Kapitoly 5-6
Transkript
Kapitoly 5-6
(QHUJHWLFNpSiV\ (QHUJHWLFNpSiV\ .RYSRORYRGLþL]RODQW 0RGHO YROQêFK HOHNWURQ$ XPRåXMH VSUiYQ SRFKRSLW DGX YODVWQRVWt NRY$ QDS PUQp WHSOR WHSHOQRX D HOHNWULFNRX YRGLYRVW PDJQHWLFNRX VXVFHSWLELOLWX 1HY\VYWOXMH YãDN UR]GtO PH]L NRY\ SRORNRY\ SRORYRGLþL D L]ROiWRU\ 8NiåHPH åH MHGQRGXFKp ]REHFQQt PRGHOX YROQêFK HOHNWURQ$ SLQiãtGDOãtFHQQpYêVOHGN\ 5R]GtO PH]L GREUêP YRGLþHP D GREUêP L]ROiWRUHP MH ]HMPê (OHNWULFNê PUQê RGSRU þLVWpKR NRYX P$åH SL WHSORW . NOHVQRXW Då QD Ω PQHKOHGQDVXSUDYRGLYRVW0UQêRGSRUGREUpKRL]ROiWRUXP$åH GRVDKRYDW KRGQRW Då Ω P äiGQi I\]LNiOQt YHOLþLQD FKDUDNWHUL]XMtFt SHYQRXOiWNXQHPiWDNYHONêUR]VDKKRGQRWGRVDKXMtFtiGRY 7DEXONDXGiYiKRGQRW\PUQpYRGLYRVWLSUR]iNODGQtW\S\SHYQêFK OiWHN 7DE5R]VDKPUQêFKYRGLYRVWtSHYQêFKOiWHN PUQiYRGLYRVW OiWND WtGD −1 Ω P Ω−1P Ω−1P Ω−1P Ω−1P Ω−1P Ω−1P Ω−1P Ω−1P P åHOH]RYL]PXW ,Q6EYODVWQt NRY\DSRORNRY\ 6LYODVWQt &G6 3( 6HDPRUIQt SDUDILQGLDPDQW GLHOHNWULND .DåGi SHYQi OiWND REVDKXMH HOHNWURQ\ 3UR HOHNWULFNRX YRGLYRVW MH SRGVWDWQpMDNêP]S$VREHPUHDJXMtHOHNWURQ\QDSLORåHQpYQMãtHOHNWULFNp SROH8YLGtPHåHHOHNWURQ\YNU\VWDOHFKVHQDFKi]HMtYHQHUJHWLFNêFKSiVHFK RGGOHQêFK RG VHEH REODVWPL HQHUJLt NWHUêP QHRGSRYtGDMt åiGQp HOHNWURQRYp YOQRYp VWDY\ ]DNi]DQp SiV\ .U\VWDO VH FKRYi MDNR L]ROiWRU SRNXG MVRX GRYROHQp HQHUJHWLFNp SiV\ ]DSOQQp QHER SUi]GQp REUF QHER" SRWRP VH HOHNWURQ\ QHPRKRX SRK\ERYDW Y SLORåHQpP HOHNWULFNpP SROL =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN .U\VWDO VH FKRYi MDNR NRY SRNXG MHGHQ QHER YtFH SiV$ MH ]DSOQQR þiVWHþQREUD .U\VWDO MH SRORYRGLþ QHER SRORNRY MHVWOLåH MVRX YãHFKQ\ SiV\ V YêMLPNRX MHGQRKR QHER GYRX ]FHOD ]DSOQQp HOHNWURQ\ 1H]DSOQQp SiV\ MVRXWpPSUi]GQpQHERQDRSDNWpP~SOQ]DSOQQp 5R]GtOPH]LL]ROiWRUHPDYODVWQtPSRORYRGLþHP]DQt]NpWHSORW\. VSRþtYiSRX]HYãtFHSiVX]DND]iQêFKHQHUJLt 7 DNRY 7! ESRORYRGLþ FL]ROiWRU 2EU6FKHPD ]DSOQQt HQHUJHWLFNêFK SiV$ SUR U$]Qp OiWN\ âHG MVRX R]QDþHQ\KODGLQ\]DSOQQpHOHNWURQ\ (OHNWURQYSHULRGLFNpPSRWHQFLiOX 0RGHOWpPYROQêFKHOHNWURQ$ $E\FKRP SRFKRSLOL UR]GtO PH]L L]ROiWRU\ D YRGLþL PXVtPH UR]ãtLW PRGHOYROQêFKHOHNWURQ$WDNDE\]DKUQRYDOSHULRGLFLWXPtåN\NU\VWDO$ 1HMG$OHåLWMãtP G$VOHGNHP MH H[LVWHQFH ]DND]iQêFK SiV$ HQHUJLt 2EMHYtVHL]FHODQRYpYODVWQRVWLHOHNWURQXYNU\VWDOX 3L S$VREHQt YQMãtKR HOHNWULFNpKR QHER PDJQHWLFNpKR SROH VH HOHNWURQ\FKRYDMtMDNRE\PO\HIHNWLYQtKPRWQRVWPNWHUi]iYLVtQDVPUX SRK\EX Y]KOHGHP N PtåFH D P$åH EêW YWãt QHER PHQãt QHå MH KPRWQRVW YROQpKR HOHNWURQX GRNRQFH L ]iSRUQi (OHNWURQ\ Y NU\VWDOLFNp OiWFH UHDJXMt QDSLORåHQpYQMãtSROHMDNRE\PO\]iSRUQêQHERNODGQêQiERMHHFRå Y\VYWOXMH]iSRUQpQHERNODGQpKRGQRW\+DOORY\NRQVWDQW\7HRULHYROQêFK HOHNWURQ$ SHGSRNOiGi åH VH HOHNWURQ SRK\EXMH Y SROL NRQVWDQWQtKR SRWHQFLiOX8 6FKU|GLQJHURYDURYQLFHSURMHGQRUR]PUQêStSDGPiSDNWYDU GΦ (( − 8)Φ = + P G[ K (QHUJHWLFNpSiV\ 3R]QiPND9GDOãtPWH[WXMH]WLVNRYêFKG$YRG$3ODQFNRYDNRQVWDQWD Kπ SViQDMHQRPMDNRK 7pWRURYQLFLY\KRYXMHHãHQtYHWYDUXURYLQQêFKYOQW\SX Φ([) = H[S [±LN[] 3RGRVD]HQtGRVWiYiPHNLQHWLFNRXHQHUJLLHOHNWURQX S ( NLQ = ( − 8 = KPN = P )\]LNiOQtVP\VOYHOLþLQ\NMHYWRPåHSHGVWDYXMHK\EQRVWHOHNWURQX GOHQRXNRQVWDQWRXK ÒSOQê WYDU YOQRYp IXQNFH ]iYLVOp QD þDVH GRVWDQHPH QiVREtPHOL IXQNFL Φ([) YêUD]HP H[S>LωW@ Y QPå ω = (K 'RVWiYiPH VNXWHþQp YOQ\ ãttFtVHSRGpORV\[ 8YDåXMPH Q\Qt R 6FKU|GLQJHURY URYQLFL SUR HOHNWURQ NWHUê VH SRK\EXMH Y MHGQRUR]PUQpP SHULRGLFNpP SRWHQFLiORYpP SROL 3RWHQFLiOQt HQHUJLHHOHNWURQXVSOXMHSRGPtQNXSHULRGLþQRVWL 8[ 8[D NGHDMHSHULRGD 6FKU|GLQJHURYDURYQLFHSDNPiWYDU G Φ P + K [( − 8([)] = G[ HãHQtPWDNRYpURYQLFHXåQHQtþLVWURYLQQiYOQDDOHPRGXORYDQiURYLQQi YOQDW]Y%ORFKRYDYOQD Φ([) = H ±LN[ X N ([) VDPSOLWXGRXXN[NWHUiPiSHULRGLFLWXPtåN\ XN[ XN[D 8NiåHPH VL åH L ]MHGQRGXãHQê PRGHO MHGQRUR]PUQpKR SHULRGLFNpKR SRWHQFLiOX.URQLJ$Y3HQQH\$YPRGHOYHGHNH[LVWHQFLSiVX]DNi]DQêFK HQHUJLt 3HGSRNOiGHMPH åH SRWHQFLiOQt HQHUJLH HOHNWURQX Y NU\VWDOX Pi WYDU SHULRGLFNpKRXVSRiGiQtREGpOQtNRYêFKMDPREU 8[ DE E D DE [ 2EU0RGHOSHULRGLFNKRSRWHQFLOXSRGOH.URQLJDD3HQQH\H =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN 3HULRGD SRWHQFLiOX MH D E 9 REODVWL [D MH 8[ Y REODVWL E[MH8[ 8 .DåGiMiPDMH]KUXEDDSUR[LPDFtSU$EKXSRWHQFLiOXY EOt]NRVWLDWRPX6FKU|GLQJHURYXURYQLFLUR]HStãHPH G Φ P( + K Φ = SUR[D D G[ R SURE[ E + K Φ = %XGHPH SHGSRNOiGDW åH HQHUJLH HOHNWURQ$ (9 'HILQXMPH GY UHiOQp YHOLþLQ\ P(8 −() α = KP( β = K 3RXåLMHPHOL WRKR åH HãHQtP PXVt EêW %ORFKRYD IXQNFH H X[ GRVWDQHPHSURX[URYQLFH GG[X + LN GX [D D G[ + (α − N )X = GX + LN GX E[ E G[ − (β + N )X = G[ HãHQtWFKWRURYQLFPDMtWYDU X = $H L(α−N)[ + %H −L(α+N)[ [D D −(β+LN)[ LN [ (β− ) X = &H E[ E + 'H GΦ G[ P((−8 ) R LN[ $%&'MVRXNRQVWDQW\NWHUpMHWHEDYROLWWDNDE\YOQRYpIXQNFHYþHWQ SUYQtFKGHULYDFtE\O\VSRMLWp GX GX 6SRMLWRVWX X G[ [= = G[ [= GX GX 3HULRGLFLWDX D X E G[ [=D = G[ [=E 3R GRVD]HQt ]D X D X ]H Y]WDK$ GRVWDQHPH VRXVWDYX þW\ KRPRJHQQtFK DOJHEUDLFNêFK URYQLF SUR NRQVWDQW\ $%&' 6RXVWDYD Pi HãHQtNG\åMHMtGHWHUPLQDQWPiQXORYRXKRGQRWX 7RYHGHNQiVOHGXMtFtSRGPtQFH β −α αβ VLQK βE VLQ αD + FRVK βE FRV αD = FRV N(D + E) $E\VHWDWRURYQLFH]MHGQRGXãLODXYDåRYDO.URQLJD3HQQH\StSDGY QPå PDMt SRWHQFLiORYp YDO\ WYDU IXQNFH GHOWD WM 8 URVWH QDGH YãHFKQ\ PH]HDEVHEOtåtNQXOHSLþHPåVRXþLQ8 E]$VWiYiNRQVWDQWQt 3RGPtQNDVH]MHGQRGXãtQDWYDU 3 α VLQ αD + FRV αD = FRV ND P8 ED NGH3 = K FKDUDNWHUL]XMHYD]EXHOHNWURQXNXUþLWpSRWHQFLiORYpMiP R R (QHUJHWLFNpSiV\ 5RYQLFHVHGiVSOQLWMHQSURWDNRYpα = WHG\WDNRYpHQHUJLH ( SUR NWHUp SUDYi VWUDQD OHåt Y LQWHUYDOX = WRKR MH YLGW åH HQHUJHWLFNp VSHNWUXP HOHNWURQ$ VH VNOiGi ] DG\ SiV$ GRYROHQêFK HQHUJLt RGGOHQêFK]DNi]DQêPLREODVWPL 9\MiGtPHOL ]iYLVORVW HQHUJLH QD YOQRYpP þtVOH N ]MLVWtPH åH QHVSRMLWRVWLYKRGQRWiFKHQHUJLHQDVWiYDMtSURN = π DQ = ± ± ± P( K Q D ( KN P SRYROHQêSiV ]DNi]DQêSiV π D π D π D π D π D π D π D N π D 2EU =iYLVORVW HQHUJLH QD YOQRYpP þtVOH SUR .URQLJ$Y3HQQH\$Y SRWHQFLiO6SRMLWiNLYNDRGSRYtGiYROQêPHOHNWURQ$P ( ( SUYQt%= π D π D GUXKi%= π D WHWt%= π D þWYUWi%= π D N π D N 2EU 9 ]iYLVORVWL ( QD N MVRX Y\]QDþHQ\ %ULOORXLQRY\ ]yQ\ YåG\ MHQ SRORYLQD GUXKi SRORYLQD MH Y REODVWL ]iSRUQpKR N +ODGLQ\ MH PRåQpSHQpVWGRSUYQt%= =iYLVORVWHQHUJLHVHREY\NOHY\QiãtYLQWHUYDOHFKYOQRYpKRþtVOD −π π HãHQt WURMUR]PUQp ~ORK\ VH ]DSRþWHQtP UHDOLVWLþWMãtKR SU$EKX SRWHQFLiOX NU\VWDORYp PtåH MH SUREOpPHP ]QDþQ VORåLWêP $OH L HãHQt ]MHGQRGXãHQpKR MHGQRUR]PUQpKR StSDGX Xå GRYROXMH SRFKRSLW åH Y NU\VWDOHFKH[LVWXMtGRYROHQpD]DNi]DQpSiV\HQHUJLtHOHNWURQ$ D D =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN .YDOLWDWLYQtSHGVWDYXRY]QLNXSiV$GRYROHQêFKD]DND]iQêFKHQHUJLt P$åHPH ]tVNDW L MLQêP ]S$VREHP 8YDåXMPH VRXVWDYX L]RORYDQêFK DWRP$ %XGHPHVOHGRYDWFRVHGMHNG\åVHDWRP\NVREQDY]iMHPSLEOLåXMtDå Y\WYRtSHYQRXOiWNX (QHUJHWLFNpSiV\ VG (H9 S V VNXWHþQiY]GiOHQRVW S PH]LMDGHUQiY]GiOHQRVWQP 2EU (QHUJLRYp KODGLQ\ DWRPX VRGtNX SHFKi]t Y SiV\ NOHViOL PH]LDWRPRYiY]GiOHQRVW3HYQêVRGtNPiPH]LDWRPRYRXY]GiOHQRVW QP 3LEOLåRYiQtP DWRP$ VH MHGQRWOLYp GLVNUpWQt HQHUJHWLFNp KODGLQ\ L]RORYDQêFKDWRP$UR]ãLXMtYSiVHQHUJLtREU6ORåtPHOL1DWRP$SDN =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN YNDåGpPGRYROHQpPSiVXEXGH1HQHUJHWLFNêFKKODGLQ9]KOHGHPNWRPX åH1MHYHONpGiVHXYDåRYDWåHHQHUJLHHOHNWURQXYSiVXVHPQtVSRMLW (QHUJHWLFNp SiV\ SHYQp OiWN\ MVRX REGRERX HQHUJHWLFNêFK KODGLQ DWRPX D HOHNWURQ Y SHYQp OiWFH P$åH PtW MHQ W\ HQHUJLH NWHUp OHåt XYQLW WFKWR HQHUJHWLFNêFK SiV$ 5$]Qp SiV\ VH PRKRX L SHNUêYDW 3RNXG VH QHSHNUêYDMt H[LVWXMH PH]L SiV\ REODVW HQHUJLt NWHUêFK HOHNWURQ QHP$åH QDEêYDWSiV]DNi]DQêFKHQHUJLt (OHNWULFNp YODVWQRVWL SHYQp NU\VWDOLFNp OiWN\ MVRX XUþHQ\ VWUXNWXURX HQHUJHWLFNêFK SiV$ W]Y SiVRYRX VWUXNWXURX D REVD]HQtP HQHUJHWLFNêFK SiV$HOHNWURQ\ 3RGtYHMPH VH QD GLDJUDP HQHUJHWLFNêFK KODGLQ VRGtNRYpKR DWRPX D HQHUJHWLFNêFKSiV$VRGtNXYSHYQpIi]LREU $WRP VRGtNX Pi MHGHQ HOHNWURQ V ~SOQi HOHNWURQRYi VWUXNWXUD MH >VVS@V7R]QDPHQiåHSiVVMHMHQ]SRORYLQ\REVD]HQêYNDåGpP SiVX MH 1 KODGLQ QD NWHUp O]H XPtVWLW 1 HOHNWURQ$ 9ORåtPHOL VRGtN GR HOHNWULFNpKR SROH HOHNWURQ\ VQDGQR ]tVNDMt GRGDWHþQRX HQHUJLL D SLWRP ]$VWiYDMt YH VYpP S$YRGQtP HQHUJHWLFNpP SiVX 7DWR GRGDWHþQi HQHUJLH MH YH IRUP NLQHWLFNp HQHUJLH D SRK\EXMtFt VH HOHNWURQ\ SLVStYDMt N HOHNWULFNpPXSURXGX -LQi VLWXDFH MH QDS Y GLDPDQWX $WRP XKOtNX Pi HOHNWURQRYRX VWUXNWXUX >VVS@ REU 0H]L SRVOHGQtP REVD]HQêP SiVHP D GDOãtP QHREVD]HQêP MH SiV ]DNi]DQêFK HQHUJLt R ãtFH H9 7R ]QDPHQi åH HOHNWURQXYNU\VWDOXGLDPDQWXMHWHEDGRGDWQHMPpQH9HQHUJLHPiOLVH MHKR NLQHWLFNi HQHUJLH ]YêãLW HOHNWURQ SHVNRþt ]DNi]DQê SiV GR GDOãtKR GRYROHQpKRSiVX VRGtN GLDPDQW S ^ V V V V V S 2EU(QHUJHWLFNpSiV\SURVRGtNDGLDPDQW H9 (QHUJHWLFNpSiV\ 'RGDW WDNRYp PQRåVWYt HQHUJLH O]H QDS ]YêãHQtP WHSORW\ $E\ SUDYGSRGREQRVWSHFKRGXE\ODGRVWDWHþQiPXVHODE\WHSHOQiHQHUJLHþiVWLF N % 7 ≈ H9 'RVD]HQtP þtVHOQêFK KRGQRW E\FKRP ]MLVWLOL åH RGSRYtGDMtFt WHSORWD7 . 