SFIWCT\INJN FJLI@FJH NC MSJFLJWCHILJ (Lrcjnj p`r Hcy Fr`. 43473)
Gjluht jluhtjn nc Licflijs nc Ifecficrâj
Csluchj _r`gcsi`fjh nc Livih-Mvlj
NI\^TIDSLI@FC\ C\^JNI\^ILJ\ CF MINT@H@EÂJ Y \S J_HILJLIÜF CF T
LÊ^CNTJ 1 MINT@H@EÂJ ECFCTJH LJ^CNTÊ^IL@ 1 Ife. IWJF JYJHJ DIQJTT@ C\^SNIJF^C 1 WCF^STJ MSJBJF, Ncffis @hivcr LILH@ Y \CLLIÜF 1 WII ‘J‒
MSJFLJWCHILJ - 4?:7
Ncnil` nc bjfcrj buy cspclijh j bis pjnrcs, yj quc chh`s s`f ch libicft` pjrj hj l`fstrulliüf nc bi vinj pr`gcsi`fjh.
:
Ncnil` nc bjfcrj buy cspclijh j bis pjnrcs, yj quc chh`s s`f ch libicft` pjrj hj l`fstrulliüf nc bi vinj pr`gcsi`fjh.
:
Âfnilc ecfcrjh IF^T@NSLLIÜF
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3 3
4. NI\^TIDSL NI\^TIDSLI@FC\ I@FC\ NC _T@DJDI _T@DJDIHIN HINJN JN 4.:. NI\^TIDSLIÜ NI\^TIDSLIÜF F H@E-F@ H@E-F@TBJH TBJH NC < _JTÊBC^T@\ JTÊBC^T@\ . . . . . . 4.:.:. Gufliüf ncfsinjn ncfsinjn . . . . . . . . . . . . . . . . . . . . . . . . . 4.:.4. Cstibjliüf Cstibjliüf nc pjrêbctr`s, pjrêbctr`s, bét`n` nc Bêxibj Bêxibj Wcrisi Wcrisibihit bihitun un . 4.:.<. Gufliüf nc nistridu nistriduliüf liüf jlubuh jlubuhjnj jnj . . . . . . . . . . . . . . . 4.:.>. _r`lcnibicft` _r`lcnibicft` pjrj ch lêhluh` y jphiljliüf jphiljliüf cf cf T . . . . . . . . 4.4. NI\^TIDSLIÜ NI\^TIDSLIÜF F EJBBJ 4 _JTÊBC^T@\ JTÊBC^T@\ . . . . . . . . . . . . 4.4.:. Gufliüf ncfsinjn ncfsinjn . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4. Gufliüf Jlubuhjnj Jlubuhjnj . . . . . . . . . . . . . . . . . . . . . . . . 4.4.<. Cstibjliüf Cstibjliüf nc pjrêbctr`s, pjrêbctr`s, bét`n` nc Bêxibj Bêxibj Wcrisi Wcrisibihit bihitun un . 4.<. NI\^TIDSLIÜ NI\^TIDSLIÜF F EJBBJ < _JTÊBC^T@\ JTÊBC^T@\ . . . . . . . . . . . . 4.<.:. Gufliüf ncfsinjn ncfsinjn . . . . . . . . . . . . . . . . . . . . . . . . . 4.<.4. Gufliüf Jlubuhjnj Jlubuhjnj . . . . . . . . . . . . . . . . . . . . . . . . 4.<.<. Cstibjliüf Cstibjliüf nc pjrêbctr`s, pjrêbctr`s, bét`n` bét`n` nc b`bcft b`bcft`s `s . . . . . . . . 4.>. NI\^TI NI\^TIDSL DSLIÜF IÜF H@E-_CJ H@E-_CJT\@F T\@F ^I_@ ^I_@ III . . . . . . . . . . . . . 4.>.:. Gufliüf ncfsinjn ncfsinjn . . . . . . . . . . . . . . . . . . . . . . . . . 4.>.4. Gufliüf Jlubuhjnj Jlubuhjnj . . . . . . . . . . . . . . . . . . . . . . . . 4.>.<. Cstibjliüf Cstibjliüf nc pjrêbctr`s, pjrêbctr`s, bét`n` bét`n` nc b`bcft b`bcft`s `s . . . . . . . . 4.3. NI\^TI NI\^TIDSL DSLIÜF IÜF ESBDCH ESBDCH . . . . . . . . . . . . . . . . . . . . . . . 4.3.:. Gufliüf jlubuhjnj jlubuhjnj . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Gufliüf ncfsinjn ncfsinjn . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.<. Cstibjliüf Cstibjliüf nc pjrêbctr`s, pjrêbctr`s, bét`n` bét`n` nc b`bcft b`bcft`s `s . . . . . . . . 4.7. NI\^TI NI\^TIDSL DSLIÜF IÜF H@E-ES H@E-ESBDC BDCH H . . . . . . . . . . . . . . . . . . . 4.7.:. Gufliüf jlubuhjnj jlubuhjnj . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4. Gufliüf ncfsinjn ncfsinjn . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.<. Cstibjliüf Cstibjliüf nc pjrêbctr`s, pjrêbctr`s, bét`n` bét`n` nc b`bcft b`bcft`s `s . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
5 5 5 5 = = :4 :4 :4 :4 :3 :3 :3 :3 := := := := 4: 4: 4: 4: 4> 4> 4> 4>
L`flhusi`fcs
4=
Didhi`erjgâj
49
4
IF^T@NSLLIÜF Hjs nistriduli`fcs cstjnâstil`s cf ch ljbp` nc hj minr`h`eâj, ticfcf ufj erjf ibp`rtjflij pjrj hj cstibjliüf nc vjh`rcs nc niscú` j uf pcri`n` nc rct`rf` sceñf jh tip` nc cstrulturj j hj quc sc pr`ycltj; j l`ftifujliüf sc prcscftj = nistriduli`fcs cstjnâstiljs, cftrc chhjs1 :. H`e-F`rbjh < pjrêbctr`s 4. Ejbbj 4 pjrêbctr`s <. Ejbbj < _jrêbctr`s >. H`e-_cjrs`f tip` III 3. Eubdch 7. H`e-Eubdch. H`s pjrêbctr`s nc hjs nistriduli`fcs yj bcfli`fjnjs pucncf scr cstibjn`s p`r vjri`s bét`n`s, h`s bês usjn`s s`f ch bét`n` nc b`bcft`s, bét`n` sibphiljn` y ch bét`n` nc bêxibj vcrisibihitun. Ch bét`n` nc b`bcft`s ticfc ufj hibitjliüf, pucs pjrj su jphiljliüf ch l`clicftc nc lcse` ticfc quc sc bjy`r j ?.34, ljs` l`ftrjri` cstc bétn`n` f` p`nrij scr jphiljn`. Ch bét`n` sibphiljn` s`h` jpr`xibj ch vjh`r nch pjrêbctr` nc p`siliüf y p`r tjft` h`s vjh`rcs nc h`s ncbês pjrêbctr`s tjbdicf s`f s`h` jpr`xibjli`fcs, y cst` l`fhhcvj j quc f` p`nrâj usjrsc pjrj jheufjs nc hjs nistriduli`fcs. Ch bét`n` nc bêxibj vcrisibihitun cs uf` nc h`s bét`n`s bês l`fjdhcs pjrj ch lêhluh` nc h`s pjrêbctr`s. Cf ch prcscftc trjdja` sc nctjhhj ch pr`lcnibicft` pjrj ch lêhluh` nc ljnj uf` nc h`s nigcrcftcs pjrêbctr`s; jncbês nc h`s jhe`ritb`s cf ch hcfeujac nc pr`erjbjliüf cstjnâstil` T.
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@DAC^IW@\ •
@ptibizjr y rcnulir ticbp` cf ch lêhluh` nc vjh`rcs nc niscú`s minr`hüeil`s c minrêuhil`s usjfn` ch hcfeujac nc pr`erjbjliüf cstjnistil` T.
•
Jbphijr l`f`libicft`s s`drc hj pr`erjbjliüf `ricftjn`s j hj minr`h`eâj.
•
_`r bcni` nch tcst nc d`fnjn nc jaustc nc K`hb`e`r`v-\birf`v, nctcrbifjr hjs gufli`fcs nc nistriduliüf nc pr`djdihinjn quc bca`r rcprcscftjf j hj scric.
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:. LJ_Â^SH@ :. SF _@L@ NC MI\^@TIJ
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Ljpâtuh` : SF _@L@ NC MI\^@TIJ :.:.
