Веленкин;Комбинаторика
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-sjkj 9 fОcХcЛК‰ fПИР¬ЗТМиК¬ЙФ К d¬НиcТЛФc dМЛФ
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΀ wvsy·qu ·zv |
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(− )Q− |
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(− )Q |
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(3 |
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(Q − ) |
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(− )Q− |
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(− )Q− |
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+ (Q − ) − |
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(Q − ) |
(Q − ) |
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Æ vin xrvirq u€ lvijkqsq wv vltvuy wvxsnltnuy ·snty v·nkql
tv ·zv ëzq ·snt€ kojqutv ytq·zv jízx¹ zjr rjr wvxsn éjxré€zq¹ xrvivr wénkéj•jízx¹ xvvzknzxzknttv k (− )Q q (− )Q− Ëv wnékj¹ xrvi
rj ² ëzv tn ·zv qtvn rjr '3 j kzvéj¹ ² tn ·zv qtvn rjr '3− Ívëzvuy
(3 = '3 + 'Q− |
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Êzjr ·qxsv wnénxzjtvkvr qo Q ·qxns Q k rvzvé€n tn k}vlqz tq vltj qo wjé (Q − Q) éjktv 'Q + 'Q−
fvkné¡nttv zjr |
n lvrjo€kjnzx¹ ·zv rvsq·nxzkv wnénxzj |
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tvkvr qo |
Q |
ësnuntzvk |
k rvzvé€n tn |
k}vl¹z |
ojljtt€n |
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U ≤ Q − wjé éjktv |
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3 |
Q |
− & 3 |
Q− |
+ & 3 |
Q− |
− + (− )U&U3 |
Q−U |
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U |
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Êtvp vzknz wvsy·qzx¹ nxsq ·qxsv ojwén•ntt€} wjé ivs•¡n ·nu Q − Íyxz• tjwéquné révun wjé
(Q − Q) k wnénxzjtvkrq tn lvs tj k}vlqz• q wjéj (Q )
èjxxy |
lj¹ zv·tv zjr n rjr q k€¡n wvsy·qu ·zv vzknz |
kێj |
jnzx¹ {véuysvp |
) |
Q |
= 3 |
Q |
− & 3 |
+ & 3 |
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− + |
(− )N&N3 |
Q−N |
+ + (− )Q− &Q− 3 |
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= |
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Q Q− |
Q Q− |
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Q |
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− |
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+ + |
(− )Q− |
= Q' |
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= Q |
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Q− |
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(Q − ) |
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Æ xjuvu lnsn k ëzvu xsy·jn ·qxsv ojwén•ntt€} wjé éjktv Q j xsy·j¹ rvmlj k wnénxzjtvkry k}vl¹z kxn Q wjé i€z• tn uv
nz Ænl• tjwéquné nxsq k wnénxzjtvkry k}vl¹z wjé€
(Q − Q) zv wnék€u ësnuntzvu ¹ks¹nzx¹ j wv xsnltqu Q q wvëzvuy wjéj (Q ) k wnénxzjtvkry tn k}vlqz Ívëzvuy wvxsnltqp ·snt k {véuysn éjknt (− )Q− &QQ− 3 j
tn (− )Q&Q3 |
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= (− )Q |
Q |
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Çjzjtqn tj rjéyxnsq
Ëj rjéyxnsq rjzjízx¹ Q éni¹z Ìtq én¡qsq wnénxnxz• ² zjr ·zvi€ y rj lvmv qo tq} xuntqsx¹ kwnénlq xql¹•qp frvs•rquq xwvxvijuq vtq uvmyz ëzv xlnsjz•"
djzéyltntqn uj vélvuj |
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êzj ojlj·j wv}v j tj én¡nttyí k€¡n ojlj·y v rjéjkjtn
Ëv znwné• ·qxsv ojwén•ntt€} wjé éjktv Q lvs t€ vzxyzxzkv kjz• wjé€ (Q − Q) q (Q ) Çévun zvmv
wnénxzjtvkrq wvsy·jnu€n léym qo léymj wnénxjlrvp éni¹z wv réymy u€ tn iylnu x·qzjz• éjosq·t€uq ² rvmlj rjéyxns•
ojréy qzx¹ q} tn éjosq·q¡• Ívëzvuy qo Q ësnuntzvk uv tv xlnsjz• sq¡• 3Q− = (Q − ) xy•nxzknttv éjosq·t€} wnénxzjtv
