Веленкин;Комбинаторика
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ÌÒÆcÒÔ
tj wévqokvs•tvu ujé¡éyzn P′ vzsq·tvu vz P qunnu évktv Q vxzj tvkvr Æ xqsy yxsvkq¹ tj P′ éjxwvsv ntv tn untnn zén} vxzjtvkvr
wéq·nu k xqsy yxsvkq¹ sq¡• vltj qo ëzq} vxzjtvkvr ¹ks¹nzx¹ k zv |
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kénu¹ vxzjtvkrvp ujé¡éyzj P |
Ætn ujé¡éyzj P′ éjxwvsv nt€ |
Q − vxzjtvkvr ujé¡éyzj P Ívrj |
nu ·zv wvuquv ëzvmv ktn P′ |
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éjxwvsv ntj n•n }vz¹ i€ vltj vxzjtvkrj tn sn j•j¹ tj P bnpxz |
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kqzns•tv wyxz• $ nxz• vltj qo Q − vxzjtvkvr ujé¡éyzj P éjxwv |
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ntt€} ktn P′ D & nxz• vltj qo vxzjtvkvr ujé¡éyzj P′ éjxwv |
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ntt€} ktn P zjrq} vxzjtvkvr qunnzx¹ tn untnn ·nu Æ xqsy |
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yxsvkq¹ xy•nxzkynz ujé¡éyz P′′ wév}vl¹•qp ·néno vxzjtvkrq $ |
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q & wéq·nu k xqsy yxsvkq¹ tj ëzvu ujé¡éyzn wvuquv $ q &
qunnzx¹ n•n wv réjptnp unén vltj vxzjtvkrj % rvzvéj¹ iylnz éjx wvsv ntj ktn P′ q ktn P ïnéno ëzy vxzjtvkry % rjr u€ otjnu qo
én¡ntq¹ wénl€ly•np ojlj·q wév}vlqz Q ujé¡éyzvk Çj l€p qo ëzq} Q ujé¡éyzvk wnénxnrjnzx¹ x ujé¡éyzvu P′ k vltvp nlqtxzknttvp zv
·rn Íéq ëzvu ·néno rj lyí vxzjtvkry ujé¡éyzj P′ wév}vlqz }vz¹
i€ vlqt ujé¡éyz xvnlqt¹í•qp nn x vxzjtvkrvp % Ívëzvuy ·qxsv vxzjtvkvr tj ujé¡éyzn P′ éjktv ·qxsy ujé¡éyzvk wév}vl¹•q} ·néno
vxzjtvkry |
% z n éjktv |
Q Çjr u€ kqlnsq wéq én¡ntqq wénl€ly•np |
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ëzvu xsy·jn |
·qxsv |
ujé¡éyzvk k€éj jnzx¹ {véuysvp |
Q(Q − ) + Òjr rjr wv yxsvkqí ëzv ·qxsv éjktv zv tjlv én¡qz• |
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yéjktntqn Q(Q − ) + = èn¡j¹ nmv tj}vlqu ·zv Q = |
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Ævouv tv éjxxuvzéqu |
tjwéquné wé¹u€} wsvxrvxzq zj |
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rq} ·zv tqrjrqn lkn qo tq} tn wjéjssns•t€ q tqrjrqn zéq tn wnén |
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xnrjízx¹ k vltvp zv·rn q iylnu x·qzjz• ·zv wé¹u€n ² ëzv jkzviyx t€n ujé¡éyz€ j zv·rq q} wnénxn·ntq¹ ² vxzjtvkrq Íéq ëzvu x rj lvp vxzjtvkrq uv tv wévn}jz• tj síiyí léymyí ino wnénxjlrq nxsq vtq sn jz tj vltvp wé¹uvp q x vltvp wnénxjlrvp nxsq vtq sn jz tj éjosq·t€} wé¹u€} cxsq lj n vziévxqz• k ëzvp x}nun vlty wé¹uyí zv kxn n vxzjtnzx¹ kvouv tvxz• wévn}jz• x rj lvp vxzjtvk rq tj síiyí léymyí xlnsjk k wyzq tn ivsnn vltvp wnénxjlrq Ëv nxsq vziévxqz• lkn wé¹u€n zv vltj vxzjtvkrj ² zv·rj wnénxn·ntq¹ ëzq} lky} wé¹u€} tn iylnz xvkxnu vixsy qkjz•x¹ vxzjk¡quqx¹ ujé ¡éyzjuq q x ëzvp vxzjtvkrq iylnz tnkvouv tv wévn}jz• tq tj rjryí léymyí vxzjtvkry
-sjkj 9,,
j bvrj nu éjkntxzkv x wvuv••í qtlyr|qq wv Q + P Íyxz• ls¹ kxn} N q V zjrq} ·zv N + V < Q + P éjkntxzkv lvrjojtv Òvmlj qunnu
XQ+P = XQ+P− + XQ+P− = (XQ− XP− + XQXP) + (XQ− XP− + XQXP− ) =
= XQ− (XP− + XP− ) + XQ(XP + XP− ) = XQ− XP + XQXP+
Ívxrvs•ry wéq P + Q = ëzv éjkntxzkv tnwvxénlxzknttv wévkné¹ nzx¹ vtv xwéjknlsqkv ls¹ síi€} P q Q