5RYQå GRGiQt HQHUJLH SRPRFt HOHNWULFNpKR SROH QHQt VQDGQp (OHNWURQ SRK\EXMtFt VH NU\VWDOHP SURGOiYi Y SU$PUX MHGQX VUiåNX QD NDåGêFK P D ]WUiFt SL Qt YWãLQX HQHUJLH NWHURX ]tVNi ] YQMãtKR HOHNWULFNpKR SROH . WRPX DE\ QD VWHGQt YROQp GUi]H O P ]tVNDO HQHUJLLH9MH]DSRWHEtSROHRLQWH]LW 9P FRåMHKRGQRWD NUiW YWãtQHåLQWHQ]LWDSROHNWHUpVWDþtNY\YROiQt]QDWHOQpKRSURXGXYVRGtNX 'LDPDQWMHWXGtåãSDWQêPYRGLþHPHOHNWULFNpKRSURXGXDDGtVHPH]L L]ROiWRU\ .HPtNPiSRGREQRXNU\VWDORYRXVWUXNWXUXMDNRGLDPDQW6WHMQMDNR X GLDPDQWX MH YUFKRO ]DSOQQpKR HQHUJHWLFNpKR SiVX RGGOHQ RG Y\ããtKR SUi]GQpKRSiVXPH]HURX=DNi]DQêSiVPiYãDNãtNXMHQH9 3L Qt]NêFK WHSORWiFK VH 6L FKRYi MDNR L]ROiWRU $OH Xå SL SRNRMRYp WHSORW Pi PDOi þiVW HOHNWURQ$ GRVWDWHþQRX NLQHWLFNRX HQHUJLL WHSHOQpKR S$YRGX DE\ PRKO\ SHVNRþLW ]DNi]DQê SiV 7\WR HOHNWURQ\ VWDþt NH Y]QLNX PDOpKRHOHNWULFNpKRSURXGXSLS$VREHQtHOHNWULFNpKRSROH (OHNWULFNêRGSRUNHPtNXMHFRGRYHOLNRVWLPH]LL]ROiWRU\DYRGLþL POXYtPHRSRORYRGLþL 6KUQXWt.RY\DL]ROiWRU\ 9 StSDG åH YDOHQþQt HOHNWURQ\ ]FHOD ]DSOXMt MHGHQ QHER QNROLN SiV$ SLþHPå RVWDWQt SiV\ ]$VWiYDMt SUi]GQp EXGH VH NU\VWDO FKRYDW MDNR L]ROiWRU 9QMãt HOHNWULFNp SROH QHY\YROi Y WDNRYpP StSDG HOHNWULFNê SURXG 3LWRP SHGSRNOiGiPH åH HOHNWULFNp SROH QHQt QDWROLN YHONp DE\ UR]UXãLOR HOHNWURQRYRX VWUXNWXUX 3URWRåH ]DSOQQê SiV MH RGGOHQê RG QHMEOLåãtKRYêãHSRORåHQpKRSiVX]DNi]DQêPSiVHPQHH[LVWXMH]S$VREMDN VSRMLW PQLW FHONRYRX K\EQRVW HOHNWURQ$ Y NU\VWDOX 9ãHFKQ\ GRVWXSQp VWDY\ MVRX Xå REVD]HQ\ 7DWR VLWXDFH MH ]FHOD RGOLãQi RG StSDGX YROQêFK HOHNWURQ$ .U\VWDO P$åH EêW L]ROiWRUHP MHQRP NG\å MH SRþHW YDOHQþtFK HOHNWURQ$ SLSDGDMtFtFK QD SULPLWLYQt EXNX VXGp þtVOR 'iOH MH YãDN QXWQp Y\ãHWLW]GDVHSiV\SHNUêYDMtþLQLNROLY3RNXGVHSiV\SHNUêYDMtSRWRP P$åHPH PtW PtVWR MHGQRKR ]FHOD ]DSOQQpKR SiVX FKDUDNWHULVWLFNpKR SUR L]ROiWRUGYDþiVWHþQREVD]HQpSiV\NWHUpRGSRYtGDMtNRYX 8DONDOLFNêFKDY]iFQêFKNRY$ /L 1D SLSDGi MHGHQ YDOHQþQt HOHNWURQ QDSULPLWLYQtEXNXWDNåHVHFKRYDMtMDNRNRY\YRGLþH =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN .RY\ DONDOLFNêFK ]HPLQ 0J%H PDMt GYD YDOHQþQt HOHNWURQ\ D PO\E\ EêW WHG\ L]ROiWRU\ -HMLFK SiV\ VH YãDN SHNUêYDMt SURWR VH FKRYDMt MDNR YRGLþHLNG\åQHStOLãGREUp 'LDPDQW NHPtN JHUPDQLXP PDMt GYD þW\PRFQp DWRP\ YDOHQþQtFK HOHNWURQ$ Y SULPLWLYQt EXFH 3URWRåH VH MHMLFK SiV\ QHSHNUêYDMt MVRX L]ROiWRU\ (QHUJHWLFNpSiV\ =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN 3RORYRGLþH 3RORYRGLþH ÒYRG 3L WHSORW DEVROXWQt QXO\ VH þLVWp GRNRQDOp NU\VWDO\ YWãLQ\ SRORYRGLþ$ FKRYDMt MDNR L]ROiWRU\ &KDUDNWHULVWLFNp SRORYRGLYp YODVWQRVWL Y]QLNDMtYG$VOHGNXWHSHOQpH[FLWDFHStPVtGHIHNW$PtåN\QHERRGFK\OHN RG MPHQRYLWpKR FKHPLFNpKR VORåHQt 3RORYRGLþH MVRX HOHNWURQRYp YRGLþH MHMLFKå PUQê HOHNWULFNê RGSRU SL SRNRMRYp WHSORW OHåt Y UR]PH]t Då ΩP WM PH]L KRGQRWRX SUR GREUp YRGLþH ( − ΩP) D SUR L]ROiWRU\ ( − ΩP) 0UQêRGSRUSRORYRGLþ$VLOQ]iYLVtQDWHSORW+RGQRW\ NRQFHQWUDFHQRVLWHO$FKDUDNWHULVWLFNpSURNRY\SRORNRY\DSRORYRGLþHMVRX ]Qi]RUQQ\QDREU $ 1 D . $V 6 E SRORNRY\ JUDILW %L SRORYRGLþH S LSRNRMRYpWHSORW NRQFHQWUDFHYRGLYRVWQtFKHOHNWURQ FP & X N R Y \ *HþLVWp 2EU .RQFHQWUDFH QRVLWHO$ Y NRYHFK SRORNRYHFK D SRORYRGLþtFK 9RGRURYQiRVDYJUDIXMHSRXåLWDMHQSURQi]RUQMãt]REUD]HQt =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN . VRXþiVWNiP Y\XåtYDMtFtP YODVWQRVWt SRORYRGLþ$ SDWt WUDQ]LVWRU\ XVPURYDþHPRGXOiWRU\GHWHNWRU\WHUPLVWRU\IRWRþOiQN\9WpWRNDSLWROH EXGHPH GLVNXWRYDW KODYQt I\]LNiOQt YODVWQRVWL NU\VWDOLFNêFK SRORYRGLþ$ ]HMPpQD NHPtNX D JHUPDQLD 'DOãtPL G$OHåLWêPL SRORYRGLYêPL NU\VWDO\ MVRX&X 26H3E7H3E66L&,Q6E*D$V 1i]YRVORYt 3RORYRGLYpVORXþHQLQ\VFKHPLFNêPY]RUFHP$%NGH$MHWURMPRFQêD% SWLPRFQêSUYHNVHQD]êYDMtVORXþHQLQDPL,,,9,Q6E*D$V 3RNXGMH$GYRMPRFQêD%ãHVWLPRFQêSUYHNQD]êYiVHWDNRYiVORXþHQLQD ,,9,=Q6&G6 .HPtNDJHUPDQLXPMVRXQD]êYiQ\QNG\SRORYRGLþLW\SXGLDPDQWSURWRåH PDMtNU\VWDORYRXVWUXNWXUXGLDPDQWX.DUELGNHPtNXMHVORXþHQLQD,9,9 9\VRFH þLVWp SRORYRGLþH Y\ND]XMt W]Y YODVWQt YRGLYRVW QD UR]GtO RG StPVRYp YRGLYRVWL PpQ þLVWêFK Y]RUN$ 3iVRYp VFKpPD NU\VWDO$ NWHUp Y\ND]XMtYODVWQtYRGLYRVWMHQDREU ( QHREVD]HQêYRGLYRVWQtSiV ]DNi]DQêSiV (J ]DSOQQêYDOHQþQtSiV 2EU3iVRYpVFKpPDSURYODVWQtYRGLYRVWSRORYRGLþH =DNi]DQê SiV (J MH HQHUJHWLFNê LQWHUYDO PH]L QHMQLåãtP ERGHP YRGLYRVWQtKRSiVXKUDQDYRGLYRVWQtKRSiVXDQHMY\ããtPERGHPYDOHQþQtKR SiVXKUDQDYDOHQþQtKRSiVX 6 U$VWHP WHSORW\ MVRX HOHNWURQ\ WHSHOQ H[FLWRYiQ\ ] YDOHQþQtKR GR YRGLYRVWQtKRSiVXREU.HOHNWULFNpYRGLYRVWLSLVStYDMtMDNHOHNWURQ\YH YRGLYRVWQtP SiVX WDN L QHREVD]HQp KODGLQ\ QHEROL GtU\ NWHUp ]$VWDO\ YH YDOHQþQtPSiVX 3RORYRGLþH NRQFHQWUDFHHOHNWURQ$FP WHSORWD. 