RUNGE KUTTA 4 Orden

 

‟HØA DC OH SEMYCZVHOMWHFMAE DC OH VHOSD‐

SEMYCZVMDHD EHFMAEHO DCO FCETZA DCO ^CZ× TC@H

@CTADA DC ZSEKC LSTTH DC 4¾ AZDCE  FHTCDZH7

 CVTHJMOMDHD DC VMVTC@HV VMVTC@HV DC

 ^ATCEFMH FHTCDZÈTMFA7 MEK. FHTCDZÈTMFA7  MEK. ^CDZA ^CDZA TAZZCV @H]TH @H]TH  METCKZHET METCKZHETCV7 CV7  H^S@H]TH CEZMXSCW  H^S@H]TH CEZMXSCW FZMVTBMHE LSLM    FCZZÖE FAEDAZ CD[HZD ICZEHEDA   FHVTHØCDH JAGAZXSCV VH@SCO   FHVTZA ZC]CV GHMZ [MOIZCDA   FAEDAZM ECVTHZCV FZMVTMHE GBAECOO   FAZAECO ^HVTZHEH CYA GBAEHTBHE    FBMBSHE HZTCHKH, GSHE FHZOAV    FÈZDCEHV AVAZMA CD[HZD OCEEME    FHVTZA AZAØH DHEMCO CDSHZDA   FBHBSH]A KHZFÀH @HZFA HETAEMA

 BSHEFH]A 84 DC GSOMA :>:>

 

MEDMFC METZADSFFMÖE............... METZADSFFMÖE ............................. ............................. .............................. ............................. ............................. ............................. ............................. ............................ .............8 8 CVTHJMOMDHD TZHEVMTAZMH @CTADA DC ZSEKC LSTTH DC 4¾ AZDCE....................... AZDCE.............................................: ......................: 8.

CO ^ZAJOC ^ZAJOC@H @H DC CVTHJMOMD CVTHJMOMDHD. HD. -......... -.............. ......... ......... .......... .......... .......... .......... .......... .......... ......... ......... .......... .......... .......... .......... ............. .........: .:

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FAEFC^T FAEFC^TA A DC CVTHJMOMD CVTHJMOMDHD HD TZHEVMTA TZHEVMTAZMH. ZMH. -........... -............... ......... .......... .......... .......... .......... .......... .......... ......... ............. .................. ..........4 .4

6.

@ÍTADA @ÍTADA DC ZSEKC ZSEKC ‖ LSTTH. LSTTH. -........ -............. .......... .......... .......... .......... .......... .......... ......... ......... .......... .......... .......... .............. .................. .................4 ........4 6.8.

@ÍTADA @ÍTADA DC ZSEKC ZSEKC ‖ LSTTH LSTTH DC 8¾ AZDCE. AZDCE. -..... -.......... .......... .......... .......... ......... ......... .......... .......... .......... .......... .......... ..........4 .....4

6.:.

@ÍTADA @ÍTADA DC ZSEKC ZSEKC ‖ LSTTH LSTTH DC :¾ AZDCE. AZDCE. -..... -.......... .......... .......... .......... ......... ......... .......... .......... .......... .......... .......... ..........3 .....3

6.6.

@ÍTADA @ÍTADA DC ZSEKC ZSEKC ‖ LSTTH LSTTH DC 4¾ AZDCE. AZDCE. -..... -.......... .......... .......... .......... ......... ......... .......... .......... .......... .......... .......... ..........9 .....9

6.4.

IAZ@SOHV DCO @CTADA DC ZSEKC LSTH DC 4 AZDCE7.................................... AZDCE7..............................................9 ..........9

CGC@^OA 87........................... 87......................................... ............................. .............................. ............................. ............................. ................................................ .................................... ...2 2 CGC@^OA :7........................... :7......................................... ............................. .............................. ............................. ............................. .............................. ............................. ....................89 ......89 Faefousmaecs7.............. Faefousmaecs7 ............................ ............................. ............................. ............................. .............................. ............................. ............................................ .............................. :8 Hecxas........................ Hecxas......... .............................. ............................. ............................. ............................. ............................. .............................. ............................. ..............................:8 ................:8 ^rakrh`h dco prajoc`h 8.............................. 8............................................. ............................. ............................. ............................. ............................. ..................... ......:8 :8 ^rakrh`h dco prajoc`h :.............................. :............................................. ............................. ............................. ............................. ............................. ..................... ......:6 :6

 

METZADSFFMÖE H `cdmdh quc co scr bu`hea y oh cfaea`àh dc oas phàscs sc bhe dcshrraoohda, oh dc`hedh dc cecrkàh bh hu`cethda faesthetc`cetc, y dcetra dc cstc `hrfa sc bh pradufmda ue frcfm`mceta m`parthetc dc oh dc`hedh cecrkàh coíftrmfh, quc bh oockhda h prcscethr ue parfcethgc m`parthetc dco faesu`a tatho dc cecrkàh ce co `ueda. Ce co ^cr÷ phrh padcr shtmsihfcr oh frcfmcetc dc`hedh, sc bh mda iarthocfmceda ue smstc`h coíftrmfa dc patcefmh quc sc tareh fa`pocga. Ce oas ÷otm`as høas, oh tcedcefmh ce oh praduffmöe dc cecrkàh coíftrmfh bh smda ddcimfmthrmh cimfmthrmh ircetc h oh dc`hedh dc oh `ms`h, par oa quc phrh padcr shtmsihfcr  dmfbh dc`hedh ueh dc ohs saoufmaecs bh smda oh metcrfaecxmöe dc smstc`hs, fae sus vcethghs y dcsvcethghs, hsafmheda kcecrhdarcs quc apcrhe ce phrhocoa y fhrkhs dcetra dc ue krhe smstc`h metckrhda. Oas smstc`hs coíftrmfas sc bhe mda tare tareheda heda fhdh vcz `ès fa`pocgas y prcscethe ue uehh vhrmcdhd dc rctas dc mekcemcràh theta ce co pohech`mceta, y faestruffmöe fa`a oh apcrhfmöe dco `ms`a. Co dmscøa tatho dco smstc`h smstc`h coíftrmfa dc patcefmh dcjc scr himr`hda sajrc co fae faetrao trao huta`ètmfa y ea sajrc oh rcspucsth oceth dco apcrhdar bu`hea. ^hrh scr vmhjoc prcdcfmr co iuefmaeh`mceta dco smstc`h, ohs eucvhs cxmkcefmhs bhe iarzhda h jusfhr smc`prc bcrrh`mceths `ès hvhezhdhs dc heèomsms y dc sàetcsms. Ce co prcscetc trhjhga, jusfh saoufmaehr oas prajoc`hs dc oh csthjmomdhd trhesmtarmh utmomzheda co `ítada dc Zuekc-Lutth dc fuhrta ardce.

