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283. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 8WLOL]DQGR OD LQWHJUDO GH 'XKDPHO [ ] [ ] [ ]       Ω−Ω      Ω− = −−Ω+−+Ω= −−+=⋅ − Ω=−= ∫∫ ∫∫ WVHQWVHQN ) W[ GWFRVGWFRV P ) W[ %$FRV%$FRV%VHQ$VHQXWLOL]DQGR G P WVHQ VHQ)GWKIW[ Q Q Q W Q W Q Q W Q Q W ωω ω ηηωηηηωη ω η ω ηω ηηηη

291. 5HVXPHQ UHSUHVHQWDFLyQ VLVWHPDV HQ GRPLQLR WLHPSR GRPLQLR IUHFXHQFLDV 'RPLQLR 7LHPSR 'RPLQLR )UHFXHQFLDV ([FLWDFLyQ LPSXOVLYD R GH LPSDFWR

292. ¢4Xp HV OD IXHU]D GH LPSDFWR 1R H[LVWH XQD GHILQLFLyQ FODUD JHQHUDOPHQWH VH OODPD D XQD ³WUDQVIHUHQFLD UiSLGD GH HQHUJtD´ R ³DSOLFDFLyQ UiSLGD GH XQD IXHU]D´ ³5iSLGD´ HV JHQHUDOPHQWH UHIHULGD D VX SHUtRGR QDWXUDO GH YLEUDU (VSHFWURV UHVSXHVWD 3DUD ILQHV GH GLVHxR HO LQJHQLHUR HVWi JHQHUDOPHQWH LQWHUHVDGR HQ HO GHVSOD]DPLHQWR Pi[LPR (O JUiILFR GH ORV HVSHFWURV GH UHVSXHVWD R GH FKRTXH HV XQ JUiILFR ( ) HVW;;N); Pi[Pi[ = GHELGR D XQD IXHU]D LPSXOVLYD GH XQ FLHUWR YDORU Pi[LPR ) GXUDFLyQ 7 YHUVXV Q77 FRQ ξ FRPR SDUiPHWUR

293. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (MHPSOR 'HWHUPLQDU ;Pi[ Pi[LPR GHVSOD]DPLHQWR GHO VLVWHPD LGHDO QR DPRUWLJXDGR

294. FXDQGR DFW~D XQ SXOVR VHQRLGDO GH DPSOLWXG ) GXUDFLyQ 7 )DVH , 0LHQWUDV DFW~D OD IXHU]D

296. 7W WVHQWVHQ N ) W[ WVHQ) 7 W VHQ)WI Q Q Q ≤≤            − − = == ω ω ω ω ω ω ω π )DVH ,, 9LEUDFLRQHV OLEUHV FRQ FRQGLFLRQHV LQLFLDOHV [7

297. 7[

301. 7FRV N ) 7[ 7VHQ N ) 7[ 7WWFRV7[WVHQ 7[ W[ Q Q Q Q Q QQ Q ω ω ω ω ωω ω ω ω ωω ω +            − −=            − −= ≥+=

302. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (VSHFWURV UHVSXHVWD

303. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (V DSDUHQWH TXH SDUD FRUWDV GXUDFLRQHV GHO LPSXOVR GLJDPRV Q77 OR LPSRUWDQWH HV HO YDORU GHO LPSXOVR OD IRUPD GHO LPSXOVR WLHQH PHQRU HIHFWR (VWR WUDH FRPR FRQVHFXHQFLD TXH HQ OD SUiFWLFD LPSXOVRV FRQ Q77 VHDQ FRQVLGHUDGRV FRPR YHORFLGDG LQLFLDO GDGR DO VLVWHPD ∫ ∫==

304. 7 7 L GWWI P GW P W, Y SDUD HO FDVR GH XQD IXHU]D LPSXOVR ∫=

305. 7 EL GWWDY SDUD XQ PRYLPLHQWR GH OD EDVH DEW

306. (O IDFWRU GH LPSDFWR UDUDPHQWH HV PDRU TXH 7HUUHPRWR GH (O HQWUR DOLIRUQLD GH 0DR GH ″6KRFN 9LEUDWLRQ +DQGERRN″ ULO 0 +DUULV ° HG SDJ

307. $

308. $FHOHUDFLyQ GHO WHUUHQR %

309. 9DULDFLyQ GH OD YHORFLGDG GHO WHUUHQR REWHQLGD SRU LQWHJUDFLyQ GH $

310. 9DULDFLyQ GHO GHVSOD]DPLHQWR GHO WHUUHQR REWHQLGR SRU LQWHJUDFLyQ GH %

311. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (VSHFWUR GH UHVSXHVWD SDUD VLVWHPDV HOiVWLFRV VXMHWRV DO WHUUHPRWR GH (O HQWUR SDUD YDULRV YDORUHV GH IDFWRU GH DPRUWLJXDPLHQWR ξ 9DORUHV Pi[LPRV GH ODV JUiILFDV GP SXOJ YP SXOJV DP J 0RYLPLHQWRV UHODWLYRV GH OD PDVD P ' DPSOLWXG GHO GHVSOD]DPLHQWR UHODWLYR 9 SVHXGR YHORFLGDG PRYLPLHQWR VXSXHVWR DUPyQLFR

312. ωQ' $ SVHXGR DFHOHUDFLyQ ωQ9 ωQ ' 1RWD 8Q VLVWHPD GH ωQ EDMR FRUUHVSRQGH D XQD PDVD JUDQGH UHVRUWH SRFR UtJLGR (Q HVWH FDVR →GP' HV GHFLU Pi[LPD GHIRUPDFLyQ HQ HO UHVRUWH Pi[LPR GHVSOD]DPLHQWR GHO VXHOR HO VXHOR VH PXHYH UHODWLYR D OD PDVD pVWD QR WLHQH WLHPSR SDUD PRYHUVH

313. (Q HO RWUR H[WUHPR SDUD ωQ DOWRV UHVRUWHV UtJLGRV OD PDVD VXHOR VH PXHYHQ GH OD PLVPD PDQHUD WLHQH LJXDO DFHOHUDFLyQ

314. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (MHPSOR 'HWHUPLQH SDUD HO VLVPR GH (O HQWUR D

