HPzAKPzA:

EtPOnPLHPArP oPFjPLnPL kPPrPA hPFiHP aArP çpPFEtPO oPFjPLinPLrP fPOlPAfPOlP EtPOnPL rPkPPmLP hPFitPO LpPOAirP -- LpPOA`ÂlPA sLPmLPAnPL,HPÿA LpPOA`ÂlPA BPOArPQ,wPOAnPL LpPOA`ÂlPA BPOArPQ (HPA HPÿA LpPOA`ÂlPA hPFAlPkPPA)P| aTPYAð oPFjPLinPLrP fPOlP 3 x 3 x 3 = 27 rPkPPmLP hPFitPO LpPOAirPP| ePrP mLPiDPFz yPEdP EtPOnPLHPArPvhPF LpPOA`ÂlPA sLPmLPAnPL TPAikPP tPOAhPFilP HUJPFiHPA iyP sLPHP imLPAhPFrPvhPF sLPmLPAnPL oPFjPLinPLrP P| aAHPArP oPFjPLnPL kPPrPArP sLPmLPbP yPEdP iKPbPAlP rPAEKP iyP EHPiSPFXP ikPPAinPLA imLPAhPFrP EtPOnPLHPArPvhPF ePkPPvhPF LpPOA`ÂlPAbP tUOlPECPF nPLA,tPOAhPFilP EtPOnPLHPArPvhPF HPÿA LpPOA`ÂlPA BPOArPQ HPA hPFAlPkPPA idPKPAiHP,eP sLPÆPFAHPnPLA dLUiWvPFAoPF HPAdP idPoPFbPA yPAbPP| HPAkPPQ rPvhPFilPA 24WvPFA fPOlPAfPOlPP| tPOArP mLPiDPFz HPAirPAWvPFA LpPOAiHPA yPEdP KLUÿitPOA imLPAhPFrP iHPSPFQ BPOArPQ hPFbP aArP HPAkPPQ HPAirPAWvPFA yPEdP oPFWvPFA hPFAlPkPPA hPFbPP|

çpPFEtPO oPFjPLinPLrP EtPOnPLEWvPF fPOlPAfPOlPikPP EtPOnPLEWvPF aÌPO EdPibP iHPAJPFAinPLA yPAkPP,ilPKPArP sFUEHPiDPFrP jPLnPLzP| LpPOA`ÂlPA sLPmLPAnPL = 0,HPÿA LpPOA`ÂlPA BPOArPQ = 1,wPOAnPL LpPOA`ÂlPA BPOArPQ = HPÿA LpPOA`ÂlPA hPFAlPkPPA = -1P| tPOAhPFilP,çpPFTPmLPHPArP oPFjPLinPL HPÿA LpPOA`ÂlPA BPOArPQ,EªPtPOQbPEWvPFitPO LpPOA`ÂlPA sLPmLPAnPL,tÖOtPOQbPEWvPFitPO wPOAnPL LpPOA`ÂlPA BPOArPQ(HPA HPÿA LpPOA`ÂlPA hPFAlPkPPA) -- ePvhPF EtPOnPLEWvPF oPFjPLinPLrP EtPOnPLEWvPF aÌPO LpPOASPFALpPOAESPF ElPKPilP sLPMKPzAEWvPF hPFiHP 10-1 P| ePmLPnPL kPPirP HPAirPAEWvPF nPLÈ÷PrP LpPOAoPFbPA iyPitPO LpPOAirP aArP aAmLPAidPrP HPAirPAEWvPF imLPAhPFrP aAiCPF -- atPOePHP çpPFEtPOEWvPF imLPAhPFirP yPEdP ePkPPEWvPF nPLÈ÷PrP lPAgPFAinPLA yPAbP tPOAhPFilP oPFjPLinPLrP fPOlPAfPOlPgFUilPA LpPOASPFALpPOAESPF sLPAEjPLibP iyP nPLÈ÷PrPEWvPF LpPOAoPFbPA yPAiHP isLPvhPF nPLÈ÷PirPrP imLPAhPFrPEWvPFvhPF KLUÿitPOA HPilP icPFnPLA yPAiHPP|

