( O; )  morhcfggk ik`k acporleoh

⎐∖ ⎖ i ⎕ ∣ Of ⎜ 9 eci i ( Of  ) 9 2 ⎙ f 9; ⎢ f ↝∖  K`cmktfyk morek`u

∂ο 6 2, ∎ I ∀ i ( O I  ) 5 ο ⇑ i ({x ∈ P k , m Y 4 j ( x ) 6 I } ) 5 ο  Hke Hke cfc cfc mork morkrt rtcc  j ( x ) ≡ I  `onukec pkak hcipufkf {x ∈ P k , m Y4 Y 4 j ( x ) 6 I }  ykfg u`urkffyk `urkfg akrc ο  . =) Icske`kf  j  jufgsc   jufgsc toru`ur ykfg acaojcfcsc`kf pkak P k   Y akf i ( {x 4 j ( x )   9 »∖} ) 9 2.  K`kf k , m  acmu`tc`kf mkhwk ∂ο  6 2  akf I , kak jufgsc soaorhkfk ϟ   sohcfggk soaoic` c`kf kf sohc sohcfg fggk gk  j ( x ) ≡ I   j ( x ) ∝ ϟ( x ) 5 ο , ∂x  soaoi Icske`kf ο  6  6 2   akf  I   acmorc`kf. Rcech  F   mcekfgkf ksec sohcfggk

∂` ∈{∝ F , ∝F + ;,… , F ∝  ;}

13

 I   F 

5 ο  . Acaojcfcsc`kf

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

 O`

 I I  9 ⎧⎨x ∈ P k , m Y 4 j ( x ) ≯ ` ⎠⎥ ∣ ⎧⎨x ∈ P k , m Y 4 j ( x ) 5 ( ` + ;) ⎠⎥  F ⎩ ⎣ F ⎩ ⎣

@krofk j  toru`ur,  toru`ur, ik`k  O` toru`ur. Acaojcfcsc`kf jufgsc ϟ  , ykctu

⎧  F ∝;  I  , x ∈ O` ⎤∛` ϟ ( x ) 9 ⎨` 9∝ F  F  ⎤2 , x ∃ O` ⎣  Dc`k x ∈P k , m Y sohcfggk  j ( x ) 5 I  ik`k x ∈ O`  uftu` suktu ` sohcfggk `

@krofk x ∈ O` ik`k ϟ ( x ) 9 `

I   F 

 I  F

≡ j ( x ) 5 ( ` + ;)

I  F 

. K`cmktfyk

 j ( x ) ∝ ϟ ( x )

5 ( ` + ;)

 I

 F

∝`

I F

9

I  F 

5 ο  

Icske`kf K soimkrkfg hcipufkf, akf acaojcfcsc`kf jufgsc `krk`torcstc` akrc hcipufkf  K, ykctu  χ  K somkgkc morc`ut4 ⎧; , x ∈ K  χ  K ( x ) 9 ⎨ ⎣2 , x ∃ K Jufgsc  χ  K  toru`ur dc`k akf hkfyk dc`k  K toru`ur. Aojcfcsc 4 Jufgsc morfcekc roke ϟ  ac`ktk`kf   ac`ktk`kf soaorhkfk ( scipeo   ), dc`k ϟ  toru`ur   toru`ur akf ioicec`c hkfyk soduiekh scipeo  morhcfggk fcekc.

 Dc`k ϟ   soaorhkfk ( scipeo   ) akf morfcekc λ ;, λ =, . . ., λ f   ik`k ϟ 9 scipeo 

f 

∛λ χ  c 9;

 Kc

c

Kc 

  aofgkf

9 {x 4 j ( x ) 9 λ c } . Rofduiekhkf akf pofgurkfgkf auk jufgsc soaorhkfk kakekh soaorhkfk. Nlftlh 4 Jufgsc tkfggk

⎧;, 2 5 x  5 ; ⎤  j ( x ) 9 ⎨=, ; 5 x  5 > ⎤>, > 5 x  5 1 ⎣ akpkt actuecs iofdkac

 j ( x ) 9 ;.χ K;

aofgkf  K;

+ =.χ K

=

1 )  akf  K= 9 (;,>) 9 ( 2, ;) ∤ ( >, 1)

 >.:.

 Qcgk Rrcfscp Ectteowlla  D. O. Ectteowlla iofgktk`kf mkhwk torakpkt tcgk prcfscp akeki tolrc jufgsc roke ykfg mkfyk` acgufk`kf, ykctu4 ;. Hkipcr sotckp hcipufkf (toru`ur) iorupk`kf gkmufgkf morhcfggk soekfg-soekfg =. Hkipcr sotckp jufgsc (toru`ur) iorupk`kf jufgsc `lftcfu >. Hkipcr sotckp mkrcskf jufgsc (toru`ur) ykfg `lfvorgof kakekh `lfvorgof sorkgki Rrcfscp portkik akf `oauk toekh actoiuc akeki poimkhkskf somoeuifyk. Moftu`-moftu` uftu` ?2

Mkm > „ _`urkf Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

prcfscp portkik akf akpkt acechkt pkak prlplscsc somoeuifyk. Nlftlh prcfscp `otcgk acmorc`kf pkak prlplscsc morc`ut cfc4 Rrlplscsc 4 Icske`kf  O ioicec`c u`urkf hcfggk, 5 j f 6 mkrcskf jufgsc toru`ur pkak  O, akf  j  jufgsc   jufgsc morfcekc roke. Dc`k uftu` sotckp x  ac  ac  O morek`u  j f ( x  ( x    6 2 ykfg acmorc`kf x )  ↝  j (  x ),   ik`k uftu` ο  6 2 akf α  6 torakpkt hcipufkf toru`ur  K ⊍  O aofgkf i (  ( K    ) 5 α  akf   akf mcekfgkf ksec F  soaoic`ckf   soaoic`ckf sohcfggk uftu` sotckp x ∃ K akf sotckp f ≯ F  morek`u  morek`u | j f ( я   )  „ j (  (я    )| 5 ο  я я   Mu`tc 4 Acmorc`kf ο , α  6 2  somkrkfg ∂f ∈   acaojcfcsc`kf Gf

9 {x ∈ O 4 j f ( x ) ∝ j ( x )  ≯ ο }

akf ∂f ∈   acaojcfcsc`kf  Of

9 ∤ G` ` ≯ f 

9 {x ∈ O 4

j f ( x ) ∝ j ( x )

≯ ο , uftu` suktu ` ≯ f}

^ohcfggk  O; ⊎ O= ⊎ O>  ↝ {Of }  turuf iofudu Rorhktc`kf mkhwk ∂x ∈ O, eci j f ( x ) 9 j ( x )  

 

∣ Of  .

f ↝∖

x ∈ O , ∂ο

6 2, ∎f2 ∈  sohcfggk uftu` f ≯ f2 ⇝ j f ( x ) ∝ j ( x ) 5  ο   Dkac, x ∈ Ofn  ⇝ x ∃ Of  . ^ohcfggk ∣ Of  9 ∏ . 2

2

 Dkac, Of  turuf iofudu ∏  . ^ohcfggk eci i( Of  ) 9 2 f ↝∖

@krofk eci i( Of ) 9 2  ik`k uftu` α  6  6 2  ac ktks, ∎ F ∈   sohcfggk uftu` f ≯ F  morek`u4 f ↝∖

i( Of ) ∝ 2

5 α ⇑ i( Of ) 5 α 

 Kimce  K 9 Of  ⊍ O toru`ur akf i( K ) 5 α  .  K n

9 Of n  9 {x ∈ O 4 j f ( x ) ∝ j ( x ) 5 ο , ∂f ≯ F }  

 Dc`k acakeki hcpltoscs prlplscsc ac ktks  j f ( x  ( x  x )  ↝  j (  x )  uftu` sotckp x , ac`ktk`kf mkhwk 5 j f 6 `lfvorgof tctc`-aoic-tctc` ( nlfvorgos   ) `o  j  pkak   pkak  O. Dc`k kak hcipufkf M ⊍  O aofgkf i (  (M    ) 9 2 nlfvorgos  plcftwcso   plcftwcso  soaoic`ckf sohcfggk j f  ↝ j tctc`-aoic-tctc` pkak  O „ M, ac`ktk`kf mkhwk j f  ↝ j k.o pkak O. Rrlplscsc 4 Icske`kf  O  hcipufkf toru`ur akf mor-u`urkf hcfggk, akf 5 j f 6 mkrcskf jufgsc toru`ur ykfg `lfvorgof `o jufgsc morfcekc roke  j   k.o pkak  O. Ik`k, uftu` uftu` ο  6 2 akf α   6 2 ykfg acmorc`kf, torakpkt hcipufkf  K ⊍  O  aofgkf i (  ( K    ) 5 α , akf mcekfgkf ksec  F   soaoic`ckf sohcfggk uftu` sotckp x ∃ K akf sotckp f ≯ F  morek`u,  morek`u, | j f ( я   )  „ j (  (я    )| 5 ο  я я  

?;

Mkm 1 Cftogrke Eomosguo 1.;.

Cftogrke ]coikff @ctk k`kf soac`ct iofguekfg `oimkec momorkpk aojcfcsc pkikrtcsckf akeki Cftogrke ]coikff. Icske`kf  j   jufgsc morfcekc roke ykfg tormktks akf toraojcfcsc pkak cftorvke P k   Y akf icske`kf P k   Y k,  m  k,  m  acpkrtcsc iofdkac f  mkgckf,  mkgckf, ykctu4 4 k 9 ζ 2 5 ζ; 5  5 ζ f  9 m  _ftu` sotckp pkrtcsc  p torsomut, acaojcfcsc`kf Duiekh Ktks4

^ 9 Duiekh Ktks 9

f 

∛ (ζ c 9;

∝ ζ c ∝; ) I c 

c

akf s 9 Duiekh Mkwkh 9

f 

∛ (ζ c 9;

acikfk,  I c  9 sup

x ∈( ζc ∝; ,ζ c  )

j ( x )  akf i c  9

cfj

x ∈( ζc ∝; ,ζ c  )

∝ ζ c ∝; )i c 

c

j ( x ) .

Akrc pofaojcfcsckf cfc torechkt mkhwk uftu` sotckp pkrtcsc  p morek`u, s ( p ) ≡ ^ ( p ) akf s ( p )  akf ^ ( p )  . @krofk pkrtcsc-pkrtcsc torsomut tcak` tufggke, ik`k akpkt acaojcfsc`kf

∠

Cftogrke Ktks ]coikff 9 ]

m 

k 

Cftogrke Mkwkh ]coikff 9 ]

j 9 cfj ^  p

∠

m 

k 

j 9 sup s   p

@krofk s ( p ) ≡ ^ ( p ) , ∂p

ik`k

sup s ( p ) ≡ ^ ( p ), ∂p sup s( p ) ≡ cfj ^ ( p ) ]

∠

m

k

j ≡]

∠

m 

k 

j  

^ohcfggk Cftogrke Ktks soekeu eomch akrc ktku skik aofgkf Cftogrke Mkwkh. Dc`k, ]

∠

m

j 9]

k

∠

m 

k 

j  

ik`k  j  acsomut  acsomut torcftogrke ]coikff ( ]coikff  Y akf iofyomut fcekc cftogrke `oaukfyk ]coikff Cftogrkmeo  ) pkak P k  k,  m  aofgkf Cftogrke ]coikff akrc  j  akf  akf acekimkfg`kf ]

∠

m 

k 

j ( x ) ax  

uftu` ioimoak`kffyk aofgkf Cftogrke Eomosguo ykfg k`kf `ctk tcfdku `oiuackf. Jufgsc Qkfggk Jufgsc tkfggk ψ   kakekh jufgsc ykfg acaojcfcsc`kf leoh ψ ( x ) 9 n c , uftu` ζ c ∝; 5 x  5 ζ c  Akrc pofaojcfcsckf cfc ik`k,

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c f 

∠ ψ ( x ) a ( x ) 9 ∛ n ( ζ m 

k 

c 9;

m

j ( x ) a ( x ) 9 cfj

c

∝ ζ c ∝; )

c

Rorhktc`kf mkhwk

∠ ]∠ ]

∠ ψ ( x ) a ( x )   j ( x ) a ( x ) 9 sup ∠ ψ ( x ) a ( x )   ψ ≯ j  k 

k m

m 

k

`krofk ψ ( x ) 9 I c , x ∈ (ζ c ∝; , ζ c  ) ⇝ ^ 9

m 

k 

ψ ≡ j 

∠

m 

ψ ( x ) a  x  akf   ϟ ( x ) 9 i c , x ∈ (ζ c ∝; , ζ c  ) ⇝ s 9

k 

m 

∠ ϟ( x ) a  x k 

 

1.=.

Cftogrke Jufgsc Qormktks @ctk iuekc poimkhkskf pkak sum-mkm cfc aofgkf iofaojcfsc`kf suktu jufgsc ykfg morfcekc ; pkak suktu hcipufkf toru`ur akf fle soekcffyk ykfg torcftogrke`kf akf ioicec`c cftogrke skik aofgkf u`urkf akrc hcipufkffyk. Jufgsc @krk`torcstc` Icske`kf  O ⊍   hcipufkf toru`ur. Acaojcfcsc`kf jufgsc `krk`torcsctc` akrc  O aofgkf kturkf ⎧; x ∈ O  χ  O ( x ) 9 ⎨ ⎣2 x ∃ O ^oakfg`kf `limcfksc ecfokr ϟ( x ) 9

f 

∛ k χ  c 9;

c

Oc 

( x )

acsomut jufgsc soaorhkfk dc`k hcipufkf  Oc  toru`ur. Nlftlh4 ϟ ( x ) 9 ;. χ( 2 , = ) + =. χ( ;, = ) + = χ ( = ,> )  

 dugk akpkt actuecs ϟ  dugk ϟ ( x ) 9 ;. χ( 2 ,;) + >.χ ( ;, = ) + = χ ( = ,> )    Dkac, roprosoftksc ϟ  tcak` tufggke.