315. 'HVSOD]DPLHQWR UHODWLYR Pi[LPR E

316. )XHU]D GH FRUWH Pi[LPR = = = ξ SLH OEN OEP +]I Q Q = × == ππ ω D

317. GHO JUiILFR J$ DP $ OJSX' GP ' =×=→≈ =×=→≈ E

318. V SLHOE V SLH OEP$N'IV =××===

319. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV DStWXOR 6LVWHPDV GH 1 JUDGRV GH OLEHUWDG (FXDFLRQHV GHO PRYLPLHQWR /RV VLVWHPDV GH 1 JUDGRV GH OLEHUWDG SXHGHQ HVFULELUVH HQ IRUPD PDWULFLDO [ ]{ } [ ]{ } [ ]{ } { } [ ] [ ] [ ] [ ] { } HQWRGHVSOD]DPL9HFWRU[ H[WHUQDVIXHU]DV9HFWRUI ULJLGH]GH0DWUL]. RYLVLHQWRDPRUWLJXDPGH0DWUL] PDVDVGH0DWUL]0 )[.[[0 FRV =++ (MHPSOR ( ) ( ) ( )WIN[N[[F[F[P WIN[N[[F[FF[P =+−−+ =−+−++       =             − − +             − −+ +            

321. WI WI [ [ NN NN [ [ FF FFF [ [ P P

322. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 9LEUDFLRQHV OLEUHV QR DPRUWLJXDGDV [ ]{ } [ ]{ } =+ [.[0 /DV VROXFLRQHV VRQ GH OD IRUPD ( ) UW LL H;W[ =

323. ( ){ } { } UWUW QQ H;W[H ; ; ; W[ W[ W[ =⇒               =              

325. (FXDFLyQ

326. HQ

327. [ ]{ } =+ ;.0U

328. /D VROXFLyQ QR WULYLDO GH HVWH VLVWHPD GH HFXDFLRQHV VH REWLHQH SDUD ORV YDORUHV GH U TXH VDWLVIDJDQ OD HFXDFLyQ FDUDFWHUtVWLFD [ ] GHW =+ .0U

329. (MHPSOR GHW = +− −+ NPUN NNPU ( )( ) =++ =−++ P N U P N U NNPUNPU MM P N U P NU MM P N U P NU ω ω ±=±=→−= ±=±=→−= UU YDORUHV SURSLRV HLQJHQYDORUHV ωω IUHFXHQFLDV QDWXUDOHV GH YLEUDU

330. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 0RGRV GH YLEUDU R YHFWRUHV SURSLRV 6RQ ORV YHFWRUHV GHWHUPLQDGRV SRU OD HFXDFLyQ

331. TXH FRUUHVSRQGHQ D FDGD YDORU SURSLR (MHPSOR [[[N[ N[N[ P N USDUD =⇒=+− =− −= [[N[N[ N[N[ P N USDUD −=⇒=−− =−− −= 1RUPDOL]DFLyQ GH ORV PRGRV /RV YDORUHV [L REWHQLGRV QR VRQ LQGHSHQGLHQWHV 6L KD [L YDORUHV H[LVWHQ L HFXDFLRQHV LQGHSHQGLHQWHV 3DUD GHWHUPLQDU XQ YDORU KD TXH DJUHJDU XQD QXHYD HFXDFLyQ OR TXH VH OODPD QRUPDOL]DFLyQ +DFHU XQD GH ODV FRPSRQHQWHV LJXDO D M L; $PSOLWXG GH PDVD L FXDQGR YLEUD FRQ HO PRGR M +DFHU VX ORQJLWXG LJXDO D XQR { } { } ( ) { } ( ) { }       − =→=−+→       =→=+→=+ =→=++= ;;; ;;;;; ;;;;;; 7 Q { } { }       − =       =⇒       =       =⇒= ; ; ; ; ; ;[6L

332. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV +DFHU HO SURGXFWR ;7 0; { } =+=             P[P[ [ [ P P [[ ( ) { } { } − −       − =       =→=→=+ P; P; P [[PP[ (FXDFLRQHV GHO PRYLPLHQWR 6L HO VLVWHPD YLEUDUD VyOR FRQ HO PRGR ( )VVUU ωω == HQWRQFHV GH HFXDFLyQ

333. ( ){ } ( ){ }V VVVVV ;WFRV%WVHQ$W[ ωω +=

334. (Q HO FDVR JHQHUDO (O VLVWHPD YLEUD FRQ XQD FRPELQDFLyQ OLQHDO GH ORV PRGRV ( ){ } ( ){ } ( ){ }∑∑ == +== 1 V V VVVV 1 V V ;WFRV%WVHQ$W[W[ ωω /D VROXFLyQ SDUD HO FDVR L VHUi ( ) ( )∑= += 1 V V LVVVVL ;WFRV%WVHQ$W[ ωω

335. SDUD VLVWHPDV DPRUWLJXDGRV ( ) ( )∑= − += 1 V V LGVVGVV W L ;WFRV%WVHQ$HW[ VV ωωωξ

336. (MHPSOR 'HWHUPLQDU [W

337. [W

338. 6L ( ) ( ) ( ) ( ) === = [[[ [[ 6H KDEtD GHWHUPLQDGR { } { }       − =       =       =       = ; ; ; ; ; ; $SOLFDQGR HFXDFLyQ

339. ( ) ( ) ( ) ( ) ( ) ( ) ;WFRV%WVHQ$;WFRV%WVHQ$W[ ;WFRV%WVHQ$;WFRV%WVHQ$W[ ωωωω ωωωω +++= +++=

340. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 8VDQGR ODV FRQGLFLRQHV LQLFLDOHV ( ) ( ) ( ) ( ) ωω ωω $$[ $$[ %%[ %%[[[ −=→= +=→= −=→= +=→= ==→ =→ =→ $$ [% [% ( ) ( ) ( ) ( )WFRVWFRV;W[ WFRVWFRV[W[ ωω ωω −= += ( ){ } ( ) ( ) { } { } ;WFRV[;WFRV[ W[ W[ W[ ωω +=       = ⇒ (O VLVWHPD YLEUD FRQ XQD FRPELQDFLyQ SRQGHUDGD

