Practica Browniana

 

Sopkrto pr± prhltdlh h± ltdlh , 3>8=)

\o ostuadhrh oe gkvdgdobtk crkwbdhbk, ob oe luhe trhthrogks eh sdguehldkb k± b ao ubh y vhrdhs phrtdluehs ao pkeob ob ad`orobtos lhsks, pkr ojogpek ob hefubk ao eks lhsks aosproldhbak eh ghsh, luhbak ehs phrtdluehs ao pkeob pksoob lhrfh oeoltrdlh. Eks dgpkrthbtos rosuethaks quo so adorkb h phrtdr ao eks gotkaks oxpordgobtheos,  jubtk lkb sus dgpedlhdkbos skcro tkak phrh oe lkbtoxtk mdstkrdlk ob oe quo so eeovhrkb h lhck, y oe rosuethak ghs dgpkrthbto dgpkrthbto quo os eh aogkstrhld aogkstrhld±kb k± b ao eh oxdstobldh ao ehs gkeoluehs. Ahak osto molmk pkaogks ahr lkgk rosuethak quo h poshr ao rohedzhr ubh sdguehld kb k± b chsthbto cuobh bk pkaogks ahr pkr molmk quo eh trhyoltkrdh ao ehs phrtdluehs so lkgpkrthrhb ao ldorth `krgh, pork sd pkaogks lkbleudr quo aocdak h eh oxdstobldh ao ehs gkeoluehs ob eks ffiudaks ehs trhyoltkrdhs ao ehs phrtdluehs ao pkeob phrolob sor guy heohtkrdhs. ^HL\ bugcors:

D.

DBQS DBQSKA_L KA_LQDKB QDKB

Oe ostuadk aoe gkvdgdobtk Crkwbdhbk os osobldhegobto vor oe lkgpkrthgdb lkgpkrthgdbotk otk ao phrtd phrtdluehs luehs su﬈ldobtogob su﬈ldobtogobto to poquo˕ poquo˕b bhs, hs, aobtrk ao ub ffiudak, pkr ojogpek ub frhbk ao pkeob dbgorsk ob hfuh, oe luh luhee mhcdh mhcdh sda sdak k kcsorv kcsorvhak hak pkr prdgor prdgorhh voz voz ob 8=36 8=3 6 pkr oe cdk cdkekf ekfk k y ckthbd ckthbdlk, lk, Skcort Skcort Crkw Crkwb, b, qud qudob ob kcsorvk pkr goadk ao su gdlrkslkpdk h hefubhs phrtdluehs htrhphahs aobtrk ao ubh lhvdaha, frhbks ao pkeob sugorfdaks ob hfuh, torgdbhah su kcsorvhld± kcsorvhldkb k± b lkbleuyk quo ehs phrtdluehs ao pkeob tobdhb trhyoltkrdhs heohtkrdhs h trhv± trhvos o± s aoe edqudak, pork oe gdsgk bk puak aotorgdbhr ehs vhrdhceos phrh proaoldr oe gkvdgdobtk ao ehs phrtdluehs aobtrk aoe ffiudak, dbleusk ob hquoeeh opklh so eeofk h pobshr quo ehs adgdbuths phrtdluehs ao pkeob osthchb vdvhs, y quo so gkvdhb pkr luhbth prkpdh aobtrk ao hefub ffiudak. Ob 84>? Hecort Odbstodb pucedlk ub hrtdluek eehghak pkr ˕ ghrhvdeeksk oe gdsgk lkgk  lkgk   oe o± e Hbbus Gdrhcdeds, os aoldr h bk ob eht eht±±ĸb, ĸb, akbao oxpedlh oxpedlh lkb gulmk aothe aotheeo eo eh oxdstobld oxdstobldhh ao poquo˕bhs phrtdluehs quo lkbstdtuyob eh ghtordh,So`.1, So`.?, ek quo mky eehghgks gkeoluehs, y ek quo mhcdh kcsorvhak ob oso obtkblos os aoldr ob 8=36, Skcort Crkwb, os quo eh trhyoltkrdh ao eks frhbks ao pkeob bk os gorhgobto heohtkrdh k quo tuvdorkb gkvdgdobtk prkpdk, sd bk quo so aocdh h su dbtorhlldkb lkb ehs gkeoluehs dbadvdauheos quo lkb`krghchb oe ffiudak. Osto molmk `uo lkgprkchak ob oe h˕bk ao 84>= pkr ooee `±ĸsdlk ĸsdlk qu±ĸĸgdlk gdlk Johb ^orrdb pkr su trhchjk trhchjk aoe oqudedcrdk oqudedcrdk ao soad soad-± b y eh adslkbtdbudaha ao ghtordh,So`.0, oe luhe dbgobthld±k gobthld kb leusk `uo fhehrakbhak pkr oe progdk bkcoe ao ` ±ĸsdlh ĸsdlh mhsth oe h˕bk bk ao 8430, lkb su trhchjk trhchjk so puak lheluehr lheluehr ao `krgh oxhl oxhlth th oe bugork ao Hvkfhark, os aoldr, oe b±u ugork gork ao gke±ooluehs luehs lkbtobdahs ob ub gke ao fhs ob lkbadldkbos bkrgheos k osthbahr. Haog±hhss ob eh opklh ob akbao so oblkbtrhchb quo os heroaoakr aoe sdf sdfek ek ]] so obl oblkbt kbthch hchb b tros tros frh frhbao baoss ro rovk vkeld eldkbo kboss ldobt±±ĸ﬈lhs, ldobt ĸ﬈lhs, eh tokrdh ao eh roehtdvd roehtdvdaha, aha, eh golhb golhbdlh dlh luhbtdlh y oe gkvdgdobtk crkwbdhbk osth uetdgh lkgk lkbsoluobldh eeovk h eh tokr±ĸĸhh aoe lhks y ao eh adbhgdlh bk edbohe, lkgk yh

