Веленкин;Комбинаторика
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cxsq wvxsnlvkjzns•tv xrsjl€kjz• ·snt€ ëzvmv é¹lj zv wvsy·jízx¹ xyuu€ êzq xyuu€ kxnmlj wvsv qzns•t€ tnvméjtq·nttv yknsq·qkjízx¹ q tn wéqisq j
ízx¹ r ·qxsy − ΀ kxzénzqsqx• zjrqu viéjovu x lkyu¹
xsy·j¹uq ïzvi€ éjosq·jz• ëzq xsy·jq kknlnu vi•nn wvt¹zqn v x}vlquvxzq q éjx}vlquvxzq ·qxsvkvmv é¹lj
Íyxz• ojljt inxrvtn·t€p ·qxsvkvp é¹l
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D + D + + DQ + |
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-vkvé¹z ·zv vt |
x}vlqzx¹ |
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·qxsy E |
nxsq éjotvxz• |
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E − (D + D |
+ + DQ) |
xzénuqzx¹ |
r |
tysí wéq |
tnvméjtq·nttvu |
yknsq·ntqq Q Êt€uq xsvkjuq rjrvn i€ ·qxsv ε > u€ tq ko¹sq vzrsvtntqn xyuu€ D + D + + DQ vz E tj·qtj¹ x tnrv
zvévmv tvunéj 1 vwénlns¹nuvmv wv ε vrj nzx¹ unt•¡n ε _ E − (D + D + + DQ) _ < ε
nxsq Q ≥ 1
Æ ëzvu xsy·jn ·qxsv E tjo€kjíz xyuuvp inxrvtn·tvmv é¹lj D + D + + DQ + q wq¡yz E = D + D + + DQ +
cxsq tn xy•nxzkynz ·qxsj E r rvzvévuy x}vlqzx¹ ljtt€p é¹l zv ëzvz é¹l tjo€kjíz éjx}vl¹•qux¹
Íévknlnttvn k€¡n qxxsnlvkjtqn wvrjo€kjnz ·zv
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k zv kénu¹ rjr é¹l + + + + N + éjx}vlqzx¹ |
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Èvsnn z•jzns•tvn qxxsnlvkjtqn |
wvrjo€kjnz ·zv nxsq |
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_ [ _ < zv é¹l + [ + [ + + [Q + x}vlqzx¹ r − [ j nxsq
_ [ _ ≥ zv |
vt éjx}vlqzx¹ ïzvi€ lvrjojz• ëzv yzkné |
lntqn |
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lvxzjzv·tv |
ojunzqz• ·zv + [ + [ + + [Q = |
− [Q+ |
q nxsq |
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Q xzénuqzx¹ r inxrvtn·tvxzq zv k€éj ntqn [Q+ xzénuqzx¹ r tysí nxsq _ [ _ < q r inxrvtn·tvxzq nxsq _ [ _ > Íéq [ =
q [ = − wvsy·jnu éjx}vl¹•qnx¹ ·qxsvk€n é¹l€ + + q− + − +
¬smniéjq·nxrqn léviq q xznwntt€n é¹l€ |
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Êzjr nxsq _ [ _ < zv k {véuysn uv tv yiéjz• otjr kvwévxj
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Ìzunzqu ·zv éjkntxzkv ² ëzv qoknxztj¹ qo ¡rvs•tvmv ryéxj ujznujzqrq {véuysj ls¹ xyuu€ inxrvtn·tv yi€kjí•np mnvunzéq·nxrvp wévménxxqq
΀ k€¹xtqsq zjrqu viéjovu xu€xs ojwqxq Ìtj wvrjo€kjnz ·zv ls¹ otj·ntqp [ sn j•q} k tnrvzvévp visjxzq
j qunttv wéq _[ _ < |
xzv¹•qp xwéjkj é¹l x}vlqzx¹ r |
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-vkvé¹z ·zv {ytr|q¹ |
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wéq _[ _ < éjosjmjnzx¹ k xznwnt |
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tvp é¹l + [ + [ + [ + + [Q +
Ònwné• y n uv tv k€¹xtqz• q ivsnn vi•qp kvwévx Íyxz• wéq lnsntqq utvmv·sntj I([) tj utvmv·snt J([) wvsy·qsx¹
xznwnttvp é¹l |
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F + F [ + F [ + + FN [N + |
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Ìrjo€kjnzx¹ ·zv |