341. GH VXV PRGRV GH YLEUDU ⇒ (O PRGR TXH PiV SDUWLFLSD HQ OD YLEUDFLyQ VHUi HO TXH HV SUHSRQGHUDQWH HQ OD GHIRUPDFLyQ LQLFLDO 6L OD GHIRUPDFLyQ LQLFLDO HV OD IRUPD GH XQ PRGR HO VLVWHPD YLEUDUi VyOR FRQ HVH PRGR ³FRQGLFLRQHV DSURSLDGDV´

342. 'HIRUPDFLyQ LQLFLDO 0RGR ^; ` [ ^; ` 0RGR ^; ` [ ^; `

343. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 9LEUDFLRQHV IRU]DGDV VLQ DPRUWLJXDPLHQWR 0pWRGR GLUHFWR 3DUD H[FLWDFLRQHV DUPyQLFDV OD UHVSXHVWD HVWDFLRQDULD SXHGH VHU GHWHUPLQDGD XVDQGR HO iOJHEUD FRPSOHMD 3DUD HVWR KD TXH UHHPSOD]DU HQ ODV HFXDFLRQHV GHO PRYLPLHQWR ( ) ( ) ( ) ( ) WM LL WM LL WM LL WM LL H;W[ HM;W[ H;W[ H)WI Ω Ω Ω Ω Ω−= Ω= = = UHVROYHU HO VLVWHPD GH HFXDFLRQHV UHVXOWDQWHV (MHPSOR 'HWHUPLQDU [W

344. [ W

345. HVWDFLRQDULRV 5HHPSOD]DQGR ( ) ( ) WM LL WM LL WM H;W[ H;W[ H)WVHQ) Ω Ω Ω Ω−= = =Ω ( ) ( ) =Ω−+− =−+Ω− ;PN.; )N;NP; ( )           −Ω           −Ω Ω− = ∆ Ω− − =⇒

349. ωω P N P NP PN)PN N) ; WVHQ)W) Ω=

350. P N P N = = ω ω =+− Ω=−+ N[N[[P WVHQ)N[N[[P

351. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV ( )( ) ωω −Ω−Ω − = ∆ − Ω− = P N)N )PN ; ( )( ) ( )( )P N P NP P N P N P NPNPN PNN NPN −Ω−Ω=       +Ω−Ω= −Ω−Ω−= Ω−− −Ω− =∆ ( ) ( ) ( )( ) WVHQ P PN) W[ Ω −Ω−Ω Ω− = ωω 6LVWHPD GH 1 JUDGRV GH OLEHUWDG WLHQH 1 IUHFXHQFLDV QDWXUDOHV XDQGR OD IUHFXHQFLD GH OD H[FLWDFLyQ Ω FRLQFLGH DSUR[LPDGDPHQWH FRQ FXDOTXLHUD GH ODV Lω VH JHQHUDQ JUDQGHV DPSOLWXGHV GH YLEUDFLyQ → 5HVRQDQFLD

352. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 0pWRGR PRGDO /DV FRRUGHQDGDV [ [ HOHJLGDV SDUD GHILQLU HO PRYLPLHQWR GHO VLVWHPD HVWiQ DFRSODGDV HQ HO VHQWLGR HQ TXH DPEDV FRRUGHQDGDV DSDUHFHQ HQ FDGD HFXDFLyQ SRU OR WDQWR VL YDUtD [ YDUtD WDPELpQ [ 6LQ HPEDUJR VLHPSUH HV SRVLEOH HQ XQ VLVWHPD QR DPRUWLJXDGR HQFRQWUDU XQ VLVWHPD GH FRRUGHQDGDV TL VLQ QLQJ~Q WLSR GH DFRSODPLHQWR R GHVDFRSODGDV OODPDGDV FRRUGHQDGDV SULQFLSDOHV 3URSLHGDGHV GH RUWRJRQDOLGDG GH ORV YHFWRUHV SURSLRV RQVLGHUHPRV GRV PRGRV FXDOHVTXLHUD L M 'H HFXDFLyQ

353. [ ]{ } [ ]{ };;. Ω−= LL U−=ω D

354. [ ]{ } [ ]{ }L L L ;0;. ω= E

355. [ ]{ } [ ]{ }M M M ;0;. ω= F

356. { } [ ]{ } { } [ ]{ }L7M L L7M ;0;;.; ω= 3UHPXOWLSOLFDQGR HFD

357. SRU { }7M ; G

358. { } [ ] { } [ ]0;.; 7M L 7M ω= 7UDQVSXHVWD GH HFXDFLyQ E

359. H

360. { } [ ]{ } { } [ ]{ }L7M L L7M ;0;;.; ω= 3RVWPXOSOLFDQGR HFG

361. SRU { }L ; ( ){ } [ ]{ } =−⇒ L7M ML ;0;ωω (FXDFLyQ F

362. H

363. 3DUD L ≠ M VL ML ωω ≠ { } [ ]{ } =L7L ;0; 5HODFLRQHV GH 2UWRJRQDOLGDG

364. { } [ ]{ } =L7M ;.; L M 1 HV GHFLU ORV YHFWRUHV SURSLRV VRQ RUWRJRQDOHV UHVSHFWR D ODV PDWULFHV [ ] [ ].0 =+− =−+ N[N[[P N[N[[P

365. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 3DUD L M { } [ ]{ } { } [ ]{ } LL L7L LL L7L ;.; ;0; γ µ = =

366. =LLµ 0DVD PRGDO FRUUHVSRQGLHQWH DO PRGR L LLγ 5LJLGH] PRGDO FRUUHVSRQGLHQWH DO PRGR L 1RWD LLLL γµ VRQ FRQVWDQWHV TXH GHSHQGHQ GH FyPR IXH QRUPDOL]DGR HO YHFWRU SURSLR { }L ; 3DUD HILFLHQFLD RSHUDFLRQDO VH GHILQH OD PDWUL] PRGDO [ ]; FRPR OD PDWUL] FXDV FROXPQDV VRQ ORV YHFWRUHV SURSLRV GH ORV GLIHUHQWHV PRGRV R VHD FRPR [ ] [ ]1 ;;;; −−−=

