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 ∝; )