Веленкин;Комбинаторика
.pdf
-sjkj 9 fОcХcЛК‰ fПИР¬ЗТМиК¬ЙФ К d¬НиcТЛФc dМЛФ
Æ xjuvu lnsn
(Q − )>(Q − ) + (Q − ) @ = (Q − )(Q − ) (Q − + ) = Q
Ívrj nu ·zv ëzv |
n éjkntxzkv knétv q ls¹ xyi{jrzvéqjsvk |
||||||||||||||||||||||||
'Q z n ·zv |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
'Q = (Q − )('Q− + 'Q− ) |
|
|
|
|
|
|
|
|
|
|
|
||||||||||||
bs¹ ëzvmv ojuntqu 'Q− q 'Q− q} éjosv |
|
ntq¹uq wv {vé |
|||||||||||||||||||||||
uysn ΀ wvsy·qu vzlns¹¹ k k€éj |
ntqq 'Q− |
wvxsnltnn |
|||||||||||||||||||||||
xsjmjnuvn ·zv |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(Q − )('Q− + 'Q− ) = (Q − )>(Q − ) + (Q − ) @ × |
|
|
|
|
|
|
|
|
|
|
|||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
− |
|
|
|
|
|
|
|
|
|
||
|
× |
− |
|
+ |
|
− |
|
|
+ + |
|
(− )Q |
|
+ (− )Q− (Q − ) |
||||||||||||
|
|
|
|
|
|
|
|
||||||||||||||||||
|
|
|
|
|
|
|
|
|
(Q − ) |
|
|
|
|
|
|
|
|
||||||||
Ëv wv {véuysn (Q − )>(Q − ) + (Q − ) @ = Q Çévun zvmv |
|
||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
− |
|
|
Q |
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
(− )Q |
|
|
|
|
|
|
|
|
|
|
||||||
(− )Q− (Q − ) = Q |
|
+ |
(− ) |
|
|
|
|
|
|
|
|
|
|
||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||
|
|
|
|
|
|
|
|
|
(Q − ) |
|
Q |
|
|
|
|
|
|
|
|||||||
Ívëzvuy |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(Q − )('Q− + 'Q− ) = |
|
|
|
|
|
|
|
− |
|
|
|
|
− |
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(− ) |
Q |
|
|
||||||
= Q − |
|
+ |
|
− |
|
+ + |
(− )Q |
|
+ |
(− )Q |
|
+ |
|
|
= ' |
||||||||||
|
|
|
(Q − ) |
(Q − ) |
|
|
|||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
Q |
|
|
|
Q |
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
bvrjojttvn xvvztv¡ntqn uv tv xsnly¹ êpsnéy k€knxzq x wv uv••í ·qxzv rvuiqtjzvét€} éjxxy lntqp èjxxuvzéqu kxn wnénxzj tvkrq k rvzvé€} kxn ësnuntz€ xun•nt€ Ínékvn unxzv k zjrq} wnén
xzjtvkrj} uv nz ojt¹z• síivp ësnuntz révun wnékvmv Òjr rjr ·qxsv vxzjk¡q}x¹ ësnuntzvk éjktv Q − zv 'Q wnénxzjtvkvr éjoiqkjnzx¹ tj
Q − méyww k ojkqxquvxzq vz zvmv rjrvp ësnuntz ojt¹s wnékvn unxzv
‰xtv ·zv kv kxn} méywwj} iylnz wvévkty ësnuntzvk
fvx·qzjnu xrvs•rv ësnuntzvk k vltvp qo ëzq} méyww xrj nu k zvp mln wnékvn unxzv ojt¹s kzvévp ësnuntz êzj méywwj éjoiqkjnzx¹ tj lkn ·jxzq zn k rvzvé€} wnék€p ësnuntz xzvqz tj kzvévu unxzn
q kxn vxzjs•t€n cxsq wnék€p ësnuntz ojt¹s kzvévn unxzv j kzvévp rjr u€ wvutqu ² wnékvn unxzv zv vxzjk¡qnx¹ Q − ësnuntzvk