kvr Ëjrvtn| k tvkvp ojlj·n uvmyz i€z• wnénxzjtvkrq k rvzvé€n k}vl¹z kxn Q wjé Òjrvp iylnz tjwéquné qx}vltj¹ wnénxzjtvkrj Ï·qz€kj¹ kxn ëzq vixzv¹zns•xzkj wvsy·jnu wv {véuysn krsí·ntqp q qxrsí·ntqp ·zv ·qxsv qxrvu€} wnén xzjtvkvr éjktv
4 |
Q |
= 3 |
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− & 3 |
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+ & 3 |
Q− |
− + (− )Q− &Q− 3 |
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+ (− )Q&Q |
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Q− |
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Q Q− |
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Énmrv wévknéqz• ·zv ëzv k€éj ntqn uv |
tv ojwqxjz• k kqln |
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4 = ' |
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− ' |
Q− |
+ ' |
Q− |
− + (− )Q− ' |
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Q |
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Q− |
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Æ xjuvu lnsn k xqsy éjkntxzkj &N − &N− |
= &N |
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xsnlynz ·zv wéq Q ≥ |
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Q |
Q− |
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Q− qo {véuys€ |
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4 + 4 |
= 3 |
Q− |
− & |
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+ & |
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− + (− )Q− |
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Q |
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Q− |
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Q− 3Q− |
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Q− 3Q− |
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j ëzv k€éj |
ntqn |
éjktv |
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'Q− xu w |
Òjrqu |
viéjovu |
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4Q + 4Q− = 'Q− Çévun zvmv qo {véuys€ |
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k€znrjnz ·zv |
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4 = ·zv q zjr v·nkqltv frsjl€kj¹ éjkntxzkj |
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4Q + 4Q− = 'Q− |
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−4Q− − 4Q− = −'Q− |
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4Q− + 4Q− = 'Q− |
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(− )Q− 4 |
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= (− )Q− ' |
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wéq}vlqu r xvvztv¡ntqí |
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djzéyltntqn uj |
vélvuj |
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Ìltj |
l€ uj vélvu rvévs¹ ¬ézyéj vitjéy |
qs ·zv r vinly |
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oj réyms€u xzvsvu wéqmsj¡ntv wjé kéj |
lyí•q} é€|jénp |
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frvs•rquq xwvxvijuq uv |
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tv éjxxjlqz• q} zjr ·zvi€ tqrj |
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rqn lkj kéjmj tn xqlnsq é¹lvu" |
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êzj ojlj·j wv}v j tj ojlj·y v xywéy nxrq} wjéj} tv vzsq·jnzx¹ vz tnn znu ·zv k ojlj·n v xywéy nxrq} wjéj} zénivkjsvx• tn zvs•rv ·zvi€ xywéymq i€sq éjolnsnt€ léym vz léymj tv q ·zvi€ é¹lvu tn xqlnsq lkvn uy ·qt qsq lkn
-sjkj 9 fОcХcЛК‰ fПИР¬ЗТМиК¬ЙФ К d¬НиcТЛФc dМЛФ
nt•qt€ Æ ojlj·n v é€|jé¹} ëzq} yxsvkqp tnz ² tjlv sq¡• ·zvi€ é¹lvu tn xqlnsq tqrjrqn lkj kéjmj ΀ én¡qu ojlj·y