zvmlj wéq lvxzjzv·tv ujs€} |
otj·ntq¹} |
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[ é¹l x}vlqzx¹ r |
I([) |
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J([) |
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èjouné€ visjxzq x}vlquvxzq ojkqx¹z vz rvétnp otjuntjzn s¹ z n ·qxns wéq rvzvé€} otjuntjzns• viéj•jnzx¹ k tys• Êunttv nxsq ëzq ·qxsj éjkt€ [ [N q U ² tjqunt•¡nn
qo ·qxns _[N _ zv é¹l x}vlqzx¹ k visjxzq _[ _ < U Ëjwéquné {ytr|q¹ − [ viéj•jnzx¹ k tys• wéq [ = q wvzvuy éjosv
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ntqn |
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knétv sq¡• wéq _ [ _ < ¬ |
{ytr|q¹ [ − [ + |
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viéj•jnzx¹ k tys• wéq [ = [ = q |
wvëzvuy éjosv ntqn |
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x}vlqzx¹ wéq _[ _ < |
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Ìzunzqu ·zv tq vlqt qo rvétnp otjuntjzns¹ tn éjknt tysí zjr rjr u€ wénlwvsv qsq ·zv xkvivlt€p ·snt otjuntjzns¹ vzsq·nt vz tys¹ q wvzvuy J( ) = E ≠ Êt€uq xsvkjuq kxnmlj
nxz• visjxz• _ [ _ < U k rvzvévp k€wvst¹nzx¹ éjkntxzkv
I([) |
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[ + + F [N + |
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-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
Æxznwntt€n é¹l€ uv tv éjosjmjz• tn zvs•rv jsmniéjq·nx rqn léviq tv q utvmqn léymqn {ytr|qq Æ ujznujzq·nxrvu jtjsqon lvrjo€kjíz tjwéquné ·zv
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VLQ [ = [ − |
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FRV [ = − |
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bs¹ tjx iylnz wénlxzjks¹z• qtznénx éjosv |
ntqn |
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H[ = + [ + |
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Êo {véuys€ kqltv ·zv |
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H = + + |
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Èné¹ lvxzjzv·tv utvmv ·sntvk é¹lj wvsy·jnu otj·ntqn ·qxsj H x zéniynuvp xznwnt•í zv·tvxzq Ínék€n lnx¹zq·t€n otjrq ·qxsj H quníz kql
è¹l€ x}vl¹zx¹ wéq kxn} otj·ntq¹} [
Ìzunzqu n•n xsnlyí•nn kj tvn yzkné lntqn
Ðytr|q¹ I([) tn uv nz qunz• lky} éjosq·t€} éjosv ntqp
k xznwntt€n é¹l€
Êt€uq xsvkjuq nxsq
I([) = D + D [ + D [ + + DN[N +
q
I([) = E + E [ + E [ + + EN[N +
zv D = E D = E DN = EN
bnpxzkq¹ tjl xznwntt€uq é¹ljuq
Ínénplnu znwné• r lnpxzkq¹u tjl xznwntt€uq é¹ljuq Íyxz• {ytr|qq I([) q J([) éjosv nt€ k xznwntt€n é¹l€
I([) = D + D [ + D [ + + DN[N + |
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J([) = E + E [ + E [ + + EN[N + |
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bnpxzkq¹ tjl xznwntt€uq é¹ljuq |
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Òvmlj
I([) + J([) = (D + D [ + D [ + + DN[N + ) +
+ (E + E [ + E [ + + EN[N + )