uv tv wnénxzjks¹z• síi€u viéjovu sq¡• i€ tq vlqt qo tq} tn ojt¹s xkvnmv unxzj êzv uv tv xlnsjz• 'Q− xwvxvijuq dtj·qz wnékj¹ ·jxz• xvxzvqz qo 'Q− wnénxzjtvkvr
Ívrj nu ·zv kzvéj¹ ·jxz• xvxzvqz qo 'Q− wnénxzjtvkvr Æ xjuvu
lnsn kv kzvéyí ·jxz• kvplyz kxn wnénxzjtvkrq k rvzvé€} wnék€p ësnuntz tn xzvqz tj kzvévu unxzn j vxzjs•t€n ësnuntz€ tn tj}vl¹zx¹ tj xkvq} unxzj} Ëv zvmlj uv tv x·qzjz• kzvévn unxzv ©ojrvtt€uª
djwénzt€n ovt€ q sjlnpt€n ·qxsj |
|
|
|
ls¹ wnékvmv ësnuntzj q wvsy·qzx¹ ·zv kxn ësnuntz€ lvs t€ xunx zqz•x¹ Òjr rjr ·qxsv ëzq} ësnuntzvk éjktv Q − zv kv kzvévp ·jxzq
'Q− wnénxzjtvkvr Ívëzvuy zvmlj kx¹ méywwj xvxzvqz qo 'Q− + 'Q−
wnénxzjtvkvr Òjr rjr kxn utv nxzkv xun•jí•q} wnénxzjtvkvr xvxzv qz qo Q − méyww€ zv k tnmv k}vlqz (Q − )('Q− + 'Q− ) wnénxzjtvkvr
Ònu xju€u lvrjojtv éjkntxzkv
Êo {véuys€ k€znrjnz ·zv
'Q − Q'Q− = − ['Q− − (Q − )'Q− ]
Ívëzvuy wéq qountntqq Q k€éj ntqn 'Q − Q'Q− zvs•rv unt¹nz
otjr Íéqunt¹¹ ëzv xvvztv¡ntqn tnxrvs•rv éjo wvsy·jnu ·zv 'Q − Q'Q− = (− )Q− (' − ' ) Ëv ' = D ' = q wvëzvuy
'Q = Q'Q− + (− )Q |
|
êzj {véuysj tjwvuqtjnz xvvztv¡ntqn Q = Q(Q − ) ls¹ {jr zvéqjsvk Æ€wq¡nu otj·ntq¹ wnék€} xyi{jrzvéqjsvk
' = |
' = |
' = |
' = |
' = |
' = |
' = |
' = |
' = |
' = |
' = |
' = |
djwénzt€n ovt€ q sjlnpt€n ·qxsj
Ònwné• u€ tnxrvs•rv vivi•qu éjxxuvzénttyí ojlj·y q éjoinénu vi¡qét€p réym ojlj· v zjr tjo€kjnu€} ©ojwénzt€} ovtj}ª
΀ otjnu ·zv rj lyí wnénxzjtvkry Q ·qxns Q uv tv qoviéjoqz• wvxzjkqk Q sjlnp tj rkjléjztyí lvxry x
Q mvéqovtzjs¹uq q Q knézqrjs¹uq ² nxsq wnénxzjtvkrj qunnz kql (D DQ) zv sjl•q ojtqujíz wnénxn·ntq¹ N } knézqrjsnp
x DN uq mvéqovtzjs¹uq Íéq ëzvu tq vltj sjl•¹ tn i•nz léymyí
zjr rjr tq tvunéj mvéqovtzjsnp tq tvunéj knézqrjsnp k wjéj} ( D ) (Q DQ) tn wvkzvé¹ízx¹
Æ ojlj·n v wjxwvézj} inéyzx¹ tn kxn wnénxzjtvkrq j sq¡• zjrqn wéq rvzvé€} tq vltv ·qxsv tn xzvqz tj xkvnu unxzn
Æ ëzvu xsy·jn tq vltj sjl•¹ tn uv nz ojt¹z• wvs¹ x tvunéjuq ( ) (Q Q) j ëzv rjr éjo wvs¹ tj lqjmvtjsq Êt€uq
xsvkjuq k ëzvp ojlj·n zéniynzx¹ tjpzq ·qxsv éjxxzjtvkvr sjlnp wéq rvzvé€} tq vltj sjl•¹ tn wvwjljnz tj lqjmvtjs•
èjoyunnzx¹ kunxzv lqjmvtjsq uv tv k€iéjz• síivn léymvn utv nxzkv ojwénzt€} wvsnp j ·qxsv sjlnp uv nz i€z• síi€u ΀ wéq}vlqu zjrqu viéjovu r xsnlyí•np ojlj·n
-sjkj 9 fОcХcЛК‰ fПИР¬ЗТМиК¬ЙФ К d¬НиcТЛФc dМЛФ
ïjxz• wvsnp wé¹uvymvs•tvp ¡j}ujztvp lvxrq x P mvéqovt zjs¹uq q Q knézqrjs¹uq vréj¡ntj k ·nét€p |knz j vxzjs• t€n ² k ins€p frvs•rquq xwvxvijuq uv tv éjxxzjkqz• tj ins€} wvs¹} lvxrq N sjlnp zjr ·zvi€ tq vltj qo tq} tn uvmsj ko¹z• léymyí"
