Веленкин;Комбинаторика
.pdf
-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
bs¹ vz€xrjtq¹ én¡ntq¹ tju tjlv éjosv |
qz• k xznwnttvp |
|||||||||||||
é¹l lévi• |
|
− [ |
|
|
Ëv − [ + [ = ( − [)( − [) wvë |
|||||||||
− [ + [ |
||||||||||||||
zvuy |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
− [ |
|
= |
|
|
− [ |
= |
$ |
+ |
% |
|||
|
− [ + [ |
( − [)( − [) |
− [ |
− [ |
||||||||||
|
|
|
|
|||||||||||
Ìxkviv |
lj¹x• vz otjuntjzns¹ wvsy·jnu |
|
|
|||||||||||
|
|
|
|
− [ = $( − [) + %( − [) |
|
|
||||||||
Ívsjmj¹ [ = tj}vlqu % = − j wvsjmj¹ [ = tj}vlqu |
||||||||||||||
$ = dtj·qz |
|
|
|
|
|
|
|
|
|
|
||||
− [ |
|
= |
|
− |
|
|
|
= ( − [)− − ( − [)− = |
||||||
− [ + [ |
− [ |
|
− [ |
|||||||||||
|
|
|
|
|
|
|
|
|||||||
= > + [+ [ + + ( [)N + @ − > + [+ [ + + ( [)N + @
vzrylj FQ = Q − Q
Êosv nttvn k€¡n wvokvs¹nz wv tvkvuy koms¹tyz• tj w
Íévqokvl¹•qn {ytr|qq q ojlj·j v éjoiqntq¹}
Íéq én¡ntqq ojlj·q v wvxsnlvkjzns•tvu éjoiqntqq u€ wéq¡sq r énryééntztvuy xvvztv¡ntqí
7Q = 7 7Q− + 7 7Q− + + 7Q− 7 |
|
mln 7 = xu w êzv xvvztv¡ntqn i€sv én¡ntv v·nt• qx
ryxxzkntt€u viéjovu ² u€ xknsq ojlj·y r ojlj·n vi v·nénlq rvzvéyí y n yunsq én¡jz• Ëv én¡ntqn ojlj·q vi v·nénlq xjuv i€sv lvkvs•tv mévuvolrqu
fnp·jx u€ wvrj nu rjr tnwvxénlxzknttv én¡qz• xvvztv ¡ntqn bs¹ ëzvmv xvxzjkqu wévqokvl¹•yí {ytr|qí
I([) = 7 + 7 [ + 7 [ + + 7Q[Q +
Ívsv |
qu |
|
|
|
|
|
|
|
)([) = [I([) = |
7 [ + 7 [ + 7 [ + + 7 [Q+ + |
|
||||||
|
|
|
|
|
|
Q |
|
|
q kvoknlnu )([) k rkjléjz ΀ wvsy·qu ·zv |
|
|
||||||
) ([) = 7 [ + (7 7 |
|
+ 7 7 )[ + + |
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
+ (7 7 |
+ 7 7 |
+ + 7 |
|
)[Q+ + |
|
|
|
|
Q− |
Q− |
|
Q− 7 |
|
|
|
|
|
|
|
Ívsqtvuqjs•tj¹ {véuysj |
|
|
|
|
|
||||||||||
|
|
|
|
|
|
|
|||||||||||||||
Ëv wv énryééntztvuy xvvztv¡ntqí |
|
|
|
|
|
|
|||||||||||||||
|
|
|
7 7Q− + 7 7Q− + + 7Q− 7 = 7Q |
|
|
|
|
||||||||||||||
dtj·qz ) ([) = 7 [ + 7 [ + + 7 [Q+ + |
|
|
|
|
|
|
|||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
Q |
|
|
|
|
|
|
|
|
|
Ívsy·ntt€p é¹l nxz• tn ·zv qtvn rjr )([) − 7 [ wvxrvs•ry |
|||||||||||||||||||||
7 = vt éjknt )([) − [ Êzjr |
|
