Веленкин;Комбинаторика
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·qxns q vwénlnsqz• &PQ ls¹ síi€} otj·ntqp Q q P zjr ·zvi€
kxn qsq }vz¹ i€ wv·zq kxn q} xkvpxzkj xv}éjtqsqx• q wvxsn zjrvmv vivi•ntq¹ êzy ojlj·y yljsvx• én¡qz• êpsnéy Ìltjrv nmv én¡ntqn vxtvk€kjsvx• tj qtznméjs•tvu qx·qxsntqq Ívrj nu xnp·jx rjr ino kx¹rq} qtznméjsvk uv tv vwénlnsqz• ·qxsj
&PQ ls¹ síi€} otj·ntqp Q tv wéq |ns€} tnvzéq|jzns•t€}
otj·ntq¹} P bs¹ ëzvmv ojwq¡nu &QP k xsnlyí•nu kqln |
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&P = |
Q(Q − ) (Q − P + ) |
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Íéjkj¹ ·jxz• éjkntxzkj qunnz xu€xs wéq síi€} otj ·ntq¹} Q ΀ wéqunu k€éj ntqn xzv¹•nn k wéjkvp ·jxzq
oj vwénlnsntqn &PU wéq síivu otj·ntqq U
&P = |
U(U − ) (U − P + ) |
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fvvztv¡ntq¹ lvrjojtt€n tjuq ls¹ ·qxns &PQ vxzjízx¹ k
ivs•¡qtxzkn xsy·jnk knét€uq q ls¹ ·qxns &PU bnsv k zvu ·zv wéjkj¹ ·jxz• éjkntxzkj ² ëzv utvmv·snt vz U fvvztv¡n tq¹ lvrjojtt€n tjuq éjtnn ls¹ ·qxns &PQ quníz kql éjkntxzk
ls¹ &PQ rjr utvmv·sntvk vz Q j k jsmnién lvrjo€kjíz ·zv lkj
utvmv·sntj vz U éjkt€n wéq kxn} tjzyéjs•t€} otj·ntq¹} U éjkt€ q wéq vxzjs•t€} otj·ntq¹} U
Ëjx iylyz k ljs•tnp¡nu qtznénxvkjz• otj·ntq¹ &PQ wéq |ns€} vzéq|jzns•t€} otj·ntq¹} Q Íyxz• Q = −U Òvmlj
P = |
(−U)(− U − ) (− U − P + ) |
= |
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&−U |
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(U + ) (U + P − ) |
P |
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= (− )P U |
P |
= (− )P &U+P− |
Ívsy·qsvx• zv n xjuvn r ·nuy u€ wéq¡sq léymqu wyznu k w
è¹l Ë•ízvtj
΀ tjokjsq rjr ëzv vi€·tv lnsjíz k ¡rvsn {véuysy ls¹ (D + [)Q iqtvuvu Ë•ízvtj êzv tjokjtqn x zv·rq oéntq¹ qxzvéqq
ujznujzqrq tnknétv Ðvéuysy ls¹ (D + [)Q }vév¡v otjsq xénltnjoqjzxrqn ujznujzqrq Ìujé ®jp¹u -q¹xëllqt q lé
è¹l Ë•ízvtj
Æ djwjltvp |
ckévwn ojlvsmv |
lv Ë•ízvtj nn otjs Èsno Íjxrjs• |
djxsymj n |
Ë•ízvtj i€sj |
k qtvu ² nuy yljsvx• vivi•qz• |
{véuysy ls¹ (D + [)Q tj xsy·jp tn|ns€} wvrjojznsnp Êunttv vt lvrjojs ·zv nxsq D ² wvsv qzns•tvn ·qxsv q _[ _ < D zv ls¹ síivmv lnpxzkqzns•tvmv otj·ntq¹ U qunnz unxzv éjkntxzkv
(D + [)U = DU + UDU− [ + |
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+ U |
(U − ) |
U− |
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(U − ) (U − N + ) |
U− |
N[N + = |
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+ + U |
N |
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= DU + & DU− [ + & DU− [ + + &NDU−N[N + |
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U |
U |
U |
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mln &NU ² kknlntt€n k€¡n {véuysvp ·qxsj Òvs•rv znwné•
wvsy·qsvx• tn rvtn·tvn ·qxsv xsjmjnu€} j inxrvtn·t€p é¹l Æ xsy·jn rvmlj U ² tjzyéjs•tvn ·qxsv U = Q tj·qtj¹ x tnrv
zvévmv unxzj kv kxn k€éj ntq¹ ls¹ &N k}vlqz xrvirj (U − Q)
U
viéj•jí•j¹ ëzq rvë{{q|qntz€ k tys• Êunttv wvëzvuy wéq tjzyéjs•t€} otj·ntq¹} U kunxzv inxrvtn·tvmv é¹lj wvsy·jnzx¹ rvtn·tj¹ xyuuj
bvrjojzns•xzkv {véuys€ k vi•nu kqln zéniynz wéquntntq¹ lq{{nént|qjs•tvmv qx·qxsntq¹ ΀ tn iylnu lvrjo€kjz• {véuysy
ls¹ kxn} otj·ntqp U q éjxxuvzéqu sq¡• xsy·jp rvmlj U ² |nsvn vzéq|jzns•tvn ·qxsv U = −Q Æ ëzvu xsy·jn wvlsn j•j¹ lvrjojzns•
