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]klúk bm Mzr}blckrmz bm Jlmkjlcz Mjhkúeljcz q CMJ]J:E
CP]KRMZECRMEÃRLJCZ PC_C AC
MJHKHEÅC L
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Rmec 3
N}kjlhkmz bm crlcoam _mca
3+3+ Ac _mjc _mca
Ahz k²}emrhz rmcamz zm p}mbmk hrbmkcr jheh ahz p}khz bm }kc rmjc+
Ahz mkmrhz phzll~hz v3! ?! :! 2! + + +u s}m z}rimk ca jhkcr! zm aaceck k²}emrhz kc}rcamz$ma jmrh zm p}mbm jhkzlbmrcr h kh jheh }k k²}emrh kc}rca! khzhrhz kh ah dcrmehz,+
Acz hpmrcjlhkmz crle²mljcz bm cbljl²hk | e}alpaljcjl²hk zm p}mbmk dcjmr bmkrh bmahz k²}emrhz kc}rcamz pmrh ac rmzc | ac bl~lzl²hk khz aam~c c lkrhb}jlr ahz zli}lmkmzk²}emrhz7
jmrh7 : ∟ : 5 0!
kmicl~hz7 ? ∟ > 5 ∟2!
nrcjjlhkmz7 : ù = 5:
=+
Phr ckh! phbmehz jaczlffijcr ahz k²}emrhz bm ac eckmrc zli}lmkm7
Kc}rcamz7 K 5 v3! ?! :! 2! + + +uMkmrhz7 W 5 v+ + + ! ∟:! ∟?! ∟3! 0! 3! ?! :! 2! + + +uma jmrh g}kh jhk ahz mkmrhz phzll~hz | kmicl~hz
_cjlhkcamz7 S 5
e
k7 e! k ∄ W! k 5 0
Lrrcjlhkcamz7 zhk ahz k²}emrhz rmcamz s}m kh zhk rcjlhkcamz! jheh phr mgmepah∗?! π! m
_mcamz7 _ 5 Lrrcjlhkcamz & _cjlhkcamz
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? N}kjlhkmz bm crlcoam _mca
Rhbhz mzhz k²}emrhz cpcrmjmk jheh zha}jl²hk bm mj}cjlhkmz7t ∟ 3 5 0 ↟ t 5 3!
t & : 5 0 ↟ t 5 ∟:!?t & 3 5 0 ↟ t 5 ∟3
?!
t? 5 ? ↟ t 5 ³∗?!
t? & 3 5 0 ↟ t 5 ³∗∟3 (∄ _ ↟ Zhk ahz k²}emrhz Jhepamghz7J 5 vc & ol 7 c! o ∄ _! l 5
∗∟3u+
E²MRHBHZ BM _CWHKCELMKRH ECRME ²CRLJH
Ac bmffikljl²hk bm ahz k²}emrhz kc}rcamz mz ac zli}lmkm7
3 ∄ K+ Zl k ∄ K! ceol²mk pmrmkmjm k & 3
^mcehz j²heh n}kjlhkc mzc bmffikljl²hk7: ∄ K ⇟ : & 3 5 2 ∄ K!:(? (∄ K ⇟ :(? & 3 5 =(? (∄ K+
Mzc bmffikljl²hk bm ahz k²}emrhz kc}rcamz khz lkrhb}jm ma prhjmzh bm lkb}jjl²hk7
Bmehzrcjl²hk phr lkb}jjl²hk
Mz }kc ²mjkljc pcrc prhocr }kc cffirecjl²hk rmcalwcbc zhorm jcbc k²}emrh kc}rca7
Ac cffirecjl²hk mz ~mrbcbmrc pcrc k 5 3
Zl ac cffirecjl²hk mz ~mrbcbmrc pcrc k ∄ K! mkhkjmz ceol²mk mz ~mrbcbmrc pcrcz} z}jmzhr7 k & 3 ∄ K
Mzh lepaljc s}m ac cffirecjl²hk mz jlmrc pcrc hbh k²}emrh kc}rca k+
Jheh mgmepah! phbmehz prhocr phr lkb}jjl²hk s}mkl53
l 5k$k & 3,
?+
Bmehzrcjl²hk Blrmjc
Ac lbmc mz prhocr ac cffirecjl²hk blrmjcemkm
Phr mgmepah! pcrc bmehzrcr s}m $c ∟ o,$c & o, 5 c? ∟ o?!z²hah mkmehz s}m hpmrcr blrmjcemkm $c ∟ o,$c & o, 5 c? ∟ oc & co ∟ o? 5 c? ∟ o?+
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3+3+ Ac _mjc _mca :
Bmehzrcjl²hk phr jhkrcbljjl²hk h _mb}jlh cb coz}rb}e
S}mrmehz prhocr }kc dlp²hmzlz
Cz}elehz ah hp}mzh c ac dlp²hmzlz |
Rmrelkcehz jhk }kc jhkrcbljjl²hk
Jhkja}lehz s}m ac dlp²hmzlz mz ~mrbcbmrc
Jheh mgmepah! ehzrcrmehz s}m∗
? mz lrrcjlhkca7∗? ∄ S ↟ ∗
? 5c
onrcjjl²hk lrrmb}jloam+
∗? 5
c
o ↟? 5
c?
o?
↟c? 5 ?o?
↟c? mz pcr
↟c pcr
↟c 5 ?r
↟c? 5 2r 5 ?o? ↟ o? pcr ↟ o pcr+
Zl c | o zhk pcrmz! mkhkjmzc
okh p}mbm zmr }kc nrcjjl²hk lrrmb}jloam ↟ jhkrcbljjl²hk#∗
? (∄ S+
BMZLI]CABCBMZ! ^CAH_ COZHA]RH
Prhplmbcbmz bm Hrbmk bm _c!o!j ∄ _
3+ Z²hah }kc bm acz zli}lmkmz cffirecjlhkmz zm ~mrlffijc7 c 1 o! c 5 o! c ; o
?+ Zl c 1 o | o 1 j! mkhkjmz c 1 j
:+ Zl c 1 o mkhkjmz c & j 1 o & j pcrc hbh k²}emrh rmca j
2+ Zl c 1 o | j ; 0 mkhkjmz cj 1 ojZl c 1 o | j 1 0 mkhkjmz cj ; oj
Prhplmbcb+ Mkrm bhz k²}emrhz rmcamz blzlkhz mtlzmk lkffiklhz k²}emrhz rcjlhkcamz mlrrcjlhkcamz+
Bmffikljl²hk 3+3+3 Ma ^cahr Cozha}h bm t ∄ _ mz7
qtq 5 t! zl t ≭ 0!