367. 6H SXHGH GHPRVWUDU TXH [ ] [ ][ ] [ ] [ ][ ];.;;0; 77 VRQ PDWULFHV GLDJRQDOHV RPR HMHPSOR FRQVLGHUHPRV XQ VLVWHPD FRQ GRV JUDGRV GH OLEHUWDG [ ] [ ][ ] { } { } [ ]{ }{ }[ ] { } [ ]{ } { } [ ]{ } { } [ ]{ } { } [ ]{ }        =         = ;0;;0; ;0;;0; ;;0 ; ; ;0; 77 77 7 7 7 [ ] [ ][ ]       = µο οµ ;0; 7 (FXDFLRQHV GHO PRYLPLHQWR HQ FRRUGHQDGDV SULQFLSDOHV ,QWURGXFLHQGR OD VLJXLHQWH WUDQVIRUPDFLyQ OLQHDO GH FRRUGHQDGDV ( ){ } [ ] ( ){ }WT;W; =

368. HQ ODV HFXDFLRQHV GHO PRYLPLHQWR SUHPXOWLSOLFDQGR SRU [ ]7 ; VH REWLHQH [ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] { } [ ]{ } [ ]{ } { }3TT I;T;.;T;0; 777 =+ =+ → ← → ← γµ

369. /DV FRRUGHQDGDV TL W

370. VH OODPDQ FRRUGHQDGDV SULQFLSDOHV JHQHUDOPHQWH QR WLHQHQ VLJQLILFDGR ItVLFR (V XQD KHUUDPLHQWD GH FiOFXOR SDUD GHVDFRSODU ODV HFXDFLRQHV GHO PRYLPLHQWR (O VLVWHPD GH HFXDFLyQ

371. VRQ 1 HFXDFLRQHV GHVDFRSODGDV

372. W3TT LLLLLLL =+ γµ

374. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV [ ] { }I;W3 7 L =

375. XDQGR ORV PRGRV VRQ QRUPDOL]DGRV UHVSHFWR D OD PDWUL] GH PDVD D YHFHV VH OODPDQ PRGRV QRUPDOHV { }L 1; ODV PDWULFHV GH

376. VH WUDQVIRUPDQ HQ [ ] [ ][ ] [ ] [ ] [ ][ ] [ ]→ ← → ← = = L1 7 1 1 7 1 ;.; ;0; ω

377. (MHPSOR 5HVROYHU SRU HO PpWRGR PRGDO ( ) ( )W[W[ ( ) ( ) ( ) ( ) ==== [[[[[VL { } { } P N P N;; ==       − =       = ωω 'HVDFRSODQGR ODV HFXDFLRQHV GHO PRYLPLHQWR { } [ ]{ } { } P P P ;0; 7 =             ==µ { } P P P =       −      −=µ { } { } N NN NN N NN NN =       −      − − −= =             − − = γ γ RPSUREDFLyQ GH UHODFLRQHV GH RUWRJRQDOLGDG YHULILFDU VL IXHURQ ELHQ FDOFXODGRV ORV { }L ;

378. { } ==−=       −      = γµ LGHPPP P P =             − − +             [ [ NN NN [ [ P P

379. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 9HULILFDFLyQ GH ODV IUHFXHQFLDV QDWXUDOHV ODV IUHFXHQFLDV QDWXUDOHV VRQ LQGHSHQGLHQWHV GHO VLVWHPD GH FRRUGHQDGDV HOHJLGR

380. P =+ TNT P =+ TNT (FXDFLyQ GHO PRYLPLHQWR HQ FRRUGHQDGDV SULQFLSDOHV HFXDFLRQHV GHO PRYLPLHQWR GHVDFRSODGDV

381. P N P N === µ γ ω P N P N === µ γ ω 6ROXFLyQ GH FDGD HFXDFLyQ ( ) ( ) WVHQ T WFRVTWT L L L LLL ωωω

382. += iOFXOR GH ODV FRQGLFLRQHV LQLFLDOHV HQ FRRUGHQDGDV ( ) ( )( ) LLL TTT XVDQGR HFXDFLyQ

383. ( ){ } [ ] ( ){ }WT;W[ = ( ) ( ) ( ) ( )            − =       WT WT W[ W[ ( ) ( ) ( )

384. [TW[W[WT =⇒+=⇒ ( ) ( ) ( )

385. [TW[W[WT =−= /XHJR ( ) ( ) WFRV[WT WFRV[WT ω ω = = 6ROXFLyQ HQ FRRUGHQDGDV TL W

386. /D VROXFLyQ HQ FRRUGHQDGDV [L W

387. WFRV[WFRV[W[ WFRV[WFRV[W[

388. ωω ωω += += 9HULILFDFLyQ SDUD W ( ) ( ) =−= =+= [[[ [[[[

389. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (MHPSOR 'HWHUPLQH ODV UHVSXHVWDV HVWDFLRQDULDV [ W

390. [ W

391. XVDQGR PpWRGR PRGDO 'H HFXDFLyQ

392. ( ){ } [ ] { }I;W3 7 L = 3RU OR WDQWR ODV HFXDFLRQHV GHVDFRSODGDV GHO PRYLPLHQWR GH DFXHUGR D HFXDFLyQ

393. VRQ WVHQ)TNTP WVHQ)TNTP Ω=+ Ω=+ XDV VROXFLRQHV SDUD HO PRYLPLHQWR HVWDFLRQDULR VRQ ( ) ( ) ( ) ( ) ( ) ( ) ( ) WVHQ P ) WT WVHQ P ) WT WVHQ ) WVHQ ) WT LLLL LL L Ω Ω− = Ω Ω− = Ω Ω− =Ω Ω− = ω ω ωµω γ /XHJR ( ) ( ) ( ) ( ) ( )      Ω− + Ω− Ω=+= ωω PP WVHQ P ) WTWTW[ ( ) ( ) ( )( ) WVHQ;WVHQ P PN) W[ Ω=Ω Ω−Ω− Ω− = ωω ( ) ( ) ( ) ( ) ( )( ) WVHQ;WVHQ P N) W[ WTWTW[ Ω=Ω Ω−Ω− − = −= ωω { }       Ω Ω =       Ω       − = WVHQ) WVHQ)WVHQ) W3L