k vi•nu kqln rvmlj wéqmsj¡ntv Q wjé kéj lyí•q} é€|jénpQ > Íéq ëzvu iylnu x·qzjz• ·zv rénxsj ls¹ mvxznp ojty
unévkjt€ xrj nu tj·qtj¹ x tnrvzvévmv wv }vly ·jxvkvp xzénsrq q xwvxvi€ éjxxjlrq éjosq·t€ nxsq }vz• vlqt é€|jé• wvwjljnz k léymvn rénxsv
bs¹ én¡ntq¹ ojlj·q kvxwvs•oynux¹ xtvkj {véuysvp krsí ·ntqp q qxrsí·ntqp Ìivotj·qu ·néno DL xvi€zqn xvxzv¹•nn
k zvu ·zv L ¹ wjéj kéj lyí•q} un ly xvivp é€|jénp xqlqz é¹lvu Ívlx·qzjnu ·nuy éjktv 1(D D DN) z n kv xrvs•rq}
xsy·j¹} é¹lvu xql¹z N wjé kéjmvk
Ínékyí wjéy uv tv yxjlqz• oj xzvs Q xwvxvijuq Q xwvxvijuq k€iéjz• unxzv ls¹ vltvmv kzvévmv wvxjlqz• tj xsnlyí•nn wv }vly ·jxvkvp xzénsrq unxzv q y·nxz• ·zv é€|j
énp uv tv wvunt¹z• unxzjuq bs¹ vxzjs•t€} é€|jénp vxzj tnzx¹ Q − unxz wéq·nu q} tjlv ojt¹z• zjr ·zvi€ kzvéj¹
zénz•¹ N ¹ wjé€ kéjmvk vrjojsqx• é¹lvu Çj lyí zjryí
wjéy é¹lvu xql¹•q} kéj lyí•q} é€|jénp uv tv x·qzjz• vltqu ©vi~nrzvuª êzq N − wjé é€|jénp q Q − N vxzjs•t€}
é€|jénp uv tv wnénxzjks¹z• léym x léymvu ( Q − N − ) xwvxv
ijuq cxsq ko¹z• vlty qo zjrq} wnénxzjtvkvr q wvxjlqz•
é€|jénp wv wvé¹lry tj xkvivlt€n unxzj wvlé¹l wv }vly ·jxvkvp xzénsrq zv k€iéjtt€n tjuq N − wjé kéjmvk vrj yzx¹
é¹lvu êzv yxsvkqn tn tjéy¡qzx¹ q k xsy·jn nxsq u€ wvunt¹nu
unxzjuq tnrvzvé€} é¹lvu xql¹•q} kéjmvk Òjr rjr zjrqn |
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wnénxjlrq uv tv xlnsjz• N− |
xwvxvijuq zv kxnmv wvsy·jnu |
Q N− ( Q − N − ) xwvxvivk |
wvxjlrq Òjrqu viéjovu |
djzéyltntqn uj vélvuj
1(D D D ) = N+ Q ( Q − N − ) ΀ }vzqu tjpzq kv xrvs•rq} |
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xsy·j¹} |
tq |
vltj |
wjéj |
kéjmvk tn |
vrj nzx¹ xvxnl¹uq z n |
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k€·qxsqz• 1(D D DQ) Íéqtquj¹ |
kv ktqujtqn ·zv N wjé |
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uv |
tv k€iéjz• &N |
xwvxvijuq wvsy·jnu wv {véuysn krsí·ntqp |
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q qxrsí·ntqp ·zv |
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Q ( Q − ) − & ( Q − ) + & ( Q − ) − |
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= 1(D D |
D ) = |
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− + (− )N N&N( Q − N − ) + + (− )Q Q(Q − ) |
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Q |
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Íéq Q = wvsy·jnu $ = = dlnx• utv qzns• xvvz
knzxzkynz kvouv tvxzq wvknétyz• kxí éjxxjlry z n wévqokvs• tv k€iéjz• unxzv vltvmv é€|jé¹ xrj nu xjuvmv xzjé¡nmv tjwévzqk tnmv lvs nt xqlnz• kéj lyí•qp x tqu é€|jé• j wv wéjkyí q wv snkyí éyry vz tnmv ² é€|jéq qo léymvp kéj lyí•np wjé€ q} uv tv wvxjlqz• xquunzéq·t€uq xwvxvijuq Òjr ·zv x zv·tvxz•í lv wvkvévzvk q xquunzéqq éjxxjlrj
é€|jénp nlqtxzknttj
Íéq Q = wvsy·jnu $ = = dlnx• utv qzns• ²
wvkvévz€ ² xquunzéq¹ vxzjnzx¹ xy•nxzknttv éjosq·t€}
xwvxvivk éjxxjlqz• é€|jénp
Íéq Q = wvsy·jnu $ =
djlj·q
Ëjlv vzmjljz• rjrqn w¹z• uvtnz lné qz k éyrn wjéztné Æ€ivé uvtnz zjrvp rvwnnr éyis• frvs•rv uv nz
i€z• ljtv tnknét€} vzknzvk" |
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¬ nxsq kxn uvtnz€ k éyrn éjosq·tvmv lvxzvqtxzkj" |
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frvs•rquq xwvxvijuq uv tv wvxjlqz• é¹lvu |
jtmsq·jt |
{éjt|yovk q zyévr zjr ·zvi€ |
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j tqrjrqn zéq xvvzn·nxzknttqrj tn xqlnsq é¹lvu |
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i tqrjrqn lkj xvvzn·nxzknttqrj tn xqlnsq é¹lvu" |
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frvs•rquq xwvxvijuq uv tv wvxjlqz• oj vlqt |
réyms€p xzvs |
jtmsq·jt {éjt|yovk q zyévr zjr ·zvi€ tqrjrqn lkj xvvzn·nxzknttqrj tn xqlnsq é¹lvu"