∟t! zl t 1 0+
Rmkmehz ac bmffikljl²hk camrkcl~c7 qtq 5∗
t?+
Ac lbmc imhe²mrljc bma ~cahr cozha}h mz ac bm blzckjlc+ Phr mgmepah! ¸s}²m p}khz mz²ckc blzckjlc : bma 08 Ac rmzp}mzc mz qtq 5 :! mz bmjlr t 5 ³:+
Bmffikljl²hk 3+3+? Zl t! | ∄ _ ac blzckjlc mkrm t m | mz
b$t! |, 5qt
∟|q+
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2 N}kjlhkmz bm crlcoam _mca
Prhplmbcbmz bma ~cahr cozha}ht! | ∄ _
3+ qtq 5 0 ⇐⇟ t 5 0
?+ q ∟ tq 5 qtq:+ qt|q 5 qtqq|q2+ qt & |q ≢ qtq & q|q=+ qqtq ∟ q|q q ≢ qt ∟ |q
Lkmr~cahz mk _
Lkmr~cah colmrh7 $c! o, zhk hbhz ahz ~cahrmz bm t ca s}m c 1 t 1 o+
Lkmr~cah jmrrcbh7 Vc! oU zhk hbhz ahz ~cahrmz bm t ca s}m c ≢ t ≢ o+
Lkmr~cah zmel%colmrh7 Vc! o, zhk hbhz ahz ~cahrmz bm t ca s}m c ≢ t 1 o+
Lkmr~cah zmel%colmrh7 $c! oU zhk hbhz ahz ~cahrmz bm t ca s}m c 1 t ≢ o+
Lkmr~cah zmel%colmrh7 Vc! ∛, zhk hbhz ahz ~cahrmz bm t ca s}m c ≢ t 1 ∛+
Lkmr~cah colmrh7 $c! ∛, zhk hbhz ahz ~cahrmz bm t ca s}m c 1 t+
Lkmr~cah zmel%colmrh7 $∟∛! oU zhk hbhz ahz ~cahrmz bm t ca s}m t ≢ o+
Lkmr~cah colmrh7 $∟∛! o, zhk hbhz ahz ~cahrmz bm t ca s}m t 1 o+
Lkmr~cah colmrh7 $∟∛! ∛, zhk hbhz ahz ~cahrmz bm t ca s}m t ∄ _+
Khc+ ∛ kh mz }k k²}emrh rmca! lkbljc s}m ma lkmr~cah zm mtlmkbm zlk a²ijelm+
Bmffikljl²hk 3+3+: Zmc C }k jhkg}kh bm k² }emrhz
Ma z}prmeh bm C mz ma emkhr mamemkh bm _ ec|hr h li}ca c hbhz ahz mamemkhzbm C+ Mz ac emkhr bm acz jhcz z}pmrlhrmz+
Ma ²ijkffieh bm C mz ma ec|hr mamemkh _ emkhr h li}ca c hbhz ahz mamemkhz bmC+ Mz ac ec|hr bm acz jhcz lknmrlhrmz+
Zl ma z}prmeh pmrmkmjm c C mz ceol²mk ma e²ctleh bm C+
Zl ma ²ijkffieh pmrmkmjm c C mz ceol²mk ma e²ijkleh bm C+
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3+?+ N}kjlhkmz mamemkcamz =
3+?+ N}kjlhkmz mamemkcamz
]kc n}kjl²hk n mk ahz rmcamz mz }kc rmiac s}m czlikc c jcbc k²}emrh t }k ²}kljhk²}emrh rmca n $t,7
t ↟ | 5 n $t,
t7 ~crlcoam lkbmpmkblmkm! cri}emkh h mkrcbc+|7 ~crlcoam bmpmkblmkm! ~cahr bm ac n}kjl²hk h zcalbc+
Bmffikljl²hk 3+?+3 Zmc B ⊏ _ }k jhkg}kh+ ]kc n}kjl²hk n jhk bhelklh B mz }kc rmiac s}m czlikc }k ² }kljh k² }emrh rmca n $t, c jcbc k² }emrh t mk B+B mz ma bhelklh bm n 7 Bhe$n , 5 B+
Ma jhkg}kh bm hbhz ahz ~cahrmz bm n nhreck ma rckih h ac lecimk bm n 7Le$n , 5 v| ∄ _ 7 ∊ t ∄ B! n $t, 5 |u+
Khcjl²hk7 n 7 B ⊏ _ ↟ _+
Pcrc ~lz}calwcr }kc n}kjl²hk mz e}| ²}la blo}gcr z} ir²cffijc7
Bmffikljl²hk 3+?+? Zmc n }kc n}kjl² hk jhk bhelklh B! ma jhkg}kh bm p}khz$t! n $t,, ∄ _? jhk t ∄ B nhreck ac ir²cffijc bm n +
¸J²heh phbmehz rmjhkhjmr ac ir²cffijc bm }kc n}kjl²hk8 Phbmehz blo}gcr a²ijkmcz ~mrl%jcamz! }kc j}r~c s}m jhrc jcbc a²ijkmc ~mrljca jheh e}jdh }kc ~mw! mz ac ir²cffijc bm }kcn}kjl²hk+
Phr mgmepah! |? 5 t lmkm bhz zha}jlhkmz | 5 ³∗t! phr ah s}m dc| bhz jhrmz jhk
jcbc a²ijkmc ~mrljca pcrc t ; 0! phr ah ckh kh mz ac ir²cffijc bm }kc n}kjl²hk+
Zl blo}gcehz a²ijkmcz ~mrljcamz! phbmehz hozmr~cr s}m ma bhelklh bm }kc n}kjl²hk mzma jhkg}kh bm t0 s}m ~mrlffijck s}m ac a²ijkmc ~mrljca t 5 t0 jhrc ac ir²cffijc+ Bm ac elzecnhrec! zl blo}gcehz a²ijkmcz dhrlwhkcamz! phbmehz ~lz}calwcr ac lecimk jheh ma jhkg}khbm |0 s}m ~mrlffijck s}m ac a²ijkmc dhrlwhkca | 5 |0 jhrc ac ir²cffijc+
]kc n}kjl²hk zm p}mbm bmffiklr emblckm }kc n²hre}ac! }kc ir²cffijc h }kc bmzjrlpjl²hk+
Bmffikljl²hk 3+?+: Zmc n 7 B ↟ _ }kc n}kjl² hk7
3+ n mz }kc n}kjl² hk lk|mjl~c zl t3 5 t? lepaljc s}m n $t3, 5 n $t?,!h! bm eckmrc ms}l~camkm! zl n $t3, 5 n $t?, mkhkjmz t3 5 t?!mzh mz! zl czlikc cri}emkhz blzlkhz c ~cahrmz blzlkhz+
?+ n 7 B ↟ _ zm bljm zhorm|mjl~c zl Le$n , 5 _+n 7 B ↟ B zm bljm zhorm|mjl~c zl Le$n , 5 B+
:+ n mz }kc n}kjl²hk ol|mjl~c h }kh%c%}kh zl mz c ac ~mw lk|mjl~c | zhorm|mjl~c+
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> N}kjlhkmz bm crlcoam _mca
3+:+ A²ijelmz
Zmc n $t, }kc n}kjl²hk bmffiklbc pcrc hbh t jmrjc bm t0! pmrh kh kmjmzcrlcemkm mk maprhplh t 5 t0! zl ma ~cahr n $t, bm n zm cprhtlec ca k²}emrh A j}ckbh t zm cprhtlec cak²}emrh t0! bmjlehz s}m A mz ma a²ijelm bm n $t, j}ckbh t lmkbm c t0+
^mcehz ma jhephrcelmkh bm ac zli}lmkm n}kjl²hk mk acz prhtlelbcbmz bm t 5 =7
n $t, 5?t? ∟ 9t & :
t ∟ :
t ?+< ?+<<<< ?+<<<<<< ↟ : ← :+000003 :+0003 :+3
n $t, 2+4 2+<<<4 2+<<<<<4 ↟ = ← =+00000? =+000? =+?a²ijet↟:
n $t, 5 =
n $t, 5 ?t
?
∟ 9t & :t ∟ : 5 $?t ∟ 3,$t ∟ :,t ∟ : 5t5:
?t ∟ 3+
Bmffikljl²hk 3+:+3 $Ymlmrzrczz! bmffikljl²hk ∟ β, Zmc n }kc n}kjl²hk bmffiklbc mk }k lkmr~cah colmrh s}m jhklmkm c t0 $mtjmph phzloamemkm mk t0, | zmc A }k k² }emrhrmca! mkhkjmz
a²ijet↟t0
n $t, 5 A!
zl ∀ ; 0! ∊β ; 0! 0 1 qt ∟ t0q 1 β ⇟ qn $t, ∟ Aq 1 ν+
Khc+ Phbmehz }lalwcr ac bmffikljl²hk ν∟
β pcrc prhocr s}m a²ijet↟:
$?t∟
3, 5 =+
Khc+ ]kc rmiac imkmrca e}| ²}la mz mzjrlolr n $t, 5 A | mtprmzcrah mk ²mrelkhz bmt ∟ t0 hbh ah s}m phbcehz! emblckm ac li}cabcb t 5 $t ∟ t0, & t0+
Khc+ Mk ac bmffikljl²hk ν∟β! A | t0 zhk k²}emrhz ffiklhz+ Rmkmehz bmffikljlhkmz zlelacrmzpcrc t ↟ ³∛ | ceol²mk zl ma ~cahr bma a²ijelm kh mz }k k²}emrh ffiklh+ Rceol²mk phbmehzbmffiklr a²ijelmz acmrcamz7 phr ac bmrmjdc bm t07 a²ije
t↟t&
0
n $t, | phr ac lws}lmrbc bm t07
a²ijet↟t
∟
0
n $t,+
Nhrecz Lkbmmrelkcbcz
∛ ∟ ∛!∛∛ ! ∛ ¹ 0!