xzky {véuysj wéqtqujnz xsnlyí•qp kql
(D + [)−Q = |
D−Q − & D−Q− [ + & |
−Q− [ − + (− )N &N |
−Q−N[N + |
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Q |
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Q+ D |
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Q+N− D |
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Êtj·n ëzv éjkntxzkv uv tv wnénwqxjz• zjr |
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[ −Q |
= − |
& [ |
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N |
[ N |
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+ D |
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Q D |
+ &Q+ D |
− + (− )N &Q+N− D |
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j ojuntqk [ |
tj −W zvmlj _W_ < wvsy·qu xsnlyí•nn éjkntxzkv |
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( − W)−Q = + & W + & |
+ & |
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+ &N |
N + |
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Q |
Q+ W |
Q+ W |
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Q+N− W |
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cmv q iylnu lvrjo€kjz• wéq _W_ < x wvuv••í qtlyr|qq wv Q |
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Íéq Q = qunnu &N |
= &N = q wvzvuy lvrjo€kjnuvn xvvztv¡n |
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Q+N− |
N |
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tqn wéqtqujnz xsnlyí•qp kql |
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= + W + W + W + + WN + |
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− W |
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Ëv ëzv ² y |
n qoknxztj¹ tju {véuysj |
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-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
Íénlwvsv qu znwné• ·zv y |
n lvrjojtv éjkntxzkv q wvrj |
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·zv qo tnmv xsnlynz éjkntxzkv |
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( − W)−Q− = + & |
+ & |
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+ & |
+ + &N |
N + |
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Q+ W |
Q+ W |
Q+NW |
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bs¹ ëzvmv yutv qu vin ·jxzq éjkntxzkj tj xvvzknzxzkyí•qn ·snt€ xvvztv¡ntq¹ ΀ wvsy·qu ·zv
( − W)−Q− = ( + & W + & |
+ & |
+ + &N |
N + ) × |
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Q |
Q+ W |
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Q+ W |
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Q+N− W |
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× ( + W + W + W + + WN + ) |
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znwné• |
éjxrévnu |
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xrvirq |
q |
kvxwvs•oynux¹ zv lnxzkvu |
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+ & + & |
+ + &N |
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k w |
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&Q− |
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énoys•zjzn r lvrjo€kjnuvuy xvvztv¡ntqí
Êzjr éjkntxzkv lvrjojtv Ëjwvutqu n•n éjo ·zv vtv xwéjknlsqkv sq¡• wéq _W _ < cxsq tnvxzvév t€p ·qzjzns•