-sjkj 9,, |
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i Íévknlnu lvrjojzns•xzkv x wvuv••í qtlyr|qq wv |
N Íéq |
N = yzkné lntqn zéqkqjs•tv Íyxz• y n lvrjojtv ·zv XVP |
lnsqzx¹ |
tj XP Êo wytrzj j qunnu X(V+ )P = XVP+P = XVP− XP + XVPXP+ q wvzvuy
X(V+ )P zjr n lnsqzx¹ tj XP Ív qtlyr|qq k€kvlqu ·zv kxn XNP ln s¹zx¹ tj XP
k Íyxz• XQ q XQ+ lns¹zx¹ tj N ≠ Òvmlj q XQ− = XQ+ − XQ ln
sqsvx• i€ tj N Íévlvs j¹ ëzv éjxxy lntqn u€ wvsy·qsq i€ ·zv X = lnsqzx¹ tj N j ëzv tnkvouv tv
Èylnu vivotj·jz• tjqivs•¡qp vi•qp lnsqzns• ·qxns D q E ·néno (D E) Êo éjkntxzkj XQ+P = XQ− XP + XQXP+ xsnlynz ·zv (XP+Q XQ) ¹k
s¹nzx¹ lnsqznsnu XQ− XP q wvxrvs•ry XQ q XQ− kojqutv wévxz€ ln
sqznsnu XP Ìiéjztv (XP XQ) ¹ks¹nzx¹ lnsqznsnu
(XP XQ) = (XP+Q XQ) Ëv zvmlj nxsq Q = NP + T zv (XP XQ) = (XP XT) Íéqunt¹¹ jsmvéqzu ckrsqlj yin ljnux¹ ·zv (XP XQ) = X(P Q) Æ
·jxztvxzq (X X ) = X =
èjxxuvzéqu wvxsnlvkjzns•tvxz• xvxzjksnttyí qo wvxsnltq} ·n z€én} |q{é ·qxns Ðqivtj··q Òjr rjr rvsq·nxzkv ·nz€én}otj·t€}
·qxns kqlj éjktv zv rvsq·nxzkv wjé zjrq}
·qxns éjktv dtj·qz xénlq wnék€} ·qxns Ðqivtj··q tjplyzx¹ lkn wjé€ (XP XP+ ) q (XQ XQ+ ) Q > P zjrqn ·zv vlqtjrvk€
wvxsnltqn ·nz€én |q{é€ XP q XQ j zjr n XP+ q XQ+ Ëv zvmlj ·qxsj XQ − XP q XQ+ − XP+ vrjt·qkjízx¹ ·nz€é•u¹ tys¹uq Òjr rjr
XQ− − XP− = (XQ+ − XP+ ) − (XQ − XP) zv q XQ− − XP− zjr n vrjt·qkj
nzx¹ ·nz€é•u¹ tys¹uq Íévlvs j¹ yunt•¡jz• qtlnrx lv}vlqu lv XQ−P − X tv X = wvëzvuy ·qxsv XQ−P
s¹uq
Íyxz• k€iéjt€ ·qxsj XQ XQ+ XQ+ XQ+ Æ€éjoqu q} ·néno
XQ q XQ+ XQ+ = XQ + XQ+ XQ+ = XQ + XQ+ XQ+ = XQ + XQ+
XQ+ = XQ + XQ+ XQ+ = XQ + XQ+ XQ+ = XQ + XQ+ fsnlvkjzns• |
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tv xyuuj ëzq} ·qxns |
éjktj XQ + XQ+ Ëv XQ+ = XQ + XQ+ |
XQ+ = XQ + XQ+ Êo |
tnéjkntxzkj XQ+ < XQ + XQ+ < XQ+ ¹xtv |
·zv XQ + XQ+ tn ¹ks¹nzx¹ ·qxsvu Ðqivtj··q |
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j Ïzkné lntqn lvrjo€kjnzx¹ wv qtlyr|qq Íéq Q = vtv v·n |
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kqltv Íyxz• vtv xwéjknlsqkv ls¹ Q = N X + X + + X N = X N+ −
Íéqijkqu r |
vinqu ·jxz¹u |
éjkntxzkj X N+ Òjr |
rjr |
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X N+ + X N+ = X N+ |
wvsy·jnu |
X + X + + X N+ = X N+ − |
Ònu |
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xju€u tj¡n yzkné |
lntqn lvrjojtv |
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Òv·tv zjr |
n wv qtlyr|qq lvrjo€kjízx¹ yzkné lntq¹ i q k |
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ÌÒÆcÒÔ
m bs¹ lvrjojzns•xzkj ojunzqu ·zv
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− X |
X |
= X |
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− X − X X |
Q+ |
= X (X |
Q+ |
− X |
Q |
) − X = − (X − X |
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XQ+ |
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Q Q+ |
Q+ |
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Q |
Q |
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Q |
Q |
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Q− XQ+ |
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− X |
X |
= (− )Q(X − X X ) = (− )Q |
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Ívëzvuy XQ+ |