8

 

CVTHJMOMDHD TZHEVMTAZMH @CTADA DC ZSEKC LSTTH DC 4¾ AZDCE 8. CO ^ZAJ ^ZAJOC OC@H @H DC DC CVTH CVTHJMO JMOMDH MDHD. D. Co prajoc`h dc csthjmomdhd csth faefcremda fae co fa`parth`mceta dc ohs `èqumehs smefröemfhs dcspuís dc bhjcr smda pcrturjhdhs. Vm oh pcrturjhfmöe ea mevaoufrh fuhoqumcr fh`jma ecta ce oh patcefmh, ohs `èqumehs dcjce rctarehr h su csthda armkmeho. Vm cxmstc ue dcsjhohefc cetrc oh patc patcefmh efmh supo supomdh mdh y oh fhrkh fhrkh,, frchd frchdaa pa parr ue ffh`jm h`jmaa ce oohh fhrk fhrkh, h, ce kcecr kcecrhfmöe, hfmöe, a ccee ohs faedmfmae faed mfmaecs cs dc oh rcd, ue eucv eucvaa csthd csthdaa dc apcrh apcrhfmöe fmöe cs ecfcsh ecfcshrma. rma. Ce fuhoq fuhoqumcr umcr fhsa tadhs

ohs `èqumehs smefröemfhs metcrfaecfthdhs dcjce pcr`hecfcr ce smefraems`a sm co smstc`h

cs csthjoc< tadhs coohs dcjce pcr`hecfcr apcrheda ce phrhocoa y h oh `ms`h vcoafmdhd. Co trhesmtarma quc smkuc h ueh pcrturjhfmöe dco smstc`h cs asfmohtarma par ehturhoczh< pcra sm co smstc`h cs csthjoc, csthjoc, csths asfmohfmaecs scrèe h`artmkuhdhs y oocvhrèe ho smstc`h h ueh eucvh faedmfmöe dc apcrhfmöe y dc cqumomjrma. Csths asfmohfmaecs, sme c`jhrka, sae rciocghdhs fa`a iouftuhfm iouf tuhfmaecs aecs ce co iou iouga ga dc patc patcefmh efmh saj sajrc rc ohs oàec oàechs hs dc trhes trhes`msmö `msmöe. e.

Vm ueh met metcrfae crfaecxmöe cxmöe

faecfth das krhedcs krupas dc `èqumehs y cxpcrm`ceth cxfcsmvhs iouftuhfmaecs dc patcefmh, csth pucdc scr dmsphrhdh par su cqumph`mceta dc pratcffmaecs, dc tho `ada, quc dcsfaecfth oas das krupas dc `èqumehs. Cstc prajoc`h rciocgh oh csthjmomdhd dc oas das krupas ddcc `èqu `èqumehs, mehs, sm oh metcrf metcrfaec aecxmö xmöee cs dms dmsphr phrhdh hdh,, oas smstc` smstc`hs hs metcrf metcrfaec aecfth fthdas das dcj dcjce ce apc apcrhr rhr fa` fa`aa èrc èrchs hs medcpcedmcetcs yh quc tmcece medcpcedcefmh apcrhtmvh. Oh csthjmomdhd dco smstc`h coíftrmfa dc patcefmh cs ue m`parthetc prajoc`h ce oh sckurmdhd dc oh apcrhfmöe dco smstc`h, oh `hyaràh dc oas faohpsas fhushdas par mecsthjmomdhd moustrhe oh oh m`parthefmh dc cstc iceö`cea. Bmstörmfh`cetc, oh mecsthjmomdhd trhesmtarmh bh smda ue prajoc`h dc csthjmomdhd da`mehetc ce oh `hyar phrtc dc smstc`hs, y bh smda oh prcafuphfmöe dc oh medustrmh rcicrcetc h oh csthjmomdhd dco smstc`h. Oh faedmfmöe ecfcshrmh phrh oh shtmsihftarmh apcrhfmöe dco smstc`h cs quc tadhs ohs `èqumehs smefröemfhs, c`pochdhs phrh oh kcecrhfmöe dc cecrkàh coíftrmfh, pcr`heczfhe ce smefraems`a. Cstc hspcfta dc csthjmomdhd cs meioucefmhda par oh dmeè`mfh dco èekuoa dco ratar dco kcecrhdar y ohs rcohfmaecs patcefmh-èekuoa patcefmh-èekuoa.. Ohs dmicrcetcs iar`hs dc mecsthjmomdhd dcpcedce dc oh faeimkurhfmöe y oh apcrhfmöe dc

oas sms smstc` tc`hs hs coíf coíftrm trmfas fas dc patce patcefmh fmh par oa quc cs ecfcs ecfcshrm hrmaa co cet cetced cedm`m m`mcet cetaa dc oh :

 

fa`pocgmdhd dc dc oh apcrhfmöe, dmscø dmscøaa y co usa ddcc eucvhs tcfea tcfeaoakàhs oakàhs y faetraocs. Hsà th`jmíe co usa faesmstcetc dc oh tcr`meaoakàh cs rcqucrmda phrh dcshrraoohr co frmtcrma dc apcrhfmöe y dmscøa dco smstc`h, dc mkuho `hecrh, ohs bcrrh`mceths hehoàtmfhs y prafcdm`mcetas prafcdm`mcetas dc cstudma. Oas smstc`hs dc patcefmh sae sa`ctmdas h ue h`poma rheka dc dmsturjmas, pcqucøas y krhedcs. ^cqucøas dmsturjmas afurrce faetmeuh`cetc ce iar`h dc fh`jmas dc fhrkh< co smstc`h pucdc scr  fhphz dc hgus hgusthrsc thrsc h ohs faed faedmfmaecs mfmaecs fh`jm fh`jmhetc hetcss y apcr apcrhr hr shtms shtmsihfta ihftarmh`c rmh`cetc. etc. Cstc pucd pucdcc th`jmíe scr fhphz dc saparthr eu`crasas dmsturjmas dc ueh ehturhoczh scvcrh, tho fa`a ue farta fmrfum fmrfumta ta sajrc ueh oàech dc trhes`msmöe a ppírdmdh írdmdh dc ue kcecrhda kcecrhdar, r, dc ueh fhrkh krhedc krhe dc a dc ueh metcrf metcrfaecxm aecxmöe öe cetrc das èrchs. Se krhe dmstu dmsturjma rjma pucd pucdcc faedufm faedufmrr h fh`jm fh`jmas as cstrufturhocs dcjmda ho hmsoh`mceta dc oas coc`cetas ihoohdas. Oh rcpucsth rcpucsth dco smst smstc`h c`h dc patce patcefmh fmh hetc ue dmst dmsturjm urjmaa pucdc mev mevaoufr aoufrhr hr h oh `hyaràh dco cqumpa. ^hrh ueh mesthefmh, ueh ihooh sajrc ue coc`ceta fràtmfa sckumda par co hmsoh`mceta dc oas rcoís dc pratcffmaecs, fhushrè vhrmhfmaecs ce iougas dc patcefmh, vaothgcs ce ohs jhrrhs dc oh rcd y vcoafmdhdcs dco ratar dc ohs `èqumehs< dcjmda h ohs vhrmhfmaecs dc vaothgc dcjcrèe hftuhr faegueth`cetc oas rckuohdarcs dc vaothgc dc kcecrhdarcs y dco smstc`h dc trhes`msmöe<  phrh ohs vhrmhfmaecs dc