|
|
|
|
|
|
|
|
||||||||||||
|
|
|
|
|
|
|
) ([) = )([) − [ |
|
|
|
|
|
|
|
|||||||
bs¹ {ytr|qq )([) wvsy·qsvx• rkjléjztvn yéjktntqn |
|||||||||||||||||||||
èn¡j¹ nmv tj}vlqu ·zv )([) |
= |
± √ − [ |
Ívxrvs•ry qo |
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
)( ) = k€iqéjnu wnénl rvétnu otjr uqtyx Ív {véuysn |
|||||||||||||||||||||
u€ qunnu |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
√ − [ = ( − [) = |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
|
|
|
= |
|
− |
[ |
− |
& [ |
|
|
[ |
|
|
|
|
Q |
Q+ |
|
|
||
|
|
|
|
|
− − |
|
|
|
− |
||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Q + & Q [ |
|
|
|||
dtj·qz |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
)([) = |
|
|
− |
− [ − |
& [ − − |
|
Q |
Q+ − |
= |
|
|
||||||||||
|
Q + & Q [ |
|
|
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
= [ + & [ + + |
|
Q Q+ − |
|
|||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
Q + & Q [ |
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
féjktqkj¹ {véuys€ q wvsy·jnu ·zv 7Q = |
|
|
Q |
||||||||||||||||||
Q + & Q |
|||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
êzv wvstvxz•í xvvzknzxzkynz én¡ntqí wvsy·nttvuy éjtnn rvu iqtjzvét€u unzvlvu w
Ívsqtvuqjs•tj¹ {véuysj
Ënxrvs•rv éjo wéqunt¹¹ {véuysy iqtvuj Ë•ízvtj uv tv wvwévivkjz• éjosv qz• q ivsnn xsv t€n k€éj ntq¹ tjwéquné zjrqn rjr ([ + \ + ]) Êunttv
([ + \ + ]) = >([ + \) + ]@ = |
|
+ & ( + |
) |
+ |
( |
+ |
) |
+ |
|
||||
= ( |
+ |
|
) + |
( + \) |
|
||||||||
[ |
\ |
|
& [ |
] |
[ |
\ ] |
|
& [ |
|
\ ] |
|
& ] |
|
èjosv qu znwné• ([ + \) ([ + \) ([ + \) wv {véuysn iqtvuj Ë•ízvtj q wéqknlnu wvlvit€n ·snt€ ΀ wvsy·qu ·zv
([ + \ + ]) = [ + \ + ] + [ \ + [ ] + [\ + \ ] + [] + + \] + [ \ + [ ] + \ ] + [ \] + [\ ] + [\]
-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
Ëv zjrvp xwvxvi xsq¡rvu xsv nt f nmv wvuv••í zéyltv i€xzév vwénlnsqz• x rjrqu rvë{{q|qntzvu k}vlqz k éjosv n tqn ([ + \ + ]) tjwéquné ·snt [ \ ] •nsjzns•tv wvëzvuy
k€knxzq {véuysy xéjoy ljí•yí éjosv ntqn ls¹ k€éj ntq¹
([ + [ + + [P)Q |
|
êzy {véuysy tnzéyltv ymjljz• Íéq lvrjojzns•xzkn {véuys€ iqtvuj Ë•ízvtj u€ kqlnsq ·zv k éjosv ntqq (D + [)Q ·snt