 Aojcfcsc (Moftu` @kflfc`)  Dc`k ϟ  kakekh  kakekh jufgsc soaorhkfk akf {k ;, k =, . . ., k f } kakekh hcipufkf fcekcfyk aofgkf k c  ≢ 2, ik`k ϟ( x ) 9

f 

∛ k χ  c 9;

c

Kc 

( x )

aofgkf  Kc 9 {x 4 ϟ ( x ) 9 k c }  acsomut roprosoftksc moftu` `kflfc` akrc ϟ . Nktktkf 4 ;. k c  soiuk moak akf k c  ≢ 2. =.  Kc  skecfg eopks >. Moftu` `kflfc` akrc suktu jufgsc soaorhkfk morscjkt tufggke Akrc scfc  K 9 {ϟ 4 ϟ jufgsc tkfggk} akf M 9 {ϟ 4 ϟ jufg  sc soaorhkfk} ⇝ K ⊍ M Aojcfcsc (Cftogrke Jufgsc ^oaorhkfk) 4

 Dc`k ϟ   kakekh jufgsc soaorhkfk akf ioipufykc moftu` `kflfc` ϟ ( x ) 9

f 

∛ k χ  c 9;

acaojcfcsc`kf

?>

c

Kc 

( x )

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c f 

∠ ϟ ( x ) ax 9 ∛ k i( K ) c 9;

c

c 

∠

Fltksc 4 ϟ  Nlftlh4 ϟ( x ) 9 ;. χ( 2 ,;) + >.χ ( ;, = ) + = χ ( = ,> )  

∠ ϟ ( x ) ax 9 ;.i( 2, ;) + >.i(;, = ) + =.i( = , >) 9 ;.; + >.; + =.; 9: Eoiik morc`ut cfc iofgktk`kf mkhwk aojcfcsc ac ktks morek`u dugk uftu` jufgsc soaorhkfk ykfg tcak` acroprosoftksc`kf akeki moftu` `kflfc`-fyk. Eoiik 4

Icske`kf ϟ 9

f 

∛ k  χ  c

c 9;

Oc 

 aofgkf  Oc ∣ O d  9 ∏  uftu` c ≢ d  . Icske`kf  Oc  toru`ur akf i( Oc  ) 5 ∖

∂c  , ik`k f 

∠ ϟ  9 ∛ k i ( O ) c 9;

c

c 

( k  kc   tcak` hkrus moak) Mu`tc 4 Acaojcfcsc`kf hcipufkf  Kk  9 {x 4 ϟ ( x ) 9 k } . Akrc pofaojcfcsckf cfc, acporleoh mkhwk  Kk 9

∤O

c 

k c  9 k 

 Dkac, soiuk  Oc   ykfg ioicec`c fcekc ykfg skik, somut k , acgkmufg `o akeki hcipufkf  Kk .  K`cmktfyk, Kk  skecfg kscfg. ^ohcfggk,

⎐ ⎕ k 9 k  ⎙

⎖ ⎜ ⎢

i( Kk ) 9 i ⎕ ∤ Oc ⎜ 9 c 

∛ i( O ) k c  9 k 

c 

Akrc scfc, ik`k

∠ ϟ( x ) ax 9 ∠ ∛ k χ  ( x )   9 ∛ ki( K ) k c  9 k 

 Kk 

k 

k c  9 k 

9 ∛ k ∛ i( Oc  ) kc 9k

k c  9k  

9 ∛ k c i ( Oc  ) c 

 Rrlplscsc 4  Dc`k ϟ   akf ψ   kakekh auk jufgsc soaorhkfk akf k , m ∈ ,  ik`k

;. =.

∠ kϟ + mψ 9 k ∠ ϟ + m ∠ψ     Dc`k ϟ ≯ ψ   (ko) ik`k ∠ ϟ ≯ ∠ψ 

Mu`tc 4 Ac`otkhuc ϟ  , ψ  jufgsc soaorhkfk. k , m ∈ .  K`kf acmu`tc`kf

∠ kϟ + mψ 9 k ∠ ϟ + m ∠ψ  .  

Icske`kf roprosoftksc `kflfc` akrc `oauk jufgsc torsomut kakekh ?1

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c f ;

∛ k  χ 

ϟ9

c 9;

c

Kc 

akf ψ 9

f =

∛ m χ  c 9;

c

Mc 

Aofgkf Kc  akf Mc  skecfg eopks, k c  soiuk mormoak akf m c  soiuk mormoak uftu` sotckp c . Icske`kf f; f =

f  { O` }; 9 { Kc ∣ M d }c 9; d 9; ( f 9 f; × f = )

 Quecs`kf ϟ9

f 

∛ k  χ  * `

` 9;

O`

 akf ψ 9

f 

∛ m  χ  ` 9;

* `

O`

aofgkf k `*  ktku m `*  iufg`cf kak ykfg skik.  Dkac, f

∠ kϟ + mψ 9 ∠ k ∛ k χ * `

` 9;

f 

O`

f

+ ∠ m ∛ m`* χ O

 

`

` 9;

f 

9 ∠ ∛ kk  χ O + ∠ ∛ km`* χ   O * `

` 9;

`

`

` 9;

f 

9 ∠ ∛ ( kk `* + mm`*  ) χ O

`

` 9;

f 

9 ∛ ( kk `* + mm`* ) i( O` ) ` 9; f

f 

9 ∛ kk i( O` ) + ∛ mm`* i( O` ) * `

` 9;

f

` 9;

f 

9 k ∛ k i( O` ) + m ∛ m`* i ( O` ) ` 9;

* `

` 9;

9 k ∠ ϟ +m ∠ψ  @oauk, `krofk ϟ ≯ ψ    (ko) ik`k i {x 4ϟ ( x ) 5 ψ ( x )}  9 2 . ^ohcfggk cftogrke-fyk tcak` acporhctufg`kf. Dkac, nu`up actcfdku uftu` ϟ ≯ ψ  . @krofk ϟ ≯ ψ  , aofgkf iofggufk`kf hksce pkak mkgckf portkik, acporleoh ϟ ∝ψ ≯ 2 ⇑ ϟ ∝ ψ ≯ 2 ⇑ ϟ ∝ ψ ≯ 2 ⇑ ϟ ≯ ψ  

∠

∠

∠

∠

∠

∠

  K`cmkt 4

 Dc`k ϟ 9

f 

∛ k  χ  c 9;

c

Oc 

 aofgkf Oc  tcak` skecfg eopks, ik`k

∠

ϟ9

f

∛ ∠ c 9;

k c χ Oc  9

f 

∛ k i( O ) c 9;

c

c

 

 Dkac rostrc`sc Eoiik Eoiik ac ktks kgkr Oc  skecfg eopks tcak` ekgc acporeu`kf. Icske`kf  O kakekh kakekh hcipuf hcipufkf kf ykfg ykfg toru`u toru`urr aofgkf aofgkf i( O ) 5 ∖ . Jufgsc  j  kakekh   kakekh jufgsc morfcekc roke ykfg tormktks akf toraojcfcsc pkak O. Mkfacfg`kf moskrkf cfj ψ ( x ) ax  akf cfj ϟ ( x ) ax 

∠

∠

ψ ≯ j   O

ϟ ≡ j   O

acikfk ψ   akf ϟ   kakekh jufgsc soaorhkfk. ??

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

Rrlplscsc 4 Icske`kf  j   kakekh jufgsc ykfg toraojcfcsc akf tormktks pkak hcipufkf toru`ur  O  aofgkf i( O ) 5 ∖ .

cfj

∠ ψ ( x ) ax 9 sup ∠ ϟ ( x ) ax ⇑ j

ψ ≯ j   O

ϟ ≡ j 

O

 kakekh jufgsc toru`ur

acikfk ψ   akf ϟ   kakekh jufgsc soaorhkfk. Mu`tc 4

⇝

@krofk  j   tormktks ik`k kak  I  6 2 sohcfggk  j ( x ) ≡ I , ∂x ∈ O . Kimce f ∈    somkrkfg. Acaojcfcsc`kf ∂` 9 ∝f ,..., f  ` ∝; ` ⎠ ⎧  O` 9 ⎨x ∈ O 4 I ≡ j ( x ) ≡ I ⎥ f f  ⎩ ⎣ ^omkgkc ceustrksc, icske`kf f  9  9 =. Ik`k ` 9 „=, . . ., = aofgkf  j  soportc  soportc pkak gkimkr morc`ut4  I `9=

> ⎧ ⎠  O∝= 9 ⎨x ∈ O 4 ∝ I ≡ j ( x ) ≡ ∝I ⎥ = ⎣ ⎩

`9;

; ⎠ ⎧  O∝; 9 ⎨x ∈ O 4 ∝ I ≡ j ( x ) ≡ ∝ I ⎥ = ⎩ ⎣

 I /= /= 2 =

;

;

`92  „I /= /= ` 9 „; 9 „;  „I ` 9 „ =  „ > I/=  I/=

; ⎧ ⎠  O2 9 ⎨x ∈ O 4 ∝ I ≡ j ( x ) ≡ 2⎥ = ⎣ ⎩ ; ⎠ ⎧ I ⎥ = ⎩ ⎣ ; ⎧ ⎠  O= 9 ⎨x ∈ O 4 I ≡ j ( x ) ≡ I ⎥ = ⎣ ⎩  O; 9 ⎨x ∈ O 4 2 ≡ j ( x ) ≡

 Qorechkt  O` toru`ur (hcipufkf mu`k),  O`  skecfg eopks akf ∤ O` 9 O . ^ohcfggk { O` }   kakekh pkrtcsc. Icske`kf hkipcrkf ktks 4 ψ f  , akf hkipcrkf mkwkh 4 ϟ f  . Roicechkf `oauk jufgsc cfc morgkftufg pkak f  (mkfyk`  (mkfyk` pkrtcsc). Akrc scfc, acaojcfcsc`kf4 f ` I  f  ψ f(x ) 9 I χ O` ( x ) 9 ` χ O` ( x )   f f  ` 9∝f ` 9∝f   akf f ( ` ∝ ;) I  f  ( ` ∝ ;)χ O` ( x )   I χ O` ( x ) 9 ϟf ( x ) 9 f f  ` 9∝f ` 9∝ f    Qorechkt ψ f ( x ) ≯ j  akf ϟ f ( x ) ≡ j  . @krofk ψ f ( x ) ≯ j  , ik`k

∛

∛

∛

∠

∛

 I  f  ψf 9 `i( O` ) ≯ cfj  ψ   . . . . . . . . .(;) ψ  ≯ j  f  ` 9∝ f 

∛

∠

@krofk ϟ f ( x ) ≡ j  , ik`k

∠

 I  f  ( ` ∝ ;)i( O` ) ≡ sup ϟ   . . . . (=) ϟf 9 f  ` 9∝ f  ϟ ≡ j 

∛

∠

@krofk

∠

∠

sup ϟ ≡ cfj  ψ  ϟ ≡ j 

ik`k, akrc (;) akf (=) acporleoh4 ?:

ψ  ≯ j 

Mkm 1 „ Cftogrke Eomosguo

∠ϟ

f

Nlipceoa my 4 @hkorlfc, ^.^c

≡ sup ∠ ϟ ≡ cfj ∠ ψ ≡ ∠ψ f .   ψ  ≯ j 

ϟ ≡ j 

 I ⇑ 2 ≡ cfj ψ ∝ sup ϟ ≡ ψ f ∝ ϟf . 9  ψ ≯  j  f ϟ ≡ j 

∠

∠

∠

∠

f

I  f  ( ` ∝ ;)i( O` ) `i( O` ) ∝ f  ` 9 ∝ f   `9∝f

∛

∛

 I  f  .9 ( ` ∝ ( ` ∝ ;)) i( O` ) f  ` 9∝ f 

∛

 I  f  .9 i( O` ) f  ` 9∝ f   I  .9 i( O ), ∂f  f  Aofgkf iofgkimce ecict-fyk acporleoh su sup ϟ ∝ cfj ψ  9 2 .

∛

ϟ ≡ j 

∠

ψ  ≯ j 

∠

⇒

∠

∠

Ac`otkhuc sup ϟ 9 cfj   ψ  . K`kf acmu`tc`kf  j  jufgsc  jufgsc toru`ur. O`ucvkeof aofgkf ioimu`tc`kf ϟ ≡ j 

ψ  ≯ j 

kak jufgsc toru`ur ψ *  sohcfggk  j  9 ψ *  (ko).

∠

∠

@krofk sup ϟ 9 cfj   ψ   ik`k ∂f ∈   kak jufgsc soaorhkfk ϟ f   akf ψ f   sohcfggk4 ϟ ≡ j 

ψ  ≯ j 

;) ϟf ≡  j  ≡ ψ f  =)

;

∠ψ ∝ ∠ ϟ  5 f  f

f 

 Qorechkt ϟ f   fkc` akf ψ f   turuf. Acaojcfcsc`kf ϟ 9 sup ϟ f   akf ψ * 9 cfj ψ f  *

f 

f 

@krofk @krofk ϟ f  ≡ j , ∂f  ik`k ϟ * 9 sup ϟ f  ≡ j  @krofk  j ≡ ψ f , ∂f  ik`k  j  ≡ supψ f  9 ψ *  Dkac, ϟ * ≡  j  ≡ ψ *  aofgkf ϟ *  akf ψ *  jufgsc toru`ur.