wvwéviynz wvsv qz• k vinq} ·jxz¹} ëzvmv éjkntxzkj W = − q k€knlnz tj ëzvu vxtvkjtqq ©ojun·jzns•tyíª {véuysy
−Q = − & + & |
− & |
+ |
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Q |
Q+ |
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zv vt xvkné¡qz xné•notyí v¡qiry ² knl• xwéjkj tjwqxjtj xyuuj |ns€} ·qxns rvzvéj¹ tqrjr tn uv nz éjkt¹z•x¹ Q
Æ ;9,,, knrn rvmlj znvéq¹ inxrvtn·t€} é¹lvk n•n tn i€sj lnzjs•tv éjoéjivzjtj wvlvit€n v¡qirq lnsjsq q qoknxzt€n ujznujzqrq Ívtjlviqsqx• lnx¹zqsnzq¹ tjwé¹ ntt€} qxxsnlv kjtqp ls¹ zvmv ·zvi€ k€¹xtqz• ·zv n zjrvn xyuuj inxrv tn·tvmv é¹lj k rjrq} xsy·j¹} vtj xy•nxzkynz rvmlj uv tv wnénutv jz• é¹l€ rjr rvtn·t€n xyuu€ j rvmlj tns•o¹ q z l
Ìrjojsvx• ·zv zjrvp é¹l rjr k {véuysn tn qunnz xyuu€qsq rjr mvkvé¹z ujznujzqrq éjx}vlqzx¹ Æwév·nu xsnlynz
xrjojz• ·zv k rvt|n ;,; knrj wvt¹zqn xyuu€ inxrvtn·tvmv é¹lj i€sv vivi•ntv q i€sq ljt€ k ·jxztvxzq zjrqn vwénln
sntq¹ wéq rvzvé€} {véuysj wéqviénzjsj xu€xs Ëv ëzq |
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kvwévx€ k€}vl¹z oj éjurq tj¡np rtqmq |
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djuntqu k W tj −W ΀ wvsy·qu éjosv |
ntqn |
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( + W)−Q = − & W + & |
− & + + (− )N&N |
N + |
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Q |
Q+ W |
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Q+ W |
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Q+N− W |
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féjktqu nmv x éjosv |
ntqnu |
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( + W)Q = & + & W + |
& W + + &N WN + + &Q WQ |
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Q |
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΀ xtvkj wéq}vlqu r k€kvly ·zv wéq vivi•ntqq xqukvsj |
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&QN tj vzéq|jzns•t€n |
otj·ntq¹ |
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tjlv |
wvsv |
qz• |
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Êoksn·ntqn rkjléjzt€} rvétnp |
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N vzéq|jzns•t€ zv tjlv |
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&−Q |
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Q+N− cxsq n otj·ntq¹ |
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wvsv |
qz• &N = wvxrvs•ry k éjosv |
ntq¹ q tn k}vl¹z |
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xsjmjnu€n x vzéq|jzns•t€uq xznwnt¹uq W Ív znu |
n xvviéj |
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ntq¹u &N = wéq ≤ Q < N |
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Q |
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Êoksn·ntqn rkjléjzt€} rvétnp
Ònwné• u€ otjnu rjr k€ms¹lqz iqtvu Ë•ízvtj tn zvs•rv ls¹ |ns€} tv q ls¹ lévit€} q lj n ls¹ qééj|qvtjs•t€}
otj·ntqp wvrjojzns¹ Ívxuvzéqu ·zv wvsy·qzx¹ wéq Q = q
Q = − |
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Íéq Q = {véuysj Ë•ízvtj wéqtqujnz kql |
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− |
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( + [) = + |
[ + |
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[ + |
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+ |
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− |
− N + |