T
Guc ncsjrr`hhjn` ifilijhbcftc p`r T`dcrt Ecfthcbjf y T`ss Imjkj nch Ncpjrtjbcft` nc Cstjnâstilj nc hj Sfivcrsinjn nc Julkhjfn cf :99<. \if cbdjre`, si sc rcb`ftj j sus djscs ifilijhcs, pucnc nclirsc quc ifiliü cf h`s Dchh Hjd`rjt`rics nc J^&^ y jm`rj Jhljtch-Hulcft cf Fucvj Acrscy l`f ch hcfeujac \. Éstc ñhtib`, uf sistcbj pjrj ch jfêhisis nc njt`s ncsjrr`hhjn` p`r A`mf Lmjbdcrs, Tilk Dclkcr, y l`hjd`rjn`rcs nigcrcftcs ncsnc fjhcs nc :95?. Hj mist`rij ncsnc cstc puft` cs prêltiljbcftc hj nch hcfeujac \. H`s niscújn`rcs ifilijhcs, Ecfthcbjf y Imjkj, l`bdifjr`f hjs g`rtjhczjs nc n`s hcfeujacs cxistcftcs, \ y \lmcbc. Cf sus pr`pijs pjhjdrjs1 ‘Ch hcfeujac rcsuhtjftc cs buy sibihjr cf jpjricflij j \, pcr` cf ch us` nc g`fn` y hj scbêftilj cs ncrivjn` ncsnc \lmcbc‒. Ch rcsuhtjn` sc hhjbü T ‘cf pjrtc jh rcl`f`libicft` nc hj iffflucflij nc \ y cf pjrtc pjrj mjlcr ejhj nc sus pr`pi`s h`er`s‒. T pr`p`rli`fj uf jbphi` jdjfil` nc mcrrjbicftjs cstjnâstiljs (b`nch`s hifcjhcs y f` hifcjhcs, tcsts cstjnâstil`s, jfêhisis nc scrics tcbp`rjhcs, jhe`ritb`s nc lhjsiljliüf y jerupjbicft`, ctl.) y erêljs. Jh ieujh quc \, sc trjtj nc uf hcfeujac nc pr`erjbjliüf, h` quc pcrbitc quc h`s usujri`s h` cxticfnjf ncficfn` sus pr`pijs gufli`fcs. Nc mclm`, erjf pjrtc nc hjs gufli`fcs nc T cstêf cslritjs cf ch bisb` T, jufquc pjrj jhe`ritb`s l`bputjli`fjhbcftc cxiecftcs cs p`sidhc ncsjrr`hhjr didhi`tcljs cf L, L++ ` G`rtrjf quc sc ljrejf nifêbiljbcftc.
LJ_Â^SH@ :. :.:. T
7
(j) Iftcrgjz T cf su vcrsiüf <.<.
(d) Itcrgjz nc Tstuni`, Sf INC pjrj T.
Gieurj :.:1 T l`b` hcfeujac nc pr`erjbjliüf.
MINT@H@EÂJ ECFCTJH
4. LJ_Â^SH@ 4. NI\^TIDSLI@FC\ NC _T@DJDIHINJN
5
Ljpâtuh` 4 NI\^TIDSLI@FC\ NC _T@DJDIHINJN 4.:.
NI\^TIDSLIÜF H@E-F@TBJH NC < _JTÊBC^T@\
Hj nistriduliüf H`e-F`rbjh nc trcs pjrêbctr`s cs sibihjr j hj nistriduliüf H`e-F`rbjh 4_ cxlcpt` quc hj vjrijdhc x cs rcstjnj cf ufj ljftinjn x ` quc rcprcscftj uf hâbitc ifgcri`r pjrj nilmj vjrijdhc, tjh quc1 y 2 hf(x ∔ x? ) 4.:.:.
Gufliüf ncfsinjn
Hj gufliüf ncfsinjn, nc hj nistriduliüf h`e-f`rbjh nc < pjrêbctr`s cs1 g (x) 2
(x
hf(x∔x? )∔³y 4 : ∔:[ R όy c 4 x? )όy 4Ϗ
∔ pjrj x ≮ x ≮ ∖
(4.:)
∑
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n`fnc1 x` 1 _jrêbctr` nc p`siliüf cf ch n`bifi` x ³y 1 _jrêbctr` nc csljhj cf ch n`bifi` x ό`4 1 _jrêbctr` nc g`rbj cf ch n`bifi` x
4.:.4.