(

)

(

)

^oekfdutfyk, k`kf acmu`tc`kf i {x 4 ϟ * (x ) ≢ ψ * ( x )} 9 i {x 4 ϟ * (x ) 5 ψ * ( x )}    9 2 . Icske,

∊ 9 {x 4 ϟ * ( x ) 5 ψ * ( x )} @krofk ϟ * ( x ) 5 ψ * ( x )  ik`k kak mcekfgkf v ∈   sohcfggk ϟ * ( x ) 5 ψ * ( x ) ∝

; v 

Icske ;⎠ ⎧ ∊v  9 ⎨x 4 ϟ * ( x ) 5 ψ * ( x ) ∝ ⎥ v ⎩ ⎣ akf

∊ 9 ∤ ∊v  v ∈

 Qorechkt ;⎠ ⎧ ;⎠ ⎧ ∊v 9 ⎨x 4 ϟ * ( x ) 5 ψ * ( x ) ∝ ⎥ ⊍ ⎨x 4 ϟf ( x ) 5 ψ f ( x ) ∝  ⎥ 9 ∊v* v⎩ ⎣ v ⎩ ⎣ akf

∠

∊v*

ψ f ∝ ϟ f 6

 Dkac, ?7

∠

∊v *

; ; 9 i ( ∊v * ) v v 

 

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

; i ( ∊v* ) 5 v

∠

∠

ψ f ∝ ϟf ≡ *

∊v 

ψf ∝

 O

∠

O

ϟ f 5 

;   f 

v  f 

⇝ i ( ∊v * ) 5 , ∂f  ⇝ i ( ∊v * ) 9 2 ⇝ i ( ∊ ) 9 2  Aojcfcsc (Cftogrke Jufgsc Qormktks) 4 Icske`kf  j  kakekh   kakekh jufgsc toru`ur akf tormktks ykfg toraojcfcsc pkak hcipufkf  O ykfg toru`ur aofgkf i( O ) 5 ∖ . Acaojcfcsc`kf Cftogrke Eomosguo akrc  j  pkak  pkak O somkgkc morc`ut4

∠

 O

 j ( x ) a ( x ) 9 cfj

ψ ≯ j 

∠ ψ ( x ) ax  O

acikfk ψ  kakekh jufgsc soaorhkfk. Aojcfcsc ac ktks mcsk actuecs Fltksc4 ;)  j ( x ) a ( x ) 9

∠

 O

∠

O

∠

 O

 j ( x ) a ( x ) 9 sup ϟ ≡ j 

∠ ϟ ( x ) ax  . O

j 

=)  Dc`k O 9 P k   Y ik`k k,  m 

∠

 O

 j 9

∠

m 

k 

j 

>)  Dc`k  j  kakekh   kakekh jufgsc ykfg toru`ur akf tormktks sortk  j  morfcekc   morfcekc fle ac eukr hcipufkf  O ykfg toru` oru`u ur aof aofgkf gkf i( O ) 5 ∖ , ik`k  j 9 j 

∠

∠

 O

1)

∠

 O

 j 9

 χ  ∠ j  χ 

 O

Humufgkf kftkrk Cftogrke ykfg acaojcfcsc`kf ac ktks aofgkf Cftogrke ]coikff ykfg acaojcfcsc`kf somoeuifyk acmorc`kf akeki prlplscsc morc`ut4 Rrlplscsc 4 Icske`kf  j  kakekh  kakekh jufgsc tormktks ykfg toraojcfcsc pkak P k   Y. Dc`k  j  torcftogrke`kf  torcftogrke`kf ]coikff pkak k,  m  P k   Y ik`k j toru`ur akf k,  m 

]

∠

m

k

j (x ) a (x ) 9

∠

m 

j ( x ) a ( x ) 

k 

Mu`tc 4 Icske`kf  K 9 `leo`sc jufgsc tkfggk M 9 `leo`sc jufgsc soaorhkfk @krofk j  torcftogrke(`kf)  torcftogrke(`kf) ]coikff, ik`k

∠

sup ϟ 9 ] ϟ ∈ K ϟ ≡ j 

@krofk  K ⊍ M  ik`k,

∠

m

k

j 9]

∠

m 

k 

∠

j  9 cfj ϟ    ϟ ∈ K ϟ ≡ j 

cfj  K ≯ cfj M  sup K ≡ sup M

 Dkac, ]

∠

m

k

∠

∠

∠

∠

j 9 sup ϟ ≡ sup ϟ ≡ cfj ϟ ≡ cfj  ϟ 9 ]  ϟ∈ K ϟ ≡ j

ϟ ∈M ϟ ≡ j 

@oscipuekf4

?8

ϟ∈M ϟ ≡ j

ϟ ∈ K ϟ ≡ j 

∠

m 

k 

j

 

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

∠

∠

sup ϟ 9 cfj   ϟ  , iofurut prlplscsc somoeuifyk  j  toru`ur  toru`ur ϟ ∈M ϟ ≡ j 

ϟ ∈M ϟ ≡ j 

akf,

∠

m

k

∠

 j 9 sup ϟ  9 ] ϟ ∈M ϟ ≡ j 

∠

m 

k 

j 

 ^cjkt Cftogrke Jufgsc Qoru`ur akf Qormktks ^cjkt-scjkt jufgsc toru`ur akf tormktks acmorc`kf akeki > prlplscsc morc`ut4 Rrlplscsc 4 Icske`kf j  akf  akf g  kakekh  kakekh jufgsc toru`ur, tormktks akf toraojcfcsc pkak hcipufkf toru`ur  O aofgkf i( O ) 5 ∖ , ik`k4

;.

mg ) 9 k ∠ ∠ ( kj + mg  O

∠

j +m

O

∠

=.  Dc`k j  9  9 g  (ko)  (ko) ik`k

 O

>.  Dc`k  j ≡ g  (ko) ik`k

O

g 

 j 9

∠

 O

∠

O

 j ≡

g 

∠

O

g  akf k`cmktfyk

1.  Dc`k  K ≡ j ( x ) ≡ M  (ko) ik`k  Ki( O ) ≡

∠

 O

∠

 O

 j ≡

∠

j 

O

j ≡ Mi( O )

?.  Dc`k  K  akf M  kakekh hcipufkf toru`ur akf  K ∣ M 9 ∏ , aofgkf i( K), i( M ) 5 ∖   ik`k

∠

 K ∤ M

 j 9

∠

K

j +

∠

M

j 

Mu`tc 4 ;.  K`kf acmu`tc`kf mkhwk4 c) kj 9 k j  

∠ ∠

∠

 O

cc)

 O

O

 j + g 9

Rortkik,

∠

 O

∠

O

j +

∠

O

g 

∠

 j  9 cfj  ϟ   aofgkf ϟ   jufgsc soaorhkfk. ϟ ≯ j 

 Dc`k k  6  6 2 ik`k

∠

kj 9 cfj

kϟ ≯ kj

∠ kϟ 9 cfj k ∠ ϟ 9 k cfj ∠ ϟ   9 k ∠

j

 

∠

kj 9 cfj

∠ kϟ 9 cfj k ∠ ϟ 9 k sup ∠ ϟ  9 k ∠

j

 

 O

kϟ ≯ kj

ϟ ≯ j

O

 

 Dc`k k  5  5 2 ik`k  O

kϟ ≯ kj

ϟ ≡ j

 

O

ϟ ≡ j 

@oauk, icske`kf  jufgsc soaorhkfk 4 ϟ  ≡ j  akf  akf ψ  ≡ g }  K 9 {ϟ , ψ  jufgsc  jufgsc soaorhkfk 4 j  +  + g ≡ ϟ } M 9 {ϟ  jufgsc Ackimce ϟ ,ψ  ∈  ∈ K  sohcfggk  j + g  ≡ ϟ +ψ  . Iofggufk`kf scjkt cfjciui akf scjkt cftogrke jufgsc soaorhkfk acporleoh  j + g 9 cfj ϟ ≡ ϟ + ψ 9 ϟ + ψ ≡ cfj ϟ + cfj ψ  9  j + g  …… (*)

∠

ϟ ≯ j + g ϟ∈M

 O

∠

∠

∠

∠

ϟ≯ j ϟ∈ K

∠

ψ ≯ j  ψ ∈ K

∠

∠

∠

O

O

@oiuackf, icske`kf  9 {ϟ , ψ  jufgsc  jufgsc soaorhkfk 4 ϟ  ≯ j  akf  akf ψ  ≯ g } N  9  9 {ϟ  jufgsc  jufgsc soaorhkfk 4 j  +  + g ≯ ϟ } A  9 Ackimce ϟ ,ψ  ∈  ∈ N   sohcfggk  j + g  ≯ ϟ +ψ  . Iofggufk`kf scjkt suproiui akf scjkt cftogrke jufgsc soaorhkfk acporleoh  j + g 9 sup ϟ ≯ ϟ + ψ 9 ϟ + ψ ≯ sup ϟ + sup ψ 9  j + g  …… (**)

∠

 O

ϟ ≡ j + g ϟ∈A

∠

∠

∠

∠

ϟ≡ j ϟ∈N

?3

∠

ψ ≡ j  ψ ∈N 

∠

∠

O

∠

O

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

Akrc (*) akf (**) acporleoh

∠

 j + g 9

∠

j +

Akrc hksce portkik akf `oauk cfc, acporleoh kj + mg 9 kj +

∠

mg 9 k

 O

∠

∠

 O

O

O

O

∠

g 

O

∠

O

j +m

∠

O

g 

=. @krofk j  9  9 g  (ko)  (ko) ik`k j „ g  9  9 2 (ko) Icske`kf ψ   jufgsc soaorhkfk aofgkf ψ  ≯  j  „  g . @krofk  j  „  g   9 2 (ko) ik`k ψ  ≯  2 (ko). @krofk ψ  ≯ 2, ik`k ψ  ≯ 2 . Leoh `krofk ctu,

∠

∠

 j ∝ g  9 cfj

ψ ≯ j ∝ g 

 O

∠ψ  ≯ 2

@oiuackf, icske`kf ϟ  jufgsc  jufgsc soaorhkfk aofgkf ϟ  ≡ j  „  „ g . @krofk j  „  „ g  9  9 2, ik`k ϟ  ≡ 2 (ko). @krofk ϟ  ≡ 2 ik`k ψ  ≡ 2 . Leoh `krofk ctu,

∠

∠

 O

∠

 j ∝ g  9 sup ψ  ≡ 2 ϟ ≡ j ∝ g 

Akrc scfc ik`k,

∠

 O

 j ∝ g  9 2

Aofgkf iofggufk`kf hksce pkak mkgckf ;, acporleoh  j ∝ g 9 2 ⇑ j ∝ g 9 2 ⇑

∠

∠

 O

∠

O

O

∠

O

j 9

∠

g 

O

>. Ac`otkhuc  j ≡ g  (ko) ik`k  j ∝ g  ≡ 2  (ko). Icske`kf ϟ  jufgsc soaorhkfk aofgkf ϟ  ≡  ≡  j ∝ g  ≡ 2 ik`k ϟ  ≡  ≡ 2 . @krofk ϟ  ≡  ≡ 2   ik`k

∠ ϟ  ≡ ≡ 2 . Iofurut aojcfcsc  O

∠

 O

@krofk

 j ∝ g  9 sup ϟ ≡ j ∝ g 

∠ ϟ  O

∠ ϟ  ≡ ≡ 2 ik`k mogctu dugk aofgkf suproiui-fyk akf aofgkf iofggufk`kf hksce  O

pkak mkgckf ;, acporleoh

∠

 O

 j ∝ g ≡ 2 ⇑

∠

@oiuackf, k`kf acmu`tc`kf mkhwk

 O

∠

O

j ∝

 j ≡

∠

O

∠

O

g ≡2⇑

∠

O

j ≡

∠

g 

O

j   ktku o`ucvkeof aofgkf iofufdu``kf

∝ ∠  j ≡ ∠ j ≡ ∠ j   O

O

O

Akrc `ofyktkkf mkhwk  j ≡ j   akf ∝  j ≡ j  ik`k iofurut hksce somoeuifyk akf mu`tc pkak mkgckf ; acporleoh  j ≡ j   akf ∝  j ≡ j ⇑ ∝ j ≡ j 

∠

∠

 O

∠

O

 O

∠

∠

O

O

∠

O

1. @krofk  K ≡ j ( x ) ≡ M  ik`k iofurut mu`tc pkak mkgckf >, acporleoh

∠  K ≡ ∠  O

O

j (x ) ≡

∠

O

M⇑ K

∠ ;≡ ∠ O

O

j (x ) ≡ M

∠

O

; ⇑ Ki ( O ) ≡

∠

O

j (x ) ≡ Mi ( O )

?. Rortkik, acmu`tc`kf mkhwk dc`k  K ∣ M 9 ∏  ik`k  χ K∤ M 9 χ K + χ M ⎧; x ∈ K ∤ M  χ  K∤ M ( x ) 9 ⎨ ⎣2 x ∃ K ∤ M @ksus C 4 _ftu` x ∈ K ∤ M ik`k  χ  K ∤M ( x ) 9 ; . @krofk x ∈ K ∤ M  ik`k x ∈ K  ktku x ∈ M Ac`otkhuc  K ∣ M 9 ∏ , sohcfggk dc`k x ∈ K  ik`k x ∃ M . K`cmktfyk4  χ K ( x ) + χ M ( x ) 9 ; + 2 9 ; 9 χ K ∤M ( x ) ^omkec`fyk, dc`k x ∃ K  ik`k x ∈ M . K`cmktfyk4 :2

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

 χ K ( x ) + χ M ( x ) 9 2 + ; 9 ; 9 χ K ∤M ( x ) @ksus CC 4 _ftu` x ∃ K ∤ M  ik`k  χ  K ∤M ( x ) 9 2 . @krofk x ∃ K ∤ M  ik`k x ∃ K  akf x ∃ M . K`cmktfyk4  χ K ( x ) + χ M ( x ) 9 2 + 2 9 2 9 χ K ∤M ( x )  Dkac tormu`tc  χ K ∤M 9 χ K + χ M . ^oekfdutfyk, acporhktc`kf mkhwk4

∠

∠ j . χ  9 ∠  j . ( χ + χ  ) 9 ∠  j  χ + j χ 

 j 9

 K ∤ M

 K ∤ M

 K

M

 K

M

Aofgkf iofggufk`kf hksce pkak mkgckf ;, acporleoh

∠

∠ j  χ + ∠ j χ  9 ∠  j + ∠ j 

 j 9

 K ∤ M

 K

M

 K

M

 Rrlplscsc (Qolroik @o`lfvorgofkf Qormktks) 4 Icske`kf { j f }  kakekh mkrcskf jufgsc toru`ur ykfg toraojcfcsc pkak hcipufkf toru`ur  O, aofgkf

i( O ) 5 ∖ . Icske`kf torakpkt  I   sohcfggk  j f ( x ) ≡ I , ∂f akf ∂x . Dc`k  j ( x ) 9 eci j f ( x ) , f ↝∖

∂x ∈ O,  ik`k

∠

 O

 j 9 eci

f ↝∖

∠

O

j f 

Mu`tc 4 Ackimce ο  6 2  somkrkfg. Iofurut prcfscp Ectteowlla , uftu` 2 5 ο ; ≡

ο 

 akf 2 5 α  ≡

= i( O )

ο 

1 I 

 torakpkt hcipufkf toru`ur

 ∈   sohcfggk  K ⊍ O  akf  F  ∈ ;. i( K ) 5 α  =.  j f ( x ) ∝ j ( x ) 5 ο ; , ∂f ≯ F , ∂x ∈ K n  Akrc scfc, ik`k

∠

 O

 j f ∝

∠

O

∠ j ≡ ∠  j 9 ∠  j

j 9

O

 O

 K

f 

∝ j 

f 

∝ j , O 9 K ∤ K n 

f

∝ j +∠

Kn 

j f  ∝ j 

..........................................(;)

@krofk  j ( x ) 9 eci j f ( x )   akf  j f ( x ) ≡ I , ∂f akf ∂x  ik`k  j ( x ) ≡ I . ^ohcfggk ∂f ∈  f ↝∖

morek`u,  j f ∝ j ≡ j f  + j ≡ I + I 9 = I 

...... ......... ...... ...... ...... ...... .......... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...(( =)  Dkac, akrc (;), (=), (=), ;, akf = uftu` sotckp f ≯ F  morek`u

∠

 O

 j f  ∝

∠

O

j ≡

∠

K

=I +

∠

ο

+

5 = I

1 I

Kn 

ο ; 9 = I .i ( K ) + ο ; .i( K n  ) 5 = Iα + ο ; .i( O  ) ο

= i( O )

i( O ) 9

ο

=

+

ο

=

 

9 ο  

Rrlplscsc 4 Jufgsc j  tormktks  tormktks pkak P k   Y, torcftogrke ]coikff ]coikff ⇑ i ({x 4 x tctc` ctc` acs`l cs`lft ftcf cfu u j } )  9 2 k,  m 

:;

Mkm 1 „ Cftogrke Eomosguo

1.>.