Ìrjo€kjnzx¹ ·zv xsjmjnu€n k wéjkvp ·jxzq éjkntxzkj uv tv wnénxzjkqz• q xméywwqévkjz• kunxzn ·snt€ x vlqtjrvk€uq xzn wnt¹uq [ ëzv yzkné lntqn xvkxnu tn zjr v·nkqltv rjr rj nz x¹ tj wnék€p koms¹l ² knl• k wéjkvp ·jxzq éjkntxzkj y tjx inxrvtn·t€n xyuu€ j k inxrvtn·t€} xyuuj} wnénxzjks¹z• xsj mjnu€n uv tv ljsnrv tn kxnmlj Ívxsn ëzvp wnénméywwqévkrq u€ wvsy·qu
I([) + J([) = (D + E ) + (D + E )[ +
+ (D + E )[ + + (DN + EN)[N +
è¹l xzv¹•qp k wéjkvp ·jxzq éjkntxzkj tjo€kjnzx¹ xyuuvp xznwntt€} é¹lvk q
Ívxuvzéqu znwné• rjr éjosjmjnzx¹ k xznwnttvp é¹l wévqo knlntqn {ytr|qp I([) q J([) ΀ qunnu
I([) J([) = (D + D [ + D [ + + DN[N + ) ×
× (E + E [ + E [ + + EN[N + )
Ìrjo€kjnzx¹ ·zv rjr q k xsy·jn utvmv·sntvk é¹l€ xzv ¹•qn k wéjkvp ·jxzq éjkntxzkj uv tv wv·snttv wnénutv qz• u€ vwyxrjnu lvrjojzns•xzkv ëzvmv yzkné lntq¹ Ëjplnu
é¹l wvsy·jí•qpx¹ wvxsn wv·snttvmv wnénutv ntq¹ fkvivlt€p ·snt ëzvmv é¹lj éjknt D E
ïsnt€ xvlné j•qn |
[ wvsy·jzx¹ lkj l€ wéq yutv |
ntqq |
D tj E [ q wéq yutv |
ntqq D [ tj E Ìtq ljíz |
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D E [ + D E [ = (D E + D E )[ |
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Òv·tv zjr n k€·qxs¹nu ·snt€ xvlné j•qn [ |
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D E [ + D E [ + D E [ = (D E + D E + D E )[ |
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Ævvi•n rvë{{q|qntz wéq [Q qunnz kql |
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D EQ + D EQ− + D EQ− + + DNEQ−N + + DQE |
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Òjrqu viéjovu |
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I([)J([) = D E + (D E + D E )[ + (D E + D E + D E )[ + + |
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+ (D EQ + D EQ− + D EQ− + + DNEQ−N + + DQE )[Q + |
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-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
è¹l xzv¹•qp k wéjkvp ·jxzq éjkntxzkj tjo€kjnzx¹ |
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wévqoknlntqnu |
é¹lvk q |
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Æ ·jxztvxzq kvokvl¹ é¹l k rkjléjz wvsy·jnu |
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I ([) = D + D D [ + (D + D D )[ + (D D |
+ D D )[ + |
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Ívxuvzéqu znwné• rjr lns¹z léym tj léymj xznwntt€n é¹l€ Íyxz• xkvivlt€p ·snt é¹lj vzsq·nt vz tys¹ Ívrj nu ·zv k ëzvu xsy·jn xy•nxzkynz zjrvp xznwnttvp é¹l
F + F [ + F [ + + FQ[Q + |
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(E + E [ + + EN[N + )(F + F [ + + FQ[Q + ) = |
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= D + D [ + D [ + + DN[N + |
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bs¹ lvrjojzns•xzkj wnénutv qu é¹l€ k snkvp ·jxzq ëzvmv éjkntxzkj ΀ wvsy·qu é¹l
E F + (E F + E F )[ + + (E FQ + E FQ− + + EQF )[Q +
bs¹ zvmv ·zvi€ ëzvz é¹l xvkwjljs x é¹lvu tnvi}vlquv q lvxzjzv·tv ·zvi€ k€wvst¹sqx• éjkntxzkj
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= D |