2EU9ODVWQtNRQFHQWUDFHHOHNWURQ$MDNRIXQNFHWHSORW\SURþLVWêNHPtN 9 SRGPtQNiFK YODVWQt YRGRYRVWL MH NRQFHQWUDFH GU URYQD NRQFHQWUDFLHOHNWURQ$ =DNi]DQêSiVHUJLt 9ODVWQtYRGLYRVWDNRQFHQWUDFHYODVWQtFKQRVLWHO$]iYLVHMtSRGVWDWQQD (J KRGQRW YêUD]X N % 7 WM QD SRPUX ãtN\ ]DNi]DQpKR SiVX N WHSORW -HOL WHQWRSRPUYHONêEXGHNRQFHQWUDFHYODVWQtFKQRVLWHO$PDOiDYRGLYRVWEXGH Qt]Ni âtN\ ]DNi]DQpKR SiVX W\SLFNêFK SRORYRGLþRYêFK PDWHULiO$ MVRX XYHGHQ\YWDEXOFH +RGQRW\( O]H]tVNDWQHMOpSHPHQtPRSWLFNpDEVRUSFH3UiKVSRMLWp RSWLFNp DEVRUSFH SL IUHNYHQFL ω XUþXMH ãtNX ]DNi]DQpKR SiVX ( Kω REU 3L SURFHVX StPp DEVRUSFH MH IRWRQ DEVRUERYiQ Y NU\VWDOX ]D Y]QLNX (J HOHNWURQXDGtU\$EVRUSþQtIUHNYHQFHω J = K J J J J =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN .U\VWDOVQHStPêP]DNi]DQêPSiVHP DEVRUSFH DEVRUSFH .U\VWDOVStPêP]DNi]DQêPSiVHP (K J Ω Kω 2EU2SWLFNiDEVRUSFHYþLVWêFKL]ROiWRUHFKSL7 . Kω (J 7DE âtND ]DNi]DQpKR SiVX PH]L YDOHQþQtP D YRGLYRVWQtP SiVHP LQHStPêGStPêSHFKRG .U\VWDO SHFKRG (JH9 .U\VWDO SHFKRG (JH9 GLDPDQW L *H L 6L . . +J7H G 3E6H G L α6Q G 3E7H ,Q$V G &G6H ,Q6E ,Q3 *D3 G &G7H =Q6 L G $O6E L 6L&KH[ 7H =Q6E G *D$V *D6E 3E6 G G &G6 =Q2 6Q7H $J&O &X2 $JO 7L2 . . G G G G G G 3RORYRGLþH 3L QHStPpP SHFKRGX MH SRKOFHQ IRWRQ ]D Y]QLNX SiUX HOHNWURQ GtUDDVRXþDVQVHHPLWXMHQHERDEVRUEXMHIRQRQDE\SODWLO]iNRQ]DFKRYiQt HQHUJLHDK\EQRVWL$EVRUSþQtIUHNYHQFH (J ω J = K ± Ω Ω MHIUHNYHQFHDEVRUERYDQpKRQHERHPLWRYDQpKRIRQRQX]PQtLK\EQRVW +RGQRWD ãtN\ SiVX ]DNi]DQêFK HQHUJLt P$åH EêW XUþHQD URYQå ] ~GDM$ R WHSORWQt ]iYLVORVWL HOHNWULFNp YRGLYRVWL QHER WHSORWQt ]iYLVORVWL NRQFHQWUDFH QRVLWHO$ Y REODVWL YODVWQt YRGLYRVWL .RQFHQWUDFH QRVLWHO$ O]H ]tVNDWUR]ERUHPPHQt+DOORYDQDSWt2SWLFNiPHQtSRVN\WXMtLQIRUPDFH RWRP]GD]DNi]DQêSiVMHStPêþLQHStPê (IHNWLYQtKPRWQRVW (OHNWURQ Y NU\VWDOX LQWHUDJXMH V NU\VWDORYRX PtåNRX D Y G$VOHGNX WpWRLQWHUDNFHMHMHKRUHDNFHQDYQMãtVtO\MLQiQHåUHDNFHYROQpKRHOHNWURQX 6HWUYDþQi KPRWQRVW Y NODVLFNp PHFKDQLFH MH GHILQRYiQD GUXKêP 1HZWRQRYêP ]iNRQHP 3RK\E YROQpKR HOHNWURQX YH VPUX RV\ [ Y HOHNWULFNpPSROLRLQWHQ]LW([ MHGiQY]WDKHP P GY [ GW = −H( [ = KOHGLVND NYDQWRYp PHFKDQLN\ MH HOHNWURQ SRSViQ SRPRFt YOQRYpKR NOXEND V YOQRYêPL þtVO\ VRXVWHGQêPL Y RNROt QMDNpKRGQRW\ N5\FKORVW SRK\EX YOQRYpKR NOXEND W]Y JUXSRYi U\FKORVW MH GHILQRYiQD Y]WDKHP Y J = GGNω =DWtPFRIi]RYiU\FKORVWURYLQQpYOQ\Y = ωN 3URWRåHIUHNYHQFHωMHYi]DQDVHQHUJLtω = (K MH Y J = K ∂∂(N =U\FKOHQtNOXEND GY J ∂ ( D J = GW = K ∂∂N (∂W = K ∂∂ N( GN GW = K ∂N ) QHER" GW = K GN GW = ) þDVRYi]PQDK\EQRVWLMHURYQDS$VREtFtVtOH 9LGtPH åH HOHNWURQ Y NU\VWDOX UHDJXMH QD YQMãt VtOX SHVQ VWHMQ MDNRE\UHDJRYDOYROQêHOHNWURQVKPRWQRVWt P ∗ = ∂K ( GS ∂N FRåMHW]YHIHNWLYQtKPRWQRVWHOHNWURQX 9StSDGYROQpKRHOHNWURQX ( = KPN ∂∂N( = KP P ∗ = P =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN HIHNWLYQtKPRWQRVWMHURYQDKPRWQRVWLVHWUYDþQp =iYLVORVWHQHUJLH(HOHNWURQXYNU\VWDOXQDYOQRYpPYHNWRUXN(NMH RYãHPYNU\VWDOXVORåLWMãtDREHFQP ∗ ≠ P (IHNWLYPtKPRWQRVWPP$åHEêWNODGQiL]iSRUQiDMHQXORYiSURN = ± QDπ 2EHFQ Pi HIHNWLYQt KPRWQRVW WHQ]RURYê FKDUDNWHU P QHER" HQHUJLH HOHNWURQX Y NU\VWDOX ( ]iYLVt YH ]YROHQpP NU\VWDORJUDILFNpP VPUX DQL]RWURSLH ∗ 3UR QNWHUp NRY\ MH SRPU PP ]QDþQ Y\VRNê 3W 1L &R 3UR&X$OHOHNWURQ\VHFKRYDMtMDNRYROQp'RVD]HQtPHIHNWLYQt KPRWQRVWL GR Y]WDKX SUR HOHNWURQRYp PUQp WHSOR VH Y\VYWOt QHVRXKODV NODVLFNpWHRULHYROQêFKHOHNWURQXVH[SHULPHQWHP (IHNWLYQt KPRWQRVW QRVLWHO$ Y SRORYRGLþtFK VH SRGDLOR VWDQRYLW PHWRGRXF\NORWURQRYpUH]RQDQFH3RORYRGLþVHYORåtGRPDJQHWLFNpKRSROH RLQGXNFL(OHNWURQ\VH]DþQRXSRK\ERYDWSRNUXKRYêFKGUDKiFKNROPêFKN YHNWRUXPDJQHWLFNpLQGXNFH%VIUHNYHQFt ω F = PH%∗ 9\VRNRIUHNYHQþQt HOHNWURPDJQHWLFNp YOQ\ FPPP VH DEVRUEXMt Y OiWFHMHOLMHMLFKIUHNYHQFHURYQDF\NORWURQRYpIUHNYHQFLHOHNWURQ$ -HOLQDS% 7DPD[LPXPDEVRUSFHQDVWiYiSLI +] Y\FKi]tP P DSRORPUGUiK\U PSLU\FKORVWL PV LM H .RQFHQWUDFHYODVWQtFKQRVLWHO$ +OHGHMPH ]iYLVORVW NRQFHQWUDFH YODVWQtFK QRVLWHO$ QD ãtFH ]DNi]DQpKR SiVX 1HMSUYH VWDQRYtPH SRþHW HOHNWURQ$ H[LWRYDQêFK GR YRGLYRVWQtFK SiV$ SL WHSORW 7 (QHUJLL ( EXGHPH PLW RG YUFKROX YDOHQþQtKRSiVXMDNMHXYHGHQRQDREU ( ( (J () I( 2EU6WXSQLFHHQHUJLHSURVWDWLVWLFNpYêSRþW\ 3RORYRGLþH 3UR REODVW WHSORW NWHUi QiV ]DMtPi P$åHPH SHGSRNOiGDW åH YH YRGLYRVWQtP SiVX SRORYRGLþH SODWt ( ( !! N 7 D )HUPLKR'LUDFRYD UR]GORYDFtIXQNFHVH]MHGQRGXãtGRWYDUX (−( I H = H[S − N % 7) 8GiYiSUDYGSRGREQRVWREVD]HQtVWDY$YRGLYRVWQtKRSiVXYSLEOtåHQt SODWQpPSURI (QHUJLHHOHNWURQXYHYRGLYRVWQtPSiVX ( N = ( J + KPNH NGHP MHHIHNWLYQtKPRWQRVWHOHNWURQX 3RGOHY]WDKXRGYR]HQpKRSURYRGLYRVWQtHOHNWURQ\YNRYXMHKXVWRWDVWDY$V HQHUJLt(URYQD P H (() 'H = π K (( − ( J ) .RQFHQWUDFHHOHNWURQ$YHYRGLYRVWQtPSiVXMH () ∞ P H N % 7 Q = ∫ ' H (()I H (()G( = P K H (J ) % H ∞ ∫ (( − ( J ) H − N (7 % (J 3RLQWHJUDFL G( ( −( J ) PN 7 Q = HπK% H N E7 =QiPHOL( MHSUREOpPY\HãHQ 2EGREQ O]H VWDQRYLW URYQRYiåQRX NRQFHQWUDFL GU 5R]GORYDFt IXQNFHI I K KROH GtUDSURWRåHGtUD]QDPHQiQHREVD]HQêHOHNWURQRYê VWDY ) K H () N%7 I K ≈ H SUR( (!!N 7 +XVWRWDGURYêFKVWDY$ P K ' K (() = π K (−() DNRQFHQWUDFHGU ( P K N % 7 − N %)7 S = ∫ 'K(()I K (()G( = πK H −∞ 6RXþLQNRQFHQWUDFt (J − N 7 QS = π%K (P H P K ) H N % 7 3L RGYR]RYiQt MVPH QLNGH QHSHGSRNOiGDOL åH MGH R YODVWQt SRORYRGLþ 9êVOHGHNSODWtVWHMQL]DStWRPQRVWLStPVL-HGLQêSHGSRNODGNWHUêMVPH ) % =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN XþLQLOLE\OåHY]GiOHQRVWL)HUPLKRKODGLQ\( RGKUDQRERXSiV$E\PO\ EêWYHONpYHVURYQiQtVN 7 3URWRåH VRXþLQ NRQFHQWUDFt HOHNWURQ$ D GU MH SL GDQp WHSORW NRQVWDQWQt D QH]iYLVt QD KXVWRW StPVt PXVt ]DYHGHQt YKRGQêFK StPVt ]Y\ãXMtFtFKQ]S$VRELW]PHQãHQtS7RPiSUDNWLFNêYê]QDPYH]QHþLãWQpP NU\VWDOX P$åHPH VQtåLW D WR QNG\ YHOPL SRGVWDWQ FHONRYRX NRQFHQWUDFL QRVLWHO$QSt]HQêPSLGiYiQtPYKRGQpStPVL 9H YODVWQtP SRORYRGLþL MH SRþHW HOHNWURQ$ URYHQ SRþWX GU QHER" WHSHOQi H[FLWDFH HOHNWURQX ]DQHFKi YH YDOHQþQtP SiVX GtUX 2]QDþtPHOL LQGH[HP L LQWULQVLF YHOLþLQ\ FKDUDNWHULVWLFNp SUR YODVWQt SRORYRGLþ GRVWDQHPH (J − N %7 Q L = S L = πK (P H P K ) H N % 7 3RURYQiQtPY]WDK$SURQDSP$åHPHXUþLWKRGQRWX( ) % H DSURSRORKX)HUPLKRKODGLQ\ P ( ) = ( J + N % 7 OQ PKH (J -HOLP P SDN( ) = D)HUPLKRKODGLQDOHåtYHVWHGX]DNi]DQpKRSiVX .URP NRQFHQWUDFH QRVLWHO$ MH G$OHåLWp ]QiW WDNp MHMLFK SRK\EOLYRVW -HGHILQRYiQDMDNRGULIWRYiU\FKORVWYMHGQRWNRYpPHOHNWULFNpPSROL µ = (Y (OHNWULFNiYRGLYRVWMHGiQDVRXþWHPHOHNWURQRYêFKDGURYêFKStVSYN$ σ = QHµ H + SHµ K 9NDSLWROHRYROQêFKHOHNWURQHFKYNRYXMVPHRGYRGLOLSURGULIWRYRX T( Hτ Hτ U\FKORVWY]WDKY = P MHWHG\µ H = PHH µ K = PKK 3RK\EOLYRVW ]iYLVt QD WHSORW H[SRQHQFLiOQ SLþHPå PRFQLWHO Pi QHYHONRXKRGQRWX7HSORWQt]iYLVORVWYODVWQtYRGLYRVWLMHRYOLYQQD]HMPpQD (J H[SRQHQFLiOQt]iYLVORVWtNRQFHQWUDFHQRVLWHO$H[S− N % 7 ([SHULPHQWiOQ ]MLãWQp KRGQRW\ SRK\EOLYRVWL SL SRNRMRYp WHSORW MVRXYWDEXOFH 7DE3RK\EOLYRVWQRVLWHO$SLSRNRMRYpWHSORW>FP 9 V @ .U\VWDO (OHNWURQ\ 'tU\ .U\VWDO (OHNWURQ\ 'tU\ ( ) N %7 H P = ( PKH ) H N % 7 ) (J K GLDPDQW *D6E *H 3E6H 6L ,Q6E 3E6 3E7H 3RORYRGLþH ,Q$V $J&O *D$V 6L& ,Q3 .%U. 3URVURYQiQtYNRYRYpPtåLMHµ H = FP 9 − V − .U\VWDO\ V ~]NêP ]DNi]DQêP SiVHP Y\ND]XMt YHONRX HOHNWURQRYRX SRK\EOLYRVW PDOi ãtND ]DNi]DQpKR SiVX Pi ]D QiVOHGHN PDORX HIHNWLYQt KPRWQRVW 3URSHGVWDYXSL.MHQ FP Y*HD FP Y6L 0UQêHOHNWULFNêRGSRUSUR*HMH ΩFP DSUR6L ΩFP L 3tPVRYiYRGLYRVW 1NWHUp StPVL D SRUXFK\ GUDVWLFN\ RYOLYXMt HOHNWULFNp YODVWQRVWL SRORYRGLþH3tGDYHNEyUXGRNHPtNXYSRPUXMHGHQDWRP%QD DWRP$ 6L ]Yêãt PUQRX YRGLYRVW NHPtNX SL SRNRMRYp WHSORW NUiW 3LGiYiQt StPVtGRSRORYRGLþHVHQD]êYiOHJRYiQtQHERGRSRYiQt 8YDåPH YOLY StPVt QD YODVWQRVWL *H D 6L 7\WR SUYN\ NU\VWDOL]XMt Y GLDPDQWRYp VWUXNWXH .DåGê DWRP Pi þW\L NRYDOHQWQt YD]E\ V QHMEOLåãtPL VRXVHGQtPLDWRP\FRåRGSRYtGiFKHPLFNpYDOHQFLþW\L -HOL GR PtåN\ PtVWR QRUPiOQtKR DWRPX YHVWDYQ SWLPRFQê DWRP StPVL QDS IRVIRU DUVHQ DQWLPRQ ]DEXGXMH VH GR PtåN\ WDN DE\ ML FR QHMPpQ QDUXãLO 9\WYiHMt VH SLWRP NRYDOHQWQt YD]E\ V QHMEOLåãtPL VRXVHG\DMHGHQYDOHQþQtHOHNWURQ]$VWDQHQDYtF 'RQRURYpVWDY\YHVWUXNWXHMVRXVFKHPDWLFN\]Qi]RUQQ\QDREU 6L 6L 6L 6L QDGE\WHþQê ]iSRUQê 6L $V 6L QDGE\WHþQê NODGQê ( (J 6L Yi]DQiGRQRURYi QiERM QiERM 6L 6L KODGLQD 6L 2EU1iERMH VRXYLVHMtFt V StPVRYêP DWRPHP DUVHQX Y NHPtNX $UVHQ Pi YDOHQþQtFK HOHNWURQ$ NHPtN HOHNWURQ\ DU]HQX XWYRt NRYDOHQWQtYD]E\VNHPtN\SiWêMHNGLVSR]LFLSURYHGHQtSURXGX 3tPVRYê DWRP XYROQLO MHGHQ YDOHQþQt HOHNWURQ D QHVH WHG\ NODGQê QiERM GRQRU 3RPRFt %RKURY\ WHRULH YRGtNRYpKR DWRPX O]H SRPUQ VQDGQRRGKDGQRXWHQHUJLLGRQRURYpStPVt =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN 9D]HEQiHQHUJLHHOHNWURQXYDWRPiUQtPYRGtNXMH H P = H9 (πε K) 9SRORYRGLþL]DPQtPHε ]Dε DP]DHIHNWLYQtKPRWQRVWHOHNWURQXPH H P H (G = (πεK) +RGQRW\LRQL]DþQtFKHQHUJLt(G SWLPRFQêFKGRQRURYêFKStPVtY*H D6LYPH9MVRXYWDE 7DE,RQL]DþQtHQHUJLHGRQRURYêFKStPVtYPH9 StPV 3 $V 6E SRORYRGLþ6L *H $NFHSWRURYp VWDY\ 7URMPRFQp StPVL MDNR MH % $O *D ,Q VH QD]êYDMt DNFHSWRU\REU&K\EMtFtHOHNWURQ]tVNDMt]YDOHQþQtKRSiVXD]DQHFKDMt WDNSRVREGtUX9\WYRHQtGtU\Y\åDGXMHGRGiQtLRQL]DþQtHQHUJLH 6L 6L 6L ( 6L 6L QDGE\WHþQê % NODGQê QDGE\WHþQê ]iSRUQê (J 6L Yi]DQiDNFHSWRURYi QiERM QiERM KODGLQD 6L 6L 6L 6L 2EU1iERMHVRXYLVHMtFtVStPVRYêPDWRPHPERUHPYNHPtNX%RUPi