Nlipceoa my 4 @hkorlfc, ^.^c

Cftogrke Jufgsc Qk` Fogktcj Aojcfcsc (Cftogrke Jufgsc tk` Fogktcj) 4 Icske`kf j  jufgsc  jufgsc toru`ur tk` fogktcj ykfg toraojcfcsc pkak hcipufkf toru`ur  O. Acaojcfcsc`kf

∠

 O

 j 9 sup h ≡ j 

∠

h 

O

aofgkf h  jufgsc  jufgsc toru`ur akf tormktks sohcfggk i ( {x 4 h( x )  ≢ 2} ) 5 ∖ . ^cjkt Cftogrke Jufgsc tk` Fogktcj ^cjkt-scjkt cftogrke jufgsc tk` fogktcj acmorc`kf akeki prlplscsc-prlplscsc akf eoiik morc`ut4 Rrlplscsc 4 Icske`kf j  akf  akf g  kakekh  kakekh jufgsc toru`ur tk` fogktcj, ik`k

;. =.

∠ ∠

 O  O

nj 9 n

∠

O

 j + g 9

j , n  6 2

∠

O

j +

∠

O

g 

>.  Dc`k  j ≡ g  (ko) ik`k

∠

 O

 j ≡

∠

O

g 

Mu`tc 4 ;. Icske`kf h   jufgsc toru`ur akf tormktks sohcfggk i ( {x 4 h( x )  ≢ 2} ) 5 ∖ . Iofurut aojcfcsc,

∠

 O

nj 9 sup nh ≡ nj

∠

O

nh 9 sup n h≡ j

∠

O

h 9 n sup

∠

h≡ j  

O

∠

h 9n

j  

O

=. Ackimce h  akf   akf ` jufgsc toru`ur akf tormktks sohcfggk h ≡ j   akf ` ≡ g . Akrc scfc acporleoh  + ` dugk iorupk`kf jufgsc toru`ur akf tormktks pkak  O. ^ohcfggk, h + ` ≡ j + g  akf   h  +

∠

 O

∠

h + ` ≡ sup

h + `≡ j + g  

O

h +` 9

∠

O

j +g  

⇑ sup h≡ j

∠

 O

h + sup `≡ g

∠  

O

∠

`≡

O

h+

∠

∠

j + g 

O

`≡

∠

O

j +g  

⇑

∠

 j +

 O

∠

g≡

O

O

^oekfdutfyk, ackimce e  jufgsc  jufgsc toru`ur akf tormktks pkak  O aofgkf i ({x 4 e ( x )  ≢ 2} ) 5 ∖  akf e ≡ j + g  Acaojcfcsc`kf jufgsc h  akf  akf ` aofgkf kturkf h 9 icf( j , e )  akf ` 9 e ∝ h  Acporleoh, h ≡ j   akf h ≡ e  . @krofk e  toru`ur   toru`ur akf tormktks akf h ≡ e   ik`k h  toru`ur   toru`ur akf tormktks. _ftu` x ∈ O  somkrkfg.  Dc`k  j ( x ) ≡ e ( x ) ik`k h( x ) 9 j ( x ) . Dkac, `( x ) 9 e ( x ) ∝ h ( x ) 9 e ( x ) ∝ j ( x ) ≡ j ( x ) + g ( x ) ∝ j ( x ) ≡ g ( x )   Dc`k  j ( x ) 6 e ( x ) ik`k h( x ) 9 e ( x )  . Dkac, `( x ) 9 e ( x ) ∝ h( x ) 9 e ( x ) ∝ e ( x ) 9 2 ≡ g ( x )  ^ohcfggk ` jufgsc toru`ur akf tormktks. K`cmktfyk e 9 h + ` 9 h + ` ≡ sup h + sup ` 9 j + g  

∠

 O

∠

∠

O

∠

O

O

h≡ j

∠

∠

O

`≡ g  

∠

O

∠

O

O

akf sup e ≡ j + g 

∠

 O

e≡

∠

O

e ≡

∠

O

j +

:=

∠

O

g⇑

∠

O

j +g≡

∠

O

j +

∠

O

g  

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

Akrc scfc ik`k

∠

 j + g 9

 O

∠

j +

O

∠

g 

O

>. Icske  jufgsc toru`ur akf tormktks  K 9 {h 4 h ≡ j } , h  jufgsc M 9 {h 4 h ≡ g }  @krofk

h ∈ K ⇝ h ≡ j ≡ g ⇝ h ≡ g ⇝ h ∈M

ik`k  K ⊍ M . Leoh `krofk ctu,

∠

 O

∠

h≡

∠

g 9 sup

∠

 j ≡

∠

g 

 j 9 sup

O

h≡ j h∈ K

O

h≡ j h∈M

∠

O  

h 

 Dkac,  O

O

 Eoiik Jktlu 4 Icske`kf {j f }  ka kakekh mkrcskf jufgsc toru`ur tk` fogktcj akf eci eci  j f ( x ) 9 j ( x )  hkipcr acikfkf ↝∖

ikfk ac O, ik`k

∠

 j ≡ eci

 O

∠

O

j f 

Mu`tc 4 Ackimce h , jufgsc toru`ur akf tormktks pkak  O somkrkfg sohcfggk h ≡ j . ⎧ h( x ), j f ( x ) ≯ h( x )  hf ( x ) 9 ⎨ ⎣ j f ( x ), j f ( x ) 5 h( x ) Akrc aojcfcsc cfc, `krofk h  akf  akf j f  toru`ur ∂f  ik`k  ik`k h f  toru`ur ∂f . Dugk acporleoh mkhwk akf eci hf  9 h  hf  ≡ h , ∂f  hf  ≡ j , ∂f  f ↝∖

@krofk h f   tormktks leoh h , ik`k h f   tormktks sorkgki. @krofk h f   tormktks sorkgki leoh h , akf eci hf  9 h  pkak O ik`k ( Qolroik Qolroik @o`lfvorgofkf Qormktks  ) f ↝∖

∠

 O

@krofk hf ≡ j f , ∂f  ik`k

∠

 O

hf ≡

∠

O

h9

∠

eci hf 9 eci

O f ↝∖

∠

f ↝ ∖ O

j f , ∂f . Akrc scfc, ∂f ,

h f 

∠

 O

h 9 eci

f ↝∖

∠

O

hf 9 eci

∠

O

hf ≡ eci

∠

O

j f   

^ohcfggk,

∠

 O

 j 9 sup h ≡ j 

∠

O

h 9 ec i

∠

O

h f ≡ ec i

∠

O

j f 

  Qolroik (@o`lfvorgofkf (@o`lfvorgofkf Ilfltlf)4 Ilfltlf)4 Icske`kf { j f } mkrcskf jufgsc to toru`ur tk` fogktcj yk ykfg ilfltlf fkc` ak akf eci  j f ( x ) 9 j ( x )  (k.o) f ↝∖

ik`k

∠  j

9 eci ∠ j f 

Mu`tc 4 Ac`otkhuc { j f } mkrcskf jufgsc toru`ur,  j f  ≯ 2, ∂f ,  j f   j    pkak  O.  O. K`kf acmu`tc`kf

∠  j

9 eci ∠ j f  .

Iofggufk`kf Eoiik Jktlu acporleoh

∠  j :>

≡ eci ∠ j f 

Mkm 1 „ Cftogrke Eomosguo

∠

Nlipceoa my 4 @hkorlfc, ^.^c

∠

∠

@krofk eci  j f ≡ eci j f  , ik`k nu`up acmu`tc`kf eci  j f  ≡

∠ j 

@krofk @krofk  j f  ≡ j , ∂f  ik`k

∠  j ≡ ∠ j

⇑ eci ∠ j f  ≡ ∠ j 

f

  K`cmkt 4 Icske`kf {u  {u f } mkrcskf jufgsc toru`ur tk` fogktcj. Icske`kf ∖

∛ u 

 j 9

f 

f 9;

ik`k ∖

∠  j 9 ∛ ∠ u 

f 

f 9;

Mu`tc 4 Acaojcfcsc`kf f 

∛ u 

 j f 9

c 

c 9;

 Qorechkt { j f f } mkrcskf fkc` ilfltlf tk` fogktcj. Akrc scfc ik`k, eci  j f 9 eci

f ↝∖

f ↝ ∖

∖

f 

∛u 9 ∛u c 9;

c

c 9;

c 

9 j 

 Dkac, eci  j f  9 j  . Morek`u tolroik `o`lfvorgofkf ilfltlf4 f ↝∖

∠  j 9 ∠ eci j f ↝∖

f

f

f 

∖

  c 9;

c 9;

9 eci ∠ j f 9 eci ∠ ∛ u c 9 eci ∛ ∠ uc 9∛ ∠ u c f ↝∖

f ↝∖

f ↝∖

c 9;

 

 Rrlplscsc 4 Icske`kf  j  jufgsc   jufgsc tk` fogktcj akf 5 Oc 6 mkrcskf hcipufkf toru`ur ykfg skecfg kscfg. Icske`kf  O 9 ∤ Oc  . Ik`k

∠

 O

Mu`tc 4 Ackimce

 j 9

∛∠

Oc 

j 

u c 9 j .χ Oc 

ik`k

∛u

c

9 j .χ O

Aofgkf iofggufk`kf hksce pkak k`cmkt ac ktks, acporleoh

∠

 O

 j 9

∠ j . χ

 O

∖

∖

9 ∛ ∠ j .χ O 9 ∛ ∠ j  c 9;

c 

c 9;

Oc 

 Aojcfcsc 4 Jufgsc toru`ur tk` fogktcj j  fogktcj  j  acsomut  acsomut torcftogrke`kf pkak hcipufkf toru`ur O toru`ur  O dc`k  dc`k

∠

 O

 j  5 ∖ .

^cjkt-scjkt jufgsc torcftogrke`kf acmorc`kf pkak auk prlplscsc morc`ut Rrlplscsc 4 Icske`kf  j   akf  g   auk jufgsc toru`ur tk` fogktcj. Dc`k  j   torcftogrke`kf pkak hcipufkf  O   O  akf  pkak O,, ik`k g  ik`k g  dugk  dugk torcftogrke`kf pkak O pkak  O,, akf  g ( x ) 5 j ( x )  pkak O :1

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

∠

 O

Mu`tc 4 Ac`otkhuc j  Ac`otkhuc j  torcftogrke`kf  torcftogrke`kf ik`k  j 9

∠

∠

 O

∠

j ∝

O

∠ ( j ∝ g)+ g 9 ∠

 O

 Dkac,

 j ∝ g 9

O

∠

O

g 

j ∝g+

O

∠

O

g  5 ∖

morkrtc g  torcftogrke`kf  torcftogrke`kf pkak O pkak  O.. ∠  g  5 5 ∖ , ykfg morkrtc g 

 j ∝ g  5 ∖ . K`cmktfyk,

 O

 Dugk,

∠

 O

 j 9

∠

O

j ∝g+

∠

g⇑

O

∠

O

j ∝

∠

O

g9

∠

O

j ∝ g 

 Rrlplscsc 4 Icske`kf  j  kakekh   kakekh jufgsc tk` fogktcj ykfg torcftogrke`kf pkak  O.  O. Ik`k uftu` somkrkfg ο  6 2 , torakpkt α  6 2  sohcfggk uftu` sotckp  K ⊍ O aofgkf i( K ) 5 α   morek`u

∠

 K

 j  5 ο 

Mu`tc 4 Ackimce somkrkfg ο  6 2 .  Dc`k j   Dc`k j tormktks tormktks akf  j  ≯ 2  ik`k kak I  kak I  6  6 2 sohcfggk  j ≡ I  . ο  Rcech α  6 2 aofgkf α  5  I   sohcfggk uftu` sotckp  K ⊍ O aofgkf i( K ) 5 α   morek`u

∠

 K

 j ≡

∠

K

ο  I 9 I .i ( K ) 5 I α 5 I  I  5 ο  .

 Dc`k j   Dc`k j  tcak`  tcak` tormktks. Acaojcfcsc`kf

 j f  9 icf( j , f ), ∂f ∈  Aofgkf pofaojcfcsckf cfc acporleoh mkhwk  j f  fkc` akf `lfvorgof `o  j ,  j f  ≡ f , ∂f ∈  , soekcf ctu  j f  ≯ 2 akf  j  ≯ 2 . Akrc scfc s cfc ik`k, (tolroik `o`lfvorgofkf ilfltlf)

∠

 O

@krofk

∠

 O

 j 9 eci

f ↝∖

∠

O

 j 9 eci

f ↝∖

∠

O

j f 

j f  ik`k uftu` ο  6 2  ac ktks, kak  F  ∈  ∈   sohcfggk uftu` f ≯ F  morek`u4

∠  j ∝ ∠  O

f 

O

j  5 ο = .

 Dc`k ackimce f  9 F   9 F ,

∠

 K

Ackimce 2 5 α  5

∠

 K

∠ 9∠

 j 9

K

 K

ο  = F 

 j F ∝

∠

K

j 9

∠

K

j ∝

∠

K

j F 5 ο= ⇝

∠

K

j ∝

∠

K

j F  5 ο =

 sohcfggk ∂ K ⊍ O aofgkf i( K ) 5 α ,  ik`k

j ∝ j F + j F   j ∝ j F +

5 ο = + ∠  F  K

∠

K

j F 

( ↛ tormktks leoh F )

5 ο = + F .i ( K ) ο  5 ο= + F . = F  5 ο= + ο = 9 ο 

 Nlftlh ^lke (Rrlmeoi 1.:) Icske`kf 5 j f 6 kakekh mkrcskf jufgsc toru`ur ykfg tk` fogktcj ykfg `lfvorgof `o  j , akf icske`kf  uftu` sotckp mcekfgkf ksec f . Mu`tc`kf mkhwk  j f  ≡ j  uftu`

∠  j

9 eci ∠ j f 

:?