0
0! ∛0! 3∛! 00+
C emkhz s}m aami}mehz c }kc lkbmmrelkcjl²hk! mkmehz acz zli}lmkmz prhplmbcbmz+
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3+:+ A²ijelmz 9
PrhplmbcbmzZ}phkicehz s}m a²ije
t↟t0
n $t, | a²ijet↟t0
i$t, mtlzmk! pcrc t0 ffiklh h ³∛
3+ a²ijet↟t0
$j3n $t, & j?i$t,, 5 j3 a²ijet↟t0
n $t, & j? a²ijet↟t0
i$t,! j3! j? ∄ _
?+ a²ijet↟t0
$n $t,i$t,, 5 a²ijet↟t0
n $t, a²ijet↟t0
i$t,
:+ a²ijet↟t0
3
n $t,
5
3
a²ijet↟t0
n $t,! zl a²ije
t↟t0
n $t, 5 0
2+ _miac bma rmmepacwh7 zl n | i jhlkjlbmk pcrc hbh t jmrjc bm t0 $kh kmjmzcrlcemkmlkja}|mkbh c t0,! mkhkjmz a²ije
t↟t0
n $t, 5 a²ijet↟t0
i$t,
=+ _miac bm ac n}kjl²hk jhep}mzc7 zl a²ijet↟t0
n $t, 5 A | a²ijet↟A
d$t, 5 d$A,! mkhkjmz
a²ijet↟t0
d$n $t,, 5 d$ a²ijet↟t0
n $t,,
A²ijelmz o²czljhz+ Zmc t0 ffiklh
c, a²ijet↟t0
j 5 j! a²ijet↟³∛
j 5 j
o, a²ijet↟t0
t 5 t0! a²ijet↟³∛
t 5 ³∛! a²ijet↟³∛
3
t5 0
j, a²ijet↟t0
tk 5 tk0
! k∄K
b, a²ijet↟t0
k
∗t 5 k
∗t0! $∀t0 mk z} bhelklh,
m, a²ijet↟t0
n $t, 5 n $t0, ↟ n}kjlhkmz rlihkhe²mrljcz mk z} bhelklh
Amec 3+:+? $Amec bma Zckbyljd, Zmc L }k lkmr~cah ca s}m t0 ∄ L + Zmck n ! i| d n}kjlhkmz bmffiklbcz mk L ! mtjmph phzloamemkm mk ma prhplh t0+ Zl a²ije
t↟t0
i$t, 5
a²ijet↟t0
d$t, 5 A |! ∀t ∄ L ! t 5 t0! d$t, ≢ n $t, ≢ i$t,+ Mkhkjmz
a²ijet↟t0n $t, 5 A+
Khc+ Phbmehz }lalwcr ma amec bma Zckbyljd pcrc prhocr s}m a²ijet↟0
zmk t
t5 3+
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4 N}kjlhkmz bm crlcoam _mca
3+2+ Jhklk}lbcb
]kc n}kjl²hk jhklk}c mz }kc n}kjl²hk pcrc ac s}m! lk}ll~cemkm! pms}mkˑhz jceolhzmk ac mkrcbc prhb}jmk pms}mkˑhz jceolhz mk ac zcalbc
Bmffikljl²hk 3+2+3 Zmc n }kc n}kjl² hk bmffiklbc mk $t0 ∟ p!t0 & p,! p ; 0
n zm bljm jhkrlk}c mk t0 ⇐⇟ a²ijet↟t0
n $t, 5 n $t0,
K²hmzm s}m ac n}kjl²hk bmom mzcr bmffiklbc mk t0 pcrc phbmr zmr jhklk}c mk mzm p}kh+ Zlac n}kjl²hk kh mz jhklk}c mk t0 bmjlehz mkhkjmz s}m mz blzjhklk}c mk bljdh p}kh7
n $t, mz blzjhklk}c mk t0 zl
a²ijet↟t0
n $t, kh mtlzm! h
ma a²ijelm mtlzm pmrh kh mz li}ca c n $t0,
Bmffikljl²hk 3+2+? ]kc n}kjl² hk n $t, mz jhklk}c mk
_! zl mz jhklk}c mk jcbc p}kh
$c! o,! zl mz jhklk}c mk jcbc p}kh bma lkmr~cah
Vc! oU! zl mz jhklk}c mk $c! o, |
n $c, 5 a²ijet↟c&
n $t,! n $o, 5 a²ijet↟o∟
n $t,
Prhplmbcbmz o²czljczZmck n! i jhklk}cz mk t0! mkhkjmz acz zli}lmkmz n}kjlhkmz zhk jhklk}cz mk t0
3+ j3n & j?i! j3! j? ∄ _?+ n i
:+3
n ! zl n $t0, 5 0
2+ N}kjl²hk jhep}mzc7 zl i mz jhklk}c mk t0 | n mz jhklk}c mk i$t0,! mkhkjmz n ◮imz jhklk}c mk t0
Cai}kcz n}kjlhkmz jhklk}czAcz zli}lmkmz n}kjlhkmz zhk jhklk}cz mk z}z bhelklhz
c, p$t, phalkhelhz!p$t,
s$t,n}kjlhkmz rcjlhkcamz! k
∗t
o, n}kjlhkmz rlihkhe²mrljcz7 zmk$t,! jhz$t,! ck$t,! crjzmk$t,! + + +
j, n}kjlhkmz dlpmro²haljcz7 zmkd$t,! jhzd$t,! + + +
b, mtp$t, | ak$t,
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3+2+ Jhklk}lbcb <
Rmhrmec 3+2+: $Rmhrmec bma ^cahr Lkmremblh, Zl n $t, mz }kc n}kjl² hk jhklk}c mk Vc! oU | F mz j}cas}lmr k² }emrh mkrm n $c, | n $o,! mkhkjmz mtlzm }k ~cahr j ∄ Vc! oU
s}m ~mrlffijc n $j, 5 F +
Rmhrmec 3+2+2 $Rmhrmec bm Ohawckh, Zl n $t, mz }kc n}kjl² hk jhklk}c mk Vc! oU | n $c, ¹ n $o, 1 0! mkhkjmz mtlzm }k ~cahr j ∄ $c! o, s}m ~mrlffijc n $j, 5 0+
Bmffikljl²hk 3+2+= ]kc n}kjl² hk n zm bljm cjhrcbc zl ma jhkg}kh bm z}z ~cahrmz mz² c cjh%cbh+ Mzh mz! zl mtlzm }k k² }emrh E ; 0 ca s}m qn $t,q ≢ E! pcrc hbh t mk z} bhelklh+
Rmhrmec 3+2+> $Rmhrmec bma ^cahr Mtrmeh, Zl n $t, mz }kc n}kjl² hk jhklk}c mk ma lkmr~cah jmrrcbh Vc! oU! mkhkjmz n cajckwc z}z ~cahrmz e²ctleh | e²ijkleh+ Mzbmjlr! mtlzmk }khz ~cahrmz te | tE mk Vc! oU ca s}m7
n $tE , ≭ n $t, ≭ n $te, pcrc hbh t ∄ Vc! oU+
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Rmec ?
J²caj}ah Blnmrmkjlca mk }kc
~crlcoam
?+3+ Bmrl~cbcz
Ac bmrl~cbc khz prhphrjlhkc }kc eckmrc bm jcaj}acr ac czc bm jceolh bm }kc n}kjl²hk
Jcaj}acehz ac ~mahjlbcb emblc jheh ac rcw²hk mkrm ac blzckjlc rmjhrrlbc mk }k lkmr~cahbm lmeph d | ac ahkil}b bma lkmr~cah bm lmeph+
~emblc 5t$r & d, ∟ t$r,
d!
mzm ~cahr mz ac pmkblmkm e bm ac rmjc s}m pczc c rc~²mz bm ahz p}khz $r! t$r,, |$r & d! t$r & d,,+ Zl s}mrmehz jcaj}acr ac ~mahjlbcb mk }k lmeph r bmomehz hecr maa²ijelm
~$r, 5 a²ijed↟0
t$r & d, ∟ t$r,
d5
b
brt$r,
Bmffikljl²hk ?+3+3 ]kc n}kjl² hk n zm bljm bmrl~coam mk t ⇐⇟
a²ijed↟0
n $t & d, ∟ n $t,
d
mtlzm | mz }k k² }emrh ffiklh+
Zl n mz bmrl~coam! mkhkjmz n $t, 5b
btn $t, 5 a²ije
d↟0
n $t & d, ∟ n $t,
dmz ac bmrl~cbc bm
n mk t+
Khc+ Ac n}kjl²hk n mtlzm mk ahz p}khz bma bhelklh bm n ca s}m ma a²ijelm mtlzm | mzffiklh+
Bmffikljl²hk ?+3+? $Bmn+ camrkcl~c,
n $t0, 5 a²ijet↟t0
n $t, ∟ n $t0,
t∟
t0
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? J²caj}ah Blnmrmkjlca mk }kc ~crlcoam
_mjc ckimkm
Ac rmjc s}m pczc phr $t0! n $t0,, jhk pmkblmkm e 5 n
$t0, mz ac rmjc ckimkm c n $t,mk t07 | 5 e$t ∟ t0, & n $t0,+
Prhplmbcbmz
3+ $j3n & j?i, 5 j3n & j?i! j3! j? ∄ _+
?+ $n ¹ i, 5 n i & n i
:+n
i
5
n i ∟ n i
i?
Rmhrmec ?+3+: $_miac bm ac Jcbmkc, Zl i mz bmrl~coam mk t | n ah mz mk i$t,!mkhkjmz ac n}kjl² hk jhep}mzc n ◮ i mz bmrl~coam mk t! | ~mrlffijc
$n ◮ i,$t, 5 n $i$t,,i$t,
Khc+ Jhk ac rmiac bm ac jcbmkc phbmehz bmehzrcr s}m
$n ∟3,$t, 53
n $n ∟3$t,,+
Phbmehz }lalwcr mzc lbmklbcb pcrc jcaj}acr acz bmrl~cbcz $crjck t,
| $ak t,
+Bmrl~cbcz o²czljcz
3+ j 5 0
?+ $tk, 5 ktk∟3
:+ $mt, 5 mt! $ahi t, 53
t
2+ $zmk t, 5 jhz t! $jhz t, 5 ∟zmk t! $ck t, 53
jhz? t
=+ $crjck t, 5 33 & t?
! $crjzmk t, 5 3∗3 ∟ t?