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Q Q+ |
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l |
Ïzkné |
lntq¹ |
l q n |
iylnu |
lvrjo€kjz• |
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xvkunxztv |
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Íéq |
Q = |
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vtq v·nkqlt€ Íyxz• vtq y |
n lvrjojt€ wéq Q = N Ív yzkné |
lntqí |
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m qunnu zvmlj |
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X X + X X + + X NX N+ + X N+ X N+ = |
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= X |
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− + X |
N+ X N+ |
= X |
N+ X N+ |
− = X |
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N+ |
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q |
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X X + X X + + X N+ X N+ + X N+ X N+ = |
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= X |
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+ X |
N+ X N+ |
= X |
N+ X N+ |
= X |
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− |
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N+ |
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fsnlvkjzns•tv ëzq yzkné lntq¹ knét€ q wéq Q = N + j wvzvuy
q wéq kxn} Q
djunzqu ·zv k xqsy j q i X + X + X + + XN+ = XN+ − Ívëzvuy nxsq qunnz unxzv éjkntxzkv wéq Q = N zv
(N + )X + NX + (N − )X + + XN + XN+ =
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= XN+ − (N + ) + XN+ − = XN+ − (N + ) |
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Òjr rjr éjkntxzkv knétv wéq Q = zv vtv knétv wéq kxn} Q |
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o êzv k€znrjnz qo éjkntxzkj |
X Q+ − |
+ X |
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= |
X Q+ − |
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Q+ |
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q bs¹ lvrjojzns•xzkj k {véuysn XQ+P = XQ− XP + XQXP+ |
wvsv |
qu |
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P = Q ΀ wvsy·qu ·zv X |
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= X |
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+ X X |
Q+ |
= X |
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− X |
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Q− XQ |
Q |
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Q+ |
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Q− Òv·tv zjr |
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n lvrjo€kjnzx¹ ·zv X Q+ |
= X + X |
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k |
zvp |
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{véuysn |
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Q |
Q+ Ívsjmj¹ |
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P = Q wvsy·jnu ·zv |
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X |
Q |
= X |
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+ X X |
Q+ |
= X (X |
− X |
) + X (X + X ) = X |
+ X − X |
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Q− X Q |
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Q− Q+ |