vcoafmdhd

dco

kcecrhdar

dcjce

hftuhr

oas

kajcrehdarcs

(rckuohda (rcku ohdarcs rcs dc vcoafm vcoafmdhd)< dhd)< dcj dcjmda mda ho fh`jma ce oh fhrkhj fhrkhjmomdh momdhdd dco smstc`h hft hftuhrèe uhrèe

oas

faetraocs dc kcecrhfmöe< oas fh`jmas ce vaothgc y ircfucefmh hicfthrèe h ohs fhrkhs ce co smstc`h ce emvcocs quc vhràhe dcpcedmceda dc sus fhrhftcràstmfhs medmvmduhocs. Hdmfmaeho`cetc, oas dmspasmtmvas utmomzhdas utmomzhdas phrh pratckcr h oas cqumpas medmvmduho`cetc padràhe rcspaedcr h vhrmhfmaecs ce ohs vhrmhjocs dco smstc`h y fhushr co dmsphra dco cqumph`mceta, dcjmomtheda ho smstc`h y pasmjoc`cetc faedufmceda h oh mecsthjmomdhd dco smstc`h. Ce fuhoqumcr smtuhfmöe dhdh, sme c`jhrka, ohs rcspucsths saoh`cetc dc ueh fhetmdhd om`mthdh dc cqumpa pucdce scr smkemimfhtmvhs. ^ar oa theta, `ufbhs faesmdcrhfmaecs sc bhfce kcecrho`cetc  phrh sm`pomimfhr co prajoc`h y phrh fcetrhrsc ce oas ihftarcs quc meioucefmhe co tmpa cspcfàimfa dc prajoc`h dc csthjmomdhd. Oh mecsthjmomdhd dc ue smstc`h th`jmíe pucdc afurrmr sme pírdmdh dc smefraems`a, par cgc`poa  pucdc oockhr h scr mecsthjoc par faohpsa dco vaothgc, `hetcecr co smefraems`a ea cs co prajoc`h ce cstc fhsa, ce su oukhr oh prcafuphfmöe rhdmfh ce oh csthjmomdhd y faetrao dc vaothgc. Csta bh frchda oh ecfcsmdhd dc rcvmshr oh dcimemfmöe y fohsmimfhfmöe dc csthjmomdhd dc smstc`hs dc  patcefmh. 6

 

:. FAE FAEFC^ FC^TA TA DC CVTHJM CVTHJMOMD OMDHD HD TZHEVMT TZHEVMTAZM AZMH. H. Oh csthjmomdhd trhesmtarmh cs oh fhphfmdhd dco smstc`h dc patcefmh dc `hetcecr co smefraems`a fuheda cs sa`ctmda h scvcrhs pcrturjhfmaecs trhesmtarmhs. 

Zcspucsth ea omecho ircetc h pcrturjhfmaecs scvcrhs



Barmzaetc dc tmc`pa dc vhrmas fmfoas

6. @Í @ÍTA TADA DA DC DC ZSEKC ZSEKC ‖ LSTT LSTTH. H. Ce heèomsms eu`írmfa eu`írmfa,, oas `ítadas dc Zuek Zuekc-Lut c-Lutth th sae ue faegu faegueta eta dc `ítad `ítadas as kceírmfa kceírmfass mtcrhtmva mtcr htmvas, s, cxpo cxpoàfmta àfmtass c m`po m`poàfmta àfmtas, s, dc rcsao rcsaoufmö ufmöee eu`ír eu`írmfh mfh dc cfuhfmaecs dmicrcefmhocs dmicrcefmhocs.. Cstc faegueta dc `ítadas iuc memfmho`cetc dcshrraoohda horcdcdar dco høa 8=>> 8=>> par  par oas `htc`ètmfas Fhro Dhvmd Tao`í Zuekc y Zuekc y @hrtme [mobco` Lutth Lutth..

6.8.

@ÍTADA DC ZSEKC ‖ LSTTH DC 8¾ AZDCE. Tadas oas `ítadas dc Zuekc - Lutth sae kcecrhomzhfmaecs dc oh iör`uoh jèsmfh dc Cuocr, ce oh quc oh iuefmöe pcedmcetc  i   sc rc`pohzh par uuee pra`cdma paed paedcrhda crhda dc pcedmce pcedmcetcs tcs ce co metcrvhoa xe  x  x  xe+8

 y e+ 80 y e + b ( w8 l 8+ w: l : + … + w` l `)

Daedc7 Ohs paedcrhfmaecs ‟wm‐7  m 0 8, :,…, `.

Vae faesthetcs quc shtmsihfce7 w8 + w: +…+ w` 0 >,

] ‟l m‐7 cs oh iuefmöe cvhouhdh ce ue pueta scocffmaehda ( x, y)  phrh  phrh co fuho  xe  x   xxe+8.

4

 

Co e÷`cra ‟`‐ sc ooh`h co ardce. Vm ta`h`as ` 0 8,  w8  0 8,  l 8  0 i (  x, x, ye),  oockh`as ho `ítada dc Cuocr. ^ar faesmkumcetc, sc dmfc quc co `ítada dc Cuocr cs ue `ítada dc Zuekc - Lutth dc prm`cr  ardce.