>+ $t, 5 jhzd! $jhzd t, 5 zmkd t
Rmhrmec ?+3+2
n bmrl~coam ⇟ n jhklk}c
Rmhrmec ?+3+= $Rmhrmec bm _haam, Zmc n om bmrl~coam mk $c! o, | jhklk}c mk Vc! oU+Zl n $c, 5 n $o,! mkhkjmz mtlzm ca emkhz }k k² }emrh j ∄ $c! o, ca s}m
n $j, 5 0+
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?+3+ Bmrl~cbcz :
Rmhrmec ?+3+> $Rmhrmec bma ^cahr Emblh, Zmc n om bmrl~coam mk $c! o, | jhklk}c mk Vc! oU! mkhkjmz mtlzm ca emkhz }k k² }emrh j ∄ $c! o, ca s}m
n $j, 5n $o, ∟ n $c,
o ∟ c!
h! ms}l~camkmemkm n $o, ∟ n $c, 5 n $j,$o ∟ c,+
Rmhrmec ?+3+9 $_miac bm A“Dhˎplca, Zmck n | i n}kjlhkmz bmrl~coamz mk $c! o,! mt%
jmph phzloamemkm mk ma p}kh t0 ∄ $c! o,+ Zl a²ijet↟t0
n $t,
i$t,mz ac lkbmmrelkcjl²hk
0
0! mk%
hkjmz
a²ijet↟t0
n $t,
i$t,5 a²ije
t↟t0
n $t,
i$t,!
zlmeprm s}m a²ijet↟t0
n $t,
i$t,mtlzc h zmc lkffiklh+
Mtmkzlhkmz
Ac rmiac bm A“Dhˎplca zm p}mbm cpaljcr mk ahz jczhz zli}lmkmz7
Zl ac lkbmmrelkcjl²hk mz ∛
∛jhk hbhz ahz zlikhz phzloamz+
Zl ma a²ijelm mz t0
↟ ³∛C ahz a²ijelmz acmrcamz
Bmrl~cjl²hk lepa²ijjlc
Zl ac mj}cjl²hk ~lmkm bcbc bm eckmrc lepa²ijjlc! N $t! |, 5 0! dc| s}m bmrl~crac rmzpmjh
c t | c pcrlr bm cd²ij homkmrb|
bt+
Bmrl~cbcz bm hrbmk z}pmrlhr
Phbmehz dcaacr ac bmrl~cbc bm }kc bmrl~cbc7
b?n
bt?5 n $t,!
b:n
bt:5 n $t,! ¹ ¹ ¹ !
bkn
btk5 n $k,$t,
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2 J²caj}ah Blnmrmkjlca mk }kc ~crlcoam
?+?+ Mtrmehz
Bmffikljl²hk ?+?+3 Zmc n }kc n}kjl² hk bmffiklbc mk }k lkmr~cah L 7
n $te, mz ma e²ijkleh iahoca $h cozha}h, bm n mk L zl n $te, ≢ n $t,! ∀t ∄ L
n $tE , mz ma e²ctleh iahoca $h cozha}h, bm n mk L zl n $tE , ≭ n $t,! ∀t ∄ L
Khc+ _mjhrbmehz s}m zl n mz jhklk}c! mk }k lkmr~cah jmrrcbh | cjhcbh Vc! oU! acn}kjl²hk zlmeprm cajckwc z} e²ctleh | e²ijkleh iahoca +
Bmffikljl²hk ?+?+? Zmc n }kc n}kjl² hk bmffiklbc mk }k lkmr~cah L ! zl mkmehz }k lkmr%~cah colmrh L 3 jhkmklmkbh c t0
n $t0, mz }k e²ijkleh ahjca bm n mk L zl n $t0, ≢ n $t,! ∀t ∄ L 3
n $t0, mz }k e²ctleh ahjca $h rmacl~h, bm n mk L zl n $t0, ≭ n $t,! ∀t ∄ L 3
Bmffikljl²hk ?+?+: Zmc n }kc n}kjl² hk bmffiklbc mk t0+ n lmkm }k p}kh jr²ijljh mk t0
zl
n $t0, 5 0 h n $t0, kh mtlzm
Rmhrmec ?+?+2 Zl n lmkm }k e² ctleh h e²ijkleh ahjca mk t0! mkhkjmz t0 mz }k p}khjr²ijljh bm n +
J²caj}ah bm mtrmehz iahocamz bm }kc n}kjl²hk mk }k lkmr~cah jmrrcbh Vc! oU
3+ Dcaac ahz p}khz jr²ijljhz bm n mk $c! o,7 n $t0, 5 0 h n $t0, kh mtlzc
?+ M~ca²}c n mk jcbc p}kh jr²ijljh bm $c! o,
:+ M~ca²}c n mk ahz mtrmehz bma lkmr~cah7 n $c, | n $o,
2+ Ma emkhr ~cahr mz ma e²ijkleh iahoca | ma ec|hr! ma e²ctleh iahoca
Bmffikljl²hk ?+?+=
n mz }kc n}kjl²hk jrmjlmkm mk }k lkmr~cah L zl
∀t3! t?
∄L jhk t3 1 t? mkmehz
s}m n $t3, 1 n $t?,+
n mz }kc n}kjl²hk bmjrmjlmkm mk }k lkmr~cah L zl ∀t3! t? ∄ L jhk t3 1 t?
mkmehz s}m n $t3, ; n $t?,+
Rmhrmec ?+?+> Zmc n }kc n}kjl²hk jhklk}c mk }k lkmr~cah jmrrcbh Vc! oU | bmrl~coammk $c! o,
3+ Zl n $t, ; 0! ∀t ∄ $c! o, mkhkjmz n mz jrmjlmkm mk Vc! oU+
?+ Zl n $t, 1 0! ∀t ∄ $c! o, mkhkjmz n mz bmjrmjlmkm mk Vc! oU+
:+ Zl n $t, 5 0! ∀t ∄ $c! o, mkhkjmz n mz jhkzckm mk Vc! oU+
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?+?+ Mtrmehz =
Jrlmrlh bm ac prlemrc bmrl~cbc
t 1 t0 t ; t0 t0 $p}kh jr²ijljh,
n $t,
∟ & e²ijkleh ahjca& ∟ e²ctleh ahjca∟ ∟ kh mtrmeh ahjca& & kh mtrmeh ahjca
Bmffikljl²hk ?+?+9 Zmc n bmrl~coam mk }k lkmr~cah colmrh L + Ac ir² cffijc bm n mz
jhk~mtc $j² hkjc~c dcjlc crrloc, mk L zl n mz jrmjlmkm+
j²hkjc~c$j² hkjc~c dcjlc cocgh, mk L zl n mz bmjrmjlmkm+
Rmhrmec ?+?+4 Zmc n }kc n}kjl² hk bmrl~coam bhz ~mjmz mk }k lkmr~cah colmrh L
Zl n $t, ; 0! ∀t ∄ L ! mkhkjmz ac ir² cffijc bm n mz jhk~mtc mk L +
Zl n $t, 1 0! ∀t ∄ L ! mkhkjmz ac ir² cffijc bm n mz j² hkjc~c mk L +
Bmffikljl²hk ?+?+< Zmc n }kc n}kjl² hk jhklk}c mk }k lkmr~cah colmrh L | zmc t0 ∄ L +
n lmkm }k p}kh bm lkmtl²hk mk t0 zl ac jhkjc~lbcb jceolc mk t0 $jhk~mtc →j² hkjc~c,+
Rmhrmec ?+?+30 Zl t0 mz }k p}kh bm lkmtl² hk bm n ! mkhkjmz n
$t0, 5 0 h n
$t0,kh mtlzm+
Rmhrmec ?+?+33 Zmc n }kc n}kjl²hk ca s}m n $t0, 5 0 | bmrl~coam bhz ~mjmz mk }k lkmr~cah colmrh jhkmklmkbh c t0
zl n $t0, ; 0! mkhkjmz n lmkm }k e²ijkleh ahjca mk t0+
zl n $t0, 1 0! mkhkjmz n lmkm }k e² ctleh ahjca mk t0+
Zl n $t0, 5 0 ma jrlmrlh kh n}kjlhkc! p}mbm zmr j}cas}lmr jhzc+
n $t0, n $t0, ir²cffijc
& ∟ jrmjlmkm! j²hkjc~c∟ ∟ bmjrmjlmkm! j²hkjc~c& & jrmjlmkm! jhk~mtc∟ & bmjrmjlmkm! jhk~mtc
0 & e²ijkleh ahjca0 ∟ e²ctleh ahjca