[NDQ−N k}vlqz x rvë{{q|qntzvu |
3(N Q − N) Îv tv wénlwvsv |
||||
qz• ·zv k éjosv |
ntqq ([ + [ + + [P)Q rvë{{q|qntzvu wéq |
||||
[N [N [NP iylnz |
3(N N |
|
N |
) ΀ lvrj nu xnp·jx ·zv |
|
|
P |
|
|
|
P |
lnsv vixzvqz qunttv zjr
Æ xjuvu lnsn ojwq¡nu ([ + [ + + [P)Q k kqln wévqokn
lntq¹ Q xvutv qznsnp q éjxrévnu xrvirq k€wqx€kj¹ kxn xvutv qznsq k wvé¹lrn q} wv¹ksntq¹ ‰xtv ·zv wéq ëzvu wvsy·jzx¹ kxnkvouv t€n éjoun•ntq¹ x wvkzvéntq¹uq xvxzjk sntt€n qo iyrk [ [ [P zjrqn ·zv k rj lvn éjoun•ntqn
k}vlqz Q iyrk Ëv tnrvzvé€n qo ëzq} éjoun•ntqp ljlyz wvlvit€n ·snt€ Òjr iylnz nxsq k wnékvn éjoun•ntqn rj lj¹
iyrkj |
k}vlqz |
xzvs•rv |
n éjo xrvs•rv q kv |
kzvévn Ívëzvuy |
||
·zvi€ |
tjpzq |
rvë{{q|qntz wéq [N [N |
[NP |
tjlv |
xvx·qzjz• |
|
|
|
|
|
P |
|
|
xrvs•rv éjoun•ntqp x |
wvkzvéntq¹uq |
xvlné |
jz N |
éjo iyrky |
||
[ N éjo iyrky [ NP éjo iyrky [P ‰xtv ·zv rj lvn
zjrvn éjoun•ntqn ¹ks¹nzx¹ wnénxzjtvkrvp x wvkzvéntq¹uq qo N iyrk [ N iyrk [ NP iyrk [P ïqxsv zjrq} wnénxzj
tvkvr u€ vivotj·jsq 3(N N NP)
Êzjr lnpxzkqzns•tv rvë{{q|qntzvu wéq [N [N [NPP k éjosv ntqq k€éj ntq¹ xsy qz 3(N N NP) mln éj
oyunnzx¹ N + N + + NP = Q Ívëzvuy ·qxsj 3(N N NP) tjo€kjíz n•n wvsqtvuqjs•t€uq rvë{{q|qntzjuq wéq P =
vtq wénkéj•jízx¹ k iqtvuqjs•t€n rvë{{q|qntz€ bvrjojttyí tjuq {véuysy uv tv ojwqxjz• zjr
([ |
|
+ [ |
|
+ + |
[ )Q = |
∑ |
3(N N N |
) [N [N [NP |
|
||||
|
|
|
|
P |
|
|
|
P |
|
P |
|
||
|
|
|
|
|
|
N +N + +N = Q |
|
|
|
|
|
||
|
|
|
|
|
|
|
|
P |
|
|
|
|
|
mln xyuuj |
éjxwévxzéjtntj |
tj kxnkvouv t€n éjoiqntq¹ |
·qxsj |
||||||||||
Q tj P |ns€} tnvzéq|jzns•t€} xsjmjnu€} N + N |
+ + NP |
||||||||||||
Ívsqtvuqjs•tj¹ {véuysj
Ëjwvutqu ·zv
|
|
|
|
|
(N + N + + N ) |
||
|
3(N N N |
) = |
|
P |
|
||
|
|
|
|||||
|
|
|
|
P |
N N NP |
|
|
|
|
|
|
|
|
||
‰xtv ·zv |
nxsq tjivé |
(V V V ) |
wvsy·jnzx¹ |
wnénxzjtvkrvp |
|||
qo tjivéj |
(N N NP) |
zv 3(V PV VP) = 3(N N NP) |
|||||
Ívëzvuy |
tjwéquné |
k éjosv ntqq |
rvë{{q|qntz€ wéq |
||||
[ \] q [\] vlqtjrvk€ êzv ojun·jtqn visnm·jnz k€wqx€kjtqn
·sntvk éjosv ntq¹ bvxzjzv·tv tjpzq rvë{{q|qntz€ ls¹ zjrq} éjoiqntqp Q = N + N + + NP ·zv N ≥ N ≥ ≥ NP j