Cstibjliüf nc pjrêbctr`s, bét`n` nc Bêxibj Wcrisibihitun
H`s prêbctr`s nc hj nistriduliüf h`e-f`rbjh nc < pjrêbctr`s, cstibjn`s p`r ch bét`n` nc bêxibj vcrisibihitun s`f1 : F ³y 2 hf(xi F i2:
όy 2
: F (hf(xi F i2:
∔x ) `
∔ x ) ∔ ³ )4 ) `
y
(4.4)
(4.<)
n`fnc1 ³y 2_jrêbctr` nc csljhj, ieujh jh pr`bcni` nc h`s hf(x ∔ x` ) ό y 2_jrêbctr` nc g`rbj, ieujh j hj ncsvijliüf cstêfnjr nc h`s hf(x ∔ x` ) x ` 2_jrêbctr` nc p`siliüf Ch pjrêbctr` nc p`siliüf
LJ_Â^SH@ 4. 4.:. NI\^TIDSLIÜF H@E-F@TBJH NC < _JTÊBC^T@\
=
sc ljhluhj bcnijftc uf pr`lcs` itcrjtiv` l`b` s`huliüf nc hj sieuicftc cxprcsiüf rcsubinj1 F
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4
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F
∔³
y
hf (xi x` ) 2? x x i ` i2:
∔ ∔
+
(4.>)
_jrj p`ncr `dtcfcr h`s vjh`rcs nc h`s pjrêbctr`s sc ticfc quc rclurrir j utihizjr h`s bét`n`s fubéril`s, cf cstc ljs` ch bét`n` nc hj scljftc1 xf+: 2 x f
∔ g (xx ) ∔∔ xg (x∔: f
f
f∔: )
f
g (xf )
(4.3)
L`b` sc pucnc vcr, cstc bét`n` fclcsitjrê n`s jpr`xibjli`fcs ifilijhcs nc hj rjâz pjrj p`ncr ifnulir ufj pcfnicftc ifilijh. 4.:.<.
Gufliüf nc nistriduliüf jlubuhjnj G (Q ) 2
n`fnc1 Q 2
: 4Ϗ
∑
hf(x
Q
∔∖
c
∔Q 4 4
nQ
∔x )∔³ `
y
(4.7) (4.5)
όy
\c utihizü hj s`huliüf cbpârilj pjrj hj gufliüf jlubuhjnj nc Jdrjb`witz y \tceuf(:973)1
∔ g (Q )(?,><7:=<7t ∔ ?,:4?:757t4 + ?,9<549=t<) : n`fnc1 pjrj Q ≣ ? t 2 :+? <<475| | G (Q ) 2 :
,
4.:.>.
Q
_r`lcnibicft` pjrj ch lêhluh` y jphiljliüf cf T
Hclturj c ibp`rtjliüf nc njt`s
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### J p h i l j t i ` f ? : 1 H `e F`rbjh <_ ### DY1 NCFFI\ WCF^STJ MSJBJF ` g w`rk g ` h n c r < # H ` lj t i` f > sctwn ( ’N1 /SFMPLIWIHPWIIPJ/MINT@H@EIJ/U 3/NI\^TIDSLI@FC\ T ’ ) rcjn . t j d h c ( ’ NjtjPZbjx( : ) . tx t ’ , m cj n cr2^TSC) 3 Njtj8 x8 s ` r t ( Njtj$Zbjx) 7 5 subbjry ( x )
∔
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∔
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Bcnijf =:=.?
Bcjf
Bjx. <=??.?