Íéq ëzvu vréjxrj tn vi¹ojzns•tv qlnz k ©¡j}ujztvuª wvé¹lrn ·nét€u uv nz i€z• ojréj¡nt síivp tjivé wvsnp tj lvxrn èjoyunnzx¹ uv tv x·qzjz• ·zv ·nét€n wvs¹ wévxzv k€énojt€ q éjxxzjks¹z• sjl•q wé¹uv tj insvwvs•tvp lvxrn rvzvéyí u€ vivotj·qu ·néno ' bs¹ lvxrq ' ·qxsv xwvxvivk wvxzjkqz• zéniynu€u viéjovu N sjlnp tjo€kjíz N u sjlnpt€u ·qxsvu q vivotj·jíz WN Æwnék€n ëzq ·qxsj kkns vzn·nxzkntt€p
ujznujzqr f c ¬é¡vt k xnénlqtn } mvlvk k lvrsjln xlnsjttvu k m tj kzvévu Æxnxvíotvu ujznujzq·nxrvu x~noln q k xzjz•n ©èn¡ntqn vltvp rvuiqtjzvétvp ojlj·qª vwyisqrvkjttvp k xivétqrn ©Îjznujzq·nxrvn wévxkn•ntqnª
ð Ívlévitvn qosv ntqn xkvpxzk ëzq} ·qxns vt }vzns xlnsjz• k rtqmn ©Ëvkj¹ rvuiqtjzvétj¹ jsmniéjª Ívxsn inokén unttvp xunézq jkzvéj vt i€s énwénxxqévkjt k m éyrv wqx• rtqmq wvmqisj Ætvk• zn n xju€n ·qxsj wéqlyujsq k
m junéqrjtxrqn ujznujzqrq Çjwsjtxrqp q èqvéljt Ìtq kknsq n•n utvmv·snt 7'([) = W + W [ + + WN[N + tjokjtt€p
sjlnpt€u utvmv·sntvu ls¹ ljttvp lvxrq
Ìi•j¹ {véuysj
Æojlj·n v xun•ntq¹} y tjx tj xjuvu lnsn lkn lvxrq ² insv q ·nétvwvs•tj¹ ·nét€u ojréj¡qkjízx¹ sq¡• wvs¹ lqj mvtjsq Éjlnptvn ·qxsv WN tjlv tjpzq ls¹ insvwvs•tvp lvxrq
Ëv vtj yxzévntj rylj xsv tnn ·nétvwvs•tvp lvxrq xvxzv¹•np qo Q tn xk¹ojtt€} léym x léymvu rkjléjzvk Ívëzvuy i€sv i€ knx•uj nsjzns•tv wvsy·qz• {véuysy xk¹o€kjí•yí sjlnpt€n ·qxsj ëzq} lky} lvxvr
Íénlkjéqzns•tv tjplnu sjlnpt€n ·qxsj ls¹ wvstvp lvxrq xvxzv¹•np qo P mvéqovtzjsnp q Q knézqrjsnp ïzvi€ wvxzjkqz• tj zjryí lvxry N sjlnp tjlv xtj·jsj k€iéjz• N mvéqovtzjsnp
·zv kvouv tv &NP xwvxvijuq j ojznu éjxxzjkqz• tj tq} N sjlnp ·zv uv tv xlnsjz• $NQ xwvxvijuq dtj·qz ls¹ zjrvp lvxrq N n sjlnptvn ·qxsv éjktv &NP $NQ
Íénlwvsv qu znwné• ·zv lvxrj P×Q éjoiqzj tj ·nétyí q insyí ·jxzq q ·zv u€ otjnu sjlnpt€n ·qxsj WN′ ls¹ ·nétvwvs•
tvp lvxrq Ívlx·qzjnu xrvs•rquq xwvxvijuq uv tv wvxzjkqz• U sjlnp tj lvxry zjr ·zvi€ tn untnn N sjlnp xzv¹sq tj ·nét€} wvs¹} bs¹ ëzvmv tjlv xtj·jsj éjxwvsv qz• N sjlnp tj ·nét€}
Ìi•j¹ {véuysj |
|
|
|
wvs¹} ·zvi€ vtq tn iqsq léym léymj êzv uv tv xlnsjz• WN′
xwvxvijuq djznu ojt¹z€n N mvéqovtzjsnp q N knézqrjsnp xsnlynz k€·nértyz• q tj vxzjk¡yíx¹ (P − N)×(Q − N) lvxry wv
xzjkqz• vxzjs•t€n U − N sjl•q tn viéj•j¹ ktqujtq¹ tj |knz
wvsnp j xsnl¹ sq¡• oj znu ·zvi€ vtq tn iqsq léym léymj
U−N U−N
êzv uv tv xlnsjz• &P−N $Q−N xwvxvijuq Ív wéjkqsy wévqokn