wvzvu wnénxzjks¹z• wvrjojznsq kxnuq kvouv t€uq xwvxvijuq Ëjwéquné k€·qxsqu ([ + \ + ]) cxsq tn y·qz€kjz• wvé¹
lvr xsjmjnu€} tjwéquné ojwqx€kj¹ q} k wvé¹lrn yi€kjtq¹ zv ·qxsv uv tv éjoiqz• tj xsjmjnu€} w¹z•í xwvxvijuq
= + + = + + = + +
= + + = + +
Ëv 3( ) = 3( ) = 3( ) = 3( ) =
3( ) = Ívëzvuy
([ + \ + ]) = [ + \ + ] + [ \ + [\ + [ ] + [] + \ ] +
+\] + [ \ + [ ] + [ \ + [ ] + \ ] + \ ] +
+[ \] + [\ ] + [\] + [ \ ] + [ \] + [\ ]
Ðvéuysj wvokvs¹nz snmrv lvrjojz• tnrvzvé€n xkvpxzkj |
||||||
·qxns 3(N N NP) Ëjwéquné |
nxsq wvsv qz• k |
ëzvp {vé |
||||
uysn [ = [ = = [P zv wvsy·qu ·zv |
|
|
|
|||
PQ = |
∑ |
3(N N NP) |
|
|
||
N +N + +N |
= Q |
|
|
|
|
|
|
|
P |
|
|
|
|
dlnx• xyuuj éjxwévxzéjtntj tj kxn éjoiqntq¹ ·qxsj |
Q |
tj |
||||
P tnvzéq|jzns•t€} |
|ns€} xsjmjnu€} Q = N + N + + NP |
q |
||||
wvé¹lvr xsjmjnu€} y·qz€kjnzx¹ |
|
|
|
|
||
bjsnn nxsq yutv qz• vin |
·jxzq éjkntxzkj |
|
tj |
|||
[ + [ + + [P wéquntqz• r snkvp ·jxzq jtjsvmq·tvn éjosv |
||||||
ntqn j k wéjkvp éjxré€z• xrvirq zv ls¹ 3(N N NP) |
||||||
wvsy·qu xsnlyí•nn énryééntztvn xvvztv¡ntqn |
|
|
|
|||
3(N N NP) = 3(N − N NP) + |
|
|
|
|||
+ 3(N N − NP) + + 3(N N NP − ) |
|
|||||
-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
cxsq n wnénutv qz• snk€n q wéjk€n ·jxzq éjosv ntqp
([ |
|
+ [ |
|
+ + [ |
)Q = |
∑ |
3(N N N |
) [N [N [N2 |
|||||||||
|
|
P |
N +N + +NP= Q |
|
|
|
|
P |
|
|
P |
||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
([ |
|
+ [ |
|
+ + [ |
)V = |
∑ |
3(O |
|
O O |
|
) [O [O [OP |
||||||
|
|
|
|
|
P |
O +O + +OP= V |
|
|
|
P |
|
|
P |
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
q xéjktqz• rvë{{q|qntz€ k vinq} ·jxz¹} wéq [U [U [UPP zv wvsy·qu zv lnxzkv
3(U U UP) = ∑ 3(N N NP) 3(O O OP) |
|
|
NL+OL=UL |
|
|
dlnx• xyuuqévkjtqn éjxwévxzéjtntv tj kxn tjivé€ |ns€} |
||
tnvzéq|jzns•t€} ·qxns (N N NP) q (O O OP) |
zjrq} |
|
·zv N + N + + NP = Q O + O + + OP = V q |
N + O = U |
|
N + O = U NP + OP = UP ΀ wénlvxzjks¹nu |
·qzjznsí wv |
|
lévitv wévknxzq xvvzknzxzkyí•qn k€rsjlrq