394. WVHQ) Ω

395. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV $QiOLVLV GH OD SDUWLFLSDFLyQ GH ORV PRGRV GH YLEUDU ( ) ( ) ( ) ( ) ( ) ( ) ( )

407. RLyQSDUWLFLSDFWTRHULyQSDUWLFLSDFWT P W) P W) WTWTW[ PRG PRG VHQ VHQ °== Ω− Ω + Ω− Ω =+= ωω 1RWDV 3DUD Ω FHUFDQDV D Lω OD UHVSXHVWD HV DSUR[LPDGDPHQWH OD GHO PRGR L 3DUD XQ VLVWHPD GH 1 JUDGRV HQ OD UHVSXHVWD D XQD H[FLWDFLyQ D Ω ORV PRGRV SUHGRPLQDQWHV VRQ DTXpOORV GRV R WUHV

408. FRQ Lω FHUFDQRV D Ω (V GHFLU XQ VLVWHPD GH 1 JUDGRV GH OLEHUWDG SXHGH VHU DQDOL]DGR HQ HVH FDVR FRPR XQ VLVWHPD GH GRV R WUHV JUDGRV GH OLEHUWDG 0pWRGR GH OD VXSHUSRVLFLyQ GH QRGRV $O XWLOL]DU HO PpWRGR PRGDO HQ HO HMHPSOR GH JUDGRV GH OLEHUWDG VH XVDURQ ORV GRV PRGRV /D SUHJXQWD HV VL HQ XQ VLVWHPD GH YDULRV JUDGRV GH OLEHUWDG (MHPSOR 1

409. VH SXHGH XWLOL]DU VROR DOJXQRV (MHPSOR Q

410. 7UDGLFLRQDOPHQWH VH KD XVDGR HO PpWRGR GH ORV GHVSOD]DPLHQWRV PRGDOHV TXH FRQVLVWH HQ FRQVLGHUDU VROR ORV PRGRV FXDV ωL HVWiQ FHUFD GH OD IUHFXHQFLD GH H[FLWDFLyQ Ω 6LQ HPEDUJR HVWH PpWRGR QR HV OR H[DFWR TXH VH UHTXLHUH +R HQ GtD VH XWLOL]D SDUD HOOR HO PpWRGR GH OD DFHOHUDFLyQ PRGDO FRQRFLGR FRPR PpWRGR GH :LOOLDPV

411. 6XSRQJDPRV XQ VLVWHPD GH 1 JUDGRV GH OLEHUWDG GH ORV FXDOHV VH YDQ D XVDU VROR Q /ODPHPRV [Ö OD VROXFLyQ FRQ Q PRGRV [ OD VROXFLyQ UHDO FRQ 1 PRGRV /D HFXDFLyQ VH SXHGH HVFULELU

414. [0I.[ WI.[[0 −= =+ − DSUR[LPDQGR [[ Ö= VH REWLHQH OD VROXFLyQ DSUR[LPDGD [0.I.[ ÖÖ −− −= FRPELQDQGR HVWD HFXDFLyQ FRQ OD HFXDFLyQ [ ] ∑= =⇒               =               = Q U U U LL Q L Q Q L T;W[ T T T T ;;; [ [ [ [ [

416. Ö ∑ ∑ ∑ ∑ = ==               = Q U U U U U Q U U U U Q T; T; T; T; [ [ [ [

419. Ö H LQWURGXFLHQGR [ ]{ } =+ [0U. { } ∑∑ ∑ ∑ = − = = − = − −             =        Ω +=               =        Ω += Ω−=−= = = =− Q U QQQQ Q Q U U L U Q U ULUL Q U Q U U U UU Q U U U U U U U U U U III III III .T;WIIW[ I I I WIT;I.[ TTFRPRT;I.[ 0. 0;.; ;0.

422. Ö Ö @ ω ω ω ω ω ω 1RWD 6L HO VLVWHPD WLHQH PRYLPLHQWR GH FXHUSR UtJLGR . HV VLQJXODU QR VH SXHGH LQYHUWLU (Q HVH FDVR HVWH PpWRGR QR SXHGH DSOLFDUVH HQ IRUPD GLUHFWD 9HU 6WUXFWXUDO 'QDPLFV GH 5R 5 UDLJ

424. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (MHPSOR RQVLGHUDQGR XQ VROR PRGR { }                  Ω− ⋅⋅ Ω +=      Ω− ⋅⋅⋅ Ω += Ω +             =       =

425. a

426. a a a a ω ω ω ω φ ω NN ) W[ N ) ) N W[ T I I II II [ [ [ P N P N P N P N == == ωω ωω { }

428. @ a WTWT WVHQ . ) WT II II N . NN NN . TI.[ UU U U U U Q U UL U Ω−= Ω      Ω− =       =      =       − − =         −= − = − ∑ ω φ ω WVHQ) Ω WVHQN ) T WVHQN ) WVHQ N ) T WVHQ)NTTP WVHQ)NTTP U UU Ω      Ω− = Ω      Ω− =Ω      Ω− = Ω=+ Ω=+ ω ωω

429. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV PNΩ ωΩ a[ 9DORU UHDO N) N) N) = − ⋅+ N) N) −= − ⋅+ $EVRUEHGRU GH YLEUDFLRQHV $O LJXDO TXH SDUD ORV VLVWHPDV GH XQ JUDGR GH OLEHUWDG FXDQGR VH SURGXFH XQ SUREOHPD GH UHVRQDQFLD VH SXHGH HYLWDU (OLPLQDQGR OD IXHQWH H[FLWDGRUD DPELDU OD PDVD R ULJLGH] $PRUWLJXDU HO VLVWHPD 6LQ HPEDUJR KD VLWXDFLRQHV GRQGH QR HV IDFWLEOH R HV PX FDUR HVWDV VROXFLRQHV (Q XQ VLVWHPD FRQ YDULDV H[FLWDFLRQHV LΩ YDULDV Lω FHUFDQDV FDPELDU N R P SDUD YDULDU XQ Lω TXH FRLQFLGtD FRQ XQ LΩ SXHGH KDFHU FRLQFLGLU RWUR MΩ FRQ RWUR Lω

430. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 2WUD DOWHUQDWLYD SDUD VROXFLRQDU XQ SUREOHPD GH DOWDV YLEUDFLRQHV HV XWLOL]DU XQ DEVRUEHGRU GLQiPLFR GH YLEUDFLRQHV 6XSRQJDPRV TXH OD PiTXLQD WLHQH DOWDV YLEUDFLRQHV SRU HMHPSOR SRUTXH OD H[FLWDFLyQ Ω HVWi FHUFD GH XQ Lω

431. $ HVWH VLVWHPD OR OODPDUHPRV HO VLVWHPD SULPDULR ( ) =+− Ω=−++ [N[N[P WVHQ)[N[NN[P WM WM WM H)WVHQ) H;WVHQ;[ H;WVHQ;[ Ω Ω Ω =Ω =Ω= =Ω= ( ) ( ) =Ω−+− =−++Ω− ;PN;N )N;;NNP ( ) ( )( ) N N ZN N ZN ) NPNPNN PN) ; −      Ω−      Ω−+       Ω− = −Ω−Ω−+ Ω− = ω

432. ( )( ) N N ZN N N ) NPNPNN N) ; −      Ω−      Ω−+ = −Ω−Ω−+ = ω

433. WVHQ) Ω WVHQ) Ω 6LVWHPD 3ULPDULR 6LVWHPD 3ULPDULR DEVRUEHGRU WVHQ) Ω WVHQ) Ω P N =ω P N=ω

434. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 'H OD HFXDFLyQ

435. VL =⇒=Ω ;ω (V GHFLU HO VLVWHPD SULPDULR PiTXLQD

436. TXHGD GHWHQLGR QR YLEUD

437. VL VH KDFH TXH OD IUHFXHQFLD QDWXUDO GHO VLVWHPD DEVRUEHGRU Ω= 8Q FDVR LPSRUWDQWH GH DOWDV YLEUDFLRQHV HV GHELGR D OD UHVRQDQFLD 3DUD HVWH FDVR ω=Ω (O DGVRUEHGRU D DJUHJDU SDUD HOLPLQDU ODV YLEUDFLRQHV GHEH WHQHU VHJ~Q OD HFXDFLyQ

438. Ω=ω VHDR N N P P P N P N ==⇒=⇒==Ω µωω 'H HFXDFLyQ

439. ( ) ( )ž +Ω=Ω−=⇒−= WVHQ N ) WVHQ N ) W[ N ) ; HV GHFLU HO DEVRUEHGRU YLEUD HQ FRQWUDIDVH FRQ WVHQ) Ω  5HVSXHVWD GHO VLVWHPD SULPDULR HVWDED HQ UHVRQDQFLD

440. ωω ==Ω − − 5HVSXHVWD FRQ DEVRUEHGRU SDUD

441. ===Ω ;ωω 'H OD ILJXUD VH REVHUYD TXH HO DEVRUEHGRU QR VHUi HIHFWLYR VL Ω QR HV FRQVWDQWH

442. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (Q HO FDVR GH TXH H[LVWD DPRUWLJXDPLHQWR QR VH WHQGUi DPSOLWXGHV →∞ HQ ODV UHVRQDQFLDV QL DPSOLWXG HQ OD DQWLUHVRQDQFLD 3DUD VLVWHPDV URWDWRULRV FRPR XQ DXWRPyYLO ORV WRUTXHV GH H[FLWDFLyQ SURSRUFLRQDOHV D VX YHORFLGDG

443. YDUtDQ DPSOLDPHQWH (Q HVWH FDVR SDUD TXH HO DEVRUEHGRU VHD HIHFWLYR VX IUHFXHQFLD QDWXUDO GHEH VHU SURSRUFLRQDO D VX YHORFLGDG FRPR HO WLSR SpQGXOR

444. (O GLVHxR GHO DEVRUEHGRU GHEH VHU WDO TXH VRSRUWH ORV HVIXHU]RV JHQHUDGRV SRU VXV YLEUDFLRQHV ; )XQFLRQHV UHVSXHVWDV SDUD VLVWHPDV GH 1 JUDGRV GH OLEHUWDG ( ) MLLM );I+ = )XQFLRQHV UHVSXHVWDV GLUHFWDV R SXQWXDOHV SRLQW

445. VL L M )XQFLRQHV UHVSXHVWDV GH WUDQVIHUHQFLD VL L≠ M )UHFXHQFLDV GH DQWLUHVRQDQFLDV R FHURV

446. 6RQ ODV IUHFXHQFLDV SDUD ODV FXDOHV ODV DPSOLWXGHV GH OD UHVSXHVWD GH XQ VLVWHPD QRDPRUWLJXDGR VRQ FHUR /DV IXQFLRQHV UHVSXHVWDV GLUHFWDV SUHVHQWDQ VLHPSUH Q

447. DQWLUHVRQDQFLDV PLHQWUDV TXH ODV GH WUDQVIHUHQFLDV UDUDPHQWH OD SUHVHQWDQ 0RYLPLHQWRV GH FXHUSR 5tJLGR /RV VLVWHPDV QR UHVWULQJLGRV D PRYHUVH SUHVHQWDQ PRYLPLHQWRV GH FXHUSR UtJLGR HO VLVWHPD VH PXHYH VLQ GHIRUPDUVH