U−N U−NW′ zéniynu€} éjxxzjtvkvr sjlnp lntq¹ wvsy·jnu &P−N $Q−N N
Ònwné• u€ y n uv nu k€knxzq vxtvktyí {véuysy kxnp znvéqq ojwénzt€} ovt xk¹o€kjí•yí sjlnpt€n ·qxsj ls¹ ·né tvp q insvp lvxvr bs¹ ëzvmv kvxwvs•oynux¹ {véuysvp krsí
·ntqp q qxrsí·ntqp djtyunéynu kxn ·nét€n wvs¹ tj lvxrn P×Q q vivotj·qu ·néno D xkvpxzkv ©ojt¹zv D n ·nétvn wvsnª
Òvmlj 1(D D DN) ² ·qxsv éjxxzjtvkvr wéq rvzvé€} ojt¹z€
·nét€n wvs¹ x tvunéjuq D D DN Æxn xsjmjnu€n kqlj 1(D D DN) k {véuysn krsí·ntqp q qxrsí·ntqp quníz
otjr (− )N cxsq xsv qz• kxn ëzq xsjmjnu€n wéq ojljttvu otj·ntqq N zv wvsy·jízx¹ kxn éjxxzjtvkrq k rvzvé€} ojt¹zv
tn untnn N ·nét€} wvsnp ¬ ·qxsv zjrq} éjxxzjtvkvr rjr u€ U−N U−N W′ djun·jzns•tv ·zv u€ tj¡sq ëzv
otjnu éjktv &P−N $Q−N N
·qxsv tn otj¹ vzlns•t€} xsjmjnu€} ² knl• ls¹ tnrvzvé€} k€ivévk ·nét€} wvsnp uv nz q tn xy•nxzkvkjz• qxrvu€} éjxxzjtvkvr
¬ znwné• wéqunt¹¹ {véuysy krsí·ntqp q qxrsí·ntqp wvsy·jnu sjlnpt€n ·qxsj ls¹ insvwvs•tvp lvxrq WU k kqln
W = &U $U W ′ − &U− U− W ′ + |
|
|
|
|
|||
U |
P Q |
P− $Q− |
|
|
|
|
|
|
+ + (− )N&U−N U−N W′ |
+ + (− )U& |
′ |
|
|||
|
|
|
P−N $Q−N N |
|
P−U $Q−U WU |
|
|
|
Íéqunt¹¹ otjr xyuu€ ëzy {véuysy uv tv ojwqxjz• k kqln |
||||||
|
|
|
U |
|
|
|
|
|
|
W = ∑ (− )N&U−N |
U−N W′ |
|
|
||
|
|
U |
N= |
P−N $Q−N N |
|
|
|
|
|
|
|
|
|
|
|
|
Òv·tv zjr |
n lvrjo€kjnzx¹ ·zv ·qxsv éjxxzjtvkvr U sjlnp |
|||||
qo rvzvé€} S xzv¹z tj ·nét€} j vxzjs•t€n ² tj ins€} wvs¹} éjktv
U−S |
|
|
|
|
|
∑ |
(− )N&N |
U−S−N |
′ |
|
|
|
S+N &P−S−N WS+N |
|
|
||
N=
-sjkj 9 fОcХcЛК‰ fПИР¬ЗТМиК¬ЙФ К d¬НиcТЛФc dМЛФ
Ðvéuysj wvokvs¹nz xknxzq wvlx·nz sjlnpt€} ·qxns ls¹ insvwvs•t€} lvxvr r wvlx·nzy zn} n ·qxns ls¹ ·nétvwvs•t€} lvxvr ¬ ëzv k tnrvzvé€} xsy·j¹} wév•n xlnsjz• Ëjwéquné nxsq lvxrj ' éjxwjljnzx¹ tj lkn lvxrq ' q ' tn quní•q}
tq vi•q} mvéqovtzjsnp tq vi•q} knézqrjsnp éqx zv xwéjknlsqkj {véuysj
7 ([) = 7 |
([) 7 |
([) ≡ (W |
|
+ W |
|
[ + )(T + T [ + ) |
|
|||
' |
' |
' |
|
|
|
|
|
|||
Æ xjuvu lnsn wvxzjkqz• N sjlnp tj lvxry ' j O sjlnp tj |
||||||||||
lvxry ' uv tv wv wéjkqsy wévqoknlntq¹ WNTO |
xwvxvijuq |
|||||||||
Ívëzvuy uv |
tv |
W TU + W TU− + + WUT |
xwvxvijuq |
wvxzjkqz• U |
||||||
sjlnp tj lvxry ' éjxxzjtvkrq rsjxxq{q|qéyízx¹ wv ·qxsy sjlnp xzv¹•q} tj lvxrn ' ¬ ëzv rjr éjo rvë{{q|qntz wéq
[U k wévqoknlntqq utvmv·sntvk 7' ([) q 7' ([)
Òjr rjr ls¹ vltvp rsnzrq sjlnpt€p utvmv·snt qunnz kql+ [ zv ls¹ lqjmvtjsq rkjléjzj xvxzv¹•np qo Q rsnzvr vt