isLPWvPFA KLUHP sLPhPFijPL kPPrPA yPAbP tÖOtPOQEbPkPP sLPMKPzArP SPFrPNP EnPLilPP| tÖOtPOQEbPkPP EkPP HPéUF tPOArP kFZOWvPFkPPcPFAilP nPLA EgPFibP EtPOnPL aiÌPOrP ePkPPWvPFA tÖOtPOQEbPkPP sLPMKPzArP dPSPFEmLPikPP dPÿAwxPFAiHP:

tÖOtPOQEbPkPP sLPMKPzArP HPÿA EdPikPP iTPikPP çpPFTPmLP aÌPO +
3 x (tÖOtPOQEbPkPP sLPMKPzArP HPÿA EdPkPP iTPikPP EªPtPOQbP aÌPO +
3 x (tÖOtPOQEbPkPP sLPMKPzArP HPÿA EdPkPP iTPikPP tÖOtPOQbP aÌPO )

aTPYAð aAmLPAidPrP icPFnPLA tÖOtPOQEbPkPP sLPMKPzA 10-1 = -1 + 3 x (0 + 3 x 1) = -1 + 9 = 8 (dPSPFEmLPkPP)

ePBPOAiHP EhPFisLPHP kPPrPilP aAmLPrPA iLpPOitPO LpPOAErP

imLPAhPFirPrP nPLAmLP		kPP	KP	gPF	GPL	cPF	CPF	jPL	JPF	WvPF	VPF	wPO	WPOL
dPSPFEmLPkPP nPLÈ÷PrP		1	2	3	4	5	6	7	8	9	10	11	12
tÖOtPOQEbPkPP nPLÈ÷PrP	*	0	0	0	0	1	1	1	1	1	1	1	1
	**	0	1	1	1	-1	-1	-1	0	0	0	1	1
	***	1	-1	0	1	-1	0	1	-1	0	1	-1	0

  

ePHPArP aAirPA ePkPPWvPFA cPFAlPAEkPP kPPrPA yPAbPP| aAmLPrPA yPEdP ePKPnPL icPF´KLULpPOQrP mLPiDPFzkPPArP iyP ikPPAinPLA 1 aiÌPOrP mLPAinPL kPPErP ePvhPF iyP,iyP éPFiÆPF isLPvhPF aÌPOEWvPF aAiCPF isLPvhPF éPFiÆPFrP imLPAhPFrPEWvPF HPÿA LpPOA`ÂlPAbP yPAiHP,-1 wPOAnPL LpPOA`ÂlPAbP aArP 0-rP aTPY ihPFAilPA iyP isLPEWvPF LpPOA`ÂlPAbP wvPOVPFiHP nPLA,tPOAhPFilP ePvhPF icPF´KLUELpPOrP çpPFEtPOEWvPF sLPAErPikPP (*,** aArP *** EdPibP iHPAJPFAinPLA hPFibPiCPF) çpPFEtPO oPFjPLinPLrP sLPmLPbP ikPPAnPL LpPOA`ÂlPAbP ikPPAnPL imLPAhPFrP itPOAlPA hPFiHP tPOArP EnPLidPYSPF EhPFisLPiHP inPLoPFbPA yPAiHPP| iyPmLPnPL ***-mLPAkPPYA sLPAErPEWvPFitPO idPKPECPF cPFArPEWvPF 1 (aTPYAð HPÿA LpPOA`ÂlPAbP cPFArPEWvPF imLPAhPFrP -- kPP ,GPL,jPL,VPF),cPFArPEWvPF -1 (wPOAnPL LpPOA`ÂlPAbP cPFArPEWvPF -- KP,cPF,JPF,wPO) ePHPM cPFArPEWvPF 0 (HPAvhPFirP cPFArPEWvPF -- gPF,CPF,WvPF,WPOL)P| tÖOtPOQbP oPFjPLinPLrP LpPOA`ÂlPAWvPFA tPOAhPFilP ePBPOAiHP sLPAjPLAinPLA iyPitPO LpPOAirPP| * aArP ** sLPAErPitPO ePkPPWvUF adPlPHPdPlP kPPirP 4-4-4 BPOAgPF kPPirP inPLoPFbPA yPAbP,hPFlPidP mLPAkPPYA idPoPFbPA GPLirP tPOA idPKPAinPLA hPFibPiCPFP| ePkPPWvPFA nPLjPLrP rPAKPitPO hPFiHP iyP ePkPPvhPF dPSPFEmLPkPP sLPMKPzA aArP tPOArP wvPOiÃFWvPFAWvPFA (inPLigPFEWvPFBPO) iyPnPL nPLA TPAikPP aAmLPAidPrP icPF´KLULpPOQitPOP| ePHPArP ikPPAinPLA EtPOnPL-oPFjPLinPLrP fPOlPAfPOlP jPLAnPLilP aAmLPAidPrP ePvhPF icPF´KLULpPOQ idPiKP KLUÿitPOA imLPAhPFrP icPFnPLA yPAiHPP| sLPrPAsLPErP EmLPlP iLpPOibP igPFilP -- BPOArPQP| yPEdP EtPOnPL-oPFjPLinPLrP fPOlPAfPOlP ePvhPF iWvPFEHPilP nPLA TPAikPP tPOAhPFilP tPOArP wvPOiÃFWvPFAWvPFA EnPLåcPFbP TPAkPPiHP,isLPi¾PPiÀeOL` hPFiHP hPFAlPkPPAP| HPlPAvhPF HPAhUFlPz 000 mLPAinPL ihPFAilPA ikPPAinPLA imLPAhPFrPvhPF KLUÿitPOA nPLbPP| kPPÈOFpOUzWvPFAirP ePvhPF iWvPFEHPlPEWvPFvhPF HPzHPhPFArP kPPrPA hPFibPiCPF!