6.:.

@ÍTADA DC ZSEKC ‖ LSTTH DC :¾ AZDCE. Trhth`as dc bhoohr uehs faesthetcs dc `ada quc oh iör`uoh  y e+ 80 y e + hl 8 + j l :

Daedc7 l 80i  ( ( x  x e , y e)

l :0i  (  ( xe + θb, ye+ κb l 8)

Faefardheda fae ue paomea`ma dc Thyoar dc krhda :, ohs faesthetcs dcjce shtmsihfcr7 8

8

:

:

w 8+ w:08 < w : θ 0 < w: κ 0

Oucka7 

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Daedc7 w :≮ >

Cgc`poa7 Csfakc`as7 w: 0 °7 

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  κ0 : w

:

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Dc daedc7  w8 0 °,   0 8,   0 8

Cetaefcs oh cfuhfmöe7  y e+ 80 y e + b ( w8 l 8+ w: l : + … + w` l `)

Vc trhesiar`h ce7  y e+ 80 y e +

( l ± ± 8 + l  ) b :

:

±

Daedc7 l 80i  ( ( x e , y e) l :0i   (( xe + b , y e + b l 8)

^ucsta quc7  x e + b0 x e + 8  y e + b l 8 0 y e + bi   (( x e , y e )  Cs mdíetmfh ho `ítada `ítada dc Cuocr `cgarhda. `cgarhda.

6.6.

@ÍTADA DC ZSEKC ‖ LSTTH DC 4¾ AZDCE. -

Se `mc`jra dc oh ih`momh dc oas `ítadas Zuekc-Lutth cs ushda the fa`÷e`cetc quc h `ceuda cs rcicrcefmhda fa`a «ZL4¼ a fa`a «co `ítada Zuekc-Lutth¼ 6.4.    

IAZ@SOHV DCO @CTADA DC ZSEKC LSTH DC 4 AZDCE7

Dcimemceda ue prajoc`h dc vhoar memfmho fa`a7

dy  0i  ( ( x , y )   <  y ( x > )0 y > dx  

Cetaefcs co `ítada ZL4 phrh cstc prajoc`h cstè dhda par oh smkumcetc cfuhfmöe7

9

 

8

 ym +80 y m + ( l 8+ : l :+: l 6+ l 4 )b 9

 

Döedc7

l 80 i ( x8 , y8 ) b

bl 8

:

:

l : 0 i  (  ( x8 + , y 8 +

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bl : b l 6 0 i   (( x 8 + , y 8 + ) :

:

l 40i   (( x 8+ b , y 8+ bl  6 ) Hsà, co smkumcetc vhoar (ym+8) cs dctcr`mehda par co prcscetc vhoar (y m) `ès co pradufta dco th`høa dco metcrvhoal (b) par ueh pcedmcetc cstm`hdh. Oh pcedmcetc csl  ue pra`cdma paedcrhda dc pcedmcetcs, daedc 8   cs oh ppcedmcetc cedmcetc ho prmefmpma dco metcrvhoa, : cs oh pcedmcetc ce co pueta `cdma dco metcrvhoa, usheda l 6

l 8

 phrh dctcr`mehr co vhoar dc y ce co pueta xe+b/: usheda co `ítada dc Cuocr .

 cs atrh vcz oh pcedmcetc dco pueta `cdma, pcra hbarh usheda

l :

l 

 phrh dctcr`mehr co vhoar dc

4   cs oh pced y< pcedmc mcet etcc ho ime imeho ho dco met metcr crvh vhoa oa,, fae fae co vhoa vhoarr dc y dctc dctcr` r`me mehd hdaa par ^ra`cdmheda ohs fuhtra pcedmcetcs, sc oc hsmkeh `hyar pcsa h ohs pcedmcetcs ce co pueta `cdma7

8

 pcedmcetc  pcedm cetc0 ( l 8 + : l : +: l 6 + l 4 ) 9

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l 6

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Csth iar`h dco `ítada dc Zuekc-Lutth, cs ue `ítada dc fuhrta ardce oa fuho smkemimfh quc co crrar  par phsa cs dco ardce dc A(b3), `mcetrhs quc co crrar tatho hfu`uohda tmcec co ardce A(b4). ^ar oa theta, oh faevcrkcefmh dco `ítada cs dco ardce dc A(b4), rhzöe par oh fuho cs ushda ce oas `ítadas fa`puthfmaecs.

CGC@^OA 87 Ce co smstc`h dc oh imkurh smkumcetc afurrc ue fartafmrfumta trmièsmfa ce oh `mthd dc oh oàech hj, medmfhda fae co pueta ^. Fuheda csta sufcdc oas metcrruptarcs hdyhfcetcs h oh ihooh sc hjrce phrh hmsohr co fartafmrfumta. Hetcs dc oh ihooh co kcecrhdar su`memstrh h oh jhrrh meimemth ueh patcefmh dc V0 8.> + g.3 pu h vaothgc ea`meho y tmcec ueh faesthetc dc mecrfmh dc B0 6.> Lw-sck/LYH. Fhofuohr7

8) Co èekuoa fràtmfa usheda co `ítada dc èrchs mkuhocs. :) Ohs cfuhfmaecs dc asfmohfmöe hetcs y dcspuís dc om`pmhr oh ihooh.

VAOSFMÖE7 8) Vc hehomzh prm`cra co csthda dco smstc`h hetcrmar h oh ihooh phrh dctcr`mehr co vaothgc '   C  dctrès dc oh rchfthefmh trhesmtarmh, co fuho pcr`hecfcrè faesthetc durhetc tada co heèomsms dco prajoc`h, y co èekuoa dc patcefmh memfmho δ  . Co csthda dco smstc`h hetcrmar h oh ihooh sc `ucstrh ce oh imkurh smkumcetc. 8

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  ^8 :.:8>

0sce√

8

  8.> :.:8>

0:9.=> ¾ 0.4; rhd

Vc hehomzh hbarh co csthda dco smstc`h durhetc oh ihooh dc `hecrh dc ajtcecr oh furvh dc patcefmh vs èekuoa dc patcefmh y oh cfuhfmöe dc asfmohfmöe ce cstc csthda. Co dmhkrh`h farrcspaedmcetc sc `ucstrh ce oh imkurh smkumcetc7

Bhfmceda ueh faevcrsmöe trmèekuoa-cstrcooh y hkrupheda oas tír`meas ce scrmc sc tmcec7

] bhfmceda dc eucva ueh faevcrsmöe cstrcooh-trmèekuoa sc ajtmcec7 =

 