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

 Dkwkm 4 Ackimce 5 j f 6 mkrcskf jufgsc toru`ur akf tk` fogktcj ykfg `lfvorgof `o  j   akf uftu` sotckp mcekfgkf ksec f , morek`u j  morek`u j f  ≡ j . Aofgkf iofggufk`kf eoiik Jktlu acporleoh (;)  j ≡ eci j f ≡ eci j f   

∠

∠

∠

@krofk j  @krofk  j f f  tk` fogktcj, akf j  akf  j f f  ≡  j   ik`k j  ik`k  j   tk` fogktcj. fogktcj. @krofk eci  j f  9 j   akf j   akf  j f f toru`ur uftu`   f ↝∖

sotckp f , ik`k j  ik`k j  toru`ur.  toru`ur. K`cmktfyk, aofgkf iofggufk`kf prlplscsc 8, `krofk  j f  ≡ j  ik`k  ik`k  j f  ≡ j 

∠

∠

^ohcfggk aofgkf iofgkimce ecict suporclrfyk acporleoh eci  j f  ≡ eci j ≡ j   

∠

∠

Akrc (;) akf (=) acscipue`kf eci  j f ≡ eci j f ≡

∠

∠

(=)

∠

∠ j ≡ eci ∠ j

∠

∠j

f

≡ eci ∠ j f

 

 K`cmktfyk eci  j f 9

9 eci ∠ j f 

 Dkac,

∠  j

9 eci ∠ j f  

1.1.

Cftogrke Eomosguo (Goforke (Goforke Cftogrke Eomosguo )

Icske`kf j  Icske`kf j  suktu  suktu jufgsc morfcekc roke. Acaojcfcsc`kf  j + ( x ) 9 ikx( 2, j ( x )) akf  j ∝ ( x ) 9 ikx( 2, ∝ j ( x )) Ik`k,  j + , j  ∝ ≯ 2 ,  j 9 j + ∝ j ∝ , akf  j 9 j + + j ∝ . ^ohcfggk acporleoh | j | + j  | j | ∝ j   akf  j  ∝ 9 .  j  + 9 = =  Dc`k j  jufgsc  jufgsc toru`ur ik`k  j + akf j  „  dugk toru`ur. Aojcfcsc 4 Jufgsc toru`ur  j   acsomut torcftogrke`kf pkak  O  dc`k  j +  akf  j  „   torcftogrke`kf pkak  O  akf acaojcfcsc`kf

∠

 O

 j 9

∠

j+∝

O

∠

O

j  ∝

Roreu acnktkt mkhwk, jufgsc  j   toru`ur akf torcftogrke`kf dc`k akf hkfyk dc`k  j +  akf  j  „  torcftogrke`kf. Dkac  j  torcftogrke`kf  torcftogrke`kf dc`k akf hkfyk dc`k  j  + 5 ∖  akf  j  ∝ 5 ∖ .

∠

∠

 O

 O

^cjkt-scjkt Cftogrke Eomosguo acmorc`kf pkak prlplscsc akf tolroik-tolroik morc`ut4 Rrlplscsc 4 Icske`kf j  akf  akf g  kakekh  kakekh jufgsc torcftogrke`kf pkak  O, ik`k4

;. Jufgsc nj  torcftogrke`kf  torcftogrke`kf pkak  O uftu` sotckp mcekfgkf roke n  akf  akf =. Jufgsc j  +  + g  torcftogrke`kf  torcftogrke`kf pkak O akf >.  Dc`k  j ≡ g  (ko) ik`k

∠

 O

 j ≡

∠

O

∠

 O

g  ::

 j + g 9

∠

O

j +

∠

O

g 

∠

 O

nj 9 n

∠

O

j 

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

1.  Dc`k  K  akf M  kakekh hcipufkf toru`ur akf  K ∣ M 9 ∏ , aofgkf  K, M ⊍ O   ik`k

∠

 K ∤ M

 j 9

∠

K

j +

∠

M

j 

Mu`tc 4 ;.  Dc`k n  6  6 2 ik`k ( nj )+ 9 ikx( 2, nj ) 9 n . ikx( 2, j ) 9 nj +   ( nj )∝ 9 ikx( 2, ∝nj ) 9 n . ikx( 2, ∝ j ) 9 nj ∝   Iofurut aojcfcsc nj 9 ( nj )+ ∝ ( nj  )∝ .

 Dkac

∠

 O

nj 9

∠ (nj ) ∝ ∠ (nj ) 9 ∠ +

∝

O

O

O

nj + ∝

∠

O

nj

∝

9 n ∠ j + ∝n ∠ j ∝ 9 n O

O

(∠

O

j+∝

∠

O

j

∝

) 9n ∠

O

j  

 Dc`k n  5  5 2 ik`k ( nj )+ 9 ikx( 2, nj ) 9 ∝n . ikx( 2, ∝ j ) 9 ∝nj ∝   ( nj )∝ 9 ikx( 2, ∝nj ) 9 ∝n . ikx( 2, j ) 9 ∝nj +   Iofurut aojcfcsc

∠

 O

nj 9

∠ (nj ) ∝ ∠ +

O

O

∠

( nj )∝ 9

O

∝nj ∝ ∝ ∠ ∝nj + 9 ∝n ∠ j ∝ + n ∠ j + 9 n O

∠

@krofk j  torcftogrke`kf,  torcftogrke`kf, ik`k

 O

O

O

(∠

O

j

+

∝∠ j O

∝

) 9n ∠

O

j  

 j  5 ∖ . K`cmktfyk

∠

 O

nj 9 n

∠

O

j  5 ∖

ykfg morkrtc nj  torcftogrke`kf  torcftogrke`kf =. Iofurut aojcfcsc

∠

 O

+

+

∠ ( j + g ) + ∠ ( j + g ) +

 j + g 9

O

∝

O

+

Rkak akskrfyk ( j + g ) ≢ j + g  . ^ohcfggk poreu iofggufk`kf mkftukf mu`tc ykfg ekcf. Icske`kf  j 9 j ; ∝ j =  aofgkf  j ; , j = ≯ 2  ik`k  j 9 j ; ∝ j =  j + ∝ j ∝ 9 j ; ∝ j =  j + + j = 9 j ; + j  ∝

∠

 O

∠  j ∠  j

+

 O

∠ +∠ j 9∠ ∝∠ j 9∠ ∠  j 9 ∠

 j + + j = 9 =

O

+

 O

O

O

∝

O

O

 O

O

j ; + j  ∝

∠ ∝∠ ∝∠

j; + j; j;

j  ∝

O

j =

O

j =

O

 Dkac,

∠

 O

∠ ( j ∝ j ) + ( g ∝ g  ) 9 ∠ (  j + g ) ∝ ( j + g  ) 9 ∠ (  j + g ) ∝ ∠ ( j + g  ) 9 ∠  j + ∠ g ∝ ∠ j ∝ ∠ g  9 ∠  j ∝ ∠ j + ∠ g ∝ ∠ g  9 ∠  j + ∠ g  5 ∖

 j + g 9

+

∝

+

∝

+

+

∝

∝

+

+

O

 O

∝

 O

O

+

+

 O

O

+

:7

∝

O

∝

 O

 O

∝

O

O

∝

O

+

O

∝

O

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

 Dkac j  +  + g  torcftogrke`kf.  torcftogrke`kf. >. @krofk  j ≡ g  ik`k  g ∝ j  ≯ 2 . Aofgkf iofggufk`kf hksce pkak =) acporleoh

∠  g ∝ j 9 ∠  O

g∝

O

∠

O

j ≯2⇑

∠

O

g≯

∠

j 

O

1. @krofk  K ∣ M 9 ∏  ik`k  χ K ∤M 9 χ K + χ M . Dkac,

∠

 K ∤ M

 j 9

∠ j . χ

9 ∠ j .χ K + ∠ j .χ M 9 ∠ j + ∠ j 

 K ∤ M

K

M

  Qolroik (Qolroik @o`lfvorgofkf @o`lfvorgofkf Eomosguo) 4 Icske`kf  g  jufgsc   jufgsc torcftogrke`kf pkak  O  akf 5 j f 6 mkrcskf jufgsc toru`ur soaoic`ckf sohcfggk  j f  ≡ g   pkak O ∂f  akf eci  j f ( x ) 9 j ( x )  (ko) ac O. Ik`k f ↝∖

∠

 j 9 eci

 O

∠

O

j f 

Mu`tc 4 @krofk  j f  ≡ g   pkak O ∂f  ik`k  g  ≯  ≯ 2  akf ∝ ≡  j f  ≡ g  . Dkac,  g ∝ j f  ≯ 2 akf  j f  + g  ≯ 2 c)

 g ∝ j f  ≯ 2 , ∂f 

eci  g ∝ j f 9 g ∝ eci j f  9 g ∝ j 

f ↝∖

Eoiik Jktlu  j ∝

∠

 O

∠

O

g9

∠

O

f ↝ ∖

j ∝ g ≡ ec i

∠

O

g ∝ j f 9 eci

∠

O

g∝

∠

O

∠

jf 9

O

g ∝ eci

∠

O

j f 

 Dkac,

∝ ∠  j ≡ ∝eci ∠ j f ⇑ ∠ j ≯ eci ∠ j f   ………………….. (*)  O

O

O

O

cc)  j f  + g  ≯ 2 , ∂f  eci  j f + g 9 eci j f  + g 9 j + g  f ↝∖

f ↝ ∖

Eoiik Jktlu

∠

 O

 j +

∠

g9

∠

 j ≡ eci

O

∠

j + g ≡ eci

O

∠

O

j f + g 9 ec i

∠

j f  +

O

∠

O

g 

 Dkac,  O

∠

j f   ………………….. (**)

O

Akrc (*), (**), akf ec i

∠

 O

 j f ≡ eci

∠

O

j f 

acporleoh eci

∠

 O

 j f ≡ eci

∠

O

jf ≡

∠

O

j ≡ ec i

∠

O

j f ≡ ec i

∠

O

j f

 

 Dkac,

∠  j  O

9 eci ∠ j f 9 eci ∠ j f 9 eci ∠ j f  O

O

O

  Qolroik ac ktks iofsykrkt`kf mkhwk mkrcskf 5 j f 6 acalicfksc leoh jufgsc totkp  g   ykfg torcftogrke`kf. Qorfyktk, akrc poimu`tckf ac ktks sykrkt cfc tcak` mogctu acporeu`kf. Dc`k `ctk iofggkftc sotckp g  pkak  pkak mu`tc ac ktks aofgkf  g f  ik`k `ctk iofakpkt`kf poruiuikf akrc Qolroik @o`lfvorgofkf Eomosguo somkgkc morc`ut4  Qolroik (Goforke (Goforke Eomosguo Nlfvorgofno Qholroi )4 )4 Icske`kf 5 g f 6 mkrcskf jufgsc torcftogrke`kf ykfg `lfvorgof `o jufgsc torcftogrke`kf  g  hkipcr   hkipcr acikfk-ikfk. Icske`kf 5 j f 6 mkrcskf jufgsc toru`ur soaoic`ckf sohcfggk | j f | ≡ g f  uftu` sotckp f  akf 5 j f 6 `lfvorgof `o  j  hkipcr  hkipcr ac ikfk-ikfk. Dc`k

∠ g 9 eci ∠ g 

f 

ik`k :8

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

∠  j

9 eci ∠ j f 

@krofk | j f | ≡ g f  uftu` sotckp f  ik`k  ik`k „   morek`u4  g f  ≡ j f  ≡ g f  uftu` sotckp f . Akrc scfc uftu` sotckp f  morek`u4 (c)  g f  + j f  ≯ 2, akf (cc)  g f  „ j f  ≯ 2.  Doeks g f  + j f akf g f  „ j f  toru`ur akf eci ( f »  j f ) 9 eci g f » eci j f 9 g  » j  f ↝∖

f ↝∖

f ↝∖

Aofgkf iofggufk`kf Eoiik Jktlu akf scjkt-scjkt ecict suporclr akf cfjorclr acporleoh ( g + j ) ≡ eci ( g f + j f )

∠

∠

⇑

∠ g + ∠ j 9 ∠ ( g + j ) ≡ ececi ∠ ( g

+ j f ) ≡ ececi ∠ g f + eci ∠ j f 9 ∠ g + eci ∠ j f

f

 

⇑

∠  j

≡ eci ∠ j f 

akf

∠ ( g ∝ j ) ≡ eci ∠ ( g

f

∝ j f )

⇑

∠ g ∝ ∠ j 9 ∠ ( g ∝ j ) ≡ eci ∠ ( g

f

∝ j f ) ≡ eci ∠ g f + eci ∠ ( ∝ j f ) 9 ∠ g ∝ eci ∠ j f

 

⇑ ∝ ∠  j ≡ ∝eci ∠ j f ⇑ ∠ j ≯ eci ∠ j f  Akrc scfc acporleoh,

∠  j ≡ eci ∠ j

 akf eci ∠  j f ≡ eci ∠ j f  ≡ ∠ j  ≡ eci ∠ j f   ak

f

^ohcfggk,

∠

∠

eci  j f ≡ eci j f ≡

∠j

≡ eci ∠ j f ≡ eci ∠ j f

 

 K`cmktfyk,

∠  j

9 eci ∠ j f 9 eci ∠ j f 9 eci ∠ j f 

Nlftlh ^lke (Rrlmeoi 1.;2)  Qufdu``kf mkhwk dc`k dc`k j  torcftogrke`kf  torcftogrke`kf pkak  O, ik`k | j | dugk torcftogrke`kf akf

∠

 O

 j ≡

∠

j 

O

 Kpk`kh dugk morek`u morek`u somkec`fyk<  Dkwkm 4 @krofk j  torcftogrke`kf,  torcftogrke`kf, ik`k  j + akf j  „  dugk torcftogrke`kf. K`cmktfyk | j | 9  j + + j  „  torcftogrke`kf pkak O akf

∠  j  O

9

∠

O

j+∝j

∝

9

∠

O

j+∝

∠

O

j∝ ≡

Mkgkcikfk aofgkf somkec`fyk<  Dc`k | j | torcftogrke`kf pkak  O, ik`k

∠

 O

∠

O

j

+

 j + ≡

 „ 

+

∠

O

j

∝

9 ∠ j + + ∠ j ∝ 9 ∠ j + + j ∝ 9 ∠ j 

∠ | j |5 ∖ O +

O

akf

O

∠

 „ 

 O

+

akf j   j

∝

+

torcftogrke`kf

9 ikx( 2, j ) 9 j 9 j + + j  ∝ :3

∠

O

 j ∝ ≡ | j | 5 ∖ . ^ohcfggk  j +

akf j  dugk torcftogrke`kf pkak  O. K`cmktfyk j  9  9 j   „ j   torcftogrke`kf. Nkrk ekcf 4  j torcftogrke`kf ⇑ j Akrc aojcfcsc,

O

O

Mkm 1 „ Cftogrke Eomosguo

Nlipceoa my 4 @hkorlfc, ^.^c

akf ∝

 j

9 ikx( 2, ∝ j  ) 9 2

ik`k,

∠

 O

 j

+

9 ∠ j + + j ∝ 9 ∠ j + + ∠ j  ∝ 5 ∖ O

O

akf,

∠

 O

 Dkac,  j

+

akf j 

∝

∝

 j  9

torcftogrke`kf.