 

imLPAhPFirPrP nPLAmLP		kPP	KP	gPF	GPL	cPF	CPF	jPL	JPF	WvPF	VPF	wPO	WPOL
dPSPFEmLPkPP nPLÈ÷PrP		1	2	-3	4	5	-6	7	-10	9	-8	11	-12
tÖOtPOQEbPkPP nPLÈ÷PrP	*	0	0	0	0	1	-1	1	-1	1	-1	1	-1
	**	0	1	-1	1	-1	1	-1	0	0	0	1	-1
	***	1	-1	0	1	-1	0	1	-1	0	1	-1	0

    

sLPADPFArPNPtPO ePvhPF DPFÿADPFArP wvPOÀoP`LrP idPoPFbPA hPFbP ePvhPF BPOAiHP:çpPFTPimLP dLUiWvPFA LpPOA`ÂlPAbP cPFArPiWvPF-cPFArPiWvPF kPPirP imLPAhPFrP EdPibP oPFjPLnPL kPPrPqnPL, yPEdP ePkPPWvPFA LpPOA`ÂlPA BPOArPQ hPFbP tPOAhPFilP . . . P| aAmLPrPA ePkPPWvUF anPLzBPOAiHP DPFÿADPFArP sLPmLPADPFAnPL kPPrPiHPA yPAitPO sLPrPAsLPErP wvPOÀoP`LrP LpPOAoPFbPA yPAiHPP| nPLQicPF aAmLPrPA ePvhPF oPFjPLinPLrP çpPFEÀePÝbPAWvPFAikPP 'kPPÈOFpOUzWvPFArP EsLPmFUilPSPFnPL' kPPirPECPFP| tPOAitPO EtPOnPLiWvPF oPFjPLinPLrP fPOlPAfPOlP jPLAnPLAilP kPPÈOFpOUzWvPFArPvhPF KLUÿitPOA imLPAhPFrP aArP BPOArPQ HPA hPFAlPkPPA HPilP idPiHPP| sLPHPiSPFiXP ePKPAinPL E`ÂFkPPkPP kPPrPilP ePkPPWvPFA HPzAKPzA idPKPA yPAiHPP|