èjoyunnzx¹ {véuys€ uv tv i€sv i€ wvsy·qz• q tn wvs•oy¹x• wévqokvl¹•np {ytr|qnp Ëv k ëzvu xsy·jn wéq¡svx• i€ wévkvlqz• tnwévxz€n éjxxy lntq¹ Íéquntntqn n wévqokvl¹•np {ytr|qq wvokvs¹nz wvsy·qz• ëzq zv lnxzkj jkzvujzq·nxrq k€wvst¹¹ sq¡• tnxsv t€n jsmniéjq·nxrqn wén
viéjovkjtq¹
djlj·q
Ëjplqzn rvë{{q|qntz wvsy·jí•qpx¹ wvxsn éjxré€zq¹ xrvivr q wéqknlntq¹ wvlvit€} ·sntvk k k€éj ntqq
j |
( + [ − [ ) |
wéq [ i ( + [ − [ ) wéq [ |
k |
( + [ − [ ) |
wéq [ |
Ëjplqzn rvë{{q|qntz€ wéq [ q [ wvxsn éjxré€zq¹ xrvivr q wéqknlntq¹ wvlvit€} ·sntvk k k€éj ntqq ( + [ + [ )
Æ rjrvu qo k€éj ntqp ( + [ − [ ) qsq ( − [ + [ )
iylnz ivs•¡n rvë{{q|qntz wéq [ wvxsn éjxré€zq¹ xrvivr q wéqknlntq¹ wvlvit€} ·sntvk"
Ëjplqzn ·qxsv éjosq·t€} vltv·sntvk wvsy·jí•q}x¹ wvxsn kvoknlntq¹ k xznwnt• k€éj ntq¹ ([ + [ + + [Q) q wéqknlntq¹
wvlvit€} ·sntvk
Ëjplqzn tjqivs•¡qp rvë{{q|qntz k éjosv ntq¹} j (D + E + F) i (D + E + F + G)
Íévqokvl¹•qn {ytr|qq q éjoiqntq¹ ·qxns |
|
|
|
Íévqokvl¹•qn {ytr|qq q éjoiqntq¹ ·qxns
Æmsjkn ,9 u€ én¡jsq éjosq·t€n rvuiqtjzvét€n ojlj·q tj éjoiqntqn ·qxns êzq ojlj·q ·jxzv snmrv én¡jízx¹ x wvuv••í wévqokvl¹•q} {ytr|qp Ìivotj·qu ·néno DQ ·qxsv xwvxvivk
éjoiqntq¹ ls¹ wjéjunzéj ojlj·q éjktvmv Q q xvxzjkqu é¹l
D + D [ + D [ + + DQ[Q + Æv utvmq} ojlj·j} yljnzx¹ xv xzjkqz• jsmniéjq·nxrvn k€éj ntqn I([) zjrvn ·zv wvxsn éjx
ré€zq¹ xrvivr k ëzvu k€éj ntqq xsjmjnuvn [Q wvkzvé¹nzx¹
évktv DQ éjo Òvmlj I([) = D + D [ + D [ + + DQ[Q + q otj·qz I([) ² wévqokvl¹•j¹ {ytr|q¹ ls¹ wvxsnlvkjzns•tvxzq
D D DQ
Íyxz• tjwéquné éjxxujzéqkjnzx¹ ywsjzj tnrvzvévp xyuu€ éjosq·t€uq uvtnzjuq ² éjoiqntqn ·qxsj 1 tj xsjmjnu€n rj lvn qo rvzvé€} éjktv vltvuy qo ·qxns Q QN Íéq ëzvu
k xyuun xsjmjnu€n tn lvs t€ wvkzvé¹z•x¹ j wvé¹lvr xsjmjn u€} tn qméjnz évsq
bs¹ tj}v lntq¹ ·qxsj xwvxvivk ywsjz€ viéjoynu k€éj n tqn ( + [Q )( + [Q ) ( + [QN) Íéq éjxré€zqq xrvivr wvsy
·jzx¹ xsjmjnu€n kqlj [QL[QM [QV mln QL QM QV ² tnrvzv
é€n qo ·qxns Q Q QN Ívëzvuy [1 kxzénzqzx¹ k xyuun
xzvs•rv éjo xrvs•rquq xwvxvijuq uv tv éjoiqz• 1 tj xsjmjn u€n zéniynu€u viéjovu
Ëjwéquné nxsq tjlv yotjz• xrvs•rquq xwvxvijuq uv tv ywsjzqz• rvwnnr uvtnzjuq wv
rvwnnr iné¹ tn ivsnn vltvp uvtnz€ rj lvmv lvxzvqtxzkj zv tjlv xvxzjkqz• k€éj ntqn