448. (OORV HVWiQ FDUDFWHUL]DGRV SRU =Lω

449. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (MHPSOR 'HWHUPLQDU ODV IUHFXHQFLDV QDWXUDOHV PRGRV GH YLEUDU GHO VLVWHPD =+== +− −+ NPUUP NPUN NNPU P N U U − = =⇒ 3DUD ( ) == ωU =       +− −+ [ [ NPUN NNPU { }       ==⇒ ;;; (V GHFLU VL OD PDVD VH GHVSOD]D OD PDVD WDPELpQ VH GHVSOD]D R VHD HO VLVWHPD VH PXHYH VLQ GHIRUPDUVH R VHD FRPR FXHUSR UtJLGR 3DUD      =−= P N P NU ω =       +−− −+− [ [ NP P NN NNP P N { }       − =−=⇒ ;;; 0DWULFHV GH IOH[LELOLGDG ULJLGH] /RV VLVWHPDV PDWULFLDOHV VRQ XQD DSUR[LPDFLyQ VLVWHPiWLFD SDUD ORJUDU FODULGDG VLPSOLFLGDG HQ HO FiOFXOR GH VLVWHPDV FRQ PXFKRV JUDGRV GH OLEHUWDG RHILFLHQWH GH LQIOXHQFLD R IOH[LELOLGDG LMI 6H GHILQH FRPR HO GHVSOD]DPLHQWR HQ L GHELGR D XQD IXHU]D XQLWDULD DSOLFDGD HQ M M L ML ) [ I = RHILFLHQWHV GH ULJLGH] LMN 6H GHILQH FRPR OD IXHU]D VRVWHQHGRUD HQ L SDUD XQ GHVSOD]DPLHQWR XQLWDULR HQ M VLHQGR QXORV ORV RWURV GHVSOD]DPLHQWRV

450. M L ML [ ) N = =−+ =−+ N[N[[P N[N[[P

451. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (MHPSOR 'HWHUPLQDU D

452. ILM E

453. NLM SDUD HO VLVWHPD GH OD ILJXUD D

454. ( ) ( ) =− =−− [[N [[NN[ ( ) ( ) =− =−− [[N N[[[N NI[ NI[ [[ == == ⇒=⇒ NI[ NI[ [[ == == ⇒=⇒ 0DWUL] GH IOH[LELOLGDG R GH FRHILFLHQWHV GH LQIOXHQFLD [ ] [ ] [ ]       − − ==⇒      = − NN NN IN NN NN I E

455. )N[ )N[N[ =− =+ )N[ )N[ = =− NN) NN) −== == ⇒ NN) NN) == −== ⇒ [ ]       − − =⇒ NN NN N 1RWD 3DUD HVWH HMHPSOR GHWHUPLQDU ODV PDWULFHV GH ULJLGH] R IOH[LELOLGDG QR SUHVHQWD YHQWDMD RSHUDWLYD SDUD GHWHUPLQDU ODV HFXDFLRQHV GHO PRYLPLHQWR 6LQ HPEDUJR HQ HO SUy[LPR SUREOHPD VL TXH VHUi GH JUDQ XWLOLGDG

456. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV (MHPSOR 'HWHUPLQDU ODV HFXDFLRQHV GHO PRYLPLHQWR SDUD ODV YLEUDFLRQHV OLEUHV HQ IOH[LyQ GH OD EDUUD GH OD ILJXUD [ ]{ } [ ]{ } =+ ;.;0 (Q HVWH FDVR HV PiV IiFLO GHWHUPLQDU HQ SULPHU OXJDU OD PDWUL] GH IOH[LELOLGDG SRU VX LQYHUVD GHWHUPLQDU OD PDWUL] [ ]N /DV FRHILFLHQWHV ILM VRQ GHWHUPLQDGRV XWLOL]DQGR ODV WDEODV GH GHIOH[LRQHV PHFiQLFD GH VyOLGRV

457. [ ]       =⇒ = = = = (, I I I I I [ ] [ ]       − −Ε == − , IN =      − − +             ⇒ ; ; P P 1RWDV 2EVHUYH TXH ODV PDWULFHV GH ULJLGH] IOH[LELOLGDG VRQ VLPpWULFDV /RV FRHILFLHQWHV GH ULJLGH] LLN GHEHQ VHU SRVLWLYRV SDUD TXH HO VLVWHPD VHD HVWDEOH N SRVLWLYR VLJQLILFD TXH OD IXHU]D HOiVWLFD VH RSRQH D ) N QHJDWLYR VLJQLILFD TXH N[ WLHQH OD PLVPD GLUHFFLyQ TXH ) HV GHFLU WLHQGH D DXPHQWDU OD GHIRUPDFLyQ → ,QHVWDEOH /RV FRHILFLHQWHV NLM SXHGHQ VHU

458. 3DUD GHWHUPLQDU ORV NLM 3DUD GHWHUPLQDU ORV ILM

459. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV 9LEUDFLRQHV DPRUWLJXDGDV *HQHUDOPHQWH HO DPRUWLJXDPLHQWR VH SXHGH LJQRUDU HQ ODV ]RQDV DOHMDGDV GH ODV UHVRQDQFLDV R DQWLUHVRQDQFLDV SHUR QR DVt HQ HVWDV ]RQDV HV GH LPSRUWDQFLD IXQGDPHQWDO

460. 9LEUDFLRQHV OLEUHV /D VROXFLyQ GHO PRYLPLHQWR HQ YLEUDFLRQHV OLEUHV [ ]{ } [ ]{ } [ ]{ } =++ [.[[0

461. 6RQ GH OD IRUPD { } { } W H;[ Ω =

462. HQ

463. [ ]{ } =++ ;.U0U

464. SDUD TXH H[LVWD VROXFLyQ [ ] GHW =++ .U0U

465. 3DUD TXH H[LVWD YLEUDFLyQ ORV YDORUHV SURSLRV U

466. GHEHQ VHU FRPSOHMRV FRQMXJDGRV HV GHFLU LLLGL LLL LLL MU ξωωβ ωξα βα −== = ±−=

467. 1RWD Lα GHEH VHU QHJDWLYR SDUD TXH HO VLVWHPD VHD HVWDEOH 5HHPSOD]DQGR

468. HQ

469. VH REWLHQHQ ORV PRGRV GH YLEUDU (VWRV YHFWRUHV SURSLRV VHUiQ FRPSOHMRV )tVLFDPHQWH VLJQLILFD TXH ODV GLIHUHQWHV PDVDV QR OOHJDQ D VXV SRVLFLRQHV H[WUHPDV DO PLVPR WLHPSR VLQR TXH GHVIDVDGDV SRU OR WDQWR D QR VH SXHGH KDEODU GH OD GHIRUPDGD R IRUPD GH YLEUDU