72

∠

O

2925∖

O

Mkm ?  Qurufkf akf Cftogrke Rkak mkgckf cfc `ctk ioikfakfg turufkf somkgkc cfvors akrc cftogrke. ^onkrk soaorhkfk, `ctk poreu iofdkwkm momorkpk portkfykkf morc`ut4 @kpkf`kh m 

∠  j '( x ) ax 9 j (m ) ∝ j ( k ) < k 

@kpkf`kf a 

∠

x 

 j ( y ) ay 9 j ( x ) < ax  k  Akrc tolrc cftogrke ]coikff toekh ac`otkhuc mkhwk humufgkf `oauk k`kf acpofuhc dc`k  j  `lftcfu  `lftcfu ac x . @ctk poreu iofufdu``kf mkhwk humufgkf cfc sonkrk uiui acpofuhc hkipcr acikfk-ikfk. ^ohcfggk turufkf iorupk`kf cfvors/`omkec`kf akrc cftogrke. Rortkfykkf portkik, dkuh eomch suect  wkekupuf iofggufk`kf iofggufk`kf Cftogrke Eomosguo, akf morfcekc mofkr mofkr hkfyk uftu` momorkpk momorkpk `oeks jufgsc. Rkak Cftogrke ]coikff, turufkf suktu jufgsc ac tctc` tortoftu iorupk`kf `oicrcfgkf (grkacof) gkrcs scfggufg ac tctc` torsomut. Rorhktc`kf ceustrksc morc`ut4 ( k  k  ` ( k+h  k+h 

( k„h  k„h 

e  k„h

_ftu` gkrcs e , i e  9

_ftu` gkrcs `, i ` 9

 j ( k + h ) ∝ j ( k ) h   j ( k ) ∝ j ( k ∝ h ) h 

k

k+h 

0 i gkrcs scfggufg ac k  9 eci

 j ( k + h ) ∝ j ( k )

h ↝ 2

0 i gkrcs scfggufg ac k  9 eci

h   j ( k ) ∝ j ( k ∝ h )

h ↝ 2

h 

?.;.

 Qurufkf Jufgsc Ilfltlf Ilfltlf Icske`kf ℛ 9 {C ;, . . .} kakekh `leo`sc cftorvke-cftorvke. @leo`sc ℛ acsomut soeciut Zctkec uftu`  O dc`k uftu` sotckp ο  6 2   akf kpkpuf x ∈ O , torakpkt cftorvke C  ∈  ∈ ℛ   soaoic`ckf sohcfggk x ∈ C    akf e ( C ) 5 ο  . Cftorvke-cftorvke cfc iufg`cf mu`k, tutup, ktku sotofgkh tutup. Skfg doeks, cftorvke torsomut tcak` mleoh hkfyk toracrc akrc sktu tctc`. Eoiik Zctkec 4  Dc`k ℛ kakekh kakekh soeciu soeciutt Zctkec Zctkec uftu` uftu` O aofgkf aofgkf i * ( O ) 5 ∖  ik`k uftu` sotckp

morhcfggk cftorvke ac ℛ ykfg skecfg eopks {C ;, C =, . . ., C  F } sohcfggk  F  ⎫ ⎡ i * ⎭ O ~ ∤ C f  ⎯ 5 ο  f 9; ⎮ ⎪

ο 

6 2  kak `leo`sc

Mkm ? „ Qurufkf akf Cftogrke

Nlipceoa my 4 @hkorlfc, ^.^c

Mu`tc4 (soo ‘]oke Kfkeyscs–, H.E. ]lyaof, > ra oa, pkgo 38„33) Akrc Eoiik ac ktks,

⎫

i * ⎭O ~

⎮

⎡ ⎐F ⎖ ⎐ F  ⎖ * ( ) * * C f ⎯ 5 ο ⇝ i O ∝ i ⎕ ∤ C f ⎜ 5 ο ⇑ i ⎕ ∤ C f  ⎜ 6 i * ( O ) ∝ ο  ∤ f 9; ⎪ ⎙ f 9; ⎢ ⎙ f 9; ⎢  F

^oekfdutfyk, akeki tudukf `ctk ioimkhks iofgofkc turufkf (aorcvktcj) akrc suktu jufgsc  j , `ctk poreu iofaojcfcsc`kf 1 iknki turufkf akrc jufgsc  j  ac  ac x  somkgkc  somkgkc morc`ut4  j ( x + h ) ∝ j ( x ) A + j ( x ) 9 eci+ h ↝ 2 h   j ( x + h ) ∝ j ( x ) A+ j ( x ) 9 eci h  h ↝ 2+  j ( x ) ∝ j ( x ∝ h ) A ∝ j ( x ) 9 eci+ h ↝ 2 h   j ( x ) ∝ j ( x ∝ h ) A∝ j ( x ) 9 eci h  h ↝ 2+ + Akrc pofaojcfcsckf ac ktks, doeks mkhwk A j ( x ) ≯ A+ j ( x )  akf A ∝ j ( x ) ≯ A∝ j ( x )  . Dc`k   ac`ktk`kf torturuf`kf ( acjjoroftckmeo   ) ac x  akf   akf `ctk A + j ( x ) 9 A+ j ( x ) 9 A ∝ j ( x ) 9 A∝ j ( x )  ≢ »∖  ik`k  j  ac`ktk`kf acjjoroftckmeo  iofuecs`kf  j– (  (x    )  somkgkc fcekc turufkf  j   ac tctc` x . Dc`k A + j ( x ) 9 A+ j ( x )    ik`k  j   ac`ktk`kf ioicec`c x turufkf `kfkf ( rcght-hkfa  ) ac x   akf iofuecs`kf  j– +( x  rcght-hkfa aorcvktos  x )  somkgkc fcekc turufkf `kfkf  j  ac x . Aoic`ckf dugk uftu` ykfg ekcf, actuecs`kf j–  „ ( x  turufkf `crc j  ac  ac x . x )  somkgkc fcekc turufkf Rrlplscsc 4  Dc`k  j  `lftcfu   `lftcfu pkak P k   Y akf skekh sktu turufkffyk tk` fogktcj pkak ( k   ) ik`k  j  kakekh   kakekh jufgsc k,  m  k,  m  tk` turuf pkak P k   Y k,  m  Mu`tc 4 Icske`kf j  `lftcfu  `lftcfu pkak P k   Y akf skekh sktu turufkffyk, turufkffyk, `ktk`kf A+ j ( x ) ≯ 2, ∂x ∈ ( k, m )  . k,  m  Akrc aojcfcsc,  j ( x + h ) ∝ j ( x ) j ( x + h ) ∝ j ( x ) 9 sup cfj A+ j ( x ) 9 eci 2 5h 5α   h h  h ↝ 2 + α  6 2 @krofk A+ j ( x ) ≯ 2, ∂x ∈ ( k , m )   ik`k  j ( x + h ) ∝ j ( x ) cfj ≯2 2 5h 5α   h  @krofk h  6  6 2, ik`k hkrusekh  j ( x + h ) ∝ j ( x ) ≯ 2  j ( x + h ) ≯ j ( x )

 Dkac, uftu` sotckp x ≡ x  +  + h  morek`u  morek`u

 j ( x ) ≡ j ( x + h )

Mu`tc ekcf4 Icske`kf  j   `lftcfu pkak P k   Y akf skekh sktu turufkffyk, `ktk`kf A + j ( x ) ≯ 2, ∂x ∈( k , m )  . k,  m  Ackimce somkrkfg x ∈ ( k , m ) . Akrc aojcfcsc,  j ( x + h ) ∝ j ( x ) A + j ( x ) 9 eci+ ≯2 h ↝ 2 h  Ackimce y  9  9 x  +  + h , acporleoh h 9 y ∝ x  akf dc`k h ↝ 2 + ik`k ∝ x 6 2 ⇑ y 6 x . Ik`k,

⎐

cfj ⎕ sup α  6 2

 j ( y ) ∝ j ( x ) ⎖

⎙ x 5 y 5 x +α   7=

 y ∝ x 

⎜≯2 ⎢

Mkm ? „ Qurufkf akf Cftogrke

 Dkac, uftu` sotckp

Nlipceoa my 4 @hkorlfc, ^.^c

x 5 y 5 x + α   sohcfggk  j ( y ) ∝ j ( x ) ≯ 2 ⇝  j ( y ) ∝ j ( x ) ≯ 2 ⇝ j ( x ) ≡ j ( y )  y ∝ x 

α   6 2  kak

Nlftlh ^lke (Rrlmeoi ?.;) 4 Icske`kf  j jufgsc ykfg acaojcfcsc`kf aofgkf  j (2) (2) 9 2 akf  j (  ( x   ) uftu` x ≢ 2. x )  9 x scf(;/x  +  „   Qoftu`kf A  j (2), (2), A + j (2), (2), A  j (2), (2), A  „  j (2) (2)  Dkwkm 4 Rortkik, poreu actufdu``kf mkhwk ecict-fyk kak. Acgufk`kf prcfscp kpct4 x ↝ 2 + , x  ≢ 2

∝; ≡ scf ( x ; ) ≡ ; ∝x ≡ x .scf ( x ; ) ≡ x  @krofk eci+ ( ∝x ) 9 2  akf eci+ x  9 2  ik`k eci+ x . scf ( x ; ) 9 2 . x ↝2

x ↝ 2

x ↝2

x ↝ 2 ∝ , x  ≢ 2

∝; ≡ scf ( x ; ) ≡ ; ∝x ≯ x .scf ( x ; ) ≯ x  @krofk eci∝ ( ∝x ) 9 2  akf eci∝ x  9 2  ik`k eci∝ x . scf ( x ; ) 9 2 . x ↝2

x ↝ 2

x ↝2

 Dkac, eci x . scf ( x; ) 9 eci∝ x . scf ( x; ) 9 eci x . scf ( x ; ) 9 j ( 2 ) 9 2 

x ↝2 +

^oekfdutfyk, ackimce soimkrkfg

x ↝2

x ↝2

α   6

2.

2 5 h 5 α   ⇝

; ; 6 h  α  

⎐ ⎐ ; ⎖⎖ ⎐ ;⎖ Ik`k sup ⎕ scf ⎜ 9 ; . K`cmktfyk cfj ⎕ sup ⎕ scf ⎜ ⎜ 9 ; . Akrc hksce cfc acporleoh, ⎙ h ⎢ ⎙ ⎙ h ⎢ ⎢ h . scf( h ; )  j ( 2 + h ) ∝ j ( 2 ) + e ci+ 9 eci+ 9 eci+ scf ( h ; ) 9 ; A j ( 2 ) 9 ec h ↝2 h ↝2 h ↝2 h h  h . scf( h ; )  j ( 2 + h ) ∝ j ( 2 ) A+ j ( 2 ) 9 ececi 9 eci 9 eci scf ( h ; ) 9 ∝; + + h h  h ↝2 h ↝2 h ↝2 +  j ( 2 ) ∝ j ( 2 ∝ h ) h . scf( ∝;h  ) ∝ A j ( 2 ) 9 ececi+ 9 eci+ 9 eci+ scf ( ∝;h  ) 9 ; h ↝2 h ↝2 h ↝2 h h  ; h . scf( ∝ h  )  j ( 2 ) ∝ j ( 2 ∝ h ) 9 ec i 9 eci scf ( ∝;h  ) 9 ∝; A∝ j ( 2 ) 9 ececi + + h h  h ↝2 h ↝2 h ↝2 + Nlftlh ^lke (Rrlmeoi ?.=) 4 k.  Qufdu``kf mkhwk mkhwk A +P„  ( x  (x    )   j (  x )Y   9 „ A  A+  j (  x + m.  Dc`k g (  (x    )  9 j („  („ x  ( x  („ x  x x ),   ik`k A  g (  x )  9 „ A  A „   j („  x )   Dkwkm 4 P ∝ j ( x + h )Y ∝ P ∝ j ( x )Y k. A + P ∝ j ( x )Y 9 eci+ h ↝2 h  j ( x + h ) ∝ j ( x ) ⎐  j ( x + h ) ∝ j ( x ) ⎖ + 9 ∝ ec i 9 ∝A+ j ( x ) A P ∝ j ( x )Y 9 eci+ ∝ ⎕ ⎜ h ↝ 2 h h  h ↝2 + ⎙ ⎢  g ( x + h ) ∝ g (x ) m. A + g ( x ) 9 eci+ h ↝ 2 h 

7>

 

Mkm ? „ Qurufkf akf Cftogrke + A j ( x  ) 9 eci+

Nlipceoa my 4 @hkorlfc, ^.^c

 j ( ∝x ∝ h ) ∝ j ( ∝ x )

h ↝2

9 eci+ ∝

j ( ∝x ) ∝ j ( ∝x ∝ h )

h ↝2

h

h

9 ∝ ec i

j ( ∝ x ) ∝ j ( ∝ x ∝ h ) h 

h ↝2 +

9 ∝ A ∝ j ( ∝x )  Qolroik 4  Dc`k j  jufgsc  jufgsc fkc` akf morfcekc roke pkak P k   Y ik`k j  torturuf`kf  torturuf`kf hkipcr ac ikfk-ikfk akf k,  m  m 

∠  j '( x ) a ( x ) ≡ j (m ) ∝ j ( k ) k 

Mu`tc 4 (Mu`tcfyk kak ;2 `ksus) Rortkik, k`kf acmu`tc`kf mkhwk A + j ( x ) 9 A+ j ( x ) 9 A ∝ j ( x ) 9 A∝ j ( x )  . K`kf acmu`tc`kf uftu` sktu `ksus, icske`kf A + j ( x ) 9 A∝ j ( x )   hkipcr acikfk-ikfk. Hke cfc o`ucvkeof aofgkf

(

)

ioimu`tc`kf i * {x 4 A + j ( x ) 6 A∝ j ( x ) } 9 2 . Icske`kf  O 9 {x ∈P k , m Y 4 A + j ( x ) 6 A∝ j ( x )}

Aofgkf iofggufk`kf k`sclik Krnhcioaos, O akpkt actuecs somkgkc  O 9 {x ∈P k , m Y 4 A + j ( x ) 6 u 6 v 6 A∝ j ( x )}