0 0 8
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> J²caj}ah Blnmrmkjlca mk }kc ~crlcoam
?+:+ Ir²cffijcz
3+ Bhelklh?+ Lkmrzmjjl²hk jhk ma mgm t ↟ n $t, 5 0
Lkmrzmjjl²hk jhk ma mgm | ↟ n $0, 5 |
:+ Zlemr²ijcz
n $∟t, 5 &n $t, ↟ pcrn $∟t, 5 ∟n $t, ↟ lepcr
Pmrlhbljlbcb ↟ n $t & R , 5 n $t,
2+ Cz²ijkhcz7
^mrljca
↟a²ijet↟t0
n $t, 5
³∛Dhrlwhkca ↟ a²ijet↟³∛
n $t, 5 D
Hoalj}c ↟ a²ijet↟³∛
n $t, ∟ $et & o, 5 0 ↟ e 5 a²ijet↟∛
n $t,
t! o 5 a²ije
t↟∛$n $t, ∟ et,
=+ Jhklk}lbcb7 a²ijet↟t0
n $t, 5 n $t0,
>+ Bmrl~cbc7 ehkhhk²ijc | p}khz jr²ijljhz
n $t, ; 0↟ jrmjlmkmn $t, 1 0 ↟ bmjrmjlmkmn $t, 5 0 h n $t, kh mtlzc ↟ p}khz jr²ijljhz
9+ E²ctlehz | e²ijklehz ahjcamz7 t0 ↟ p}kh jr²ijljh
n $t0, 5 0! n $t0, ; 0 ↟ e²ijkleh ahjcan $t0, 5 0! n $t0, 1 0 ↟ e²ctleh ahjca
n $t, 7 ∟ ↟ & ↟ e²ijkleh ahjca
n $t, 7 & ↟ ∟ ↟ e²ctleh ahjca
4+ Jhkjc~lbcb
n $t, ; 0 ↟ jhk~mtcn $t, 1 0 ↟ j²hkjc~c
<+ P}khz bm lkmtl²hk+ Ac jhkjc~lbcb jceolc+ n $t0, 5 0 h n $t0,
30+ E²ctlehz | e²ijklehz iahocamz
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?+2+ Phalkhelh bm Rc|ahr 9
?+2+ Phalkhelh bm Rc|ahr
Ac lbmc mz cprhtlecr }kc n}kjl²hk n $t, phr }k phalkhelh P $t,+ Ma phalkhelh bmRc|ahr mz ma phalkhelh s}m emghr cprhtlec c }kc n}kjl²hk mk }k p}kh t0
Zl cprhtlecehz n $t, phr }kc jhkzckm ↟ P $t, 5 n $t0,
}kc rmjc ↟ P $t, 5 n $t0, & n $t0,$t ∟ t0,
Bmffikljl²hk ?+2+3 Zl n mz bmrl~coam k ~mjmz mk t0! mkhkjmz ma phalkhelh
P k$t, 5 n $t0, & n $t0,$t ∟ t0, &n $t0,
?#$t ∟ t0,? & ¹ ¹ ¹ n $k,$t0,
k#$t ∟ t0,k
mz ma phalkhelh bm Rc|ahr bm ircbh k bm n jmkrcbh mk t0+
Khc+ Pcrc t0 5 0 ma phalkhelh ceol²mk zm aacec Phalkhelh bm Ecj Ac}rlk+
M__H_
Ma phalkhelh mz }kc cprhtlecjl²hk bm n $t,! phr ah s}m zm jhemm }k mrrhrq_k$t,q 5 qn $t, ∟ P $t,q+ Mtlzmk e}jdcz n²hre}acz pcrc mzm mrrhr! pmrh ac lbmc mz s}mhbcz maacz ~mrlffijck
a²ijet↟t0
n $t, ∟ P k$t,
$t ∟ t0,k5 0+
↟ _k$t, 5 h
$t ∟ t0,k
+ Khcjl²hk7 n $t, 5 h
i$t,
j}ckbh t ↟ t0 ⇐⇟
a²ijet↟t0
n $t,
i$t,5 0+
Mk ma zli}lmkm mhrmec mkmehz }kc n²hre}ac pcrc ma mrrhr q_k$t,q7
Rmhrmec ?+2+? Zmc n $t, }kc n}kjl² hk bmrl~coam k & 3 ~mjmz mk }k lkmr~cah colmrh L !mkhkjmz ∀t0! t ∄ L mkmehz s}m
n $t, 5 P k$t, & _k$t, 5 n $t0, & n $t0,$t ∟ t0, & ¹ ¹ ¹ n $k,$t0,
k#$t ∟ t0,k & _k$t,!
_k$t, 5 n $k&3,$λ,
$k & 3,#$t ∟ t0,$k&3,! λ mz }k p}kh mk ma lkmr~cah colmrh bmffiklbh phr t0 | t+
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Rmec :
Z}jmzlhkmz | Zmrlmz
:+3+ Z}jmzlhkmz bm k²}emrhz rmcamz
Bmffikljl²hk :+3+3 ]kc z}jmzl²hk bm k²}emrhz rmcamz vcku mz }kc cpaljcjl² hk s}mczlikc c jcbc k ∄ K }k k² }emrh rmca7
ck 7 K↟ _
c3! c?! c: + + + zhk ahz ²mrelkhz bm ac z}jmzl²hk+
ck mz ma ²mrelkh imkmrca+Ac z}jmzl²hk ceol²mk p}mbm mepmwcr mk k 5 07 c0! c3! c? + + ++
Bmffikljl²hk :+3+? ]kc z}jmzl² hk vcku zm bljm jhk~mrimkm zl a²ijek↟∛
ck 5 A! pcrc A
ffiklh+
Ma a²ijelm bm }kc z}jmzl² hk vcku mz A zl pcrc hbh ; 0 ∊ K ∄ K ca s}m zl k ; K ⇟qck ∟ Aq 1 ν+ $Dc| }kc bmffikljl² hk camrkcl~c pcrc A lkffiklh+,
Zl ac z}jmzl²hk kh mz jhk~mrimkm! bmjlehz s}m mz bl~mrimkm+Acz prhplmbcbmz bm ahz a²ijelmz bm z}jmzlhkmz zhk acz elzecz s}m acz prhplmbcbmz bm ahza²ijelmz bm n}kjlhkmz+
Pcrc dcaacr ma a²ijelm bm }kc z}jmzl²hk phbmehz }lalwcr cai}kcz ²mjkljcz jheh7
Ma jhkjmph bm a²ijelm bm }kc n}kjl²hk7
Zmc n $t, }kc n}kjl²hk | vcku ac z}jmzl²hk n $k, 5 ck+
Zl a²ijet↟∛
n $t, 5 A! mkhkjmz a²ijek↟∛
ck 5 A
Phbmehz }lalwcr hbcz acz dmrrcelmkcz pcrc jcaj}acr ma a²ijelm bm }kc n}kjl²hk!jheh! phr mgmepah! ac rmiac bm A“Dhˎplca+
Ma amec bma Zckbyljd pcrc z}jmzlhkmz7
Zl a²ijek↟∛
ck 5 a²ijek↟∛
ok $ffiklc h lkffiklc, | vjku ~mrlffijc s}m
ck ≢ jk ≢ ok! ∀ k ∄ K!
mkhkjmz a²ijek↟∛
jk 5 a²ijek↟∛
ck 5 a²ijek↟∛
ok+
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? Z}jmzlhkmz | Zmrlmz
Bmffikljl²hk :+3+: ]kc z}jmzl² hk vcku zm bljm7
3+ cjhcbc z}pmrlhremkm zl ∊
J ∄_ ca s}m ck
≢J +
?+ cjhcbc lknmrlhremkm zl ∊ J ∄ _ ca s}m ck ≭ J +
:+ cjhcbc zl mz cjhcbc z}pmrlhremkm m lknmrlhremkm$ ∊ J 3! J ? ∄ _! + s+ J 3 ≢ ck ≢ J ?,+
Bmffikljl²hk :+3+2 ]kc z}jmzl² hk vcku zm bljm7
3+ ehk² hhkc jrmjlmkm zl ck 1 ck&3 $kh bmjrmjlmkm zl ck ≢ ck&3,+
?+ ehk² hhkc bmjrmjlmkm zl ck ; ck&3 $kh jrmjlmkm zl ck ≭ ck&3,+
:+ ehk²hhkc! zl mz }kh bm ahz jczhz prm~lhz+
Rmhrmec :+3+=
vcku ehk² hhkc | cjhcbc ⇟ vcku jhk~mrimkm
Rmhrmec :+3+> $Jrlmrlh bm Zhaw, Zl acz z}jmzlhkmz vcku | voku ~mrlffijck }kh bmahz zli}lmkmz cpcrcbhz7
3+ voku mz ehk² hhkc jrmjlmkm jhk a²ijek↟∛
ok 5 ∛!