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E F + E F = D |
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êzq éjkntxzkj ljíz inxrvtn·tyí xqxznuy yéjktntqp ls¹ vz€x rjtq¹ rvë{{q|qntzvk F F FQ
Êo wnékvmv yéjktntq¹ xqxznu€ wvsy·jnu F = |
D |
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Ívlxzjkqu wvsy·nttvn otj·ntqn kv kzvévn yéjktntqn ΀
wvsy·qu yéjktntqn E F = D − E DE qo rvzvévmv tj}vlqu ·zv
D E − E D
F = E Ævvi•n nxsq y n tjplnt€ rvë{{q|qntz€ F
F FQ− zv ls¹ vz€xrjtq¹ FQ qunnu yéjktntqn
E FQ = DQ − E FQ− − − EQF
êzv yéjktntqn éjoén¡quv wvxrvs•ry E ≠
Íéquntntqn xznwntt€} é¹lvk ls¹ lvrjojzns•xzkj zv lnxzk |
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Êzjr u€ lvrjojsq xy•nxzkvkjtqn é¹lj ylvksnzkvé¹ í•nmv xvvztv¡ntqí è¹l tjo€kjíz ·jxzt€u wéq lnsntqq é¹lvk q Îv tv lvrjojz• ·zv vt wvsy·jnzx¹ wéq éjosv ntqq {ytr|qq JI(([[))
Òjrqu viéjovu xznwntt€n é¹l€ uv tv xrsjl€kjz• yutv jz• q lnsqz• wvxsnltnn ² wéq yxsvkqq ·zv xkvivlt€p ·snt lnsqzns¹ vzsq·nt vz tys¹ êzq lnpxzkq¹ xvvzknzxzkyíz lnpxz
kq¹u tjl éjosjmjnu€uq {ytr|q¹uq
Ònwné• u€ uv |
nu qtj·n qxzvsrvkjz• xu€xs éjosv ntq¹ |
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D + D [ + + DQ[Q |
= F |
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+ F [ + |
+ F [N + |
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E + E [ + + EP[P |
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Ìtv votj·jnz |
·zv é¹l F + F [ + + FN[N + wvsy·jnzx¹ |
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wéq lnsntqq rvtn·tvmv xznwnttvmv é¹lj D + D [ + + DQ[Q tj
rvtn·t€p xznwnttvp é¹l E + E [ + + EP[P Êt€uq xsvkjuq ëzv éjkntxzkv votj·jnz ·zv
(E + E [ + + EP[P)(F + F [ + + FN[N + ) =
= D + D [ + + DQ[Q
mln wévqoknlntqn k snkvp ·jxzq éjkntxzkj vwénlns¹nzx¹ wv {vé uysn
Íéquntntqn xznwntt€} é¹lvk ls¹ lvrjojzns•xzkj zv lnxzk
f wvuv••í xznwntt€} é¹lvk uv tv lvrjojz• utvmqn zv
lnxzkj bs¹ ëzvmv inéyz tnrvzvéyí {ytr|qí q lkyu¹ xwvxvijuq éjosjmjíz nn k xznwnttvp é¹l Ívxrvs•ry {ytr|q¹ uv nz i€z•
wénlxzjksntj sq¡• nlqtxzkntt€u viéjovu k kqln xznwnttvmv
é¹lj zv rvë{{q|qntz€ wéq vlqtjrvk€} xznwnt¹} [ |
k vivq} |
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é¹lj} lvs t€ xvkwjljz• êzv q wéqkvlqz r |
lvrjo€kjnuvuy |
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èjxxuvzéqu tjwéquné qoknxztvn tju éjosv |
ntqn |
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− [
Ævokvl¹ vin ·jxzq ëzvmv éjosv ntq¹ k rkjléjz wvsy·jnu
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= + [ + [ + [ + + (Q + )[Q + |
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-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
cxsq ojuntqz• olnx• [ tj −[ zv wvsy·qu ·zv
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= − [ + [ − [ + + (− )Q(Q + )[Q + |
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( + [) |
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Ínénutv |
qk éjosv ntq¹ q k€kvlqu ·zv |