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Q Q |
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Q− |
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Íyxz• XQ ≤ 1 ≤ XQ+ Òvmlj ≤ 1 − XQ < XQ− |
q wvzvuy tjplnzx¹ |
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zjrvn |
V < Q − |
·zv |
XV ≤ 1 − XQ < XV+ Ëv zvmlj |
≤ 1 − XQ − XV < XV− |
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wéq·nu V − < Q − |
ïnéno |
tnxrvs•rv |
¡jmvk |
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wvsy·qu |
·zv |
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1 = XQ |
+ XV |
+ XS |
+ + XU wéq·nu |
xvxnltqn |
qtlnrx€ |
Q |
V |
S |
U vz |
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sq·jízx¹ léym vz léymj wv réjptnp unén tj |
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ïzvi€ éjxrvlqévkjz• xsvkv tjlv wénlxzjkqz• ·qxsv k kqln xyuu€ xsjmjnu€} rj lvn qo rvzvé€} éjktv qsq Ævotqrjnz wvxsnlvkjzns•tvxz• Ðqivtj··q Ívëzvuy qunnu xwvxvij wév·qzjz• ljttvn xsvkv qo otjrvk
-sjkj 9,, |
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j èn¡j¹ }jéjrznéqxzq·nxrvn yéjktntqn U − U + = tj}v lqu rvétq U = U = Ívëzvuy vi•nn én¡ntqn qunnz kql
DQ = & Q + & Q i Òv·tv zjr |
n wvsy·jnu DQ = & Q + & (− )Q |
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k Êunnu DQ = & ( + L)Q + & ( − L)Q m DQ = & ( L)Q + & (− L)Q |
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l U = U = − Ívëzvuy DQ = (& + & Q)(− )Q |
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n ®jéjrznéqxzq·nxrvn yéjktntqn zjrvkv |
U − U + U − = |
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cmv rvétq U = U = U = Ívëzvuy DQ = & Q + & Q + & Q |
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U = U = U = − Ívëzvuy DQ = (& + & Q + & Q )(− )Q |
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o ®jéjrznéqxzq·nxrvn yéjktntqn qunnz kql U + = cmv rvétq |
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U = ± L U = − ± L Ívëzvuy |
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DQ = & ( + L)Q + & ( − L)Q + & (− + L)Q + & (− − L)Q |
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q D |
Q |
= |
Q |
Q + & (− )Q + & Q |
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j èn¡j¹ }jéjrznéqxzq·nxrvn yéjktntqn U − U + = wvsy·j |
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nu ·zv |
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U = |
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U = |
q |
wvzvuy DQ |
= & Q + & Q Ívsjmj¹ Q = q |
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Q = |
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wvsy·jnu |
ls¹ |
vz€xrjtq¹ |
& |
q |
& xqxznuy yéjktntqp |
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& + & = & + & = − Êo tnn tj}vlqu & = & = − q wvzvuy |
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D |
Q |
= Q − Q+ |
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i Êunnu DQ |
= (& + & Q) Q Ívsjmj¹ Q = q Q = wvsy·jnu xqx |
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znuy |
yéjktntqp |
& + & = & + & = |
qo |
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rvzvévp k€kvlqu ·zv |