YDOHQþQt HOHNWURQ\ D P$åH VYp WHWUDHGULFNp YD]E\ GRSOQLW MHQ RGHEUiQtPHOHNWURQX]YD]E\6L6L7R]DQHFKiYiYHYDOHQþQtPSiVX NHPtNXGtUX 7DE,RQL]DþQtHQHUJLHDNFHSWRURYêFKStPVtYPH9 StPV % $O *D ,Q SRORYRGLþ6L *H 8YHGHQpWDEXON\XND]XMtåHLRQL]DþQtHQHUJLHGRQRU$DDNFHSWRU$Y6L D *H MH SL SRNRMRYp WHSORW VURYQDWHOQi V KRGQRWRX N%7 PH9 WDNåH 3RORYRGLþH WHSHOQi LRQL]DFH GRQRU$ QHER DNFHSWRU$ EXGH SL SRNRMRYp WHSORW KUiW SUR HOHNWULFNRXYRGLYRVW*HD6LG$OHåLWRX~ORKX 3HYDåXMHOL NRQFHQWUDFH GRQRU$ EXGH YRGLYRVW RYOiGiQD HOHNWURQ\ D S$MGHRPDWHULiOW\SXQ 3HYOiGDMtOLDNFHSWRU\YRGLYRVWEXGHXUþHQDGtUDPLPDWHULiOW\SXS =QDPpQNR+DOORYDQDSWtMHMHGQRGXFKêPWHVWHPNH]MLãWQtW\SXYRGLYRVWL 7HSHOQiLRQL]DFHGRQRU$DDNFHSWRU$ 3L GRVWDWHþQ Qt]Np WHSORW NG\ SODWt N%7 (G MH NRQFHQWUDFH YRGLYRVWQtFKHOHNWURQ$SRFKi]HMtFtFK]LRQL]RYDQêFKGRQRU$ Q ≈ (Q 1 G ) H ( − N G7 % PN 7 NGH Q = HπK% MHHIHNWLYQtNRQFHQWUDFHHOHNWURQ$YHYRGLYRVWQtPSiVX D1G MHNRQFHQWUDFHGRQRU$ 3RGREQSURDNFHSWRURYpVWDY\ ( − $ S ≈ (Q 1 $ ) H N % 7 NGHQR MHHIHNWLYQtNRQFHQWUDFHGUYHYDOHQþQtPSiVX P K N % 7 D1 MHNRQFHQWUDFHDNFHSWRURYêFKDWRP$ $ Q = πK 7HUPRHOHNWULFNpMHY\YSRORYRGLþtFK 8YDåXMPH SRORYRGLþ XGUåRYDQê QD NRQVWDQWQt WHSORW NWHUêP SL YORåHQpP HOHNWULFNpP SROL SURFKi]t SURXG R KXVWRW MT -HOL SURXG QHVHQ SRX]HHOHNWURQ\MHPRåQpKXVWRWXWRNXQiERMHY\MiGLWY]RUFHP M T = −QHµ H ( WRNSURWLVPUXSROH 6WHGQtHQHUJLHSHQiãHQiHOHNWURQHPMHURYQDY]WDåHQRN)HUPLKRKODGLQ (( F − ( ) ) + N % 7 NGH (F MHHQHUJLHHOHNWURQXQDKUDQYRGLYRVWQtKRSiVX (QHUJLLPtPHRG)HUPLKRKODGLQ\QHER"UR]GtOQpYRGLþHMHåMVRXYH Y]iMHPQpP NRQWDNWX PDMt VWHMQRX )HUPLKR KODGLQX 1DãH YêVOHGN\ SRXåLMHPHNHãHQtSUREOpPXNRQWDNWX 7RNHQHUJLHVSRMHQêVWRNHPQiERMHMHURYHQ M Y = Q ( F − ( ) + N % 7 (−µ H )( 3HOWLHU$YNRHILFLHQW Π MHGHILQRYiQY]WDKHP M Y = ΠM T 0iYê]QDPHQHUJLH SHQiãHQpMHGQRWNRYêPQiERMHP 3URHOHNWURQMH Π H = − − ( F −( ) + N % 7 H =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN ]iSRUQêQHER"WRNHQHUJLHPiRSDþQêVPUQHåWRNQiERMH 3RGREQSURGtU\M T = SH µ H ( M K = S − ( ) − ( 9 + N % 7 µ K ( ( MH HQHUJLH QD KUDQ YDOHQþQtKR SiVX 3HOWLHU$Y NRHILFLHQW SUR GtU\ MH NODGQê $EVROXWQt WHUPRHOHNWULFNi VtOD4 MH GHILQRYiQD SRPRFt HOHNWULFNpKR SROH Y RWHYHQpPREYRGXY\YRODQpKRWHSORWQtPJUDGLHQWHP ( 4JUDG7 3HOWLHU$YNRHILFLHQWVRXYLVtVWHUPRHOHNWULFNRXVLORXSRGOHY]RUFH Π = 4 7 .HOYLQ$YY]WDK 0HQt ]QDPpQND QDSWt QD Y]RUNX MHKRå MHGHQ NRQHF MH ]DKtYiQ MH SLEOLåQiDSRKRWRYiPHWRGDNH]MLãWQt]GDY]RUHNMHW\SXSQHERQREU Y Π W\SS YODVWQt SRORYRGLþ W\SQ 2EU 3HWWLHU$Y NRHILFLHQW NHPtNX W\SX S D Q MDNR IXQNFH WHSORW\ 3L WHSORWiFKQDG.VHY]RUN\VWiYDMtYODVWQtPLSRORYRGLþL 7. 3RORNRY\ 9SRORNRYHFKPiKUDQDYRGLYRVWQtKRSiVXMHQRPiORQLåãtHQHUJLLQHå KUDQDYDOHQþQtKRSiVX0DOêSHNU\YYRGLYRVWQtKRDYDOHQþQtKRSiVXPi]D QiVOHGHN PDORX NRQFHQWUDFL GU YH YDOHQþQtP SiVX D HOHNWURQ$ YH YRGLYRVWQtP SiVX = SRORNRY$ WL DUVHQ DQWLPRQ ELVPXW MVRX ] SiWp VNXSLQ\SHULRGLFNpWDEXON\ 7DE 3RORNRY Q FP Q FP DUVHQ DQWLPRQ ELVPXW JUDILW H K 3RORYRGLþH 3RORNRY\ VWHMQ MDNR SRORYRGLþH PRKRX EêW OHJRYiQ\ YKRGQêPL StPVHPLDE\VH]PQLO\Y]iMHPQpSRPU\NRQFHQWUDFtGUDHOHNWURQ$ $EVROXWQt NRQFHQWUDFH QRVLWHO$ O]H URYQå PQLW S$VREHQtP WODNX QHER" V WODNHP VH PQt SHNU\Wt SiV$ 9\XåtYi VH X SRORYRGLþRYêFK WHQ]RPHWU$FLWOLYRVWYWãtQHåRGSRURYêFKWHQ]RPHWU$ $PRUIQtSRORYRGLþH $PRUIQt SRORYRGLþH PRKRX EêW SLSUDYHQ\ MDNR WHQNp YUVWY\ QDSDRYiQtP QHER QDSUDãRYiQtP QHER Y QNWHUêFK StSDGHFK URYQå SHFKOD]HQtP WDYHQLQ\ NG\ PRKRX Y]QLNQRXW VNOD &R VH VWDQH V SiVRYêP PRGHOHPSHYQpOiWN\QHQtOLYQtSUDYLGHOQpNU\VWDOLFNpXVSRiGiQt" 7HEDåH HOHNWURQRYp VWDY\ QHPRKRX EêW SRSLVRYiQ\ GREH GHILQRYDQêPL KRGQRWDPL N GRYROHQp D ]DNi]DQp SiV\ SHVWR H[LVWXMt D SU$EKKXVWRW\VWDY$MHþDVWRXUþRYiQWpPYêOXþQORNiOQtPXVSRiGiQtP YD]HE 9 DPRUIQtP SRORYRGLþL PRKRX YpVW SURXG MDN HOHNWURQ\ WDN L GtU\ DOH QRVLWHOp SRGOpKDMt LQWHQ]LYQtPX UR]SW\OX Y QHXVSRiGDQp VWUXNWXH 6WHGQt YROQi GUiKD MH iGRY VWHMQ YHONi MDNR SU$PUQi PH]LDWRPRYi Y]GiOHQRVW7\WROiWN\VHSURMHYXMtWpPMDNRYODVWQtSRORYRGLþHV)HUPLKR KODGLQRXXPtVWQRXYHVWHGX]DNi]DQpKRSiVX 1HMG$OHåLWMãtP PDWHULiOHP V DPRUIQt VWUXNWXURX VH VWDO Y SRVOHGQtFK OHWHFKVHOHQSRXåtYDQêY[HURJUDILFNêFKNRStURYDFtFKVWURMtFK $PRUIQtSRORYRGLþRYpPDWHULiO\H[LVWXMtMDNMHGQRDWRPRYp6HWDNL MDNRELQiUQtWHUFLiUQtLY\ããtVOLWLQ\REVDKXMtFt6LD*H]HþWYUWpVNXSLQ\3 $V 6E %L ] SiWp VNXSLQ\ D FKDONRJHQ\ 2 6 6H 7H ] ãHVWp VNXSLQ\ SHULRGLFNpWDEXON\QDS&G$V *H ,GHiOQt NRYDOHQWQt VNOR MH GHILQRYiQR MDNR FKDRWLFNi Vt" EH] XVSRiGiQt QD YHONRX Y]GiOHQRVW DOH V LGHiOQtP XVSRiGiQtP QD NUiWNRX Y]GiOHQRVW 9ãHFKQ\ DWRP\ PXVHMt PtW YãHFKQ\ YD]E\ QDV\FHQp 7DNRYRX SRGPtQNXVSOXMtDPRUIQtYUVWY\6LD*HSLSUDYHQpQDSDRYiQtPYHYDNXX QDVWXGHQRXSRGORåNX âtND]DNi]DQpKRSiVXMHRQFRQLåãtQHåXNU\VWDOLFNpKRXVSRiGiQt =iYLVORVW HOHNWURQRYp GULIWRYp SRK\EOLYRVWL D PUQp YRGLYRVWL QD WHSORW MH QDREU$NWLYDþQtHQHUJLHY\]QDþHQpXJUDI$RGSRYtGDMt]iYLVORVWL ε µ = $ H[S − N %X7 Ω P UQiYRGLYRVWP GULIWRYiSRK\EOLYRVWP9V =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN 7. 