470. $PRUWLJXDPLHQWR SURSRUFLRQDO 6H OODPD DO DPRUWLJXDPLHQWR TXH HV SURSRUFLRQDO D OD PDWUL] GH PDVD R PDWUL] GH ULJLGH] HV GHFLU [ ] [ ] [ ].E0D += D E FRQVWDQWHV

471. XDQGR HO DPRUWLJXDPLHQWR HV SURSRUFLRQDO /RV PRGRV GH YLEUDU VRQ UHDOHV LJXDOHV DO VLVWHPD FRQVHUYDWLYR DVRFLDGR 6H SXHGHQ GHVDFRSODU ODV HFXDFLRQHV GHO PRYLPLHQWR (Q HIHFWR [ ]{ } [ ]{ } [ ]{ } { })[.[[0 =++

473. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV RQ OD WUDQVIRUPDFLyQ { } [ ]{ }T;[ = SUHPXOSOLFDQGR SRU [ ]7 ; [ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] { }T;T;.;T;;T;0; 7777 =++

474. VL [ ] [ ] [ ].E0D += [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] 'LDJRQDO;.;E;0;D;; 'LDJRQDO 7 'LDJRQDO 77 ⇒+=

475. OD HFXDFLyQ

476. TXHGD GHVDFRSODGD [ ]{ } [ ][ ] [ ]{ } { }3TTTL =++ → ← → ← → ← γϕµ

477. 'RQGH { } [ ]{ }L7L LL ;;=ϕ

478. (MHPSOR D

479. RQ DPRUWLJXDPLHQWR SURSRUFLRQDO NPFF == =             − − +             − − +             [ [ NN NN [ [ NPNP NPNP [ [ P P 2EVHUYH[ ] SURSRUFLRQDO D [ ]. [ ] =++++ =      ++−− −−++ =++ P N P N P N P N UNPUU NUNPPUNNP NNPNUNPPU GHW .U0UGHW

481. βα βα M P N MU M P N MU ±=±−= ±=±−= ⇒ 8VDQGR HFXDFLyQ

482. === === ξωω ξωω P N P N P N P N G G 8VDQGR HFXDFLyQ

483. [ ]{ } =++ ;.U0U SDUD ( ) βα MU ±= { }       = ; SDUD ( ) βα MU ±= { }       − = ; 1RWD 6H REVHUYD TXH ORV PRGRV GH YLEUDU VRQ UHDOHV H LGpQWLFRV D ORV PRGRV VLQ DPRUWLJXDPLHQWR E

484. RQ DPRUWLJXDPLHQWR QR ± SURSRUFLRQDO NPF NPF = = =             − − +             − − +             [ [ NN NN [ [ NPNP NPNP [ [ P P GHW[ ] =++ .U0U MU MU ±−= ±−=⇒ 8VDQGR HFXDFLyQ

485. === ξωωG === ξωωG 8VDQGR HFXDFLyQ

486. [ ]{ } =++ ;.U0U

487. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV SDUD βα MU ±= { }       =       ± = ± ž HM ; SDUD βα MU ±= { }       − =       ± = ± ž HM ; 0pWRGR SVHXGR PRGDO 6L HO VLVWHPD WLHQH 1 JUDGRV GH OLEHUWDG VH XVD XQ VLVWHPD VyOR GH Q JUDGRV GH OLEHUWDG GH PRGR TXH Pi[Ω QL

488. ω Pi[LPD IUHFXHQFLD GH H[FLWDFLyQ

489. IUHFXHQFLD QDWXUDO GHO PRGR Q

490. 3DUD OR FXDO HV QHFHVDULR GHWHUPLQDU Q Q ;;; ωωω 6H UHVXHOYH HO VLVWHPD [ ]{ } { } [ ]               =               ++ → ← → ←

492. W3 W3 W3 T T T T;;T Q 7 γµ 6L HO DPRUWLJXDPLHQWR HV QRSURSRUFLRQDO SHTXHxR VH SXHGH DSUR[LPDU OD VROXFLyQ QR FRQVLGHUDQGR ORV WpUPLQRV GLDJRQDOHV GH PDWUL] ;; 7 +LSyWHVLV GH %DVLOH

493. GHVDFRSODQGR HO VLVWHPD GH HFXDFLRQHV 0pWRGR GH UHVROXFLyQ GH YDORUHV YHFWRUHV SURSLRV XDQGR HO DPRUWLJXDPLHQWR @ QR HV SURSRUFLRQDO HV LQDSURSLDGR XVDU XQ PpWRGR GH UHVROXFLyQ GH YDORUHV SURSLRV HVWiQGDU HQ ODV HFXDFLRQHV [ ]{ } [ ]{ } [ ]{ } =++ [.[[0

494. 3DUD WUDQVIRUPDU

495. D XQ SUREOHPD HVWiQGDU VH OH VXPD OD HFXDFLyQ

496. [ ]{ } [ ]{ } =− [.[.

497. REWHQLHQGR GHVSXpV GH RUGHQDU =             +             − [ [ . . [ [ . 0

499. $QiOLVLV GH 6LVWHPDV 'LQiPLFRV OODPDQGR [ ] [ ] { }       =      =      − = [ [ . . % . 0 $

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726. FXDQGR L WRGRV ORV RWURV VRQ FHUR 5HFRUGDQGR TXH NLM )XHU]D VRVWHQHGRUD HQ L SDUD SURGXFLU GHVSOD]DPLHQWR XQLWDULR HQ M (VWDV IXHU]DV SXHGHQ VHU GHWHUPLQDGDV SRU HO SULQFLSLR GH ORV WUDEDMRV YLUWXDOHV

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728. FRQVLGHUHPRV HO GHVSOD]DPLHQWR YLUWXDO LQGLFDGR N:( δ= :, 7UDEDMR GH ORV PRPHQWRV LQWHULRUHV DVRFLDGRV D DFWXDQGR VREUH ODV FXUYDWXUDV YLUWXDOHV ( )         == ∫∫ // G[G[G[0G[G[G(,8

729. [ ] ∫∫ ⋅= ∂ ∂ = / L / , G[[[[(,G[[ [ [0:

734. δψψδ :( :, ML / MLLM / NG[[[[(,N G[[[[(,N == = ∫ ∫

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