∤

u , v ∈

Icske`kf, uftu` suktu u , v   Ou ,v  9 {x ∈P k , m Y 4 A + j ( x ) 6 u 6 v 6 A∝ j ( x )} ik`k,

∤O

 O 9

u ,v 

u ,v ∃

 Dkac, nu`up acmu`tc`kf acmu`tc`kf i * ( Ou ,v  ) 9 s  9 2 . Akrc A + j ( x ) 6 u 6 v 6 A∝ j ( x )  actcfdku `ksus morc`ut c)

∂x ∈ Ou ,v  ⇝ A∝ j ( x ) 5 v    j ( x ) ∝ j ( x ∝ h )

5 v  h  @krofk suproiui, ik`k kak h  nu`up  nu`up `once sohcfggk pkak P x „ h , h   Y morek`u  j ( x ) ∝ j ( x ∝ h ) (;) 5 v ⇝ j ( x ) ∝ j ( x ∝ h ) 5 vh …………………….   h  @krofk i * ( Ou ,v  ) 5 ∖  ik`k kak hcipufkf tormu`k L sohcfggk  Ou ,v  ⊍ L  akf eci

h ↝2 +

i * (L ) 5 s + ο 

@krofk i * ( Ou ,v  ) 5 ∖  ik`k iofurut Eoiik Zctkec, kak `leo`sc morhcfggk cftorvke { C ;, C =, …, C  F } ykfg skecfg eopks akf  K ⊍ Ou ,v   sohcfggk

∤

k)  K ⊍ C f  m) i * ( K ) 6 s ∝ ο  Icske`kf C f 9 P x f ∝ hf , x f Y ,  f 9 ;, ;, =, ..., F   Akrc (;),  j ( x f ) ∝ j ( x f ∝ hf ) 5 vh f . ^ohcfggk    F

F

F 

∛  j ( x ) ∝ j ( x ∝ h ) 5 ∛ vh 9 v ∛ h  @krofk ∤ C ⊍ L  ik`k i ( ∤ C ) ≡ ∛ e ( C ) 9 ∛ h 5 i( L ) . Dkac, ∛  j ( x ) ∝ j ( x ∝ h ) 5 vi(L ) 5 v( s + ο )  ……………………(=) f

f 9;

f 

f

f

f

 F 

f 9;

f

f

f 

71

f

f 9; f 

f

f 9;

f

 

Mkm ? „ Qurufkf akf Cftogrke

Nlipceoa my 4 @hkorlfc, ^.^c

cc) ∂x ∈ K ⇝ A + j ( x ) 6 u    j ( x + h ) ∝ j ( x )

6 u  h  @krofk cfjciui, ik`k kak h  nu`up  nu`up `once sohcfggk pkak P x   Y morek`u x,  x + h   j ( x + h ) ∝ j ( x ) (>) 6 u ⇝ j ( x + h ) ∝ j ( x ) 6 uh …………………….   h  @krofk i * ( K ) 5 ∖  ik`k iofurut Eoiik Zctkec, kak `leo`sc morhcfggk cftorvke { D ;, D =, …,  D  I } ykfg skecfg eopks akf M ⊍ K  sohcfggk eci+

h ↝2

k) M ⊍

 I 

∤ D 

c 

c 9;

m) i * ( M ) 6 s ∝ =ο     D c ⊍ C f  n) @krofk x ∈ M ⇝ x ∈ D c uftu` suktu c ⇑ x ∈ K ⇝ x ∈ C f  uftu` suktu f  ik`k

9 P  y f , y f + ` f Y ,  f 9 ;, =, ..., I  Akrc (>),  j ( y f + `f ) ∝ j ( y f ) 6 u`f . ^ohcfggk   Icske`kf

f

 I

∛  j ( y f 9;

f

I

I 

f 9;

f 9;

+ `f ) ∝ j ( y f ) 6 ∛ u`f 9 v ∛ `f 6 u  ( s ∝ =ο )  ……………(1)

^o`krkfg, icske`kf  D ; , D = , D > ⊍ C ;

x;  ) „ j (  x ( x  (x    ;  +h ; )

 D =

;

x ; „  h;   „ h 

=

 D >  y >

 y 1

x ; C ;

 Dkac, u( s ∝ =ο  ) 5

 I 

∛ j( y f 9;

f

+ `f ) ∝ j ( y f ) 

 F 

≡ ∛  j ( x f ) ∝ j ( x f ∝ h f  ) f 9;

≡ v ( s + ο ),

∂ο  6 2

Ik`k, u ( s ∝ = f; ) ≡ v ( s + f ; ),

∂f ∈ 

us ≡ vs  

( u ∝ v )s  ≡ 2 @krofk u  „  „ v  6  6 2, ik`k s ≡ 2. Rkakhke s ≯ 2. Dkac hkrusekh s  9  9 2. @oauk, k`kf acmu`tc`kf mkhwk

m 

∠  j '( x ) a ( x ) ≡ j ( m ) ∝ j ( k ) . k 

Icske`kf  g ( x ) 9 eci h ↝ 2

 j ( x + h ) ∝ j ( x ) h 

Acaojcfcsc`kf 7?

9 j '( x )  toraojcfcsc (ko)

Mkm ? „ Qurufkf akf Cftogrke

Nlipceoa my 4 @hkorlfc, ^.^c

 j ( x + f ; ) ∝ j ( x )

 g f ( x ) 9

akf  j ( x ) 9 j ( m )  uftu` x  6  6 m

; f 

 Dkac, eci  g f  9 g  (ko) akf `krofk  j  jufgsc  jufgsc fkc` ik`k  g f  ≯ 2. Eoiik Jktlu, f ↝∖

∠

m

k

∠

 j ' 9

m

k

g ≡ eci

∠

m

f ↝∖ k

g f . 9 eci f f ↝ ∖

∠

m

 

j ( x + f ; ) ∝ j ( x )

k

 

(∠ . 9 eci f ( ∠ . 9 eci f ( ∠ . 9 eci f ( ∠ . 9 eci f f ↝∖

m

k

j ( x + f ; ) ∝

; m + f 

j (x )∝

; k + f 

f ↝∖

m

j (x ) +

k + f;

f ↝∖

f ↝∖

m + f;

j (x )∝

m

∠

∠

k 

j (x )  

m + f;

j (x ) ∝

m

∠

)

j (x )  

)

m 

k 

∠

m 

; k + f 

k 

∠

; k + f 

k

j (x ) ∝

∠

m

; k + f 

 

)

j (x )    

)

j (x )  

@krofk j  jufgsc  jufgsc fkc`, ik`k ∂x ∈P k , k + f ; Y morek`u  j ( k ) ≡ j ( x ) ⇑ ∝ j ( x ) ≡ ∝ j ( k ) . K`cmktfyk

∠  j ' ≡ eecci f ( ∠ m

k

f ↝∖

m + f;

m

j (m ) ∝

∠

; k + f 

k 

)

j ( k ) 9 eci f ( j ( m ).( m + f; ∝ m ) ∝ j ( k ).( k + f ; ∝ k ) ) 9 j ( m ) ∝ j ( k ) f ↝ ∖

?.=.

Jufgsc Morvkrcksc Qormktks Icske`kf j  jufgsc  jufgsc morfcekc roke ykfg acaojcfcsc`kf pkak cftorvke P k   Y akf icske`kf k  9  9 x 2 5 x ; 5 . k,  m  . . 5 x ` 9 m  iorupk`kf  iorupk`kf somkrkfg pkrtcsc akrc P k   Y. Acaojcfcsc`kf k,  m   p 9

`

∛ P j ( x ) ∝ j ( x  c 9;

f9

`

∛P j ( x ) ∝ j ( x c

c 9;

t 9 p+f 9

`

∛

+

c ∝;

c

)Y

∝

  )Y

c ∝;

j ( x c ) ∝ j ( x c ∝ ; )

c 9;

acik acikfk fk r + 9 ikx(2, kx(2, r ) , r ∝ 9 ikx( 2, ∝r ) , a akkf |r |9 r + + r ∝ . Akrc pofaojcfcsckf cfc acporleoh,  p ∝ f 9

`

∛P j ( x ) ∝ j ( x c

c 9;

+

c ∝;

)Y ∝

`

∛ P j ( x ) ∝ j ( x  c 9;

c

c ∝;

∝

)Y

 

`

9 ∛  j ( x c ) ∝ j ( x c ∝; ) c 9;

9  j ( x ` ) ∝ j ( x 2 ) 9  j (m ) ∝ j ( k ) @oiuackf, acaojcfcsc`kf  9 sup p, R  9  9 sup f ,  F  9  9 sup t , Q  9 ykctu iofgkimce suproiui akrc soiuk pkrtcsc-pkrtcsc ykfg iufg`cf pkak P k , m   Y. @krofk,  p 9

`

∛P j ( x ) ∝ j ( x c 9;

ik`k,

c

+

c ∝;

)Y ≡

`

∛ c 9;

j ( x c ) ∝ j ( x c ∝; ) 9  t 

sup p ≡ sup t 9 sup( f + p ) ≡ sup p + sup f 

 K`cmktfyk

R ≡Q ≡R +F    ikscfg-ikscfg acsomut plsctcj, fogktcj, akf vkrcksc tltke akrc  j  pkak  pkak P k   Y. Zkrcksc tltke akrc  j  R ,  F , akf Q  ikscfg-ikscfg k,  m  7:

Mkm ? „ Qurufkf akf Cftogrke

Nlipceoa my 4 @hkorlfc, ^.^c

pkak P k   Y actuecs Qk m ( j  )  ktku Q k m  . k,  m  Nlftlh 4  Dc`k j  jufgsc  jufgsc toru`ur akf fkc` ilfltlf pkak P k   Y, ik`k k,  m   p 9

`

∛P j ( x ) ∝ j ( x c

c 9;

f9

c

c 9;

t9

c ∝;

`

∛P j (x ) ∝ j ( x `

∛ c 9;

`

∛ j (x ) ∝ j (x

+

)Y 9

c

c 9;

∝

  )Y 9

c ∝;

j ( x c ) ∝ j ( x c ∝; ) 9

c ∝;

) 9 j ( x ; ) ∝ j ( x 2 )... + j ( x ` ) ∝ j ( x `∝; ) 9 j ( m ) ∝ j ( k )

`

∛2 9 2 c 9;

`

∛ j (x ) ∝ j (x c 9;

c

c ∝;

) 9  j ( m ) ∝ j ( k )  

 9 sup t  9  9 j (  (m  (k  Q  9 m  )  „ j (  k  )   9 sup f  9  9 2  F  9  9 sup p 9 j (  (m  (k  R  9 m  )  „ j (  k  )   9 F  +  + R  Q  9 Aojcfcsc (Jufgsc Morvkrcksc Qormktks)4 Jufgsc j  ac`ktk`kf  ac`ktk`kf morvkrcksc tormktks pkak P k   Y dc`k Q k m  5 ∖  akf acfltksc`kf aofgkf  j ∈ MZ  . k,  m  Eoiik 4  Dc`k j  morvkrcksc  morvkrcksc tormktks pkak P k   Y ik`k k,  m  Qkm 9 Rkm + F k m 

akf  j ( m ) ∝ j ( k ) 9 Rkm ∝ F k m 

Mu`tc 4 Ackimce somkrkfg pkrtcsc pkak Pk, mY. Iofurut aojcfcsc,  p ∝ f 9 j ( m ) ∝ j ( k ) ⇑ p 9 f + j ( m ) ∝ j ( k ) Akrc scfc, ik`k sup p 9 sup( f + j ( m ) ∝ j ( k )) 9 sup f + j ( m ) ∝ j ( k )

⇑ R 9 F + j (m ) ∝ j ( k )  ⇑ R ∝ F 9 j (m ) ∝ j ( k )   Dugk, t 9 p + f 9 p + p ∝ { j ( m ) ∝ j ( k )} 

Ik`k, sup t 9 sup( p + p ∝ { j ( m ) ∝ j ( k )} ) 9 sup = p ∝ { j ( m ) ∝ j ( k )} 

⇑ Q 9 =R ∝ { j (m ) ∝ j ( k )}  ⇑ Q 9 =R ∝ R + F 9 R + F    Qolroik 4 Jufgsc  j   morvkrcksc tormktks pkak P k   Y dc`k akf hkfyk dc`k  j   iorupk`kf soecsch auk jufgsc k,  m  ilfltlf akf morfcekc roke pkak P k   Y k,  m  Mu`tc 4 ⇝ ) Ac`otkhuc  j ∈ MZ  . K`kf acmu`tc`kf mkhwk  j 9 g ∝ h  aofgkf g akf h jufgsc ilfltlf.

@krofk  j ∈ MZ  ik`k Qk m ( j  ) 5 ∖ . Akrc Eoiik ac ktks,  j ( x ) 9 Rkx ∝ F k x  + j ( k )

Ackimce 77

Mkm ? „ Qurufkf akf Cftogrke

Nlipceoa my 4 @hkorlfc, ^.^c

 g ( x ) 9 R k x  akf h( x ) 9 F k x 

 Qorechkt mkhwk, ∂x 5 y ⇝ Rkx ≡ Rk y   ykfg morkrtc  g ( x ) 9 R k x   tk` turuf0 akf ykfg mor morkrtc krtc h( x ) 9 F k x   tk` turuf ∂x 5 y ⇝ F kx ≡ F k y   ykfg  9 g  „  „ h  aofgkf  aofgkf g  akf  akf h  jufgsc  jufgsc ilfltlf. K`kf acmu`tc`kf  j ∈ MZ ⇑ Q k m  5 ∖ ⇒ ) Ac`otkhuc j  9 t km ( j ) 9

`

∛

j ( x c ) ∝ j ( x c ∝ ; )

c 9; `

9 ∛ ( g ( x c ) ∝ h( x c ) ) ∝ ( g ( x c ∝; ) + h( x c ∝; ) )   c 9; `

`

c 9;

c 9;

≡ ∛  g ( x c ) ∝ g ( x c ∝; ) + ∛ h( x c ) + h( x c ∝; )

 

@krofk g (  (x    )  akf h (  (x    )  jufgsc ilfltlf, ilfltlf, ik`k x x t k m ( j ) ≡ g ( m ) ∝ g ( k ) + h( m ) ∝ h( k )  Akrc scfc ik`k, Qk m ( j ) ≡ g ( m ) ∝ g ( k ) + h( m ) ∝ h( k )  5 ∖  Dkac,  j ∈ MZ   K`cmkt 4  Dc`k j  jufgsc  jufgsc morvkrcksc tormktks pkak P k   Y ik`k j– (  (x    )  kak hkipcr ac ac ikfk-ikfk pkak P k   Y k,  m  x k,  m  Mu`tc 4 jufgsc ilfltlf ilfltlf  j ∈ MZ . ⇝ j 9 g ∝ h , g akf h  jufgsc

. ⇝  j ' 9 ( g ∝ h ) ' 9 g ' ∝ h ' (tolroik turufkf jufgsc ilfltlf )  Dkac, j–  kak  kak (ko) pkak P k   Y. k,  m  ?.>.