7. 2EU7HSORWQt ]iYLVORVW GULIWRYp SRK\EOLYRVWL D PUQp YRGLYRVWL SUR DPRUIQtNHPtN +RGQRW\ PUQp YRGLYRVWL L SRK\EOLYRVWL MVRX X DPRUIQtFK 6L NUiW PHQãtQHåYNU\VWDOLFNpP6L ýiVW SiVRYp VWUXNWXU\ YDOHQþQt D YRGLYRVWQt SiV\ DPRUIQtKR 6L MVRX SRURYQiQ\VNU\VWDOLFNêPQDREU ( NU\VWDO DPRUIQt ( (F (Y '( '( 2EU+XVWRWDVWDY$MDNRIXQNFHHQHUJLHYEOt]NRVWL]DNi]DQpKRSiVXSUR NU\VWDOLFNê D DPRUIQt SRORYRGLþ ( GQR YRGLYRVWQtKR SiVX ( YUFKROYDOHQþQtKRSiVX F Y ,SHVWRåHYDPRUIQtOiWFHH[LVWXMtY]DNi]DQpPSiVXHQHUJLtREVD]HQp HQHUJHWLFNp VWDY\ FKDUDNWHU ]DNi]DQpKR SiVX VH ]DFKRYiYi -H WR SR]RURYDWHOQpQDRSWLFNpDEVRUSFLNWHUiSUXGFHY]U$VWiMHOLHQHUJLHIRWRQ$ Kω!( ( (OHNWURQ\VHQHUJLt( (( VHFKRYDMtORNDOL]RYDQDPDMt WHG\QLåãtSRK\EOLYRVWQHåHOHNWURQ\YHYRGLYRVWQtPSiVX 6FKRSQRVWDPRUIQtKRSRORYRGLþHYpVWSURXG]iYLVtSHGHYãtPQDSRþWX HOHNWURQ$VHQHUJLtY\ããtQHå( DGUVHQHUJLtPHQãtQHå( &KDONRJHQQt VNOD WM VOLWLQ\ )H $V 6L *H VH MHYt MDNR VOLEQp SUR NRQVWUXNFL VStQDþ$ D SDP"RYêFK HOHPHQW$ 3L HOHNWULFNpP SROL PHQãtP F Y Y F F Y 3RORYRGLþH QHå 9P MVRX QHYRGLYp V PUQêP RGSRUHP ΩP MHOL SROH Y\ããt PUQêRGSRUVHVQtåtiGRYQDΩP 3HFKRG\SQ &RVHVWDQHGiPHOLGRY]iMHPQpKRNRQWDNWXGYDNXV\SRORYRGLþH MHGHQ]QLFKW\SXSDGUXKêW\SXQ"1DVWUDQSH[LVWXMtYROQpGtU\DVWHMQp NRQFHQWUDFH]iSRUQLRQL]RYDQêFKDNFHSWRURYêFKStPVRYêFKDWRP$NWHUp ]DMLã"XMtQiERMRYRXQHXWUDOLWX1DVWUDQQH[LVWXMtYROQpHOHNWURQ\DVWHMQê SRþHWNODGQLRQL]RYDQêFKGRQRURYêFKStPVRYêFKDWRP$ 0DMRULWQtPLYWãLQRYêPLQRVLWHOLMVRXWHG\QDVWUDQSSHFKRGXGtU\ DQDVWUDQQHOHNWURQ\ W\SQ W\S S NRQFHQWUDFHG U NRQFHQWUDFH $ HOHNWURQ HOHNWURVWDWLFNêSRWHQFLiO 2EU 3HFKRG SQ MH Y\WYRHQ ] PRQRNU\VWDOX Y QPå MVRX Y\WYRHQ\ GYREODVWL9MHGQpMVRXSLPtVHQ\DNFHSWRURYpDWRP\REODVWW\SX S D PDMRULWQtPL QRVLWHOL MVRX GtU\ 9H GUXKp MVRX GRQRURYp DWRP\ REODVW W\SX Q D PDMRULWQtPL QRVLWHOL MVRX HOHNWURQ\ 3RGpO SQ SHFKRGX VH PQt NRQFHQWUDFH QRVLWHO$ QiERMH L HOHNWURVWDWLFNê SRWHQFLiO 9WHSHOQpURYQRYi]HVPDMRULWQtPLQRVLWHOLMVRXLQRVLWHOpPLQRULWQtFK W\S$ StWRPQt Y PDOêFK NRQFHQWUDFtFK 'tU\ VRXVWHGQp QD VWUDQ S SHFKRGX E\ FKWO\ GLIXQGRYDW D URYQRPUQ ]DSOQLW NU\VWDO (OHNWURQ\ VH VQDåtGLIXQGRYDW]REODVWLQ =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN -DNPLOHVHGLIX]tXVNXWHþQtSHFKRGPDOpKRPQRåVWYtQiERMH]$VWDQH QDVWUDQSSHE\WHN]iSRUQLRQL]RYDQêFKDNFHSWRURYêFKDWRP$DQDVWUDQ QSHE\WHNNODGQêFKLRQL]RYDQêFKDWRP$GRQRU$ 7\WRGYQDELWpREODVWLY\YROiYDMtHOHNWULFNpSROHVPXMtFtRGW\SXQ GR S NWHUp ]DEUDXMH GDOãt GLIX]L D XGUåXMH VWDY NG\ YROQp QiERMH RERX GUXK$]$VWiYDMtRGVHEHRGGOHQ\ 9 G$VOHGNX H[LVWHQFH WpWR GYRMYUVWY\ Y\ND]XMH SU$EK HOHNWURVWDWLFNpKRSRWHQFLiOXSRGpOY]RUNXVNRNYREODVWLSHFKRGX&HONRYê WRN HOHNWURQ$ D GU MH QXORYê .RQFHQWUDþQt JUDGLHQW SHVQ UXãt JUDGLHQW HOHNWURVWDWLFNpKRSRWHQFLiOXGYRMYUVWY\ 8VPURYiQt 3HFKRG S Q P$åH S$VRELW MDNR XVPURYDþ 3RNXG N SHFKRGX SLSRMtPH QDSWt Y MHGQRP VPUX SUREtKi YHONê SURXG SRNXG Pi QDSWt VPU RSDþQê SRWHþH SURXG MHQ YHOPL PDOê -HOL QD SHFKRG SLSRMHQR VWtGDYpQDSWtEXGHSURXGSURFKi]HWSHYiåQMHQYMHGQRPVPUXSURXG EXGHSHFKRGHPXVPURYiQREU , P$FP 8 9 2EU&KDUDNWHULVWLNDSHFKRGXSQ 3L]iYUQpPSHGSWtMH]iSRUQpQDSWtSLORåHQRQDREODVWSDNODGQp QDSWt QD REODVW Q þtPå VH ]Yêãt UR]GtO SRWHQFLiO$ PH]L REPD þiVWPL SHFKRGX 1\Qt QHPRKRX SUDNWLFN\ åiGQp HOHNWURQ\ Y\VWRXSLW ] REODVWL Qt]Np SRWHQFLiOQtHQHUJLHQDKRUX3URXGHOHNWURQ$ - H (9 ]DYHUQH ) = - H ()H[S − NH9 %7 %ROW]PDQ$Y IDNWRU XUþXMH SRþHW HOHNWURQ$ MHMLFKå HQHUJLH MH SRVWDþXMtFt N SHNRQiQtEDULHU\ 3RORYRGLþH -HOLQDSHFKRGSLSRMHQRQDSWtYSURSXVWQpPVPUXSURXGY]URVWH SURWRåHSRWHQFLiORYiEDULHUDVHVQtåt - H (9 SURSXVW ) = - H ()H[S NH97 % 3URXGGUSURFKi]HMtFtSHFKRGHPVHFKRYiSRGREQMDNRHOHNURQRYê SURXG SUREtKi RSDþQêP VPUHP Y]KOHGHP N HOHNWURQRYpPX SURXGX (OHNWULFNpSURXG\Y\YRODQpGUDPLLHOHNWURQ\VHVþtWDMt &HONRYêSURSXVWQêSURXG − , = , 6 H[S NH9 %7 , MHVRXþHWW]YJHQHUDþQtFKSURXG$HOHNWURQRYpKRDGURYpKR , 6 = - H () + - K () 9ROWDPSpURYi FKDUDNWHULVWLND SQ SHFKRGX SDN Pi WYDU XYHGHQê QD REU V 6OXQHþQtþOiQN\DIRWRYROWDLFNpGHWHNWRU\ 3RVYL"PHQDSQSHFKRGNQPXåQHQtSLSRMHQRåiGQpYQMãtQDSWt .DåGê DEVRUERYDQê IRWRQ ]S$VREXMH