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& = & = q wvzvuy DQ = Q |
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k D |
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= |
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[(− + L√ )Q + (− − L√ )Q] m D |
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= Q + Q − Q |
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Q+ |
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®jéjrznéqxzq·nxrvn yéjktntqn |
U − U FRV ϕ + = x rvét¹uq |
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U = FRV ϕ ± L VLQ ϕ |
DQ |
= & (FRV ϕ + L VLQ ϕ )Q + & (FRV ϕ − L VLQ ϕ )Q Ív |
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sjmj¹ Q = q Q = wvsy·jnu xqxznuy yéjktntqp |
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(& + & ) FRV ϕ + (& − & ) L VLQ ϕ = FRV ϕ |
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(& + & ) FRV ϕ + (& − & ) L VLQ ϕ = FRV ϕ |
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Ìzxílj & |
= & |
= D |
= |
[(FRV ϕ |
+ L VLQ ϕ )Q + (FRV ϕ − L VLQ ϕ )Q] Æ |
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Q |
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xqsy {véuys€ Îyjkéj DQ |
= FRV Qϕ |
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Æ€znrjnz qo zvmv ·zv }jéjrznéqxzq·nxrvn yéjktntqn |
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UN − |
& UN− + & UN− − + (− )N&N = |
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N |
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N |
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uv tv ojwqxjz• k kqln (U − )N = Ìtv qunnz rvént• U = réjztvxzq N Ívëzvuy DQ = QW wéq W = N − ² én¡ntq¹ énryééntztvmv
ÌÒÆcÒÔ
Ìivotj·qu ·qxsv éjoun•ntqp zjrvmv kqlj ·néno -(QN) êzq éjo
un•ntq¹ éjovi•nu tj lkj rsjxxj Ç wnékvuy vztnxnu tj·qtjí•qnx¹ tj j rv kzvévuy ² kxn vxzjs•t€n cxsq éjoun•ntqn tj·qtjnzx¹ tjzv k€·znu qo kxn} k}vl¹•q} k tnmv ·qxns q vziévxqu xzv¹•qp
kwnénlq tys• tjwéquné wnénplnz wéq ëzvu xtj·jsj k
j wvzvu k Ívsy·qzx¹ (N − ) éjoun•ntqn zvmv n zqwj tv xv xzv¹•nn qo ·qxns Q − Ívëzvuy ·qxsv éjoun•ntqp wnékvmv
(N− ) Çj lvn éjoun•ntqn kzvévmv rsjxxj tj·qtjnzx¹ x rsjxxj éjktv -Q−
tn·nztvmv ·qxsj ivs•¡nmv z n q ivsnn Æ€·znu qo kxn} ·qxns |
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k}vl¹•q} k zjrvn éjoun•ntqn ·qxsv ΀ wvsy·qu |
N éjoun•ntqn |
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zvmv n zqwj k rvzvévn k}vl¹z ·qxsj Q − Ívëzvuy ·qxsv |
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(N) Òjrqu viéjovu xwéjknlsqkj |
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énryééntztj¹ {véuysj -(N) = -(N− ) + -(N) Ívsv qu |
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bs¹ ëzvmv ojunzqu |
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Q + zv n nlqtxzkntt€u viéjovu uv |
tv k€iéjz• Q ·qxns k xvvzknz |
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xzkqq x wvxzjksntt€uq yxsvkq¹uq Ívëzvuy q -Q+ |
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Q q N qunnz unxzv éj |
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-sjkj 9,,,
dtj·ntqn utvmv·sntj wéq [ = \ = ] = éjktv rjr éjo xyuun rv