?+ voku mz ehk² hhkc bmjrmjlmkm! jhk ok 5 0 pcrc jcbc k ∄ K |
a²ijek↟∛ck 5 a²ijek↟∛ok 5 0+
Zlmeprm s}m a²ijek↟∛
ck&3 ∟ ck
ok&3 ∟ ok5 A! mtlzc pcrcr A ffiklh h lkffiklh! mkhkjmz
a²ijek↟∛
ck
ok5 a²ije
k↟∛
ck&3 ∟ ck
ok&3 ∟ ok
Rmhrmec :+3+9 $N²hre}ac bm Zlralki,
a²ijek↟∛
k#
kkm∟k∗
?πk5 3
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:+?+ Zmrlmz bm k²}emrhz rmcamz :
:+?+ Zmrlmz bm k²}emrhz rmcamz
]kc zmrlm mz ac z}ec bm }kc z}jmzl²hk bm ²mrelkhz
Phr mgmepah ↟ ac z}ec imhe²mrljc7K k50
rk 5rK &3 ∟ 3
r ∟ 3+
Bmffikljl²hk :+?+3 Zmc vcku }kc z}jmzl² hk! }kc zmrlm$lkffiklc, mz ac z}ec bm hbhz z}z²mrelkhz7
∛k53
ck 5 c3 & c? & c: & c2 & ¹ ¹ ¹ +
Ac z}ec pcrjlca bm k ²mrelkhz mz Z k 5 c3 & c? & c: & ¹ ¹ ¹ ck+Zl ac z}jmzl² hk vZ ku bm z}ecz pcrjlcamz jhk~mrim ca a²ijelm Z ! mkhkjmz bmjlehz s}m ac zmrlm∛
k53 ck jhk~mrim! | Z mz ac z}ec bm ac zmrlm7
Z 5 a²ijek↟∛
Z k 5 c3 & c? & c: & c2 & ¹ ¹ ¹ +
Mk hrh jczh! bmjlehz s}m ac zmrlm bl~mrim+
Prhplmbcbmz
3+
ck |
ok jhk~ ⇟
$j3ck & j?ok, 5 j3
ck & j?
ok jhk~+
?+ a²ijek↟∛
ck 5 0 ⇟
ck bl~+
:+
ck jhk~ ⇟ a²ijek↟∛
ck 5 0+ $Pmrh a²ijek↟∛
ck 5 0
ck jhk~,
Rmhrmec :+?+? Ac z}ec imhe²mrljc jhk~mrim zl 0 1 qrq 1 3! mk mzm jczh
∛k50
rk 53
3 ∟ r+
Rmhrmec :+?+: Ac zmrlm mamzj²hpljc $ ck 5 ok ∟ ok&3,
∛k53
$ok ∟ ok&3, 5 $o3 ∟ o?, & $o? ∟ o:, & $o: ∟ o2, & $o2 ∟ o=, & ¹ ¹ ¹
~mrlffijc s}m Z k 5 o3 ∟ ok&3+Mzc zmrlm jhk~mrim ⇐⇟ a²ije
k↟∛ok 1 ∛ |
Z 5 o3 ∟ a²ijek↟∛
ok+
Rmhrmec :+?+2 Ac p%zmrlm $ p 5 3 mz ac zmrlm cre² hkljc,
∛k53
3
k p5
3
3 p&
3
? p&
3
: p&
3
2 p& ¹ ¹ ¹ +
3+ jhk~mrim zl p ; 3+
?+ bl~mrim zl 0 1 p
≢3+
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2 Z}jmzlhkmz | Zmrlmz
J_LRM_LHZ BM JHK^M_IMKJLC PC_C ZM_LMZ
3+ Jrlmrlh bm jhepcrcjl²hk blrmjc7
vck
u|
vok
ubhz z}jmzlhkmz bm ²mrelkhz
phzll~hz
0 1 ck ≢ ok! ∀ k ∟↟
ok jhk~ ⇟
ck jhk~
ck bl~⇟
ok bl~
?+ Jrlmrlh bm jhepcrcjl²hk mk ma a²ijelm7 vcku | voku bhz z}jmzlhkmz bm ²mrelkhzphzll~hz
a²ijek↟∛
ck
ok5 A! A ffiklh | phzll~h
⇚ck |
ok lmkmk ma elzeh jhephrcelmkh
ceocz jhk~mrimk h ceocz bl~mrimk
:+ Jrlmrlh bm ac rc²ijw7 vcku z}jmzl²hk bm ²mrelkhz phzll~hz
a²ijek↟∛
k
∗ck 1 3 ⇟
ck jhk~
a²ijek↟∛
k
∗ck ; 3 ⇟
ck bl~
a²ijek↟∛
k∗ck 5 3 ma jrlmrlh kh jhkja}|m
2+ Jrlmrlh bma jhjlmkm7 vcku z}jmzl²hk bm ²mrelkhz phzll~hz
a²ijek↟∛
ck&3
ck1 3 ⇟
ck jhk~
a²ijek↟∛
ck&3
ck; 3 ⇟
ck bl~
a²ijek↟∛
ck&3
ck5 3 ma jrlmrlh kh jhkja}|m
=+ Jrlmrlh bm Amloklw pcrc zmrlmz camrkcbcz7vcku z}jmzl²hk bm ²mrelkhz phzll~hz
Zl ck&3 ≢ ck | a²ijek↟∛
ck 5 0
⇚Ac zmrlm camrkcbc
$∟3,kck jhk~mrim jhkbljlhkcaemkm
$∟3,k&3ck
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:+:+ Zmrlmz bm phmkjlcz =
Bmffikljl²hk :+?+=
CJ+ ck mz cozha}cemkm jhk~mrimkm zl qck
qmz jhk~mrimkm+
JJ+ Zl
ck mz jhk~mrimkm pmrh qckq mz bl~mrimkm! mkhkjmz ac zmrlm mz jhkbl%
jlhkcaemkm jhk~mrimkm+
Jhk~mrimkjlc cozha}c 5⇟ Jhk~mrimkjlc jhkbljlhkca
Kh jhk~mrimkjlc jhkbljlhkca 5⇟ Kh jhk~mrimkjlc cozha}c
M__H_
J}ckbh cprhtlecehz ac z}ec bm }kc zmrlm camrkcbc phr z}z prlemrhz k ²mrelkhz!
mkhkjmz
Z 5 Z K & _K 5K k53
$∟3k,ck & _K ⇟ q_K q ≢ cK &3
Khc+ Phbmehz bmrl~cr h lkmircr }kc zmrlm lkffiklc pcrc homkmr hrc zmrlm+
:+:+ Zmrlmz bm phmkjlcz
Bmffikljl²hk :+:+3 ]kc zmrlm bm phmkjlcz jmkrcbc mk t0 mz }kc zmrlm lkffiklc bm ac nhrec
n $t, 5
∛k50
ck $t ∟ t0,k 5 c0 & c3$t ∟ t0, & c?$t ∟ t0,? & c:$t ∟ t0,: & ¹ ¹ ¹
Rmhrmec :+:+? $Jhk~mrimkjlc bm acz zmrlmz bm phmkjlcz,]kc zmrlm bm phmkjlcz mk t0 ~mrlffijc }kh | z² hah }kh bm ahz zli}lmkmz cpcrcbhz7
3+ Ac zmrlm jhk~mrim z² hah mk t0
?+ Mtlzm }k k² }emrh rmca σ ; 0 ca s}m ac zmrlm mz
cozha}cemkm jhk~mrimkm mk qt ∟ jq 1 σ
bl~mrimkm mk
qt
∟j
q; σ
:+ Ac zmrlm mz cozha}cemkm jhk~mrimkm pcrc hbh t ∄ _
Khc+ σ mz ma rcblh bm jhk~mrimkjlc bm ac zmrlm bm phmkjlcz+ σ 5 0! σ 1 ∛ ²h σ 5 ∛+Ma jhkg}kh bm hbhz ahz ~cahrmz bm t pcrc ahz j}camz ac zmrlm jhk~mrim mz ma lkmr~cah bmjhk~mrimkjlc bm ac zmrlm+ Ma rcblh zm p}mbm homkmr emblckm acz zli}lmkmz nhre}acz7
3
σ5 a²ije z}p
k↟∛
k
qckq
3
σ5 a²ije
k↟∛
ck&3
ck
! zl ma a²ijelm mtlzm
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> Z}jmzlhkmz | Zmrlmz
Rmhrmec :+:+: Zl ac zmrlm bm phmkjlcz n $t, 5∛
k50 ck $t ∟ t0,k lmkm rcblh bm jhk%~mrimkjlc σ ; 0! mkhkjmz n $t, mz jhklk}c! bmrl~coam m lkmircoam mk $t0 ∟ σ! t0 & σ,+
Ac bmrl~cbc | ac lkmirca zm jcaj}ack ²mrelkh c ²mrelkh+ Ceocz lmkmk ma elzeh rc%blh bm jhk~mrimkjlc s}m n + Ma lkmr~cah bm jhk~mrimkjlc p}mbm zmr blnmrmkm! bmolbh ca jhephrcelmkh mk ahz mtrmehz $ t 5 t0 ³ σ,+
Prhplmbcbmz+Zmck n $t, 5
∛
k50 cktk | i$t, 5∛
k50 oktk+
3+ n $ft, 5∛
k50 ckfktk
?+ n $tK , 5∛
k50 cktKk
:+ j3n $t, & j?i$t, 5∛
k50$j3ck & j?ok,tk
Bmffikljl²hk :+:+2 Zl mtlzmk hbcz acz bmrl~cbcz bm n mk t0! mkhkjmz ac zmrlm
∛k50
n $k,$t0,
k#$t ∟ t0,k
zm aacec ac zmrlm bm Rc|ahr bm n jmkrcbc mk t0 $pcrc t0 5 0 ceol²mk zm aacec ac zmrlm bm Ecj Ac}rlk bm n ,+
Rmhrmec :+:+= Zl mtlzmk hbcz acz bmrl~cbcz bm n mk }k lkmr~cah colmrh L s}m jhk%lmkm c t0 mkhkjmz
n $t, 5∛
k50
n $k,$t0,
k#
$t
∟t0,k
zl | zhah zl mtlzm λ mkrm t | t0 ca s}m
a²ijek↟∛
_k$t, 5 a²ijek↟∛
n $k&3,$λ,
$k & 3,#$t ∟ t0,k&3 5 0! ∀ t ∄ L+
ZM_LMZ BM RC\AH_
mt 5 3 & t &t?