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( + [) = + > (− ) + @[ + > + (− ) + @[ + |
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+ + > (− )Q (Q + ) + (− )Q− Q + + (Q + ) @[Q + |
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Ì·nkqltv ·zv rvë{{q|qntz€ wéq tn·nzt€} xznwnt¹} viéj•j |
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ízx¹ k tys• rj lvn xsjmjnuvn lkj |
l€ k}vlqz k ëzq rvë{{q |
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|qntz€ x wévzqkvwvsv t€uq otjrjuq Çvë{{q|qntz |
n wéq |
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[ Q éjknt |
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( Q + ) − Q + ( Q − ) − + ( Q + ) |
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Ëv {ytr|qí |
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tv éjosv qz• k xznwnttvp |
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é¹l q qt€u |
viéjovu ΀ |
qunnu |
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¬ éjosv ntqn |
ls¹ |
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wvsy·jnzx¹ qo éjosv |
ntq¹ |
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( − [ ) |
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nxsq ojuntqz• k tnu [ tj [ |
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= + [ + [ + [ + + (Q + )[ Q + |
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΀ otjnu ·zv tqrjrj¹ {ytr|q¹ tn uv nz qunz• lky} éjosq·t€} éjosv ntqp k xznwnttvp é¹l Ívëzvuy rvë{{q|qntz
wéq [ Q k éjosv ntqq lvs nt éjkt¹z•x¹ rvë{{q|qntzy
wéq [ Q k éjosv ntqq Ìzxílj k€znrjnz xsnlyí•nn zv lnxzkv
( Q + ) − Q + ( Q − ) − + ( Q + ) = Q +
Íévqokvl¹•qn {ytr|qq
¬znwné• u€ y n uv nu wnénpzq r vxtvktvp znun ëzvp msjk€ ² wvt¹zqí wévqokvl¹•np {ytr|qq Íyxz• ljtj tnrvzv éj¹ wvxsnlvkjzns•tvxz• ·qxns D D DQ Ìiéjoynu
xznwnttvp é¹l D + D [ + D [ + + DQ [Q +
cxsq ëzvz é¹l x}vlqzx¹ k rjrvp zv visjxzq r {ytr|qq I([) zv {ytr|qí I([) tjo€kjíz wévqokvl¹•np ls¹ wvxsnlvkj
zns•tvxzq ·qxns D D DQ
Íévqokvl¹•qn {ytr|qq q iqtvuqjs•t€n rvë{{q|qntz€ |
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Ëjwéquné qo {véuys€ |
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= + [ + [ + + [Q + k€ |
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znrjnz ·zv {ytr|q¹ |
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¹ks¹nzx¹ wévqokvl¹•np ls¹ wvxsn |
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lvkjzns•tvxzq ·qxns ¬ {véuysj wvrjo€ kjnz ·zv ls¹ wvxsnlvkjzns•tvxzq ·qxns Q
wévqokvl¹•np ¹ks¹nzx¹ {ytr|q¹
( − [)
Ëjx iylyz qtznénxvkjz• wévqokvl¹•qn {ytr|qq ls¹ wvxsn lvkjzns•tvxznp D D DQ zjr qsq qtj·n xk¹ojtt€} x
rvuiqtjzvét€uq ojlj·juq f wvuv••í ëzq} {ytr|qp yljnzx¹ wvsy·jz• xju€n éjot€n xkvpxzkj ëzq} wvxsnlvkjzns•tvxznp Çévun zvmv u€ éjxxuvzéqu rjr xk¹ojt€ wévqokvl¹•qn {ytr |qq x én¡ntqnu énryééntzt€} xvvztv¡ntqp
Íévqokvl¹•qn {ytr|qq q iqtvuqjs•t€n rvë{{q|qntz€
Ëjplnu wévqokvl¹•yí {ytr|qí ls¹ rvtn·tvp wvxsnlvkj zns•tvxzq iqtvuqjs•t€} rvë{{q|qntzvk &Q &Q &QQ