|
Q |
|
|
éjknt ( + [)Q = ∑ |
&N |
[N dtj·qz ls¹ lqjmvtjsq sjlnpt€n ·qxsj |
|
|
N= |
Q |
|
|
|
|
|
quníz kql W′ |
= &N |
|
|
N |
Q |
|
|
Ònwné• u€ uv |
nu lvknxzq lv rvt|j k€·qxsntqn 'Q z n |
||
·qxsj xwvxvivk éjxxzjtvkrq Q sjlnp tj rkjléjztvp lvxrn zjr
·zvi€ tq vltj qo sjlnp tn xzv¹sj tj lqjmvtjsq Æ ëzvu xsy·jn Q = P = U q u€ wvsy·jnu wv {véuysn
|
|
|
|
Q |
|
|
' |
Q |
≡ W |
Q |
= ∑ |
(− )N&N (Q − N) |
|
|
|
|
Q |
|
||
|
|
|
|
N= |
|
|
Æ ·jxztvxzq nxsq Q = zv qunnu
' |
|
= & |
− & |
+ & |
− & + & |
− & = |
|
|
|
|
|
|
|
|
|
dtj·qz nxz• xwvxvij éjxwénlnsntq¹ wjxwvézvk wv rvt knézju wéq rvzvé€} tq vlqt wjxwvéz tn wvwjljnz k xvvz knzxzkyí•qp rvtknéz ojlj·j w
dj vinlntt€u xzvsvu
Çxywéymju Énvtz•nk€u tj wéjoltq·t€p vinl wéq¡sq k mvxzq xywéy nxrq} wjé èn¡qsq xnxz• oj vinlntt€p xzvs zjr ·zvi€ uy ·qt€ q nt•qt€ ·nénlvkjsqx• wéq·nu tq vltj wjéj xywéymvk tn xqlnsj é¹lvu léym x léymvu frvs•rquq xwvxvijuq uvmyz ëzq wjé xnxz• oj xzvs"
dj vinlntt€u xzvsvu |
|
|
|
èqx |
|
èqx |
èqx |
|
|
Ívxjlqu xtj·jsj |
oj xzvs |
nt•qt |
®vo¹prj lvuj |
uv nz |
|
k€iéjz• síivn qo unxz Ívxsn ëzvmv ls¹ vxzjs•t€} |
nt•qt |
||||
vxzjnzx¹ unxz knl• lkn |
nt•qt€ tn lvs t€ xqlnz• é¹lvu |
||||
léym x léymvu Ê} |
uv |
tv |
wvxjlqz• |
tj ëzq unxzj |
= |
xwvxvijuq Æxnmv wvsy·qsvx• = xwvxvivk ¬ znwné•
iylnu éjxxj qkjz• uy ·qt Ínéntyunéynu ojt¹z€n unxzj wv ·jxvkvp xzénsrn |q{éjuq mln ² unxzv ojt¹zvn Énvtz•nkvp éqx j xkvivlt€n unxzj ² équxrquq |q{éjuq , ,, ,,, ,9 9 q 9, mln , ² unxzv xsnlyí•nn oj unxzvu }vo¹prq lvuj Òvmlj }vo¹qt lvuj uv nz xnxz• tj síivn unxzv révun unxz , q 9, vtq tj}vl¹zx¹ é¹lvu x nmv xywéymvp uy mvxz•q ojt¹k¡np unxzv ² tj síivu qo unxz révun , q ,, q z l Êt€uq xsvkjuq éjxxjlqz• mvxznp ² ëzv kxn éjktv ·zv éjxxzjkqz• sjl•q tj lvxrn qoviéj nttvp tj éqx
Ìivi•j¹ ëzy ojlj·y wéq}vlqu r zjrvp wvxzjtvkrn kvwévxj frvs•rquq xwvxvijuq uv tv wvxzjkqz• Q sjlnp tj (Q×Q)
lvxry zjr ·zvi€ vtq tn iqsq léym léymj q ·zvi€ tq vltj sjl•¹ tn xzv¹sj tj wvs¹} kqlj (N N) ≤ N ≤ Q (N N + )
≤ N ≤ Q − q (Q )"
cxsq i€ u€ otjsq ·qxsv xwvxvivk WN′ wvxzjkqz• N sjlnp tj
ojwénzt€n wvs¹ zjr ·zvi€ vtq tn iqsq léym léymj zv ojlj·j én¡jsjx• i€ wv {véuysn
WQ = Q − (Q − ) W ′ +
+ (Q − ) W′ − + (− )N(Q − N) WN′ + + (− )Q WQ′
djlj·j xknsjx• r wvlx·nzy ·qxsj xwvxvivk wvxzjkqz• N sjlnp tj ojwénzt€n wvs¹ j ëzv snm·n zjr rjr q} unt•¡n ·nu lvwyxzqu€} wvsnp djwq¡nu Q ojwénzt€} wvsnp wv réymy k€wqxjk xtj·jsj wvs¹ wnékvp mvéqovtzjsq wvzvu² kzvévp q z l cxsq wvxzjkqz• nlqtq|€ tj wvs¹} ojt¹z€} sjl•¹uq j