ë{{q|qntzvk Ívlxzjks¹¹ ëzq otj·ntq¹ wvsy·jnu ·zv xyuuj éjktj
j èjoun•ntq¹ uvmyz xvxzv¹z• qo Q wénlunzvk Ívëzvuy wvstvn ·qxsv éjoun•ntqp éjktv
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f léymvp xzvévt€ |
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ÌÒÆcÒÔ
féjktqkj¹ éjosv ntq¹ q wvsy·jnu ·zv DN = D Q−N djuntqu znwné• [ tj −[ ΀ wvsy·qu ·zv
( − [ + [ )Q = D − D [ + D [ − D [ + + D Q[ Q |
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Ínénutv j¹ éjosv ntq¹ q k€kvlqu ·zv |
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( + [ + [ )Q = ∑ (− )N >D DN − D DN− + + (− )NDND @ [N |
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j ‰xtv ·zv éjosv ntqn snkvp ·jxzq éjkntxzkj xvlné qz sq¡• ·snt€ x ·nzt€uq xznwnt¹uq [ j wvzvuy rvë{{q|qntz wéq [ Q− éjknt tysí Ëv k wéjkvp ·jxzq rvë{{q|qntzvu wéq [ Q− ¹ks¹nzx¹
−(D D Q− − D D Q− + − D Q− D ) = −(D D − D D + − D Q− D Q)
i djunzqu znwné• ·zv éjosv ntqn uv tv wénlxzjkqz• wv {vé uysn k kqln ( + [ + [ )Q = D + D [ + D [ + + D Q[ Q Ìzxílj
xsnlynz ·zv rvë{{q|qntz wéq [ Q k ëzvu éjosv ntqq éjknt DQ f léymvp xzvévt€ wv {véuysn vt éjknt
D |
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+ D |
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− + D |
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= D − D + D − + (− )QD |
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k Ínénwq¡nu éjkntxzkv k kqln |
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Ìzxílj k€znrjnz ·zv |
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− & [ + |
& [ − + (− )Q&Q[ Q = |
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Q(D |
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cxsq U tn lnsqzx¹ tj zv k snkvp ·jxzq ëzvmv éjkntxzkj rvë{ |
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{q|qntz wéq [U éjknt tysí Æ wéjkvp |
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[U éjknt |
D |
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− + (− )U&UD |
dtj·qz ëzv |
kێj |
ntqn |
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éjktv tysí nxsq U tn lnsqzx¹ tj q éjktv (− )N&N nxsq U = N
Q
m cxsq [ = k éjosv ntqq zv D + D + D + + D Q = Q cxsq [ = k éjosv ntqq zv D − D + D − + D Q = frsjl€kj¹ q k€ ·qzj¹ ëzq éjkntxzkj wéq}vlqu r lvrjo€kjnu€u xvvztv¡ntq¹u
j ΀ qunnu
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Ívëzvuy rvë{{q|qntz wéq [N éjknt |
N + nxsq ≤ N ≤ Q − q |
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Q − N − nxsq Q ≤ N ≤ Q − Ìzknz uv |
tv ojwqxjz• xsnlyí•qu vi |
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j Ívrjojzns• xznwntq uv nz i€z• |
xsnlyí•quq xwvxvijuq |
xvxzjksnt qo wvrjojznsnp q = + + + = + + êzv votj
·jnz ·zv nxsq vivotj·qz• [ ·néno \ j [ ·néno ] zv qxrvu€p rvë{ {q|qntz éjknt xyuun rvë{{q|qntzvk wéq \ q \(−]) = \] k éjosv n
tqq ( + \ − ]) Ív {véuysn kvoknlntq¹ utvmv·sntj k xznwnt• ëzvz rvë{{q|qntz éjknt 3( ) + 3( ) = u€ sqiv qo xrvivr inénu \ qo xrvivr inénu −] q qo xrvivr inénu sqiv qo xrvirq inénu \ qo xrvivr inénu −] q qo xrvivr inénu
i ¬tjsvmq·tv tj}vlqu = = + = + tv znwné• tjlv y·qz€kjz• otjrq vzknz 3( ) − 3( ) + 3( ) = −
k 3( ) − 3( ) + 3( ) = − znwné• tjlv y·q
z€kjz• q rvë{{q|qntz€