frvs•rquq xwvxvijuq uv tv éjxxzjkqz• k é¹l jtmsq·jt{éjt|yovk q zyévr zjr ·zvi€ rj l€p jtmsq·jtqt xzv¹s un ly {éjt|yovu q zyérvu tv tqrjrqn {éjt|yo q zyévr tn xzv¹sq é¹lvu" èn¡qzn zy n ojlj·y ls¹ jtmsq·jt {éjt|yovk q zyévr
j frvs•rquq xwvxvijuq uv tv wvxzévqz• k é¹l Q xywéy nxrq} wjé zjr ·zvi€ tq vltj wjéj tn xzv¹sj é¹lvu"
i bjt€ Q tjivévk k rj l€p qo rvzvé€} k}vlqz T vlqtjrvk€} ësnuntzvk wéq·nu ësnuntz€ éjosq·t€} tjivévk éjosq·t€ Æv xrvs• rq} wnénxzjtvkrj} ëzq} QT ësnuntzvk tnz tq vltvmv tjivéj kxn ësnuntz€ rvzvévmv qlyz wvlé¹l"
k èn¡qzn zy n ojlj·y wéq yxsvkqq ·zv ësnuntz€ éjxwvsjmjízx¹ wv vréy tvxzq
-sjkj 9 fОcХcЛК‰ fПИР¬ЗТМиК¬ЙФ К d¬НиcТЛФc dМЛФ
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Ëjplqzn sjlnpt€n utvmv·snt€ ls¹ lvxvr |
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qoviéj ntt€} tj éqx |
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frvs•rquq xwvxvijuq uv tv iévxqz• éjo |
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lkn qméjs•t€n |
rvxzq zjr ·zvi€ rj lj¹ rvxz• |
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k€wjsj rj lvp xzvévtvp wv éjoy tv tq éjoy |
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tn kxzénzqsqx• rvuiqtj|qq |
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q " |
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bvrj qzn |
·zv P + Q wénlunzvk uv tv |
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wnénxzjkqz• zjr ·zvi€ évktv Q vxzjsvx• tj |
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unxzn |
(P + Q) |
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Çvsvlj qo Q rjéz zjxynzx¹ xsnlyí•qu |
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viéjovu nn lns¹z wvwvsju q rjéz€ wnékvp |
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wvsvkqt€ éjxwvsjmjíz wv vltvp un ly rjézjuq |
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kzvévp wvsvkqt€ k zvu n wvé¹lrn Ëjwéquné |
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rjézj x tvunévu Q + xzjtvkqzx¹ wnékvp wnékj¹ |
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èqx |
² kzvévp x tvunévu Q + ² zénz•np kzvéj¹ ² |
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·nzknézvp q z l bvrj qzn ·zv ·néno U éjo rjézj wnékvtj·js•tv tj}vlqk¡j¹x¹ tj S u unxzn wvwjlnz tj unxzv [ mln [ ² vxzjzvr vz lnsntq¹ US tj Q +
j bvrj qzn ·zv nxsq k yxsvkqq wénl€ly•np ojlj·q rvsvlj |
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xvlné |
qz P + rjéz zv rjéz€ x tvunéjuq P + q P + iylyz |
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kxn kénu¹ unt¹z•x¹ unxzjuq |
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cxsq wéq zvu |
n yxsvkqq rvsvlj xvlné qz |
P + rjéz zv |
wvxsn |
zén} zjxvkvr |
yrjojtt€u xwvxvivu rjéz€ |
P + ( P + ) |
( P + ) ( P + ) ( P + ) ( P + ) knétyzx¹ tj qx}vlt€n unxzj
cxsq wéq zvu n yxsvkqq [ − lnsqzx¹ tj Q + zv rvsvlj qo Q rjéz wvxsn [ zjxvkvr wéqlnz k wnékvtj·js•tvn wvsv ntqn
j Çvsvlj rjéz zjxynzx¹ xsnlyí•qu viéjovu xtj·jsj inénzx¹
wnékj¹ rjézj kzvéj¹ rjézj rsjlnzx¹ tj tnn zénz•¹ ² wvl tnn q z l bvrj qzn ·zv nxsq rvsvlj xvlné qz Q − rjéz zv rjézj Q vxzjnzx¹
tj unxzn
i Ïrjojttvp k€¡n zjxvkrn wvlknémjízx¹ rjéz€ bvrj qzn ·zv rjézj kxn kénu¹ vxzjnzx¹ tj unxzn q unt¹ízx¹ unxzjuq j
wnén}vl¹z léym k léymj wv réymy
k bvrj qzn ·zv wéq zvu n yxsvkqq rvsvlj k rjéz wéqlnz k wnékvtj·js•tvn wvsv ntqn wvxsn zjxvkvr rvsvlj k rjéz€ ² wvxsn rjéz€ ² wvxsn q rjéz ² wvxsn
rjéz ² wvxsn q rjéz€ ² wvxsn rjéz ² wvxsn
rjéz ² wvxsn rjéz ² wvxsn rjéz ² wvxsn
rjéz ² wvxsn
-sjkj 9,
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