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+ + |
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Íénviéjoy¹ ëzv k€éj |
ntqn wvsy·jnu |
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( + [) = + |
[ − |
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[ + |
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+ |
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− + (− )N− ( N − ) |
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Òv·tv zjr |
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n wéq Q = − k€kvlqu ·zv |
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− |
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( + [) = − |
[ + |
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− |
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− |
[ + + (− )N |
( N − ) |
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Ívsy·ntt€n {véuys€ uv tv ojwqxjz• qtj·n bs¹ ëzvmv
ojunzqu ·zv |
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( N − ) |
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( N − ) |
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N− |
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N |
− |
((N− ) ) |
N N |
− & N− |
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N N |
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-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
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( N − ) = |
( N) |
= |
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&N |
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N |
N(N ) |
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Ívëzvuy |
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( + [) = + |
[ − |
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& [ + |
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(− )N− |
N |
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N |
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+ |
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& [ − + |
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N N− & N− |
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( + [) |
− |
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& [ + |
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= − |
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& [ − + (− )N |
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&N |
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N |
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êzq k€éj |
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ntq¹ x}vl¹zx¹ k visjxzq _ [ _ < f q} wvuv••í |
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uv tv tj}vlqz• x zéniynuvp xznwnt•í zv·tvxzq rkjléjzt€n rvétq vméjtq·qkj¹x• k {véuysj} q tnxrvs•rquq wnék€uq xsjmjnu€uq Ëjwéquné
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√ = √ + = √ + = ( + ) ≈ |
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≈ + − |
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zv·tvn otj·ntqn √ = Ìltjrv tjx iylyz qtznénxv
kjz• tn wéqsv ntq¹ yrjojtt€} {véuys r qoksn·ntqí rvétnp qo ·qxns j xvvztv¡ntq¹ un ly iqtvuqjs•t€uq rvë{{q|qnt zjuq rvzvé€n k€znrjíz qo wvsy·ntt€} éjosv ntqp