tj vxzjs•t€} wvs¹} tjwqxjz• tysq zv wvsy·qu réymvkyí éjxxzjtvkry N nlqtq| q Q − N tysnp Íéq ëzvu xvxnltqn wvs¹
-sjkj 9 fОcХcЛК‰ fПИР¬ЗТМиК¬ЙФ К d¬НиcТЛФc dМЛФ
tj vréy tvxzq xvvzknzxzkyíz tj lvxrn wvs¹u xzv¹•qu tj
vltvp mvéqovtzjsq qsq vltvp knézqrjsq Ívëzvuy lkn nlqtq|€ tn uvmyz xzv¹z• é¹lvu z n WN′ éjktv ·qxsy xwvxvivk tjwqxjz•
wv réymy N nlqtq| q Q − N tysnp zjr ·zvi€ tqrjrqn lkn nlqtq|€ tn xzv¹sq é¹lvu ¬ ëzv uv tv xlnsjz• rjr u€ y n
N |
+ &N− |
= |
Q |
N |
|
Q − N & Q−N xwvxvijuq |
|||||
xvx·qzjsq k w & Q−N |
Q−N− |
|
|||
Ívlxzjks¹¹ ëzv otj·ntqn k {véuysy wvsy·jnu vzknz
bqjméjuu€ ìtmj
Íéq k€·qxsntqq sjlnpt€} utvmv·sntvk vrjo€kjnzx¹ wv snotvp xsnlyí•j¹ {véuysj
7 ([) = 7 ([) + [7 ([) |
|
||
' |
' |
' |
|
dlnx• ·néno ' vivotj·ntj |
lvxrj wvsy·nttj¹ |
qo lvxrq ' |
|
k€·nérqkjtqnu tnrvzvévmv wvs¹ D j ·néno ' ² lvxrj wvsy·jnuj¹ qo ' k€·nérqkjtqnu tn zvs•rv xjuvmv wvs¹ D tv q mvéqovtzjsq q knézqrjsq tj wnénxn·ntqq rvzvé€} wvsn D tj}vlqzx¹ ïzvi€ lvrjojz• {véuysy lvxzjzv·tv ojunzqz• ·zv wéq síivp éjxxzjtvkrn N sjlnp wvsn D vrjo€kjnzx¹ sqiv xkvivlt€u sqiv ojt¹z€u ïqxsv éjxxzjtvkvr wéq rvzvé€} ëzv wvsn xkvivltv xvkwjljnz x ·qxsvu éjxxzjtvkvr sjlnp tj lvxrn
' ls¹ zjrq} éjxxzjtvkvr wvsn D uv tv inoivsnotnttv ylj sqz• cxsq n wvsn D ojt¹zv zv y n tq vltv wvsn knézqrjsq q mvéqovtzjsq tj rvzvé€} éjxwvsv ntv wvsn D ojtqujz• tns•o¹ Ívëzvuy q} tjlv k€·nértyz• j vxzjk¡qnx¹ sjl•q
éjxwvsjmjz• tj lvxrn ' dtj·qz ·qxsv éjxxzjtvkvr kzvévmv kqlj éjktv ·qxsy éjxxzjtvkvr N − sjlnp tj lvxrn '
Êzjr rvë{{q|qntz wéq [N k utvmv·sntn 7'([) éjknt xyuun rvë{{q|qntzj wéq [N k 7' ([) q rvë{{q|qntzj wéq [N− k
7' ([) êzv yzkné lntqn q k€éj ntv {véuysvp
èn¡qu x wvuv••í {véuys€ zjryí ojlj·y
frvs•rquq xwvxvijuq uv tv éjxxzjkqz• tj ¡j}ujztvp lvxrn N insvwvs•t€} xsvtvk zjr ·zvi€ vtq tn uvmsq iqz• léym léymj"
bs¹ én¡ntq¹ ëzvp ojlj·q wvknétnu ¡j}ujztyí lvxry tj° q vxzjkqu tj tnp sq¡• ins€n wvs¹ Òjr rjr xsvt€ }vl¹z
wv lqjmvtjs¹u zv wvxsn wvkvévzj lvxrq én·• iylnz qlzq y n v knézqrjs¹} q mvéqovtzjs¹} Ëv lvxrj iylnz yunt•¡nttvp zjrvp rjr tj éqx Êzjr tju tjlv tjpzq ·qxsv xwvxvivk éjxxzjkqz• sjl•q tj ëzvp lvxrn zjr ·zvi€ vtq tn iqsq léym
bqjméjuu€ ìtmj |
|
|
|
èqx èqx
léymj f wvuv••í wnénxzjtvkvr knézqrjsnp q mvéqovtzjsnp wénkéjzqu ëzy lvxry k qoviéj nttyí tj éqx
Òjrj¹ lvxrj ¹ks¹nzx¹ wénlxzjkqznsnu vi¡qétvmv rsjxxj
lvxvr |
y |
rvzvé€} wnékj¹ |
xkné}y |