?#&
t:
:#& ¹ ¹ ¹ &
tk
k#¹ ¹ ¹ ! ∟∛ 1 t 1 ∛
zmk t 5 t ∟ t:
:#& t
=
=#∟ t
9
9#& ¹ ¹ ¹ & $∟3,k t
?k&3
$?k & 3,#¹ ¹ ¹ ! ∟∛ 1 t 1 ∛
jhz t 5 3 ∟ t?
?#&
t2
2#∟ t>
>#& ¹ ¹ ¹ & $∟3,k
t?k
$?k,#¹ ¹ ¹ ! ∟∛ 1 t 1 ∛
crjck t 5 t ∟ t:
:&
t=
=∟ t9
9& ¹ ¹ ¹ & $∟3,k
t?k&3
$?k & 3,¹ ¹ ¹ ! ∟3 ≢ t ≢ 3
3
3 ∟ t5 3 & t & t? & t: & ¹ ¹ ¹ & tk ¹ ¹ ¹ ! ∟3 1 t 1 3
ak $3 & t, 5 t ∟ t?
?&
t:
:∟ ¹ ¹ ¹ & $∟3,k&3
tk
k¹ ¹ ¹ ! ∟3 1 t ≢ 3
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Rmec 2
Lkmircjl²hk
2+3+ Prlell~cz
AC LKRMI_CA BMNLKLBC
Zl n $t, mz }kc n}kjl²hk jhklk}c | kh kmicl~c bmffiklbc mk ma lkmr~cah t ∄ Vc! oU! mkhkjmzac lkmirca bmffiklbc o
cn $t,
rmprmzmkc ma ²crmc ocgh ac ir²cffijc bm ac n}kjl²hk n $t, | zhorm ma mgm t mk ma lkmr~caht ∄ Vc! oU+
Zl n $t, mz }kc n}kjl²hk imkmrca! ac lkmirca bmffiklbc rmprmzmkc ac z}ec bm ²crmczjhk zlikh mkrm ac n}kjl²hk | ma mgm t+ Ac lbmc pcrc jcaj}acr ac lkmirca bmffiklbc mzbl~lblr ma lkmr~cah mk k z}olkmr~cahz | cprhtlecr ac n}kjl²hk phr }kc jhkzckm $maemkhr h ma ec|hr ~cahr! ma ~cahr bma p}kh emblh h j}cas}lmr ~cahr bm ac n}kjl²hk mk malkmr~cah,+ Mkhkjmz! jcaj}acehz ma ²crmc bm k rmj²cki}ahz! s}m mz e}jdh e²cz zmkjlaah+Zl cprhtlecehz n n $tl, mk ma lkmr~cah l%²mzleh! jhk hbhz ahz lkmr~cahz bm ac elzecahkil}b! ∎t! mkhkjmz phbmehz cprhtlecr ac lkmirca jheh
oc
n $t, n $t3,∎t & n $t?,∎t & ¹ ¹ ¹& n $tk,∎t+
Zl cprhtlecehz ac n}kjl²hk mk ma lkmr~cah phr ahz zli}lmkmz ~cahrmz! homkmehz ahz s}mzm aacec! pcrc bljdc pcrljl²hk bma lkmr~cah
Ma emkhr ~cahr mk jcbc z}olkmr~cah ↟ z}ec lknmrlhr bm n Ma ec|hr ~cahr mk jcbc z}olkmr~cah ↟ z}ec z}pmrlhr bm n J}cas}lmr ~cahr mk jcbc z}olkmr~cah ↟ z}ec bm _lmeckk bm n
Ca hecr ma a²ijelm ∎t ↟ 0 zl hbcz acz z}ecz jhlkjlbmk! bmjlehz s}m ac n}kjl²hk mzlkmircoam _lmeckk+
Khc+ J}cas}lmr n}kjl²hk jhklk}c c rhwhz mz lkmircoam+
Rmkmehz ac zli}lmkm bmffikljl²hk bm lkmirca bmffiklbc7
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? Lkmircjl²hk
Bmffikljl²hk 2+3+3 Bcbc }kc n}kjl²hk lkmircoam n $t, mk ma lkmr~cah Vc! oU! zm bl~lblehz
ma lkmr~cah mk k z}olkmr~cahz bm li}ca ahkil}b ∎t! malilmkbh j}cas}lmr p}kh tl mk
jcbc z}olkmr~cah! mkhkjmz zm bmffikm ac lkmirca bmffiklbc h ac lkmirca bm _lmeckk bmn $t, bm c c o jheh o
cn $t,bt 5 a²ije
k↟∛
kl53
n $tl ,∎t+
Khc+ Ahz e²mhbhz k}e²mrljhz zm oczck mk mzm jhkjmph bm jcaj}acr }kc lkmirca cprhtl%eckbh ma ~cahr bm ac n}kjl²hk mk jcbc z}olkmr~cah phr }kc jhkzckm h phr }k phalkhelh!e}jdh e²cz zmkjlaah bm lkmircr+
Prhplmbcbmz bm ac lkmirca
3+ oc j3n & j?i 5 j3
oc n & j?
oc i
?+
oc
n 5
jc
n &
oj
n
:+
oc
n 5 ∟ co
n
2+
cc
n 5 0
=+ oc n i 5 o
c n oc i
>+ n ≭ i ⇟ oc
n ≭ oc
i
9+ n ≭ 0 ⇟ oc
n ≭ 0
zl n ≢ 0 ⇟ oc
n ≢ 0
4+ oc
n ≢
ocqn q
<+ e ≢ n $t, ≢ E! ∀t ∄ Vc! oU ⇟e$o∟c, ≢ o
c n $t, ≢ E $o∟c,
AC LKRMI_CA LKBMNLKLBC
Imhe²mrljcemkm! ac bmrl~cbc z}rim ca dcaacr ac pmkblmkm bm }kc j}r~c | ac lkmircaca jcaj}acr ma ²crmc ocgh }kc j}r~c! pmrh Kmyhk bmzj}orl²h! cbme²cz! s}m bmrl~cr mlkmircr zhk prhjmzhz lk~mrzhz7
Bmrl~cjl²hk7 bcbc }kc n}kjl²hk N $t,! dcaac }kc n}kjl²hk n $t, s}m zclzncic
bN $t,bt
5 n $t,+
Lkmircjl²hk7 bcbc }kc n}kjl²hk n $t,! dcaac }kc n}kjl²hk N $t, s}m zclzncic
bN $t,
bt5 n $t,+
]kc n}kjl²hk N $t, s}m rmz}ma~m ma ²}aleh prhoamec zm aacec }kc prlell~c! cklbmrl~c%bc h lkmirca lkbmffiklbc bm n $t,+Ma prhoamec bm bmrl~cjl²hk zlmeprm lmkm zha}jl²hk pmrh ma bm lkmircjl²hk kh zlmeprm lmkmzha}jl²hk |! mk imkmrca! z}mam zmr e}jdh e²cz jhepaljcbh+
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2+3+ Prlell~cz :
R²MJKLJCZ BM LKRMI_CJL ²HK
Cs}²ij ~mrmehz ac ²mjkljcz e²cz jhe}kmz pcrc jcaj}acr ac lkmirca $bmffiklbc h lkbmffiklbc,
bm }kc n}kjl²hk+
Prlell~cz o²czljcz
tk 5
tk&3
k & 3& j! k 5 ∟3
bt
t5 ak qtq& j
mct 5
3
cmct & j
zmk t 5 ∟ jhz t & j
jhz t 5 zmk t & j
3
jhz? t5 ck t & j
3
zmk? t5 ∟ jh t & j
3
t? & c?5
3
ccrjck
t
c
& j
3∗
c? ∟ t?5 crjzmk
tc
& j
zmkd t 5 jhzd t & j
jhzd t 5 zmkd t & j
Lkmircjl²hk emblckm jceolh bm ~crlcoamz h phr z}ozl}jl²hk $J^,
Lkmirca bmffiklbc i$o,i$c,
n $t,bt 5
oc
n $i$,,i$,b
Lkmirca lkbmffiklbc n $t,bt 5
n $i$,,i$,b
ca ffikca! dc| s}m bmzdcjmr ma jceolh
Lkmircjl²hk phr pcrmz $LPP,
Lkmirca bmffiklbc oc
n $t,i$t,bt 5 n $t,i$t,oc∟ oc
n $t,i$t,bt
Lkmirca lkbmffiklbc n $t,i$t,bt 5 n $t,i$t,∟
n $t,i$t,bt
}b~ 5 }~ ∟
~b}
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2 Lkmircjl²hk
Lkmircjl²hk bm n}kjlhkmz rcjlhkcamz7 bmzjhephzljl²hk mk nrcjjlhkmz zlepamz
P $t,
S$t,bt ↟ P! S phalkhelhz+
Zl ma ircbh bm P ≭ ircbh bm S ⇟ bmomehz bl~lblr ahz phalkhelhz7P $t, 5 S$t,J $t, & _$t, ↟
P $t,
S$t,bt 5
J $t, &
_$t,
S$t,bt+
_$t,
S$t,bt jhk ircbh$_$t,, 1 ircbh$S$t,,7
l, Prlemrh! bmomehz jheprhocr s}m ac lkmirca kh zmc lkemblcc! mz bmjlr! zlmz bma lph7
lph ak ↟
?t & :
t? & :t & 4bt 5 ak qt? & :t & 4q & j+
lph crjck ↟
bt
t? & 45
3∗4
crjckt∗
4& j+
ll, Zl kh ah mz ↟ Dcrmehz bmzjhephzljl²hk mk nrcjjlhkmz zlepamz7
Ncjhr mk ma bmkhelkcbhr R²mrelkh mk ac bmzjhephzljl²hk mk nrcjjlhkmz zlepamz
t∟ oC
t∟ o
$t∟ o,fC3
t∟ o&
C?