( + [ )( + [ )( + [ )( + [ )( + [ )( + [ )( + [ )( + [ )
éjxré€z• xrvirq q tjpzq rvë{{q|qntz wéq [ êzvz rvë{{q |qntz iylnz rjr éjo zjrqu xrvs•rv nxz• xwvxvivk wénlxzjkqz•
[ k kqln wévqoknlntq¹ lvwyxzqu€} xznwntnp [ [ [ [ [
[ [ [ [ [ q z w
¬ znwné• én¡qu x wvuv••í wévqokvl¹•q} {ytr|qp ivsnn xsv tyí ojlj·y
frvs•rquq xwvxvijuq uv tv ywsjzqz• rvwnnr uvtnzj uq wv q rvwnnr"
Æ ëzvp ojlj·n tjlv tjpzq ·qxsv xwvxvivk éjoiqz• ·qxsv tj xsjmjnu€n éjkt€n q wéq·nu wvé¹lvr xsjmjnu€} tn qunnz otj·ntq¹ q q} rvsq·nxzkv tn vméjtq·ntv Êt€uq
-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
èqx
xsvkjuq tju tjlv tjpzq ·qxsv tnvzéq|jzns•t€} én¡ntqp yéjktntq¹ P + Q = bs¹ ëzvmv xvxzjkqu k€éj ntqn
I([) = ( + [ + [ + + [ P + ) ( + [ + [ + + [ Q + )
Ívrjojznsq wéq [ k wnékvp xrvirn wévinmjíz kxn tnvzéq|j zns•t€n ·qxsj réjzt€n j kv kzvévp xrvirn ² kxn tnvzéq |jzns•t€n ·qxsj réjzt€n ‰xtv ·zv wvxsn éjxré€zq¹ xrvivr [ kvplnz x rvë{{q|qntzvu éjkt€u ·qxsy én¡ntqp yéjktntq¹P + Q = Æ ·jxztvxzq rvë{{q|qntz wéq [ ljxz vzknz tj
tj¡y ojlj·y
Æunxzv éjxré€zq¹ xrvivr uv tv wvxzywqz• zjr Ævxwvs•oy nux¹ {véuysvp ls¹ inxrvtn·tvp mnvunzéq·nxrvp wévménxxqq Òvmlj k€éj ntqn wnénwqx€kjnzx¹ k kqln
I([) = |
|
|
|
= |
|
|
|
|
|
||||
|
− [ − [ − [ − [ + [ |
|
||||
¬ znwné• éjolnsqu ·qxsqzns• tj otjuntjzns• wv wéjkqsy lnsntq¹ utvmv·sntvk zvs•rv iylnu éjxwvsjmjz• utvmv·snt€ tn wv yi€kjí•qu j wv kvoéjxzjí•qu xznwnt¹u [ Ëj·jsv ëzvmv lnsntq¹ wvrjojtv tj éqx Íévlvs j¹ lnsntqn tjplnu
qxrvu€p rvë{{q|qntz wéq [
Ìi•j¹ ojlj·j olnx• zjrvkj
Ëjpzq xrvs•rquq xwvxvijuq uv tv éjoiqz• ·qxsv 1 tj xsjmjnu€n éjkt€n xvvzknzxzknttv D E P wéq·nu wvé¹lvr xsjmjnu€} tn y·qz€kjnzx¹
Æ ëzvu xsy·jn wévqokvl¹•j¹ {ytr|q¹ qunnz kql
I([) = ( + [D + [ D + + [ND + ) ( + [E + [ E + + [VE + ) × ×
× ( + [P + [ P + + [TP+ ) = |
|
|
|
( − [D) ( − [E) ( − [P) |
|
Íévqokvl¹•qn {ytr|qq q éjoiqntq¹ ·qxns |
|
|
|
Ëjwéquné k ojlj·n v éjountn méqknttqrj w tjlv xvxzj kqz• wévqokvl¹•yí {ytr|qí
I([) = |
|
|
|
( − [) ( − [ ) ( − [ ) ( − [ ) |
|
Ínénutv j¹ k€éj |
ntq¹ xzv¹•qn k otjuntjznsn léviq wvsy |
|||
·jnu I([) = |
|
|
Æ€wvst¹¹ lnsntqn |
|
− [ − [ + [ + [ − [ − [ + [ |