mvéqovtzjs• |
xvlné |
qz |
|
Q qly•q} léym oj léymvu wvsnp kzvéj¹ ² Q |
zjr |
n |
|||||
éjxwvsv |
|
ntt€} wvsnp P ¹ xzévrj ² QP zjrq} |
n wvsnp |
||||
wéq·nu |
Q ≤ Q ≤ ≤ QP j |
kxn |
wnék€n |
wvs¹ xzévr |
xzv¹z |
tj |
|
vltvp q |
zvp n knézqrjsq |
k |
wnékvu |
xzvsi|n Òjrqn lvxrq |
|||
tjo€kjíz lqjméjuujuq ìtmj Ìtq kxzén·jízx¹ kv utvmq}
visjxz¹} ujznujzqrq tjwéquné k znvéqq wénlxzjksntqp méyww Ëjovknu wvxsnlvkjzns•tvxz• (Q Q QP) rvlvu lqjméju
u€ ìtmj j ·néno 7(Q Q QP [) vivotj·qu xvvzknzxzkyí •qp sjlnpt€p utvmv·snt
bs¹ síivmv wvs¹ D wnékvp mvéqovtzjsq lvxrj ' qunnz wéq Q > rvl (Q − Q QP) j wéq Q = rvl (Q QP)
lvxrj ' wéq Q > qunnz rvl (Q − Q − QP − ) Ívë zvuy wv {véuysn wvsy·jnu ·zv wéq Q >
7(Q Q QP [) =
= 7(Q − Q QP [) + [7(Q − Q − QP − [) Íéqunt¹¹ n•n Q − éjo ëzy {véuysy r wnékvuy xsjmjnuv
uy wvsy·qu ·zv wéq Q >
7(Q Q QP [) =
= 7(Q QP [) + Q [7(Q − Q − QP − [)
Æ xsjmjnu€} xzv¹•q} k wéjkvp ·jxzq ëzvmv éjkntxzkj ·qxsv mvéqovtzjsnp y n éjktv P − Íévlvs j¹ ëzvz wév|nxx xknlnu
kxn r lqjméjuun ìtmj quní•np vlty mvéqovtzjs• j ls¹ zjrvp lqjméjuu€ qunnu 7(Q [) = + Q[ Òjr rjr rvl lqjméjuu€
-sjkj 9 fОcХcЛК‰ fПИР¬ЗТМиК¬ЙФ К d¬НиcТЛФc dМЛФ
ìtmj tj éqx éjknt zv wvxsnlvkjzns•tv wvsy·qu
7( [) = 7( [) + + [7( [) = =
= + [ + [ + [ + [ + [ + [ + [
Ëjwéquné zén} xsvtvk uv tv wvxzjkqz• xwvxvijuq
Çjéjkjt k wyxz€tn
ïjxzv k ojlj·j} én·• qlnz v xvxnlxzkn vltvp qsq tnxrvs•rq} wjé ësnuntzvk q ojwénzy wvlsn qz tnrvzvévn utv nxzkv zjrq} wjé êzq ojlj·q kv utvmq} xsy·j¹} yljnzx¹ én¡qz• rsjxxq{q |qéy¹ ëzq wnénxzjtvkrq wv ·qxsy k}vl¹•q} k tq} ojwén•ntt€} wjé q wéquntqk {véuysy krsí·ntqp q qxrsí·ntqp
Ív wyxz€tn qlnz rjéjkjt qo knéisílvk Íyzn¡nxzkqn lsqzx¹ utvmv ltnp q tjrvtn| kxnu tjlvnljnz kqlnz• kwnénlq xni¹ vltvmv q zvmv n knéisílj frvs•rquq xwvxvijuq uv tv wnénxzjkqz• knéisílvk zjr ·zvi€ kwnénlq rj lvmv knéisílj ¡ns qtvp knéisíl ·nu éjt•¡n"
Òjrqn wnénxzjtvkrq tjknét¹rj xy•nxzkyíz Ëjwéquné uv tv wnénxzjkqz• kxn} knéisílvk k viéjztvu wvé¹lrn zjr ·zv wvxsnltqp vrj nzx¹ wnék€u q z l Ænl• rjr msjxqz jéjixrj¹ wvxsvkq|j ©rvmlj rjéjkjt wvkvéj·qkjnz tjojl }évuvp knéisíl
vrjo€kjnzx¹ kwnénlqª
bs¹ én¡ntq¹ ojlj·q wnéntyunéynu knéisílvk k wnékv tj·js•tvu wvé¹lrn vz rvt|j rjéjkjtj r tj·jsy ·qxsjuq
Ëju ty tv tjpzq kxn wnénxzjtvkrq ëzq} ·qxns k rvzvé€} tnz tq vltvp qo wjé
fvx·qzjnu kv xrvs•rq} wnénxzjtvkrj} ëzq} ·qxns y·jxzkynz wjéj ΀ uv nu x·qzjz• k ëzq} wnénxzj tvkrj} zjryí wjéy oj vlqt ësnuntz Ívëzvuy vi•nn ·qxsv wnén xzjks¹nu€} ësnuntzvk iylnz y n tn j q q} ·qxsv éjktv