ïzvi€ wvsy·qz• ëzq xvvztv¡ntq¹ kvoknlnu vin ·jxzq éj kntxzkj k rkjléjz Ív wéjkqsy yutv ntq¹ xznwntt€} é¹lvk
wvsy·jnu ·zv rvë{{q|qntz wéq [N k wéjkvp ·jxzq qunnz kql
(− )N |
(&N |
+ & &N− |
+ & &N− |
+ |
+ &N |
) |
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N |
N |
N− |
N− |
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Æ snkvp n ·jxzq éjkntxzkj wvsy·jnu |
( + [)− |
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+ [ |
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Ëv u€ otjnu ·zv |
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= − [ + [ − + (− )N[N + |
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+ [ |
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féjktqkj¹ rvë{{q|qntz€ wéq xznwnt¹} [N k vinq} ·jxz¹} éjkntxzkj wéq}vlqu r zv lnxzky
&N |
+ & &N− |
+ & &N− |
+ + &N = N |
N |
N− |
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xwéjknlsqkvn wéq N ≥
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èjxrévnu xrvirq k wéjkvp ·jxzq ëzvmv éjkntxzkj ΀ wvsy ·qu xznwnttvp é¹l wéq·nu qo éjkntxzkj snkvp ·jxzq xsnlynz ·zv kxn rvë{{q|qntz€ ëzvmv é¹lj révun xkvivltvmv ·sntj éjkt€ tysí Ìzxílj wvsy·jnu zv lnxzkv
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èjxré€kj¹ xrvirq k vinq} ·jxz¹} ëzvmv éjkntxzkj q xéjk
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-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
Íévqokvl¹•qn {ytr|qq q énryééntzt€n xvvztv¡ntq¹
΀ y n mvkvéqsq ·zv znvéq¹ wévqokvl¹•q} {ytr|qp znxtv xk¹ojtj x énryééntzt€uq xvvztv¡ntq¹uq Ænétnux¹ xtvkj r lnsntqí utvmv·sntvk
Íyxz• I([) = D + D [ + +DQ[Q q J([) = E + E [ + + EP[P ² lkj utvmv·sntj wéq·nu E ≠ ΀ iylnu révun zvmv
wénlwvsjmjz• ·zv Q < P z n ·zv jsmniéjq·nxrj¹ lévi• I([) ²
J([)
wéjkqs•tj¹ k wévzqktvu xsy·jn u€ kxnmlj uv nu k€lnsqz• qo tnn |nsyí ·jxz•
΀ otjnu ·zv nxsq
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zv
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èjxrévnu xrvirq k wéjkvp ·jxzq ëzvmv éjkntxzkj q xéjktqu
rvë{{q|qntz€ wéq vlqtjrvk€} xznwnt¹} [N |
xsnkj q xwéjkj |
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N = P − |
E FP− + E FP− + + EP− F = DP− |
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u€ wvsjmjnu |
DN = wéq Q < N ≤ P − ¬ ljs•¡n kxn xvvztv |
¡ntq¹ quníz |
vlqt q zvz n kql |
N ≥ E FN+P + E FN+P− + + EPFN =
knl• k I([) tnz ·sntvk xvlné j•q} [P [P+ q z l
Òjrqu viéjovu rvë{{q|qntz€ F F FN é¹lj
ylvksnzkvé¹íz énryééntztvuy xvvztv¡ntqí Çvë{{q|qnt z€ ëzvmv xvvztv¡ntq¹ ojkqx¹z sq¡• vz otjuntjzns¹ léviq ïqxsqzns• n léviq ty nt ls¹ tj}v lntq¹ wnék€} P ·sntvk
F F FP− énryééntztvp wvxsnlvkjzns•tvxzq
Ìiéjztv nxsq ljtv énryééntztvn xvvztv¡ntqn q ojljt€ tj·js•t€n ·snt€ F F FP− zv u€ uv nu wv {véuysju
èjosv ntqn tj ësnuntzjét€n léviq |
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k€·qxsqz• otj·ntq¹ D D DP− ¬ zvmlj jsmniéjq·nx rj¹ lévi•
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¹ks¹nzx¹ ls¹ wvxsnlvkjzns•tvxzq ·qxns F F FN wév
qokvl¹•np {ytr|qnp
Ëj wnék€p koms¹l rj nzx¹ ·zv u€ tq·nmv tn k€qméjsq wéq ojuntn énryééntztvmv xvvztv¡ntq¹ wévqokvl¹•np {ytr|qnp knl• kxn éjktv wéqlnzx¹ lnsqz• ·qxsqzns• tj otjuntjzns• j ëzv wéqknlnz r zvuy n xjuvuy énryééntztvuy xvvztv¡n tqí Ëv lnsv k zvu ·zv tjl lévi•í uv tv k€wvst¹z• tnrvzvé€n jsmniéjq·nxrqn wénviéjovkjtq¹ j ëzv visnm·qz vz€x rjtqn ·qxns FN