Oh patcefmh trhesicrmdh ce cstc csthda cs 8.3>6∛8.> sceδ 0.;88 sceδ   ^j0 :.886

^ar ÷otm`a sc hehomzh co csthda dco smstc`h dcspuís dc om`pmhr oh ihooh dc `hecrh dc ajtcecr oh furvh dc patcefmh vs èekuoa dc patcefmh y oh cfuhfmöe dc asfmohfmöe dc cstc csthda. Co dmhkrh`h farrcspaedmcetc sc `ucstrh ce oh imkurh smkumcetc

' 

 R 6 0 g .:+ g .>2 + g .9 + g .80 g .=2  ^f 0

∛

8.3>6 8.> .=2

sce δ 08.364 sceδ 

√ 8   8.>

δ 80 ώ √ sce

8.364

0 :.46: rhd 0 86=.689 ¾

Ohs krèimfhs dc patcefmh vs èekuoa dc patcefmh dc ohs trcs smtuhfmaecs hetcrmarcs sc `ucst `u cstrhe rhe ce oh im imku kurh rh sm smkum kumce cetc tc.. Fae Fae cs csth thss krèim krèimfh fhss sc fh fhof ofuo uohe he oh ohss èr èrch chss dc hfcocrhfmöe y dc dcshfcocrhfmöe, oa fuho pcr`mtc fhofuohr co èekuoa fràtmfa.

8>

 

Co èrch dc hfcocrhfmöe cs δ f

.;88

.4;>

8.>

( δ f √

 H80

)√ ∬ δ  .4;>

sceδdδ 

± δ f √.4;> + .;8 0 δ f + .;88 fasδ √8.8>4 .;88 8 fas δ ± f

.4;>

Co èrch dc dcshfcocrhfmöe cs :.46:

 H:0

 ∬ 8.364 sceδ dδ √8.> (:.46:√δ  ) f

δ f :.46:

± √8.36:fas δ ±δ  + δ f 0√8.:92 + 8.36 8.364 4 fas δ f + δ f Mkuhoheda èrchs  H 0 H .;88fas δ f √ 8.8>4 + δ f 0 8.36 8.364 4 fas δ f √8.:92 + δ f f

8

:

ö2:6fas δ f ..2:6fas

0.894

Dc daedc δ f 0 8.6; rhd 0;2.3:: ¾ t fr 0

∗

( √8.6; )   0>.8262 : ώ ( 9> Bz )( 8 )

 4  x 6 x :.46:

:) Ohs cfuhfmaecs dc asfmohfmöe sae Hetcs dc om`pmhr oh ihooh7

88

 

:

d δ  9>∛ ώ  ( 8.> √.;88 sceδ )0 9:.26:√44.9;4 sceδ   0 : 6 d t 

Dcspuís dc om`pmhr oh ihooh7 d δ  9>∛ ώ  ( 8.> √8.364 sceδ ) 09:.26:√=9.624 sceδ   0 :

:

6

d t 

@ÍTADA DC ZSEKC LSTTH Ssheda co `ítada dc Zuekc Lutth, rcsaovcr oh cfuhfmöe dc asfmohfmöe hetcs dc om`pmhr oh ihooh, cefaetrhdh ce co cgcrfmfma hetcrmar. :

d δ   09:.26:√ 44.9;4 sceδ  : d t 

Fae δ  0:9.=> ¾0.4; rhd   y δ '  0>.> >

>

Ce co metcrvhoa > ? t ? .43 sckuedas y ∉ t 0 >.>3 sck.

VAOSFMÖE7 Vc tmcec d δ   0 u 0k ( δ ) dt  :

d δ  du  0  0 i  ( ( δ )0 9:.26: √44.9;4 sceδ  : dt  d t 

Co smstc`h dc cfuhfmaecs cs cetaefcs δ m+80 δ m + ∉ t u m+ ∉ t ± um +80um + ±

Ce daedc `>0 ∉ ti ( δ m ) `80 ∉ ti   (( δ m + ∉ t um / : ) `:0 ∉ ti   (( δ m + ∉ t um / : + ∉ t `> / 4 ) `60 ∉ ti ( δ m + ∉ t u m+ ∉ t `8 / :)

^ZM@CZH MTCZHFMAE rhd .0:9.=> ¾, u 0> H t 0> 7 δ  0.4;> rhd >

>

m 0 ( t 0

>

.>3

sck. ) 8:

 

`>0.>3 ( 9:.26:√ 44.9;4 sce .4;> )0:.8== `80.>3 ( 9:.26:√ 44.9;4 sce .4;> )0:.8== ` :0.>3 \ 9:.26:√ 44.9;4 sce \ .4;> + .>3 ( :.8:== ) / 4 P P0 :.>;;6 ` 60.>3 \ 9:.26:√ 44.9;4 sce \ .4;> + .>3 (:.8:== ) / : PP0 :.>:34 δ 80.4;> + .>3 ( >.> ) + .>3 ( :.8:== + :.8:== + :.>;;6)/ 9

± .3::2 rhd0:9.=34: ¾ u80 >.> +\ :.8:== + : ( :.8:== ) + : ( :.>;;6 )+ :.>:34 P/ 9 0:.>=3>

VCKSEDH MTCZHFMAE m 08 ( t 0.8> sck. ) ` >0.>3 ( 9:.26:√ 44.9;4 sce .3::2 )0:.>:9: ` 80.>3 \ 9:.26:√ 44.9;4 sce \ .3::2 + .>3 ( :.>=3> ) / : P P0 8.=:94 `:0.>3 \ 9:.26:√ 44.9;4 sce \ .3::2 + .>3 ( :.>=3> ) /: + .>3 ( :.>:9:)/ 4 P P

± 8.2;=4 ` 60.>3 \ 9:.26:√ 44.9;4 sce \ .3::2+ .>3 ( :.>=3 )+ .>3 ( 8.=:94 ) / : P P0 8.;443 δ :0.3::2 + .>3 ( :.>=3 ) + .>3 ( :.>:9: + 8.=:94 + 8.2;=4 )/ 9

± .9;;8 rhd 062.;488 ¾ u:0 :.>=3+\ :.>:9: + : ( 8.=:94 )+ : ( 8.2;=4 )+ 8.;443 P/ 9 06.==:8

TCZFCZH MTCZHFMAE m 0: ( t 0.83 sck. ) `>0.>3 ( 9:.26:√ 44.9;4 sce .9;98 ) 08.;46= ` 80.>3 \ 9:.26:√ 44.9;4 sce \ .9;98 + .>3 ( 6.==:8 ) / : P P08.3;;6