∖ Oeoltrkbdl haaross: euds.hecortkNldobldhs.ubhg.gx haaross:  euds.hecortkNldobldhs.ubhg.gx

gobldkbhgks hbtos quo `uo ldthah pkr Odbstodb ob su hrtdluek9 ao osths tros rovke rovkeuldkb uldkbos os osth uetdgh `uo eh gobks hlehghah pkr eh lkgubdaha ldobtd﬈lh, pork lhco aosthlhr quo `uo eh ao ghykr dgphlhtk puos bk skek lhusk aoshfrhak pkr ek ` ±ĸsdlks ĸsdlks ao eh  ±opklh opklh sd bk thgcdob pkr eks qudgdlks y eks cdkekfks, ahak quo hm lkbauldak h eh dbtrkaulld kb k± b ao eh gkeoluehrdzhld±kb, kb, phrh pkaor aosl aoslrdcdr rdcdr lkgpeothg lkgpeothgobto obto luhequdo luhequdorr `obkgobk bhturhe, osto molmk `uo roshethak pkr oe `±ĸĸsdlk sdlk hgordlhbk Sdlmhra @oybghbb ob sus eolldkbos, So`.3. Eh mdstkrdh ao lkgk ogpoz±k oe ostuadk y eh dgpkrthbldh aoe gkvdgdobtk crkwbdhbk `uo lkb eh phrhakjh obtro eh tokr±ĸĸhh ldb±otdlh otdlh y eh Qorgkadb±hhgdlh, gdlh, eh luhe eeovk he aochto ﬈eksk﬈lk obtro E. Ckezghbb y eks oborfotdldsths, obtro eks quo aosthlhchb O. Ghlm y V. Kstwhea phrh roskevor eh phrhakjh ao eh rovorsdcdedaha/drrovorsdcdedaha, quo so ekfrk h trhv±ooss ao eh dbtorprothld± kb ostha± ost ha±ĸĸstdlh stdlh aoe gkvdgdobtk gkvd gdobtk ao ehs phrt±ĸluehs ĸluehs dbadvdauheos9gdobtrhs quo Odbstodb quo dfbkrhch eks oxprdgobtks aoe gkvdgdobtk crkwbdhbk, `uo   oe o± e qudob roskevd± roskevdk± ob 84>? eh ± adslrophbldh, ahbak eh oxpedlhld± oxpedlhldkb kb ao eh advorfobldh obtro eks vhekros ao ehs voekldahaos goadhs kcsorvhahs, quo ob rohedaha orhb eks aozpehzhgdobtks goadks ob ldortk dbtorvheks ao tdogpk, y eks vhekros hetdsdgks (`hltkros aoe kraob ao 8>>>) ao eh tokr± tokr±ĸĸhh ldb ldb±otdlh, o± tdlh, puos Odbstodb so chs k± ob quo eh voekldaha ±s ao eh phrt± phrt±ĸlueh ĸlueh crkwbdhbh crkwbdhbh bk orh ubh ghfbd ghfbdtua tua ao﬈bdah, g g±hhs cdob quo eh phrt±ĸlueh ĸlueh lmklh ubh guetdtua ao volos lhgcdhbak su voekldaha ao adrolld±kb kb y gkauek, eeofhbak h hbuehrso ob guetdpeos klhsdkbos. Eks pubtks prdbldpheos phrh eh skeuld± skeuldkb ± ao Odbstodb `uorkb: 8) Oe gkvdgdobtk ao eks frhbks ao pkeob osthch prkvklhak pkr oe frhb b±u ugork gork ao dgphltks, os aoldr, eh `roluobldh ao lkedsdkb aobtrk aoe ffiudak, lkb ehs gke ooluehs ± luehs aoe ffiudak. 3) Oe gkvdgdobtk ao ehs gke±ooluehs luehs os thb lkgpedlhak quo skek k± ek so puoao aoslrdcdr prkchcde±ĸstdlhgobto ĸstdlhgobto ob t±orgdbks orgdbks ao ub frhb frh b bu ugork ± gork ao dgphltks dbaopobadobtos Ob ao﬈bdtdvh, lhah phrt±ĸlueh ĸlueh ojoluth ub gkvdgdobtk quo so hsugo dbaopobadobto aoe ao tkahs ehs aog±hhss y haog±hhss aks gkvdgdobtks ao eh gdsgh phrt±ĸlueh ĸlueh ob dbtorvheks ad`orobtos ao tdogpk skb thgcd± thgcdob o± b prklosks dbaopobadobtos (1). ^kr ek thbtk aoldgks quo Odbstodb `uo oe luephceo ao aosoblhaobhr vhrdks ao eks hvhblos ghs dgpkrthbtos ob oe hgcdtk ldobtd﬈lk lkgk eh ` ±ĸsdlh ĸsdlh ht± htkgdlh ± gke gke±oluehr, o± luehr, hs± hs±ĸ lkgk ao eh ghtog±htdlh ghtog h± tdlh purh ao eks prklosks ostkl hstdlks h± stdlks k he hzhr9 eks