||||
|
|
|||
tj}vlqu ·jxztvn
I([) = + [ + [ + [ + [ + [ + [ + [ +
+ [ + [ + [ + [ + [ + [ + [ +
Çvë{{q|qntz wéq [ ljnz vzknz tj wvxzjksntt€p kvwévx èjoyunnzx¹ k€wvst¹z• lnsntqn vi€·t€u viéjovu olnx• lv
kvs•tv xsv tv Æunxzv ëzvmv uv tv kvxwvs•ovkjz•x¹ unzvlvu tnvwénlnsntt€} rvë{{q|qntzvk djwq¡nu énoys•zjz lnsntq¹ k kqln inxrvtn·tvmv é¹lj x tnqoknxzt€uq wvrj rvë{{q|qntzjuq
|
= $ |
+ $ [ + $ [ + + $Q[Q + |
|
− [ − [ + [ + [ − [ − [ + [ |
|||
|
|
Ïutv qu vin ·jxzq éjkntxzkj tj otjuntjzns• q wéqéjkt¹nu rvë{{q|qntz€ wéq [Q xwéjkj q xsnkj
Q = = $
Q ≥ = $Q − $Q− − $Q− + $Q− + $Q− − $Q− − $Q− + $Q−
Ëj·js•t€n yxsvkq¹ quníz kql $N = wéq N < $ = j wéq Q ≥ rvë{{q|qntz€ $Q lvs t€ ylvksnzkvé¹z• énryééntztvuy xvvztv¡ntqí
$Q = $Q− + $Q− − $Q− − $Q− + $Q− + $Q− − $Q−
Ív ëzqu yxsvkq¹u snmrv tjpzq vxvinttv tj rvuw•íznén k kqln ësnrzévttvp zjisq|€ wv wvé¹lry kxn rvë{{q|qntz€ $N
èjxxuvzéqu n•n wévisnuy jiqzyéqntzj qo w Æ tnp tjlv i€sv tjpzq xrvs•rquq xwvxvijuq uv tv wénlxzjkqz• ·qxsj vzlv k kqln xyuu€ ·nz€én} xsjmjnu€} wéqtqují•q} otj·ntq¹ wéq·nu wvé¹lvr xsjmjnu€} quns otj·ntqn bs¹ ëzvp ojlj·q k rj·nxzkn wévqokvl¹•np {ytr|qq tjlv ko¹z• I([) = ([ + [ + [ ) ² wéq éjxré€zqq xrvivr rj lvn xsjmjnuvn
[1 kxzénzqzx¹ xzvs•rv éjo xrvs•rquq xwvxvijuq 1 éjoiqkjnzx¹
-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
tj xyuuy ·nz€én} xsjmjnu€} wéqtqují•q} otj·ntq¹
Íéq ëzvu kxzénz¹zx¹ q ·snt€ wvsy·jnu€n léym qo léymj
wnénxzjtvkrvp xsjmjnu€} k wvrjojznsn tjwéquné [ [ [ [ q [ [ [ [ èjxré€zqn xrvivr k k€éj ntqq ([ + [ + [ ) =
= [ ( + [ + [ ) uv tv wévqoknxzq tjwéquné |
wv |
|
wvsqtvuq |
|
js•tvp {véuysn Ëv wév•n qtvp xwvxvi |
|
|
|
|
΀ ojunzqu ·zv + [ + [ = − [ |
|
|
I |
([) uv tv |
− [ Ívëzvuy |
|
|||
ojwqxjz• k kqln |
|
|
|
|
I([) = [ ( − [ ) = [ ( − [ ) ( − [)− |
|
|
||
( − [) |
|
|
|
|
Ív {véuysn iqtvuj Ë•ízvtj qunnu |
|
|
|
|
( − [ ) = − [ + [ − [ + [ |
|
|
|
|
j wv {véuysn é¹lj Ë•ízvtj w {véuysj |
|
|||
( − [)− = + [ + [ + [ + + &N |
|
N + |
||
N+ [ |
|
|
|
|
Ívëzvuy
I([) = [ ( − [ + [ − [ + [ ) ×
× ( + [ + [ + [ + [ + [ + [ + [ + [ + )
Ínénutv j¹ wv·snttv ëzq éjosv ntq¹ k xrvirj} tju ty