`:0.>3 \ 9:.26:√ 44.9;4 sce \ .9;98 + .>3 ( 6.==:8 ) / : + .>3 ( 8.;46= )/ 4 P P 86

 

± 8.34:= ` 60.>3 \ 9:.26:√ 44.9;4 sce \ >.9;98+ .>3 ( 6.==:8 ) + .>3 ( 8.3;;6 ) / : P P08.6;88 δ 60.9;98 + .>3 ( 6.==:8 )+ .>3 ( 8.;46= + 8.3;;6 + 8.34:=) / 9

± .=896 rhd 03:.4=== ¾ u60 6.==:8+\ 8.;46=+ : ( 8.3;;6 ) + : ( 8.34:= ) + 8.6;88P/ 9 03.3388

Ce oh smkumcetc thjoh sc `astrhrè oas rcsuothdas ajtcemdas dc cstc prajoc`h7

Yhoar dco èekuoa ce krhdas Zuekc Lutth :9.=>>> :9.=34: 62.;488 3:.4=== ;>.8=8= =>.=88; 884.688= 84>.=942 8;:.;49;

t .>> .>3 .8> .83 .:> .:3 .6> .63 .4>

Thjoh 8. Zcsuothdas ce co metcrvhoa > ? t ? .43 Vaoufmöe fae prakrh`hfmöe ce @htohj7

Co krhimfa Hekuoa - patcefmh hetcs durhetc y dcspuís dc oh ihooh

84

 

Cfuhfmöe dc asfmohfmöe durhetc oh ihooh fae ue hefba dc phsa dc >.>3 s utmomzheda `ítada Zuekc lutth.

83

 

CGC@^OA :7 Se kcecrhda kcecrhdarr sme smefrö fröemf emfaa h 9> Bz, pas pascc cc ueh fae faesth sthetc etc dc mecrf mecrfmh mh dc  B 0 3@G/@YH y ueh rchfthefmh trhesmtarmh dc cgc dmrcfta  R”d 0 >.6 p.u, cstè faecfthda h ueh jhrrh dc patcefmh meimemth h trhvís dc ue fmrfumta purh`cetc rcsmstmva fa`a sc `ucstrh ce oh smkumcetc imkurh.

Co kcecrhdar cetrckh ueh patcefmh rcho dc  ^cocf 0 >.2 p.u, y Xcocf 0 >.>;4 p.u h oh jhrrh dc patcefmh meimemth fuya vaothgc cs 8.> p.u. Seh ihooh trmièsmfh afurrc ce oh `mthd dc ueh oàech, y oh fuho cs dcspcghdh par oh pucsth iucrh dc scrvmfma dc oh oàech, `cdmhetc oh hpcrturh sm`uothech dc h`jas cxtrc`as dc oh oàech. Vm sc faesmdcrh quc oh ihooh cs sastcemdh par >.6 sckuedas. Ajtcecr oh saoufmöe eu`írmfh dc oh cfuhfmöe dc asfmohfmöe, c`pocheda co `ítada dc Zuekc-Lutth dc 4ta ardce, fae ue phsa dc tmc`pa  Δt 0 >.8 sckuedas. Cicftuí oh sm`uohfmöe eu`írmfh bhsth t 0 >.6 \3 ^tsP.

VAOSFMÖE7

89

 

Oh m`pcdhefmh dc trhesicrcefmh cetrc co vaothgc kcecrhdar y oh jhrrh dc patcefmh meimemth Hetcs dc quc afurrh oh ihooh, oh rchfthefmh cetrc oas puetas  H y  J  pucdc scr cefaetrhdh par oh fa`jmehfmöe scrmc phrhocoa7  R 8 0>.6> + >.:> +

>.6> :

0 >.93 p . u

oh patcefmh hphrcetc cstè dhdh par7 ±

 V >.2 + >.>;4   g  M 0 ± 0 0>.2√>.>;4  g p.u 8.> ∮ > Y 

Co vaothgc metcrea dc oh `hqumeh  C” cs dhda par7  C ” 0Y  +  + gR 8 M 08.> + ( g >.93 ) ( >.2+ >..;4  g )08.8; ∮ :9.62; ¾ p. u

Oh cfuhfmöe dc patcefmh coíftrmfh hetcs dc oh pcrturjhfmöe qucdh dhdh par7

8

 ^cocf 0

| C ”|| C  CJ|  R 8

8

 ^cocf 0

sme δ  

∛

8.> 8.8; >.93

8

 ^cocf 08.2>sme δ HETCV

Vmceda co èekuoa dc apcrhfmöe csthjoc memfmho7 >

 ^cocf 0 >.208.2>sme δ > 

>.4444

δ >0 :9.6;2 0>.49>33 rhd

0sme δ    >

Durhetc oh ihooh sc tmcec, quc oh oàech ihoohdh sc dmvmdc ce das, rcsuotheda co dmhkrh`h dc m`pcdhefmhs fa`a ce oh Imkurh smkumcetc.

8;

sme δ 

 

Vc cicft÷h ueh trhesiar`hfmöe dc cstrcooh h dcoth y sc ajtmcec oa smkumcetc7

Oh rchfthefmh cqumvhocetc cetrc co kcecrhdar y oh jhrrh meimemth rcsuoth scr7  R : 0

∛

>.3 >.6

+ >.3∛>.83 + >.6∛.>83 >.83

08.2> p . u

Oh cfuhfmöe dc patcefmh coíftrmfh hetcs dc oh pcrturjhfmöe qucdh dhdh par7 :

 ^cocf 0

| C ”|| C  CJ|  R :

:

 ^cocf 0

sme δ  

∛

8.> 8.8; 8.2>

sme δ 

 ^:cocf 0 >.93sme δ DSZHETC

Fuheda oh ihooh cs dcspcghdh, oh oàech ihoohdh cs pucsth iucrh dc scrvmfma, rcsuotheda co dmhkrh`h dc rchfthefmhs fa`a oh smkumcetc krèimfh7

Daedc oh rchfthefmh cqumvhocetc cetrc co kcecrhdar y oh jhrrh meimemth rcsuoth scr7  R 6 0>.6 + >.: + .>6 0>.2 p . u

Dc tho `ada quc oh patcefmh coíftrmfh dcspuís dc oh ihooh qucdh dhdh par7 6

 ^cocf 0

| C ”|| C  CJ|  R 6

6

 ^cocf 0

sme δ  

6

 ^cocf 08.49 8.49:3 :3 sme δ DCV^SCV

Vc prafcdc h pohetchr ohs cfuhfmaecs dc asfmohfmöe, dc oh iar`h7 :

d δ 

ώi 

d t :

 B 

 0  ( ^ √ ^ ) `cf

cocf

82

∛

8.> 8.8; >.2

sme δ 

 

B03@G/@YH, i09>Bz.