$t∟ o,?& ¹ ¹ ¹ Cf
$t∟ o,f! f 5 3! ?! :! ¹ ¹ ¹
$t∟ c,? & o?Ct & O
$t∟ c,? & o?
$t∟c,?&o?
f C3t & O3
$t∟ c,? & o?&
C?t & O?$t∟ c,? & o?
? &¹ ¹ ¹& Cft & Of$t∟ c,? & o?
f ! f 5 3! ?! :! ¹ ¹ ¹
Pcrc jcbc ncjhr mk ma bmkhelkcbhr bmomehz ckˑcblr ma ²mrelkh jhrrmzphkblmkm bm accoac | jcaj}acr acz lkj²hiklcz $C!O!C3! O3! C?! O?! ¹ ¹ ¹ , li}cackbh ahz bmkhelkcbhrmz+Bmzp}²mz dc| s}m jcaj}acr ac lkmirca bm jcbc ²mrelkh+
C pcrlr bm cs}²ij! _ 5P
Sbmkhc }kc n}kjl²hk rcjlhkca mk z}z cri}emkhz! P! S zhk
phalkhelhz+
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2+3+ Prlell~cz =
N}kjlhkmz lrrcjlhkcamz h lkmircamz jhk rc²ijjmz
Dcrmehz }k jceolh bm ~crlcoamz pcrc malelkcr acz rc²ijjmz+ _
ct & o
jt & b
p3(s3! ¹ ¹ ¹ !
ct & o
jt & b
pr(sr↟ e 5ct & o
jt & b! e 5 eje$s3! ¹ ¹ ¹ ! sr,+
eje↟ e²ijkleh jhe²}k e²}alpah+
Lkmircamz bm n}kjlhkmz rlihkhe²mrljcz
zmk?k t!
jhz?k t ↟ n²hre}acz bm ²cki}ah bhoam7 jhz ?t 5 jhz? t ∟ zmk? t
zmk?k&3 t 5 zmk?k t zmk t 5 $3
∟jhz? t,k zmk t
jhz?k&3 t 5
jhz?k t jhz t 5
$3 ∟ zmk? t,k jhz t zmk et jhz kt ↟ n²hre}acz rlihkhe²mrljcz
_$zmk t! jhz t, ↟
_ lepcr mk zmk t ↟ 5 jhz t_ lepcr mk jhz t ↟ 5 zmk t_ pcr mk jhz t | zmk t ↟ 5 ck t_mzh bm prhoamecz ↟ 5 ck t(?!
zmk t 5?
3 & ?! jhz t 5
3∟ ?
3 & ?! bt 5
?
3 & ?b
Cai}khz jceolhz bm ~crlcoamz
3+
_$t!
t? & c?, ↟ t 5 c ck
?+
_$t!
t? ∟ c?, ↟ t 5
c
jhz
:+
_$t!
c? ∟ t?, ↟ t 5 c zmk
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> Lkmircjl²hk
2+?+ Ma Rmhrmec N}kbcemkca bma J²caj}ah
Zmc n lkmircoam mk Vc!oU! N $t, 5 tc
n $,b mz }kc prlell~c bm n $t, bmffiklbc mk Vc! oU+
Rmhrmec 2+?+3n lkmircoam mk Vc! oU ⇟ N jhklk}c mk Vc! oU
Rmhrmec 2+?+? $Ma Rmhrmec N}kbcemkca bma J²caj}ah! RNJ,
Zmc n lkmircoam mk Vc! oU | N $t, 5
tc
n $,b! bmffiklbc ∀t ∄ Vc! oU+
Zl n mz jhklk}c mk j ∄ Vc! oU ⇟ N mz bmrl~coam mk j | N $j, 5 n $j,+
Zl n mz jhklk}c ∀t ∄ Vc! oU ⇟ N mz bmrl~coam ∀t ∄ Vc! oU | N $t, 5 n $t,+
Rmhrmec 2+?+: $_miac bm Ocrrhy, Zmck n | i jhklk}cz mk Vc! oU | i bmrl~coam mk $c! o,! bm nhrec s}m i$t, 5 n $t,! ∀t ∄ $c! o,! mkhkjmz
oc
n 5
oc
i 5 i$o, ∟ i$c,+
Rmhrmec 2+?+2 $RNJ imkmrcalwcbh, Zmc N $t, 5
tc
n ! jhk n lkmircoam
Zmc D $t, 5 N $i$t,, 5
i$t,c
n ! mkhkjmz zl i mz bmrl~coam! mkmehz s}m
D $t, 5 N $i$t,,i$t, 5 n $i$t,,i$t,+
Zmc D $t, 5
i$t,a$t,
n ! mkhkjmz zl i | a zhk bmrl~coamz! mkmehz s}m
D $t, 5 n $i$t,,i$t, ∟ n $a$t,,a$t,+
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2+:+ Cpaljcjlhkmz bm ac Lkmirca 9
2+:+ Cpaljcjlhkmz bm ac Lkmirca
²C_MCZ
²Crmc mkrm ac ir²cffijc bm }kc n}kjl²hk | ma mgm t! mkrm c | o7
C 5
ocqn qbt
²Crmc mkrm acz ir²cffijcz bm bhz n}kjlhkmz n! i! mkrm c | o7
C 5
ocqn ∟ iqbt
²Crmc jhk mj}cjlhkmz pcrce²mrljcz7 ma ²crmc mkrm ac ir²cffijc bm t 5 t$,! | 5 |$,| ma mgm t mkrm 5 0 | 5 3 mz7
C 5
30
|$,t$,b
²Crmc mk jhhrbmkcbcz phacrmz7 ma ²crmc bm ac ir²cffijc bm r 5 r$δ, mkrm δ 5 γ |δ 5 ο mz
C 5
ο γ
3
?r?$δ,bδ
^HA²]EMKMZ
^ha}emk phr zmjjlhkmz pcrcamacz7 zl C$t, mz ma ²crmc bm acz zmjjlhkmz pcrcamaczc ah acrih bm hbc ac ahkil}b bm }k z²halbh! ma ~ha}emk bma z²halbh cz²ij bmffiklbh mkrmt 5 c | t 5 o mz
^ 5
oc
C$t,bt
Ma e²mhbh bm blzjhz7 ma ~ha}emk bm }k z²halbh bm rm~ha}jl²hk homklbh ca rhcrqn $t,q carmbmbhr bma mgm t mkrm t 5 c | t 5 o mz
^ 5
oc
π$n $t,,?bt
Ma e²mhbh bm jcpcz7 ma ~ha}emk bm }k z²halbh bm rm~ha}jl²hk homklbh ca rhcrn $t, ≭ 0 carmbmbhr bma mgm | mz
^ 5 ?π
oc
tn $t,bt
5/8/2018 Apuntes de Matemáticas para la Economía I - slidepdf.com
http://slidepdf.com/reader/full/apuntes-de-matematicas-para-la-economia-i 31/31
4 Lkmircjl²hk
AHKILR]BMZ
Ac ahkil}b bma crjh bm j}r~c n $t, mkrm t 5 c | t 5 o mz
A$n , 5
oc
3 & $n $t,,?bt
Zl ac j}r~c ~lmkm bcbc mk nhrec pcrce²mrljc! ac ahkil}b mz
A 5
30
$t$,,? & $|$,,?b
LKRMI_CA LEP_HPLC
Bmffikljl²hk 2+:+3 Ac zli}lmkm lkmirca
∛
cn $t, 5 a²ije
K ↟∛
K
cn $t,!
zm aacec }kc lkmirca leprhplc bm n + Zl ma a²ijelm mz ffiklh bmjlehz s}m ac lkmirca
jhk~mrim mk hrh jczh! blrmehz s}m ac lkmirca bl~mrim+
Rmhrmec 2+:+? $Jrlmrlh lkmirca pcrc zmrlmz, Jhkzlbmrmehz n ≭ 0 }kc n}kjl² hk
ehk² hhkc bmjrmjlmkm bmffiklbc mk t
≭3+ Zmc ck 5 n $k,! mkhkjmz
∛k53
ck |
∛
3n $t, bt!
lmkmk ma elzeh jhephrcelmkh! h ceocz jhk~mrimk h ceocz bl~mrimk+