Çjéjkjt k wyxz€tn
3 = Òvz n énoys•zjz wvsy·jnu q ls¹ vxzjs•t€} ojwén•nt
t€} wjé
Ònwné• éjxxuvzéqu wnénxzjtvkrq ojwén•ntt€n wv réjptnp unén lkj l€ z n xvlné j•qn lkn ojwén•ntt€n wjé€ Ævo uv t€ lkj xsy·j¹ ² lkn ojwén•ntt€n wjé€ sqiv tn quníz vi•q} ësnuntzvk sqiv quníz vlqt vi•qp ësnuntz
Æwnékvu xsy·jn vi~nlqtqu kunxzn ësnuntz€ rj lvp wjé€
Ïtjx vxzjtnzx¹ kxnmv wnénxzjks¹nu€} ësnuntzvk wjé€ q
ësnuntzvk tn wvwjk¡q} tq k vlty qo ëzq} wjé ïqxsv zjrq} wnénxzjtvkvr éjktv 3 = Æv kzvévu xsy·jn zéq ësnuntzj
k}vl¹•qn k lkn ojwén•ntt€n wjé€ lvs t€ qlzq wvlé¹l Ëjwéquné nxsq k wnénxzjtvkry k}vl¹z wjé€ q zv k tnp wvlé¹l qlyz ësnuntz€ Ìi~nlqtqu ëzq zéq ësnuntzj kunxzn Òvmlj wnénxzjtvkrju wvlsn jz xtvkj ësn
untzvk ² isvr qo zén} ësnuntzvk q ësnuntzvk tn kv¡nl¡q} k isvr ïqxsv zjrq} wnénxzjtvkvr vw¹z• éjktv 3 =
fvkné¡nttv zjr |
n lvrjo€kjnzx¹ ·zv rjrqn i€ |
N oj |
|
wén•ntt€} wjé u€ tq ko¹sq ·qxsv wnénxzjtvkvr xvlné |
j•q} |
||
kxn ëzq N |
wjé éjktv |
3 −N = ( − N) Ëv N wjé qo |
uv tv |
k€iéjz• &N |
xwvxvijuq Ív {véuysn krsí·ntqp q qxrsí·ntqp |
||
|
|
|
|
wvsy·jnu ·zv rvsq·nxzkv wnénxzjtvkvr tn xvlné j•q} tq vltvp ojwén•nttvp wjé€ éjktv
3 |
|
− & 3 |
|
+ & 3 |
− & 3 |
|
+ & 3 |
|
− & 3 + & 3 − & 3 |
|
+ & 3 |
= |
|
||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||
|
|
|
|
|
|
= |
|
− |
|
+ |
|
− |
|
+ |
|
− |
|
|
+ |
|
− |
|
+ |
|
|
= |
|||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||
|
|
¬tjsvmq·tv lvrjo€kjnzx¹ |
·zv rvsq·nxzkv |
wnénxzjtvkvr |
|||||||||||||||||||||||||||||||||||
qo ·qxns Q tn xvlné |
j•q} tq vltvp qo wjé |
||||||||||||||||||||||||||||||||||||||
(Q − Q) k€éj |
jnzx¹ {véuysvp |
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||||
( |
Q |
= 3 |
Q |
− & |
|
|
+ & |
|
|
|
|
− & |
|
|
|
|
+ + (− )Q− Q− 3 |
|
= |
||||||||||||||||||||
|
|
|
|
Q− 3Q− |
|
|
|
Q− 3Q− |
|
|
Q− 3Q− |
|
|
|
|
|
|
|
|
&Q− |
|
|
|||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
− |
|
|
|
− |
|
|
|
|
− |
|
|
|
|
|
|
|
− |
|
|
|||||||
|
|
|
= (Q − ) |
|
|
− |
Q |
+ |
Q |
− |
Q |
+ + |
(− )Q |
|
|
|
|
||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||
|
|
|
|
|
|
|
Q |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(Q − ) |
|
|||||||||||
Æ€éjoqu wvsy·ntt€p vzknz ·néno xyi{jrzvéqjs€ bs¹ ëzvmv éjov i•nu rj lvn xsjmjnuvn k wéjkvp ·jxzq tj lkj
(− )N(Q − N) |
= |
(− )NQ |
+ |
(− )N− |
|
N |
N |
(N − ) |
|||
|
|