èjosv ntqn tj ësnuntzjét€n léviq
΀ wvrj nu xnp·jx rjr x wvuv••í jsmniéjq·nxrq} wénvi éjovkjtqp wévqokvl¹•np {ytr|qq uv tv én¡jz• énryééntzt€n xvvztv¡ntq¹ Íénlwvsv qu ·zv otjuntjzns• léviq éjosv
nt tj utv qznsq wnékvp xznwntq
J([) = EP ([ − [ )U ([ − [N)V
Ìzunzqu ·zv ls¹ ëzvmv tjlv wénlkjéqzns•tv én¡qz• yéjktntqn J([) = E + E [ + + EP [P = z n }jéjrznéqxzq·nxrvn yéjktntqn xv
vztv¡ntq¹
Òvmlj ¹xtv ·zv lévi• wvsy·qsjx• k énoys•zjzn wéqkn lntq¹ r vltvuy otjuntjznsí xsnlyí•q} ësnuntzjét€} lévinp
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Ëju tnqoknxzt€ olnx• sq¡• rvë{{q|qntz€ $ $NV ïzvi€
tjpzq ëzq rvë{{q|qntz€ tjlv yutv qz• vin ·jxzq éjkntxzkj
tj otjuntjzns• EP ([ − [ )U ([ − [N)V éjxré€z• xrvirq q
xéjktqz• rvë{{q|qntz€ wéq vlqtjrvk€} xznwnt¹} [ Êo wvsy ·qk¡npx¹ xqxznu€ U + + V yéjktntqp u€ q tj}vlqu qxrvu€n
rvë{{q|qntz€
Êtvmlj yljnzx¹ vivpzqx• q ino én¡ntq¹ xqxznu€ yéjktntqp
Íyxz• tjwéquné tjlv éjosv qz• lévi• [ − [ + [ + Òjr
[ − [ +
rjr [ − [ + = ([ − )([ − ) = ([ − )([ + )([ − )([ + ) zv ëzv éjosv ntqn lvs tv qunz• kql
[ − [ + [ + |
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Íéqkvl¹ r vltvuy otjuntjznsí wvsy·jnu ·zv
[ − [ + [ + = $([ + )([ − )([ + ) + + %([ − )([− )([+ ) + &([− )([+ )([ + ) + '([− )([+ )([− )
êzv éjkntxzkv lvs tv k€wvst¹z•x¹ wéq kxn} otj·ntq¹} [ Ëv wéq [ = kxn xsjmjnu€n k wéjkvp ·jxzq éjkntxzkj révun
wnékvmv viéj•jízx¹ k tys• q u€ wvsy·jnu − $ = Ívëzvuy
$ = − Òv·tv zjr |
n wvsjmj¹ |
[ = − [ = [ = − tj}vlqu |
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% = − & = ' = Òjrqu viéjovu |
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wv {véuysn Ë•ízvtj Ëjwéquné |
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èjosv ntqn tj ësnuntzjét€n léviq
= ( + [+ [ + + [Q+ ) − |
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·zv rvë{{q|qntz wéq [Q k€éj |
jnzx¹ {véuysvp |
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΀ y n otjnu ·zv ojlj·j v éjosv ntqq jsmniéjq·nxrvp léviq k xznwnttvp é¹l éjktvxqs•tj ojlj·n v én¡ntqq tnrvzv évmv énryééntztvmv xvvztv¡ntq¹ wéq ojljtt€} tj·js•t€} yx svkq¹} Òjrqu viéjovu x wvuv••í éjosv ntq¹ lévinp tj ësnuntzjét€n q wvxsnlyí•nmv éjosv ntq¹ wvsy·ntt€} ësnunt zjét€} lévinp k xznwntt€n é¹l€ uv tv én¡jz• sqtnpt€n énryééntzt€n xvvztv¡ntq¹ x wvxzv¹tt€uq rvë{{q|qntzjuq
Êzjr nxsq ojljtv énryééntztvn xvvztv¡ntqn q otj·n tq¹ F FP− zv xtj·jsj tjlv wv {véuysju tjpzq
otj·ntq¹ D DP− Ìtq ljíz rvë{{q|qntz€ utvmv·sntj I([) xzv¹•nmv k ·qxsqznsn léviq
I([) |
= F |
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J([) |
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j otjuntjzns• J([) ëzvp léviq éjknt E + E [ + + EP[P Ëjp
I([)
lnttyí lévi• J([) tjlv éjosv qz• tj ësnuntzjét€n léviq
wvxsn ·nmv rj lyí ësnuntzjétyí lévi• éjosv qz• k xznwnttvp é¹l wv {véuysn Ë•ízvtj q wéqknxzq wvlvit€n ·snt€ Çvë{
{q|qntz wéq [N k wvsy·nttvu é¹ln q ljnz otj·ntqn FN èn¡qu tjwéquné énryééntztvn xvvztv¡ntqn
FN+ − FN+ + FN = wéq tj·js•t€} yxsvkq¹} F = F = −
dlnx• E = E = − E = ëzv wéq P = wvëzvuy
J([) = − [ + [ Êo {véuys wvsy·jnu ·zv D = E F = D = E F + E F = − wvëzvuy I([) = − [ Ònu xju€u u€ y n
− [ = − [ +
− [ + [