΀ qunnu = + + j tn éjoiqkjnzx¹ tj xyuuy wvsv qzns•t€} xsjmjnu€} réjzt€} q Ívëzvuy [ k}vlqz x rvë{{q
|qntzvu & & = j [ ² x rvë{{q|qntzvu tys• |
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+ Ívëzvuy ·snt |
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΀ qunnu = + = + = |
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x [ qunnz rvë{{q|qntz − & |
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tqq ( + [ − [ ) q − |
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( − [ + [ ) ‰xtv ·zv kzvévp rvë{{q|qntz ivs•¡n |
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Êunnu & |
·sntvk kqlj [ $ = & |
·sntvk kqlj [ [ |
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& ·sntvk kqlj |
[ [ |
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Ëjqivs•¡qu rvë{{q|qntzvu k wnékvu éjosv |
ntqq ¹ks¹nzx¹ |
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rvë{{q|qntz wéq D E F q jtjsvmq·t€} Ìt éjknt 3( ) =
Æv kzvévu éjosv ntqq tjqivs•¡qu ¹ks¹nzx¹ rvë{{q|qntz 3( ) wéq D E F G
êzv ·qxsv éjktv rvë{{q|qntzy wéq [P k utvmv·sntn
([O + [O+ + + [Q)S = [OS( − [Q−O+ )S( − [)S
Íéqunt¹¹ {véuysy iqtvuj Ë•ízvtj wvsy·jnu ·zv ëzvz rvë{{q|qntz |
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P |
− & &P+O−Q− |
+ & &P+ (O−Q− ) |
− |
éjknt &P−(O− )S− |
S P−(O− )(S− )−Q− |
S P−(O− )(S− )− Q− |
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j ïqxsj k€éj jí•qn lsqt€ wvsvx rj lvmv |knzj viéjoyíz wénlxzjksntqn ·qxsj k kqln xyuu€ xsjmjnu€} wéqtqují•q} tjzyéjs•t€n otj·ntq¹ vz lv wéq·nu qméjnz évs• wvé¹lvr xsjmjnu€} ïqxsv zjrq} éjoiqntqp tj N xsjmjnu€} éjktv rvë{{q|q
ntzy wéq [ k éjosv ntqq k€éj ntq¹
([ + [ + + [ )N = [ N( − [ )N( − [)−N =
= [ N |
− & [ + & [ − + (− )N[ N |
× |
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+ & [ + & |
+ & |
+ + & |
+ |
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ÌÒÆcÒÔ |
Òjr rjr rvsq·nxzkv wvsvx rj lvmv |knzj wéq ljttvu xwvxvin vr éjxrq vlqtjrvkv j lsqty wvsvx éjosq·t€} |knzvk uv tv rvuiqtq évkjz• wévqokvs•tv wvsy·jnu + + + + = xwvxvivk
vréjxrq
i cxsq wvxsnltqu ¹ks¹nzx¹ xqtqp |knz zv xy•nxzkynz xwv xvivk vréjxrq
cxsq vréjxrj ojrjt·qkjnzx¹ réjxt€u |knzvk zv vt kxzén·jnzx¹ tj vlqt éjo ·j•n ·nu ins€p q xqtqp Æ ëzvu xsy·jn ·qxsv xwvxvivk éjktv + + + + =
Òv·tv zjr n nxsq wvxsnltqp |knz ins€p zv ·qxsv xwvxvivk vr éjxrq éjktv + + + + =
Æxnmv qunnu xwvxvivk
k cxsq tq vltj wvsvxj tn unt•¡n xu zv ojlj·j xkvlqzx¹ r wvlx·nzy ·qxsj wénlxzjksntqp k kqln xyuu€ N tjzyéjs•t€} xsjmj nu€} wéqtqují•q} otj·ntq¹ vz lv
Íéq N = qunnu vltv wénlxzjksntqn wéq N = ² w¹z• wénlxzjk sntqp wéq N = ² zéq wénlxzjksntq¹ Ívëzvuy + + = éjoj vréjxrj ojrjt·qkjnzx¹ xqtqu |knzvu + + = éjo réjxt€u |knzvu q ins€u |knzvu + + = éjo
Ìivotj·qu ·néno [ \ ] rvsq·nxzkv rtqm wnékvmv kzvévmv q zénz•nmv kqlj wvsy·ntt€} wnék€u y·jxztqrvu Ív yxsvkqí ojlj·q qunnu [ + \ + ] = wéq·nu ≤ [ ≤ ≤ \ ≤ ≤ ] ≤ ïqxsv
|ns€} én¡ntqp yéjktntq¹ ylvksnzkvé¹í•q} ljtt€u tnéjkntxzkju éjktv rvë{{q|qntzy wéq W k éjosv ntqq wévqoknlntq¹
( + W + + W )( + W + + W )( + W + + W )
êzv wévqoknlntqn uv tv wnénwqxjz• zjr
( − W )( − W )( − W ) =
( − W)
= ( − W − W − W + W + )( + W + W + W + W + + W + )
Ívxsn éjxré€zq¹ xrvivr u€ wvsy·qu wéq W rvë{{q|qntz