HETCV dc oh pcrturjhfmöe rcsuoth7 :

d δ  9> ώ  ( >.2√8.2>sme δ )08: ώ ( >.2 √8.2>sme δ )0 6>.83=:=√9;.2  0 9;.2324 324 sme δ   : 3 d t 

DSZHETC oh pcrturjhfmöe rcsuoth7 :

d δ  9> ώ  ( >.2√>.93sme δ )08: ώ ( >.2 √>.93sme δ )06>.83=:=√:4.3>44: sme δ    0 : 3 d t 

DCV^SCV dc oh pcrturjhfmöe rcsuoth7 :

d δ  9> ώ  ( >.2√8.49  0 33.864= 64=3 3 sme δ   8.49:3 :3 sme δ ) 0 8: ώ ( >.2 √ 8.49:3sme δ )0 6>.83=:= √33.8 : 3 d t 

Vm sc faesmdcrh quc oh ihooh cs sastcemdh par >.6 sckuedas. Ajtcecr oh saoufmöe eu`írmfh dc oh cfuhfmöe dc asfmohfmöe, c`pocheda co `ítada dc Zuekc-Lutth dc 4ta ardce, fae ue phsa dc tmc`pa  Δt 0 >.>3 sckuedas. Cicftuí oh sm`uohfmöe eu`írmfh bhsth t 0 >.6 \3 ^tsP ^hrh ueh ihooh sastcemdh oh cfuhfmöe dc asfmohfmöe h rcsaovcr qucdh dhda par7 d : δ   06>.83=:=√:4.3 :4.3>44:sme >44:sme δ t ∂ ± : d t 

Bhooheda Dcoth `hxm`a y Dcoth fràtmfa7 √8

δ `hx 0 ώ √ sce

>.2 √8 fas δ fr

0 8.2

(

  >.2 8.49:3

)0

:.39:2 rhd

0849.24 ¾

( :.396√ >.49 ) + 8.49:3 . fas ( 849.24 )√ >.93 fas  ( :9.62;2) 8.2

8.2

8.49:3 8.2

t fr 0

∗

√

>.93

0=2.;= ¾

8.2

(

√>.49 )   0>.6;6 : ώ ( 9> Bz )( >.2)

 4 x 3 x :.396

Co krhimfa dc patcefmh-Hekuoa dco kcecrhdar hetcs, durhetc y dcspuís dc oh ihooh.

8=

 

M`poc`cetheda oas dhtas ajtcemdas ce co prakrh`h dc @htohj ajtcec`as oas smkumcetcs rcsuothdas7

Vc pucdc vcr ce co smkumcetc krhimfa co hu`ceta dco Hekuoa dco ratar rcspcfta ho tmc`pa dc durhfmöe dc oh ihooh fae ue phsa dc >.>3s

:>

 

Faefousmaecs7 Co `ítada dc Zuekc -Luh cs uea dc oas `ítadas `ès uomzhdas par su cxhftud phrh dcshrraoohr cfuhfmaecs dmicrcefmhocs. dmicrcefmhocs. Ho hpomfhr co `ítada ce oh cfuhfmöe dc asfmohfmöe ajscrvh`as ajscrvh`as fa`a co Hekuoa dco ratar hu`ceth durhetc co c`pa quc durh oh ihooh, oh durhfmöe dc csth ihooh dcpcedcrè dc oas coc`cetas dc pratcffmöe dco smstc`h. Vm co Hekuoa dco ratar sajrcphsh dcoth Hekuoa frmfa a, cetaefcs eucstra smstc`h ea scrè csthjoc. Co c`pa phrh dcspcghr oh ihooh dcjcràh scr `cear ho c`pa frmfa dc dcspcgc, phrh `hetcecr csthjoc eucstra smstc`h hetc ueh pcrturjhfmöe, cstc c`pa frmfa sc fhofuoh `cdmhetc oh furvh dc asfmohfmöe. Vm eucstra smstc`h dcspcgh oh ihooh ce ue c`pa `cear quc co c`pa fràfa, co smstc`h th`jmíe rcspaedcrè dc `hecrh rèpmdh h ohs eucvhs faedmfmaecs dc apcrhfmöe past ihooh, cetaefcs co eucva Hekuoa dc apcrhfmöe dcspuís dc oh ihooh sc csthjocfcrè ce ue c`pa `cear.

:8

 

Hecxas ^rakrh`h dco prajoc`h 8 fochr hoo hoo< < fof< %Mekrcshr oas vhoarcs phrh co Zuekc Lutth e08< m08< b0>.>3< %Dcoth DC t phsa dco tmc`pa dcoth0:9.=< %dcoth memfmho S0>< %DCZMYHDH MEMFMHO ^HZH V CKSEDA KZHDA t0>< %tfr0'>.8262 sckuedas'< dcothQf0;2.3::< %Hekuoa Frmtmfa dcothQm0dcoth< %Dcoth `hxm`a dcothQ`hx086=.689< w0zcras(mtcrhfmaecs-8,;)< dmsp(' dmsp( ' t(s) l8 l: l6 l4 Dcoth w ')< dmsp('------------------------------------------------------------------dmsp( '-------------------------------------------------------------------------')< -------' )< wbmoc e?mtcrhfmaecs wbmoc e?mtcrhfmaecs

           

y>0cvhouhr(dcoth*(pm/82>))< `>0b*y>< y80cvhouhr(dcoth*(pm/82>)+>.3*b*S)< `80b*y8< y:0cvhouhr(dcoth*(pm/82>)+>.3*b*S+>.:3*b*`>)< `:0b*y:< y60cvhouhr(dcoth*(pm/82>)+b*S+b*>.3*`8)< `60b*y6<

     

w(e,8)0t< w(e+8,:)0`>< w(e+8,6)0`8< w(e+8,4)0`:<

         

w(e+8,3)0`6< w(e,9)0dcoth< w(e,;)0S< dcoth0dcoth+(b*S+b*(`>+`8+`:)/9)*(82>/pm)< S0S+((`>+:*`8+:*`:+`6)/9)<

e0e+8< t0t+b<

    ced   cvQdcothf0w(7,9)<   `ht0w(87mtcrhfmaecs-8,7)< dmsp(`ht)<

::