 

3 hvhblos ob eh ldobldh oxpordgobthe bk quoahrkb htr± htrhs h± s puos so eeof±k he osthceoldgdobtk drro`uthceo ao eh oxdstobldh ao eks ht±kgks, kgks, lkgk so mhcdh gobldkbhak ubk ao eks oblhrfhaks ao eeovhr h lhck ostks oxpordogtks `uo oe bkgcrhak Johb ^orrdb (oe ostuadk ao oqudedcrdks ao soadgobthld kb, k± b, goadahs guy oxhlths aoe bugork u± gork ao Hvkfhark, otl., ob 84>=), ao eh gdsgh `krgh orh pruoch ao eh oxdstobldh ao ehs gke±ooluehs. luehs. ^hrh ^hrh oe ostuad ostuadk k ao osto tdpk ao sdtogh sdtoghs, s, os aoldr aoldr,, ao eh adb adb±hgdlh h± gdlh ao ubh phrt± phrt±ĸlueh ĸlueh ob ub goadk k ffiudak aobsk eeov k± h ^hue Ekbfovdb h osthceolor ubh oluhld±kb, hmkrh lkbkldah lkgk eh oluhld±kb kb ao Ekbfovdb, eh luhe os ubh oluhld±k kb b ad`orobldhe obld he ostkl±hstdlh quo aoslrdco oe gkvdgdobtk crkwbdhbk ob ub pktobldhe. Oluhld±kb ao Ekbfovdb So`.6

∓ ∆_ ( x) ∓ μ   x ˔ +

 

3μ  I  Ic  Q S(t ) 7   >

(8)

@DF. @DF. 8: Eh fr fr±h﬈lh h± ﬈lh guostrh eks rosuehtaks ao hpedlhr oe hefkrdtgk ao Oueor phrh eh oluhld±k kb b ao Ehbfovdb, guostrh quo mhy poquo˕bks bks gkvdgdobtks heroaoakr ao su pkldsd±k kb b krdfdbhe.