wvëzvuy éjolns uv nz i€z• xvkné¡nt xwvxvijuq
ïqxsv xwvxvivk éjosv qz• Q éjosq·t€} wénlunzvk wv N
¹•qrju éjktv N ÍN êzv ·qxsv éjktv rvë{{q|qntzy wéq [Q k éjo |
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Ìzxílj xsnlynz ·zv Q ( − Í + Í − |
Í + ) ¹ks¹nzx¹ rvë{ |
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{q|qntzvu wéq [Q k éjosv |
ntqq xyuu€ é¹lj |
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Òjr rjr [ − [ + [ − [ + = OQ ( + [) zv xyuuj ëzvmv é¹lj |
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éjktj OQ ( + (H[ − )) = [ Ívëzvuy wéq Q > qx}vltvn k€éj ntqn éjktv tysí
-sjkj 9,,, |
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êzv ·qxsv xwvxvivk éjktv rvë{{q|qntzy wéq [V k éjosv ntqq k€éj ntq¹
( + [ + + [S)( + [ + + [T)( + [ + + [U) =
= ( − [S+ )( − [T+ )( − [U+ )( − [)− =
= ( − [S+ − [T+ − [U+ + [T+U+ + )( + [ + [ + + & N + )
N+ [
Òjr rjr S < T + U zv U ≤ T ≤ S < V tv T + U > V q ëzvz rvë{{q|qntz qunnz kql
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− & |
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− & |
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&V+ |
V−S+ |
V−T+ |
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= |
(V + )(V + ) |
− |
(V − S + )(V − S) |
− |
(V − T + )(V − T) |
− |
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èjxrévnu xrvirq q wéqunu kv ktqujtqn ·zv S + T + U = V Ívxsn |
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wénviéjovkjtqp wvsy·qu V + V + − S + T + U |
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cxsq T + U < S |
zv |
U ≤ T < V tv |
S ≥ V |
ojzv |
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T + U + ≤ V |
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Òjr rjr = + = + = + = + = + + |
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= + + zv x wvuv••í yrjojtt€} |
éjotvknxvr uv |
tv |
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síivp |ns€p knx vz |
lv um |
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zjr |
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knxj |
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k€éj jnu€n lnx¹zrjuq xvzt¹uq uqssqméjuuvk q z l |
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Æ€¹xtqu xtj·jsj rjrvkj tjqivs•¡j¹ lsqtj |nwq zjrvp ·zv |
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wvxsn éjxrvk€kjtq¹ |
N oknt•nk uv |
tv wvsy·qz• síivp knx vz lv Q |
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èjxxuvzéqu ls¹ ëzvmv rjrvkv tjqivsnn k€mvltvn éjxwvsv |
ntqn éjx |
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rvkjtt€} oknt•nk Òjr rjr ·qxsv éjxrvkjtt€} oknt•nk éjktv N q ujxxj rj lvmv okntj m zv x q} wvuv••í uv tv wvsy·qz• síivp knx vzlv N Ëv knx N + u€ y n tn xuv nu wvsy·qz• nxsq y tjx tn iylnz
n•n vltvp ·jxzq qo ·qxsj vxzjk¡q}x¹
‰xtv ·zv k€mvltnn kxnmv ·zvi€ ëzj ·jxz• xvxzv¹sj qo N + okn t•nk zvmlj u€ xuv nu wvsy·qz• síivp knx vz lv N + bjsnn tju wvtjlvi¹zx¹ ·jxzq knxvu (N + ) (N + ) N(N + ) f q} wvuv••í uv tv iylnz wvsy·qz• síivp knx vz lv
Q = N + >(N + ) + (N + ) + (N + ) + + N(N + )@ =
= N + (N + )( N+ − ) = N+ (N + ) −
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