Nistriduliüf cbpârilj pr`pucstj p`r Ucihduhh1 _ c 2
4 : 4 < > 3 7 5
b f+:
f8 h c f e t m ( x ) _PC 8 l ( ) g ` r ( i i f : 1 f) { _PC[ i R 8 i / (f+:) } _PC
∔
∔
∔
MINT@H@EÂJ ECFCTJH
LJ_Â^SH@ 4. 4.:. NI\^TIDSLIÜF H@E-F@TBJH NC < _JTÊBC^T@\ [:R [7R [::R [:7R [4:R [47R
?.?<<<<<<< ?.4??????? ?.<7777775 ?.3<<<<<<< ?.5??????? ?.=7777775
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?.:7777775 ?.<<<<<<<< ?.3??????? ?.77777775 ?.=<<<<<<<
p h ` t ( x , _PC , p lm 2 := , l ` h2" d h u c " , t y p c2"d " , h t y 2: , y hj d 2 "_" , bjif 2 " C bp i ri l jh N i s t r i d u t i ` f " ) 4 < e r i n ( ) 7 :
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### _ j r j b ct c r s l j h l u h j t i ` f . l j h l u h j t i ` f dy ` g bjxibub h i k c h i m ` ` n b ct m`n 4 # _ j rj b ct c rs t m c s c l j f t bct m` n < # S s if e : > x`8 x [ : R :? 3 x:8 x` 8 sub ( h ` e ( x x` , cxp ( : ) ) ) /f 7 bu x` , cxp (:) ) bu) 4 /f ) ? . 3 5 siebj8 sub ( ( h ` e ( x :
= 9 :? :: :4 :< :> :3 :7 :5
∔ ∔ ∔ ∔ ∔ ∔ ∔ ∔ ∔ ∕∕ ∕∕ C`8∔sub ( ( h ` e ( x∔x` , cxp ( : ) ) ) / ( x∔x ` ) ) +sub ( : / ( x∔x ` ) ) ∕ ( s i e b j ∕∕4∔bu) w m i h c ( jds (C`)62:C∔ 3) { bu8∔sub ( h ` e ( x∔x: , cxp ( : ) ) ) /f siebj8∔sub ( ( h ` e ( x∔x: , cxp (: ) )∔bu) ∕∕ 4/f ) ∕∕ ? . 3 C:8∔sub ( ( h ` e ( x∔x: , cx p ( : ) ) ) / ( x∔x : ) ) +sub ( : / ( x∔x : ) ) ∕ ( s i e b j ∕∕4∔bu) x48∔x: ∔(C: ∕ ( x` ∔x : ) ) / ( C`∔C: ) x`8∔x: x:8∔x4 MINT@H@EÂJ ECFCTJH
LJ_Â^SH@ 4. 4.:. NI\^TIDSLIÜF H@E-F@TBJH NC < _JTÊBC^T@\
:=
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C`8 C:
∔
} 4? x`8 x: bu , s i eb j ) 4: njtj . grjbc ( x ` , :9
∔
x` bu siebj : 473.4395 7.44>?5: ?.5=45?:5
Hj gufliüf nc nistriduliüf jlubuhjnj nc hj nistriduliüf H`e-F`rbjh<_, usjfn`
> : 4 < > 3 7 5 = 9 :? :: :4 :<
# ^mc l ub uh jt i vc n i s t r i d u t i ` f g u f l t i `f ` g H`e F `r bj h< _ N i s t r i d u t i ` f GPJ8 l ( ) g ` r ( i i f : 1 f) { z8 ( h ` e ( x [ i R x` ) bu) / siebj g z 8 :/ ( s q r t ( 4 p i ) ) cx p ( ?.3 zV4) w 8 : . / (:+?.<<475 jds ( z ) ) GPJ[ i R8 : g z ( ? . > < 7 : = < 7 w ?.:4?:757 w 4+?.9<54=? w < ) i g ( z8?){ GPJ[ i R8 : GPJ[ i R } } GPJ
∔
∔
∔ ∔ ∔
[:R [5R [:
∔
∔∔ ∕ ∔∔
?.?:749<57 ?.43??<3>7 ?.>7=579?? ?.7459==99 ?.597?7:3:
∔ ∕
∕
∕
∔ ∕ ∕ ∔
?.?4?<54<4 ?.45>>7?<3 ?.3?9??347 ?.7:<> ?.=34453:<
∕ ∕∕
?.?7<<34:: ?.<:74?43> ?.7>?<7?9: ?.9?497>=9
?.:9::>>:= ?.<4<3>7:? ?.33:77<73 ?.5?44:779 ?.97:3?>9?
∕ ∕∕
?.:9<=5594 ?.<5>47?=< ?.33:77<73 ?.5>93?=44 ?.99<3>454
?.4>:=>:55 ?.>?>>=:9: ?.3974?3>9 ?.57<<:939
p h ` t ( x , GPJ , t y p c2" h " , p lm 2 4? , l ` h2" r c n " , h t y 24 , x h j d2"" , yhjd2" L ub uh jt iv c p r ` d j d i h i t y " , bjif2"" ) 4 _PC , p lm 2 := , l ` h2" d h u c " , t y p c2"d " , h t y 2 :) < h i f c s ( x , F `rb jh <_ N i s t r i d u t i ` f ’ , ’ C b p i r i l j h > h c e c f n ( ’ d`tt` briem t ’ , h c e c f n2l ( ’H`e N i s t r i d u t i ` f ’ ) , h t y 2 : 14 , l ` h2l ( ’ r c n ’ , ’ dhu c ’ ) , l cx 2 ? . = , t c x t . g`ft 2>) 3 7 e r i n ( ) :
∔
MINT@H@EÂJ ECFCTJH