Oe dbtoros os ostuadhr ob `±ĸsdlh ĸsdlh y qu±ĸĸgdlh gdlh osto tdpk ao lkgpkrthggdobtks os quo ob eh bhturheozh klurrob `obkgobks quo sofub ehs eoyos bk so hpedlhb h obtos dbadvdauheos sd bk h ehs pkcehldkbos, pkr ek thbtk skb dgpkrthbtos ehs eoyos ghs foborh orheos eos ao eh ostha osthadst dstdlh dlh,, ob eufhr eufhr ao tobor tobor olu oluhld hldkbo kboss adb adbhgd hgdlhs lhs phrh lhah ubh ao ehs phrt±ĸluehs, ĸluehs, os gojkr mhlor ub ostuadk ao eh ovkeuld±kb kb togpkrhe ao prkchcdedaha ao oblkbtrhr ubh phrtdlueh

pkr eh togporhturh y ubh `ubld±k kb b heohtkrdh tdpk Fhussdhbh, ± ad`orkctobogks phrh eh oluhld± oluhldkb k± b ao Ehbfovdb, ubh oluhldkb obldhe edbohe ao prdgor kraob, dgpeogobthbak oe hefkrdtgk ao Oueor ob ubh adgobsd±k kb b phrh roskevor adlmh oluhld±k kb b kctobogks quo buos buostrh trh phrt± phrt±ĸlueh ĸlueh ao pkeo pkeob b vdcrh heroaoak heroaoakrr ao su pksdld±kb krdfdbhe, os aoldr, akbao so aojk lhor. ^hrh buostrh sofubah sdguehld± sdguehldkb k± b hfrofhrogks ub pktobldhe

ob ldorth rofd±k kb b aoe osphldk, h phrtdr aoe lkbkldgdobtk ao ehs prkchcdedahaos lkbadldkbheos. Mky a±ĸĸh, h, oe ostuadk ao eh adb hhggdlh ± ggdlh crkwbdhbh ao ub sdstogh ao gulmhs phrt±ĸĸluehs luehs so mhlo cdob lkb eh aoslrpldk kb ± b ao Edkuvdeeo-v Edkuv deeo-vkb kb Bow Bowghb ghb lkbs lkbsdaorh daorhbak bak he sdstogh sdstogh tkthe, os ao± ao Ekbfovdb roogldr, sdstogh + obtkrbk, k lkb eh aoslrpldkb pehzhbak oe obtkrbk pkr `uorzhsh ao adsdphld±kb y h˕b bhadobak hadobak `uorzhs ao hzhr quo prkvklhb ffiultuhldkbos ob oe sdstogh ostuadhak.

± Eh prdgorh sdguehld± sdguehldkb,os oe ostuadk aoe lkgpkrthgdobtk ao ubk phrt± phrt±ĸĸlueh lueh ao pke pkeob ob quo so aojh lhor ob ub ffiudak ffiudak,, phr phrhh buostrk buos trk lhsk os hfuh h lkbadldkbo lkbadldkboss bkrgh bkrgheos9p eos9puos uos eh tog-

frvdhthldkbhe lorlh ao eh Qdorrh, phrh ubh phrt±ĸlueh ĸlueh ao pkeob, buovhgobto sor±h ob oe lhsk aoe hfuh lkb ehs gdsghs lkbadldkbos quo phrh eh prdgorh sdguehldk kb, ± b, skekhgobto quo osth ± ao Ehbfovdb, sor± ± ad`orvoz buostrh oluhldkb sorh± ubh oluhld± oluhldkb obldhe edbohe kradbhrdh skek quo hmkrh os ao lhrhltor voltkrdhe,eh √  luhe quoah ao eh sdfudobto ghborh ∓∆_ ( x  y  z) ∓ μ   x ˔ + 3μ  I I c  Q S(t ) 7 > ob akba±o hmkrh _ ( x  y  z) 7 4 =8 ∖ z, phrh roskevor osth buovh oluhld±kb kb dgpeogobthgks dgpeogobthgks oe gdsgk g ootkak ± tkak quo utdedzhgks hbtordkrgobto skek quo ek gkad﬈lhgks oe hefkrdtgk phrh b-adgobsdkbos, y ek quo kcsorvhgks os quo he hfrofhr oe pktobldhe frhvdthtkrdk eh phrt±ĸlueh ĸlueh vh lhyobak aobtr trk k ao aoee ffiuda ffiudak k po pork rk bk lho lho ob e±ĸĸboh boh ro rolt lthh sd bk g±hs h s cd cdob ob quo quo mh mhlo lo gulmhs trhyoltkrdhs ad`orobtos hbtos ao lhor he `kbak, lkgk so guostrh ob eh @DF.3. ^hrh buostrh torlorh sdguehld±k kb b buovhgobto tkghgks ehs gdsghs lkbadldkbos quo ob eks prkceoghs hbtordkros, skek quo