t€ sq¡• ·snt€ lv [ tjplnu qtznénxyí•qn tjx ·snt€
+ [ + [ + [ + [ + dtj·qz ijssvk uv tv
tjiéjz• xwvxvijuq ² xwvxvijuq ² ·nz€é•u¹ q kxn ² nlqtxzkntt€u xwvxvivu
Ævvi•n nxsq tjlv tjpzq xrvs•rquq xwvxvijuq uv tv éjoiqz• ·qxsv 1 tj N xsjmjnu€} wéqtqují•q} otj·ntq¹ Q QV wéq·nu wvé¹lvr xsjmjnu€} y·qz€kjnzx¹ zv ·zvi€
k wvrjojznsn xznwntq wvsy·qsqx• zjrqn xsjmjnu€n tjlv kvokv lqz• k xznwnt• xyuuy [Q + [Q + + [QV j zjr rjr ·qxsv
xsjmjnu€} éjktv N zv q kvokvlqz• tjlv k N í xznwnt• Ívëzvuy wévqokvl¹•j¹ {ytr|q¹ qunnz kql
I([) = ([Q + [Q + + [QV)N
djlj·j ywév•jnzx¹ nxsq ·qxsj Q QV viéjoyíz jéq{un
zq·nxryí wévménxxqí ² k ëzvu xsy·jn [Q [QV viéjoyíz
mnvunzéq·nxryí wévménxxqí j ëzv wvokvs¹nz ywévxzqz• k€éj ntqn ls¹ I([)
Íévqokvl¹•qn {ytr|qq q éjoiqntq¹ ·qxns |
|
|
|
Ëjplnu tjwéquné xrvs•rquq xwvxvijuq uv tv wvsy·qz•v·rvk iévxj¹ rvxznp dlnx• tjlv viéjovkjz• wévqokvl¹•yí {ytr|qí
I([) = ([ + [ + [ + [ + [ + [ )
Ív {véuysn ls¹ xyuu€ mnvunzéq·nxrvp wévménxxqq ëzy {ytr|qí uv tv ojwqxjz• k xsnlyí•nu kqln
I([) = [ ( − [ ) = [ ( − [ ) ( − [)−
( − [)
Ònwné• éjosv qu ( − [ ) wv {véuysn iqtvuj Ë•ízvtj j ( − [)− ² wv {véuysn é¹lj Ë•ízvtj ΀ wvsy·qu
I([) = [ ( − [ + [ − [ + [ − [ + [ − [ ) ×
× ( + [ + [ + [ + [ + [ + )
Êo ëzvmv k€éj ntq¹ k kqln xyuu€ xsjmjnu€} tj}vlqu rvë{ {q|qntz wéq [ Ìt q ljnz vzknz tj wvxzjksnttyí ojlj·y
èjtnn k w u€ éjxxuvzénsq ojlj·y v tj}v lntqq ·qxsj
xwvxvivk wénlxzjkqz• ·qxsv 1 k kqln xyuu€ Q xsjmjnu€} rj lvn qo rvzvé€} éjktv vltvuy qo ·qxns P −
q vivotj·qsq ëzv ·qxsv xwvxvivk &P(Q 1) Ívt¹ztv ·zv wév qokvl¹•j¹ {ytr|q¹ ls¹ tnmv qunnz kql
I([) = ( + [ + [ + + [P− )Q =
=&P(Q ) + &P(Q ) [ + &P(Q ) [ + + &P(Q 1) [1 +
Êolnx• wévqokvl¹•yí {ytr|qí ls¹ ·qxns &P(Q 1) uv tv
ojwqxjz• wv léymvuy nxsq kvxwvs•ovkjz•x¹ {véuysvp ls¹ xyuu€ mnvunzéq·nxrvp wévménxxqq
I([) = ( + [ + [ + + [P− )Q = |
( − [P)Q = ( − [P)Q( − [)−Q |
|||||
|
|
|
|
( − [)Q |
|
|
èjosjmj¹ ( − [P)Q q ( − [)−Q wv {véuysju iqtvuj q é¹lj |
||||||
Ë•ízvtj wvsy·jnu |
|
|
|
|
||
I([) = |
− & [P + & [ P − + (− )N&N[NP + + (− )Q&Q[QP × |
|||||
|
Q |
Q |
|
Q |
Q |
|
|
|
× |
+ & [ + & |
+ + &S |
S + |
|
|
|
|
Q |
Q+ [ |
Q+S− [ |
|