Y]QLN SiUX HOHNWURQGtUD -DNPLOH WLWR QRVLWHOp SHMGRX GLIX]t GR REODVWL SHFKRGX RGGOt VH Y G$VOHGNX H[LVWHQFH YQLWQtKRHOHNWULFNpKRSROHSHFKRGXHOHNWURQVHGRVWDQHGRREODVWLQGtUD GR REODVWL S 2GGOHQt QRVLWHO$ Y\YROiYi Y]QLN QDSWt QD EDULHH 1DSWt MH RULHQWRYiQR Y SURSXVWQpP VPUX SURWRåH HOHNWULFNp SROH QRVLWHO$ JHQHURYDQêFKIRWRQ\MHRSDþQpY$þLYQLWQtPXSROLSHFKRGX 9]QLN QDSWt RULHQWRYDQpKR Y SURSXVWQpP VPUX QD RVYWOHQpP SHFKRGXVHQD]êYiIRWRYROWDLFNê MHY 2VYWOHQêSHFKRG P$åH GR YQMãtKR REYRGXGRGiYDWYêNRQ 9HONRSORãQp NHPtNRYp SQ SHFKRG\ PRKRX EêW SRXåLW\ N SHPQ IRWRQ$VOXQHþQtKR]iHQtQDHOHNWULFNRXHQHUJLL 6FKRWWN\KREDULpUD -HOL SRORYRGLþ Y NRQWDNWX V NRYHP Y]QLNi Y SRORYRGLþL EDULpURYi YUVWYD]QtåMVRXQRVLWHOp]YHONpþiVWLRGþHUSiYDQL7DWREDULpURYiYUVWYDVH WDNpQNG\QD]êYiRFKX]HQRXQHERY\þHUSDQRXYUVWYRX.RQWDNWSRORYRGLþH W\SXQVNRYHPMH]Qi]RUQQQDREU =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN KODGLQDYDNXD GQRYRGLYRVWQtKRSiVX () () () () YDOHQþQt SiV NRY SRORYRGLþ 2EU 8VPUXMtFt EDULpUD PH]L NRYHP D SRORYRGLþHP 9OHYR SHG YSUDYRSRVSRMHQt 3R SHFKRGX HOHNWURQ$ ] YRGLYRVWQtKR SiVX SRORYRGLþH GR NRYX MH )HUPLKR KODGLQD QD RERX VWUDQiFK NRQWDNWX YH VWHMQp YêãL 9 WpWR REODVWL WpP ]EDYHQp YROQêFK HOHNWURQ$ ]$VWiYDMt MHQ NODGQ QDELWp GRQRURYp LRQW\ *XQQRY\RVFLOiWRU\ 9URFHREMHYLO-%*XQQåHYGHVWLþFH*D$VQDQtåMHYORåHQR VWHMQRVPUQpQDSWtQNROLNDWLVtFYROW$Y]QLNDMtPLNURYOQQpRVFLODFH 3RPRFt *XQQRYêFK RVFLOiWRU$ E\O\ GRVDåHQ\ IUHNYHQFH Då *+] 9]QLN RVFLODFt VRXYLVt VH ]iSRUQRX KRGQRWRX GLIHUHQFLiOQt SRK\EOLYRVWL YRGLYRVWQtFK HOHNWURQ$ Y *D$V -DN MH ]Qi]RUQQR QD REU GULIWRYi U\FKORVWURVWHVU$VWHPSROHGRNXGHOHNWURQ\QHGRViKQRXU\FKORVWL PV YSROL9FP 9HY\ããtFKHOHNWULFNêFKSROtFKU\FKORVWHOHNWURQ$NOHVi YG FPV ( N9FP 2EU 'ULIWRYi U\FKORVW HOHNWURQ$ Y *D$V SL SRNRMRYp WHSORW MDNR IXQNFHLQWHQ]LW\SLORåHQpKRHOHNWULFNpKRSROH 3RORYRGLþH 9 WpWR REODVWL MH GLIHUHQFLiOQt SRK\EOLYRVW ]iSRUQi 'LIHUHQFLiOQt HOHNWULFNi YRGLYRVW P$åH EêW URYQå ]iSRUQi 3URXGRYi KXVWRWD MH GiQD VRXþWHPStVSYN$GYRXYRGLYRVWQtFKSiV$QLåãtKRNGHHOHNWURQ\PDMtQLåãt KPRWQRVWDY\ããtKRVYWãtHOHNWURQRYRXKPRWQRVWt M = σ(()( = (Q / Hµ / + Q + Hµ + )( LQGH[/OLJKWMHStVSYHNOHKNêFKHOHNWURQ$ +KHDY\MHStVSYHNWåNêFKHOHNWURQ$ 'LIHUHQFLiOQtYRGLYRVWMHURYQD GM GQ + GQ / G( = (Q / Hµ / + Q + Hµ + ) + G( µ / + G( H( -HOLSRþHWQRVLWHO$NRQVWDQWQt]DGDQpWHSORW\ GQ GQ Q / + Q + = NRQVW =〉 G(/ + G(+ = GM GQ + G( = (Q / Hµ / + Q + Hµ + ) − (µ / − µ + ) G( H( 9êUD]P$åHEêW]iSRUQêMHVWOLåHGUXKêþOHQQDSUDYpVWUDQSHYDåXMH QDGþOHQHPSUYQtP7DWRSRGPtQNDMHVLOQSRGSRURYiQDVLOQêPHOHNWULFNêP SROHP(0$åHQDVWDWMHOLµ / 〉〉µ + QHERP / 〈〈P + 3R]RUXMHVHåHYUHåLPXQHJDWLYQtYRGLYRVWLQHQtKRPRJHQQtUR]ORåHQt HOHNWULFNpKR SROH YH Y]RUNX VWDELOQt 9UVWY\ V QDKURPDGQêP QiERMHP VH SRK\EXMt NU\VWDOHP U\FKORVWt VKRGXMtFt VH V GULIWRYRX U\FKORVWt QRVLWHO$ .PLWRþHW RVFLODFt I = GY NGH GULIWRYi U\FKORVW Y a FPV SL WORXã"FH NU\VWDOXGa FPMHIa +] 6KUQXWt 9ODVWQtSRORYRGLþHVHY\]QDþXMtSiVHP]DNi]DQêFKHQHUJLtYHOHNWURQRYp VWUXNWXHRãtFHNROHPH9SL57 .RQFHQWUDFH QRVLWHO$ MH XUþHQD WHSHOQRX H[FLWDFt ] YDOHQþQtFK GR YRGLYRVWQtFKSiV$Q S aH[S( N 7 $NFHSWRURYpStPVLYHGRXNQDGE\WNXGURSURWLHOHNWURQ$P'RQRURYp StPVL Y\WYiHMt SHE\WHN HOHNWURQ$ 9H YODVWQtP SRORYRGLþL MH SRþHW HOHNWURQ$URYHQSRþWXGU (OHNWURQ\ Y NU\VWDOHFK XPtVWQêFK GR YQMãtKR HOHNWULFNpKR QHER PDJQHWLFNpKR SROH VH SRK\EXMt WDN MDNRE\ MHMLFK KPRWQRVW E\OD URYQD L J L ∂ ( % HIHNWLYQtKPRWQRVWL P ∗ = K (OHNWURQYNU\VWDOXVHFKRYiMDNRYOQDMHMtåYOQRYêYHNWRUMHGiQHãHQtP 6FKU|GLQJHURY\ URYQLFH V SHULRGLFNêPL RNUDMRYêPL SRGPtQNDPL -HKR HQHUJLH ]iYLVt QDN ( (N 3U$EK IXQNFH ]iYLVt QD VWDYX YH NWHUpP VH HOHNWURQQDFKi]t ∂N =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN 3RORYRGLþH
Podobné dokumenty
PDF dokument
MHGQDN YROQp HOHNWURQ\ YH YRGLYRVWQtP SiVX NDåGê HOHNWURQ SHQiãt QiERM T
MHGQDNSRK\EOLYpGtU\YHYDOHQþQtPSiVXQiERMT
Gs3+ 26 A+++ / Gs3+30 A+++ Shiver 26 A+++ - Kuchyn
.G\åYR]LGORQHQtY\EDYHQRSURYR]Qt]iVXYNRX 3ĜHG XVNODGQČQtP MtGHO Y ]DĜt]HQt QHFKWH WHSOi MtGOD QHMGĜtYH
SĜLSRMWHQDSiMHFtNDEHOSURQDSČWt9 GR]i- ochladit.
8PtVWČWHFKO...
Kapitoly 1-4
XUþXMt SURVWRURYRX PtåNX NU\VWDOX 1iVOHGNHP VWUXNWXUQt RSDNRYDWHOQRVWL
PiNDåGêPtåNRYêERGVWHMQêSRþHWVRXVHGQtFKERG$WMVWHMQpNRRUGLQDþQt
þtVOR9WãLQDSUYN$DMHGQRGXFKêFKFKHPLFN...
Elektromagnetické pole a zdravotní rizika V
DWpPČĜRåHEUDþHQVHSDN7HVODYČQRYDOSRURFHMLQêPRERUĤPY\QDOp]iQtWHUPRG\QDPLFNêPRWRUSRþtWDGOD
HOHNWULFNpKR SURXGX PČĜLþH U\FKORVWL D GDOãt NGH PČO YHOLNp ~VSČFK\ D YãH...