Ívsjmj¹ k iqtvun Ë•ízvtj w j = q éjxwvsjmj¹ xsjmjnu€n k wvé¹lrn kvoéjxzjtq¹ xznwntnp [ wvsy·jnu
( + [)Q = & + & [ + & [ + + &N [N + + &Q [Q |
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ls¹ ·qxns &N N = Q f wvuv••í ëzvp wévqokvl¹•np
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{ytr|qq uv tv xéjktqzns•tv wévxzv lvrjojz• utvmqn xkvpxzkj ·qxns &NQ rvzvé€n i€sq wvsy·nt€ éjtnn x wvuv••í lvkvs•tv
}qzé€} éjxxy lntqp
N = &N + &N− bs¹ ëzvmv lvxzjzv·bvrj nu xtj·jsj ·zv &Q+ Q Q
tv yutv qz• vin ·jxzq éjkntxzkj tj + [ ΀ wvsy·qu ·zv
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Q tj Q + Ívëzvuy rvë{{q|qntzvu wéq [N iylnz &NQ+ Æ wéjkvp
n ·jxzq wéq éjxré€zqq xrvivr ·snt xvlné j•qp [N wv¹kqzx¹ lkj l€ wéq yutv ntqq &NQ [N tj q wéq yutv ntqq &NQ− [N− tj [ Ívëzvuy rvë{{q|qntz wéq [N k wéjkvp ·jxzq
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Ìin xyuu€ rvtn·t€ q vié€kjízx¹ rvmlj P xvvzknzxzknttvP + xzjtnz ivs•¡n Q
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(− )P = & + & L + & L + & |
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Énmrv lvrjojz• x wvuv••í wévqokvl¹•np {ytr|qq q |
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bévit€n wénlunz€
( + [)P = & |
+ & [ + & [ + + &W [W + + &P [P |
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wnénutv qz• snk€n q wéjk€n ·jxzq ëzq} éjkntxzk q wvlx·qzjz•
rvë{{q|qntz wéq [V
ïjxzt€u xsy·jnu ëzvmv éjkntxzkj wéq V = P = Q ¹ks¹nzx¹
&Q = (& ) + (& ) + + (&Q) |
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tjwvutqu ·zv &N = &Q−N
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( [ − \ + ] ) (\ − \ + \ + ]) + ( [ + \ − )
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wv Q ¹ks¹nzx¹ isq jp¡qu |ns€u r H Q − |
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éjoun•ntq¹ zv ·qxsv |
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rj lvmv wénlunzj ¹ks¹nzx¹ isq |
jp¡qu |ns€u r H(Q − )(Q − ) |
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xrvivr k kێj ntqq ( + [)N + ( + [)N+ + + ( + [)Q q |
ywév•ntq¹ |
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Íyxz• ( + [ + [ )Q = D + D [ + D [ + + D Q[ Q bvrj qzn ·zv j D D − D D + D D − D D + − D Q− D Q =
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D − D + D − + (− )QD |
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Ëjplqzn rvë{{q|qntz wéq [N wvsy·jí•qpx¹ wvxsn éjxré€zq¹ xrvivr q wéqknlntq¹ wvlvit€} ·sntvk k k€éj ntqq
j ( + [ + [ + + [Q− ) i ( + [ + [ + + [Q− )W
bévit€n wénlunz€
΀ vwénlnsqsq &PQ rjr ·qxsv xwvxvivk k€iéjz• P wénlunzvk
qo éjosq·t€} Q dlnx• ·qxsj P q Q lvs t€ i€z• tjzyéjs•t€uq qsq wvxrvs•ry u€ lvwyxrjnu q wyxz€n utv nxzkj k réjptnu xsy·jn tys¹uq Ëq lévit€uq tq vzéq|jzns•t€uq vtq i€z• tn uvmyz ² tns•o¹ mvkvéqz• ·zv k€iéjtv wénlunzvk qsq
wénlunzj
Ëv y ·qxns &PQ xsq¡rvu utvmv ojun·jzns•t€} xkvpxzk Ívëzvuy }vznsvx• i€ k€pzq oj znxt€n éjurq tjzyéjs•t€}