ççpPFTPimLP ePvhPF HPAirPAWvPFA imLPAhPFirPrP iTPikPP mLPinPL mLPinPL iyP ikPPAinPLA ePkPPWvPFA imLPAhPFrP EVPFkPP kPPrPqnPL iyPWvPFA KLUÿitPOA ePHPM ikPPmLPnPL KLUÿitPOA -- BPOArPQ nPLA hPFAlPkPPAP| (yPEdP cPFAnPL tPOAhPFilP ikPPAinPLA imLPAhPFirPvhPF KLUÿð inPLvhPF -- ePWvPFAoPF EVPFkPP kPPrPitPO LpPOAirPnPLP|) aALpPOEnPL kPPQ EVPFkPP kPPrPilPnPL tPOA kPPÈOFpOUzWvPFArPikPP HPlPitPO hPFiHP nPLAP| ePHPArP 'ãOFWvPFAWvPFY' iHPAtPOAmLPWvPFA EWvPFiLpPO EsLPmFUilPSPFnPL SFUrPq kPPrPArP LpPOrP kPPÈOFpOUzWvPFArP EtPOnPLHPArP oPFjPLnPLdPÿAEwxPFWvPFA EBPOnÔP EBPOnÔP BPOAiHP sLPAEjPLibP aALpPOnPLAikPP idPKPAiHPP| aALpPOnPLAikPP SFUDFU jPLAnPLAitPO hPFiHP iyP aALpPOEnPL yPA iBPOiHPiCPFnPL tPOArP oPFLpPOrP EnPLBPOYrP kPPirP oPFjPLinPLrP kPPQ fPOlPAfPOlP hPFiHPP| EtPOnPLHPArP oPFjPLinPLrP LpPOrP kPPÈOFpOUzWvPFArP aALpPOnPLAikPP jPLAEnPLibP idPiHP aALpPOEnPL ikPPAnPL imLPAhPFrPWvPFAikPP KLUÿitPOA iBPOiHPiCPFnPL aArP kPPQ tPOArP oPFjPLnPLP| kPPQBPOAiHP ePWvPFA kPPrPitPO hPFiHP tPOA nPLQicPF idPoPFbPA ihPFAilPAP|

'ãOFWvPFAWvPFY'iHPAtPOAmLPWvPFA EWvPFLpPOilP LpPOA`ÂlPA BPOrPA hPFiHP çpPFTPmLP oPFjPLinPLrP jPLnPLzPP| aALpPOnPLArP HPACPFA imLPAhPFirPrP aHPëPOAinPLrP oPFLpPOrP EnPLBPOYrP kPPirP iyP LpPOA`ÂlPA BPOArPQ hPFiHP isLPvhPF LpPOA`ÂlPAbP (LpPOA`ÂlPA sLPmLPAnPL TPAkPPilP oPFjPLnPL dPÿAEwxPFrP mLPiDPFzrP dPÕPOWvPFAbP) E`ÂFkPPkPP kPPrPqnPLPP| ePWvPFA ePikPPHPAirP EVPFkPP hPFoPFbPA dPrPkPPArP,tPOAvhPF BUOlP LpPOA`ÂlPA EWvPFiLpPO TPAkPPilP EVPFkPP kPPirP EdPnPLPP| ePHPArP fPOlPAfPOlPWvPFA kPPÈOFpOUzWvPFArPikPP jPLAnPLAnPL 'çpPFTPmLP' iHPAtPOAmLPWvPFA EWvPFiLpPOPP| tPOKPnPL 'EªPtPOQbP' iHPAtPOAmLP aArP EªPtPOQbP oPFjPLinPLrP LpPOA`ÂlPA EHPnPLzAsLP idPKPA yPAiHPPP| aAigPFrP mLPitPOA kPPirP yPAnPLPP| 'tÖOtPOQbP' iHPAtPOAmLP EWvPFLpPOHPArP LpPOrP kPPÈOFpOUzWvPFArP aALpPOnPLArP HPACPFA KLUÿitPOA imLPAhPFrP aArP BPOArPQ HPA hPFAlPkPPA jPLAEnPLibP idPiHPPP| wvPOdPAhPFrPNP: DPFrPqnPL aALpPOEnPL iHPiCPFiCPFnPL hPFAlPkPPA 'kPP' -- tPOAhPFil PçpPFTPmLP oPFjPLinPLrP sLPmLPbP idPKPiHPnPL iyP 'kPP' ikPPAinPLA LpPOA`ÂlPAitPOvhPF inPLvhPF atPOePHP çpPFTPmLP oPFjPLinPLrP fPOlP hPFiHP 'sLPmLPAnPL',aTPYAð aALpPOEnPL mLPiDPFzrP dPÕPOWvPFA E`ÂFkPPkPP kPPrPiHPnPL; aAHPArP yPEdP iHPiCPF TPAikPPnPL hPFAlPkPPA 'CPF' tPOAhPFilP idPKPiHPnPL HPÿAEdPikPPrP LpPOA`ÂlPAbP 'CPF' imLPAhPFrPEWvPF aAiCPF,aTPYAð wPOAnPLEdPikPPrP LpPOA`ÂlPAbP E`ÂFkPPkPP kPPrPitPO hPFiHPPP|