 

x0`ht(7,8)< y0`ht(7,9)<   iar iar m08787mtcrhfmaecs-8  m08787mtcrhfmaecs-8 mi cvQdcothf(m)5dcothQf  cvQdcothf(m)5dcothQf   mi   dmsp('Co dcoth sajrcphsa co dcoth Fràtmfa ce co tmc`pa dc dmsp('Co dcspcgc')< dcspcgc' )<   dmsp(cvQdcothf(m))<   jrchl jrchl< <   ced   ced   imkurc(8)< poat(x,y)< krmd ae ae< < tmtoc('Krèimfh tmtoc( 'Krèimfh hekuoa vs Tmc`pa' Tmc`pa')< )< 'Tmc`pa (s)')< xohjco('Tmc`pa xohjco( (s)')< 'Krhdas (¾) ')< yohjco('Krhdas yohjco( ')<   imkurc(:)< hekuoa0>73782>< ^hi0:.:8*smed(hekuoa)< ^i0>.;88*smed(hekuoa)< ^pi08.364*smed(hekuoa)< poat(hekuoa,^hi,hekuoa,^i,hekuoa,^pi)<   baod ae ae< < stc`(dcothQm,:.:8*smed(dcothQm))< stc`(dcothQf,:.:8*smed(dcothQf))< stc`(dcothQ`hx,8.364*smed(dcothQ`hx))< 'r')< )< poat(\:,82>P,\8,8P,'r' poat(\:,82>P,\8,8P, %ockced('^8','^:','^6','^`','DQ>','Df','DQ`hx')< krmd ae  

%cfuhfmae cgcrfmfma : y09:.26:-44.9;4*sme(dcothQc)< %cfuhfmae cgcrfmfma 8 y06>.83=:=-:4.3>44:*sme(dcothQc)< iueftmae y0cvhouhr(dcothQc) iueftmae  y0cvhouhr(dcothQc) y09:.26:-44.9;4*sme(dcothQc)< %cfuhfmae Ce IHOOHHHH   ced  ced  

^rakrh`h dco prajoc`h : fochr hoo hoo< < fof< %Mekrcshr oas vhoarcs phrh co Zuekc Lutth

:6

 

e08< m08< b0>.>3< %Dcoth DC t phsa dco tmc`pa dcoth0:9.62;2< %dcoth memfmho S0>< %DCZMYHDH MEMFMHO ^HZH V CKSEDA KZHDA t0>< %tfr0'>.6;6 sckuedas'<   dcothQf0=2.;=< %Dcoth frmtmfa dcothQm0dcoth< dcothQ`hx0849.24< w0zcras(mtcrhfmaecs-8,;)< ' t(s) l8 l: l6 l4 Dcoth w dmsp(' dmsp( ')< dmsp('------------------------------------------------------------------dmsp( '-------------------------------------------------------------------------')< -------' )< wbmoc e?mtcrhfmaecs wbmoc e?mtcrhfmaecs

 

y>0cvhouhr(dcoth*(pm/82>))< `>0b*y>< y80cvhouhr(dcoth*(pm/82>)+>.3*b*S)<

         

`80b*y8< y:0cvhouhr(dcoth*(pm/82>)+>.3*b*S+>.:3*b*`>)< `:0b*y:< y60cvhouhr(dcoth*(pm/82>)+b*S+b*>.3*`8)< `60b*y6<

               

w(e,8)0t< w(e+8,:)0`>< w(e+8,6)0`8< w(e+8,4)0`:< w(e+8,3)0`6< w(e,9)0dcoth< w(e,;)0S< dcoth0dcoth+(b*S+b*(`>+`8+`:)/9)*(82>/pm)< S0S+((`>+:*`8+:*`:+`6)/9)<

e0e+8<   t0t+b<   ced   cvQdcothf0w(7,9)<   `ht0w(87mtcrhfmaecs-8,7)< dmsp(`ht)< x0`ht(7,8)< y0`ht(7,9)<   iar iar m08787mtcrhfmaecs-8  m08787mtcrhfmaecs-8

:4

 

  mi mi cvQdcothf(m)5dcothQf  cvQdcothf(m)5dcothQf   dmsp('Co dcoth sajrcphsa co dcoth Fràtmfa ce co tmc`pa dc dmsp('Co dcspcgc')< dcspcgc' )<   dmsp(cvQdcothf(m))< jrchl<   jrchl<   ced   ced   imkurc(8)< sujpoat(:,8,8)< poat(x,y)< krmd ae ae< < 'Krèimfh hekuoa vs Tmc`pa' Tmc`pa')< )< tmtoc('Krèimfh tmtoc( xohjco('Tmc`pa xohjco( (s)')< 'Tmc`pa (s)')< 'Krhdas (¾) ')< yohjco('Krhdas yohjco( ')<   sujpoat(:,8,:)< hekuoa0>73782>< ^hi08.2>*smed(hekuoa)< ^i0>.93*smed(hekuoa)< ^pi08.49:2*smed(hekuoa)< poat(hekuoa,^hi,hekuoa,^i,hekuoa,^pi)< 'Krèimfh Hekuoa')< )< tmtoc( tmtoc('Krèimfh ^atcefmh vs Hekuoa' yohjco('^ATCEFMH yohjco( (p.u)' '^ATCEFMH (p.u)')< )< 'Krhdas (¾) ')< xohjco('Krhdas xohjco( ')<   baod ae ae< < stc`(dcothQm,8.2>*smed(dcothQm))< stc`(dcothQf,8.2>*smed(dcothQf))< stc`(dcothQ`hx,8.49:2*smed(dcothQ`hx))< 'r')< )< poat(\:,82>P,\>.2,>.2P,'r' poat(\:,82>P,\>.2,>.2P, %ockced('^8','^:','^6','^`','DQ>','Df','DQ`hx')< krmd ae  

%cfuhfmae cgcrfmfma : y09:.26:-44.9;4*sme(dcothQc)< %cfuhfmae cgcrfmfma 8 y06>.83=:=-:4.3>44:*sme(dcothQc)< iueftmae y0cvhouhr(dcothQc) iueftmae  y0cvhouhr(dcothQc) y06>.83=:2-:4.3>44:*sme(dcothQc)< %cfuhfmae Ce IHOOHHHH   ced  ced  

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