 Qurufkf Cftogrke Cftogrke  Dc`k  j  jufgsc   jufgsc torcftogrke`kf pkak P k   Y acaojcfcsc`kf cfaojcfct cftogrke (cftogrke tk` toftu) akrc  j k,  m  ykctu J  ykfg  ykfg acaojcfcsc`kf pkak P k   Y aofgkf kturkf k,  m  J(x ) 9

∠

m 

k 

j ( t ) at  

Akeki sum-mkm cfc `ctk k`kf ioechkt mkgkcikfk turufkf akrc cfaojcfct cftogrke akrc suktu jufgsc torcftogrke`kf kakekh skik aofgkf cftogrkefyk hkipcr acikfk-ikfk. @ctk iuekc poimkhkskf cfc aofgkf ioechkt momorkpk eoiik. Eoiik 4  Dc`k j  torcftogrke`kf  torcftogrke`kf pkak P k   Y ik`k jufgsc J  aofgkf  aofgkf k,  m  J(x ) 9

∠

m 

k 

j ( t ) at  

kakekh jufgsc `lftcfu akf morvkrcksc tormktks pkak P k , m   Y. Mu`tc 4  Kimce somkrkfg n ∈P k , m Y . K`kf acmu`tc`kf J   `lftcfu ac n . O`ucvkeof aofgkf ioimu`tc`kf ∂ο 6 2, ∎α   6 2 sohcfggk | x ∝ n |5 α ⇝ j ( x ) ∝ j ( n )  5 ο  . Ackimce somkrkfg ο  6 2 . @krofk j  torcftogrke`kf  torcftogrke`kf pkak P k   Y ik`k | j | torcftogrke`kf pkak P k   Y k,  m  k,  m  @krofk | j | ≯ 2 akf torcftogrke`kf pkak P k   Y ik`k kak α  6 2 sohc sohcfg fggk gk uftu uftu`` sotc sotckp kp  K ⊍ P k , m Y k,  m  aofgkf i( K ) 5 α   ik`k

∠ | j |5 ο  .  K

 Kimce  K 9 {x ∈P k , m Y4 Y 4| x ∝ n |5 α  / >} 78

Mkm ? „ Qurufkf akf Cftogrke

 „ α   n  α/>   

Nlipceoa my 4 @hkorlfc, ^.^c

n +α  />

n 

n 9k 

n +α  />

n„ α   α/>   

n9m

 Dkac, sonkrk uiui i( K ) ≡

= >

α

5 α  . ^ohcfggk

| x ∝ n |5 α ⇝ J ( x ) ∝ J ( n ) 9

∠

x

k

j (t ) ∝

∠

n

j (t ) 9

k

∠

n

x

j (t ) ≡

∠

n

x

 

j (t ) 5

∠

K

j ( t )  5 ο 

^oekfdutfyk, k`kf acmu`tc`kf  j ∈ MZ  . O`ucvk O`ucvkeof eof aofgkf aofgkf ioimu` ioimu`tc` tc`kf kf Qk m ( J ) 5 ∖ t km ( J ) 9

`

∛ J(x ) ∝ J(x

 )

c ∝;

c

c 9` `

9 ∛ ∠  j ( t ) ∝ ∠ c 9` `

xc

x c ∝;

k

k 

j ( t )

9 ∛ ∠  j ( t ) x c 

x c ∝;

c 9` `

≡ ∛∠ c 9`

xc

x c ∝;

 j ( t ) 9

∠

x;

x2

j (t ) +

∠

x=

x;

j ( t ) + ... +

∠

x`

x ` ∝;

j (t ) 9

∠

m

k

j ( t )

 

 

@krofk  j   torcftogrke`kf ik`k | j | torcftogrke`kf. @krofk ruks `kfkf tcak` morgkftufg pkak pkrtcsc, ik`k Qkm ( J ) 9 sup t k m ( J ) ≡

^oekfdutfyk, aofgkf iofaojcfcsc`kf J ( x ) 9

∠

m 

k 

∠

m 

k 

j ( t )  5 ∖

j ( t ) at ,  ∂x ∈P k , m Y   torfyktk acporleoh mkhwk

jufgsc J  `lftcfu   `lftcfu akf morvkrcksc tormktks pkak P k   Y. @ctk k`kf ioechkt mkhwk J  ioipufykc   ioipufykc turufkf. k,  m   Krtcfyk J– (  ( x  morek`u4 x )  kak. ^ohcfggk dugk morek`u4 a  m  ;) J '( x ) 9 j ( t ) at 9 j ( x )  ax  k 

∠

=)

m 

∠  j (t ) at 9 j (m ) ∝ j ( k ) k 

_ftu` ctu, `ctk poreu iofcfdku momorkpk eoiik morc`ut4 Eoiik 4  Dc`k j  torcftogrke`kf  torcftogrke`kf pkak P k   Y akf k,  m 

∠

x 

k 

 j ( t ) at  9 2 , ∂x ∈P k , m Y

ik`k j (  ( )  t   Y. t  9 2 (ko) ac P k  k,  m  Mu`tc 4 (soo ‘]oke Kfkeyscs–, H.E. ]lyaof, > ra oa, pkgo ;2?-;2:). Eoiik 4  Dc`k j  jufgsc  jufgsc tormktks akf toru`ur pkak P k   Y akf k,  m  J(x ) 9

∠

x 

k 

j ( t ) at + J ( k ) 

ik`k J– (  (x    )  9 j (  ( x   Y. x x )  (ko) pkak P k  k,  m  Mu`tc 4 (soo ‘]oke Kfkeyscs–, H.E. ]lyaof, > ra oa, pkgo ;2:-;27).  Qolroik 4  Dc`k j  jufgsc  jufgsc torcftogrke`kf pkak P k   Y akf k,  m  73

Mkm ? „ Qurufkf akf Cftogrke

Nlipceoa my 4 @hkorlfc, ^.^c

J(x ) 9

x 

∠

j ( t ) at + J ( k ) 

k 

ik`k J– (  (x    )  9 j (  ( x   Y. x x )  (ko) pkak P k  k,  m  Mu`tc 4 Ac`otkhuc j  torcftogrke`kf  torcftogrke`kf pkak P k   Y akf J ( x ) 9 k,  m 

∠

Akrc auk eoiik ac ktks, nu`up acmu`tc`kf

x

∠

x 

k 

(x    )  9 j (  ( x  j ( t ) at + J ( k ) . K`kf acmu`tc`kf J– (  x x ) 

J '( t ) at 9

k

∠

x 

k 

j ( t ) at  , ∂x ∈P k , m Y .

@krofk j  9  9 j + „ j  „  aofgkf j +, j  „ ≯ 2 (tkfpk iofgurkfgc `ouiuikf mu`tc) ik`k j ≯ 2. Acaojcfcsc`kf mkrcskf { j f } somkgkc morc`ut4 ⎧ j ( x ),), j ( x ) 5 f   j f ( x ) 9 ⎨ j (x ) ≯ f   ⎣ f, Akrc pofaojcfcsckf cfc acporleoh  j f  ≯ 2 , ∂f ∈  0  j f   j  0  j f  ≡ f , ∂f ∈  akf j f  tormktks akf toru`ur. Iofurut eoiik `oauk ac ktks, ik`k a  x   j f ( t ) at 9 j f ( x ) , ∂f ∈  ax  k  @oiuackf acaojcfcsc`kf ∂f ∈  Gf 9 j ∝ j f  ≯ 2 ^ohcfggk,

∠

x5 y⇝

∠

x

k

∠

j ∝ jf ≡

x 

k 

j f ∝ j ⇑ Gf ( x ) ≡ Gf ( y )    

 Dkac Gf  ilfltlf tk` turuf. K`cmktfyk ; Gf ' ( x ) ≯ 2 , ∂x ∈ ( k , m ) . Akrc hcpltoscs a  x  J '( x ) 9 j f ( t ) at   ax  k  a  x  9 ( j ∝ j f ) + j f  ax  k  a x a  x   j ∝ j f ) + j f  9 ( ax k ax k   9 Gf' ( x ) + j f ( x ) 

∠ ∠

∠

∠

≯ 2 +  j f ( x ) 9  j f ( x ), ∂f   Dkac,

), ∂x ∈P k , m Y, ∂f ⇑ j f ≡ J ', ∂f ⇑ eci j f  ≡ J ' ⇑ j ≡ J '  j f ( x ) ≡ J '( x ), f ↝∖

Akrc hcpltoscs acporleoh J ( x ) ∝ J (k ) 9

@krofk  j  ≯ 2  ik`k

∠

x

k

j ≡

∠

x 

k 

J '   ……………………… (;)

x5 y⇝

∠

x

k

j ≡

∠

y 

k 

j

 

 Dkac J  tk`  tk` turuf, k`cmktfyk

∠

x

k

J ' ≡ J ( x ) ∝ J (k ) 9

∠

x 

k 

Akrc (;) akf (=) acporleoh

; Echkt

prlplscsc ac hkekikf 7=. 82

(=) j ………………………  

Mkm ? „ Qurufkf akf Cftogrke

Nlipceoa my 4 @hkorlfc, ^.^c

∠

x

k

 j ( t ) at 9

∠

x 

k 

J '(t ) at 9 J ( x ) ∝ J (k )

 K`cmktfyk,

∠

x

k

 j ( t ) at ∝

∠

x

k

J '(t ) at 9 2 ⇑

∠

x 

k 

P j (t ) ∝ J 't )Y at 9 2, ∂x 

Acporleoh (  ) ( )  t  ( ) (t   ),  j (  t  t  „ J– (  t  9 2 ⇑ j (  t  t   9 J– (  t   ∂t . Akrc tolroik tork`hcr cfc, tordkwkm suakh portkfykkf-portkfykkf ykfg aciufnue`kf ac kwke mkm cfc. Skctu mkhwk Qolroik Akskr @ke`ueus C akf CC morek`u4 ;) J '( x ) 9 j ( x )  =)

∠

m 

k 

J '( t ) at at 9 J (m ) ∝ J ( k ) 

Ektchkf (^lke [ucs) 4 Icske`kf j  jufgsc  jufgsc tk` fogktcj akf torcftogrke`kf (ko) pkak P k   Y. Acaojcfcsc`kf k,  m  J(x ) 9

∠

x 

k 

j ( t ) at  

mu`tc`kf mkhwk

∠

m

k

J '( t ) at ≡

∠

m 

k 

j (t ) at  

 Dkwkm 4 Rortkik actufdu``kf mkhwk J  tk`  tk` turuf. @krofk  j ≯ 2, ik`k x 5 y ⇝ Pk, x ) ⊍ Pk, y ) ⇑

∠

x

j ≡

k

∠

y 

k 

j ⇑ J (x ) ≡ J ( y )  

 Dkac, J  tk`  tk` turuf. Ik`k=

∠

m

k

J '( t ) at ≡ J (m ) ∝ J ( k ) 9

∠

m

k

j (t ) at ∝ 2 9

∠

m 

k 

j (t ) at  

Nlftlh 4 Ac`otkhuc j (  ( x  x )  9 scf x , x ∈ P2, ς  Y. Kpk`kh  j ∈ MZ   pkak P2, ς  Y<  Dkwkm 4 Icske`kf E  somkrkfg  somkrkfg pkrtcsc akrc P2, ς  Y. Ik`k E 4 2 9 x 2 5 x ; 5 ... 5 x ` 9  ς  . Actcfdku auk `ksus. @ksus C4  Dc`k ς = ∈ E , ik`k ∎f2 , 2 ≡ f2 ≡ f  sohcfggk t2 ( j ) 9 ς 

f 2

∛ c 9;

j ( x c ) ∝ j ( x c ∝; ) +

9

`

∛ c 9 f 2 +;

 f 2

9 ∛  j ( x c ) ∝ j ( x c ∝; ) + c 9;

j ( x c ) ∝ j ( x c ∝;  )

 

 `

∛

c 9 f 2 +;

j ( x c ∝; ) ∝ j ( x c )

 

9  j ( x f ) ∝ j ( x 2 ) + j ( x f ) ∝ j ( x ` ) 2

2

9  j ( ς= ) ∝ j ( 2 ) + j ( ς = ) ∝ j (ς  ) 9 scf ς= ∝ scf 2 + scf ς = ∝ scf ς  9= @ksus = 4  Dc`k ς = ∃ E , ik`k ∎f; , 2 ≡ f; ≡ `  sohcfggk x f; ∝; 5 ς = 5 x f ; . = Echkt

Qolroik hkekikf 71 8;

Mkm ? „ Qurufkf akf Cftogrke

t2 ( j ) 9 ς 

f ; ∝;

∛ c 9;

Nlipceoa my 4 @hkorlfc, ^.^c

j ( x c ) ∝ j ( x c ∝; ) + j ( x f; ) ∝ j ( x f; ∝; ) +

`

∛ c 9 f ; +;

j ( x c ) ∝ j ( x c ∝;  )

f ; ∝;

9 ∛  j ( x c ) ∝ j ( x c ∝; ) + j ( x f ) ∝ j ( = ) + j ( = ) ∝ j ( x f ∝; ) + ς

;

c 9;

`

ς 

;

c 9 f ; +;

f ; ∝;

≡ ∛  j ( x c ) ∝ j ( x c ∝; ) + j ( x f ) ∝ j ( = ) + j ( = ) ∝ j ( x f ∝; ) + ς

ς 

;

c 9;

;

 9 9  j ( ς= ) ∝ j ( 2 ) + j ( ς = ) ∝ j (ς  ) 9 scf ς= ∝ scf 2 + scf ς = ∝ scf ς  9=



 Dkac,

⎧ =, t 2 ( j  ) 9 ⎨ ⎣≡ =, ς 

∈ E  ς  = ∃ E  ς 

=

^ohcfggk Q2ς  ( j  ) 9 = 5 ∖ . Dkac tormu`tc  j ∈ MZ   pkak P2, ς  Y

8=

∛ `

∛

c 9f ; +;

 

j ( x c ) ∝ j ( x c ∝; ) j ( x c ) ∝ j ( x c ∝; )