porhturh porht urh y vdslk vdslkldah ldaha a skb phrhgotrk phrhgotrkss dgpkr dgpkrthbto thbtoss phrh aotorgdbhr eh trhyoltkrdh ob eh adbhgdlh crkwbdhbh9 ostks frh˕ quo luhbak obbks k phrt± phrt±ĸluehs ĸluehs ao pkeob skb thb poquo˕ poquobhs trhb trhb ob lkbthl lkbthltk tk lkb oe hfu hfuh, h, sus trhyo trhyoltk ltkrdh rdhss osthb osthb h`o h`ollthahs pkr ehs gke±oluehs oluehs aoe ffiudak quo osth lkbstdtudak, phrh mhlorr ubh aoslrpl mhlo aoslrpld± d±k kb b cuobh cuobh aoe sds sdstog toghh roquor roquordgk dgkss ao eh adb±hbdgdlh hbdgdlh crkwbdhbh, lkgk gobldkbhgks lkb hbtordkrdaha ± ao Ehbfovdb 8, utdedzhgks eh oluhld± oluhldkb Ehbfovdb  8, phrh  phrh pkaor mhlor adlmh aoslrdpld±kb aoslrdpld k± b ao buostrh phrt± phrt±ĸlueh ĸlueh ao pkeob supkbogks quo buostrk strk frhbk frhbk ao pkeob pkeob os thb poquo˕ poquo˕b bk k quo eh ghsh tdobao h >, y ob prdgorh prdgorh hprkxd hprkxdghl ghld± d±k kb b mhlogks quo buostrk pktobldhe soh√  >, pkr ek thbtk eh oluhld±kb kb h roskevor os eh sdfudobto ∓μ   x ˔ + 3μ  II c  S(t ) 7 > ubh voz quo kctobogks tkaks buostrks phrhgotrks phrh oe hfuh h togporhturh hgcdobto 3> frhaks Loesdus, oe lko﬈ldobto ao vdslkldaha quo yh osth aotorgdbhak

osth voz buostrh phrt± phrt±ĸlueh ĸlueh ao pkeob osthr± osthr h± lhrfhah bofhtdvhgobto, y buostrh oluhld± oluhldkb k± b ao Ekbfovdb 8 Ekbfovdb  8  sor  sor±h± eh gdsgh skek quo buostrk pktoblhe quoah ao eh sdfudobto ghborh _ ( x  y  z) 7 q   o , chjk chjk ub pktob pktobldh ldhee ao Lkuek Lkuekgc gc ao ktr ktrhh 1ώ ο   x3 + y3 + z3 phrt±ĸlueh ĸlueh ghsdvh, ghsdvh, lhrfhah pksdtdvh pksdtdvhgobto,ob gobto,ob buostrk lhsk roogpehzhgks pehzh gks oe pktob pktobldhe ldhe frhvdth frhvdthtkrdk tkrdk pkr oe Lkue Lkuekgcdh kgcdhbk, bk, ao buovk dgpeogobthgks oe gotkak ao Oueor phrh vhrdhs adgobsdkbos, ahbak lkgk rosuethak eh trhyoltkrdh gkstrhah ob eh @DF. phrt± phrt±ĸluehs ĸluehs ao pkeob, y lkb eh lkbadld±k kb b ao quo luhbak eeofu±o mhcor hefubh

H.

Os Ostha thak k aoe Hrto Hrto

DD. DD.

GK GKAO AOEK EK

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@DF. 3: Ob osth fr± fr hh﬈lh ± ﬈lh so guostrh oe rosuehtak kctobdak he hpedlhr oe gotkak ao Oueor b-adgobsdkbhe phrh eh oluhld±k kb b ao Ehbfovdb, sdguehbak ub lhvdaha lkb hfuh y eh trhyoltkrdh ao lkekr hzue phrh eh phrt±±ĸlueh phrt ĸlueh ao pkeob.

@DF. @DF. 1: Kc Kcso sorv rvhg hgks ks ob eh fr fr±hh﬈lh ± ﬈lh oe lkgpkrthgdob lkgpkrthgdobtk tk phrh lhah ubh ao ehs phrt±ĸĸluehs luehs ao pkeob quo vhb lhyobak aobtrk aoe hfuh, hubquo ob hefub dbsthbto mhyh hefubh lkedsd±kb kb ehs eh s phrt±ĸĸluehs luehs sofudrhb gkvdobakso chjk eh adb± adbhgdlh h± gdlh crhwbdhbh, lhah lkekr os ubh phrt± phrt±ĸlueh. ĸlueh.

bkgcroo ek adlo os ubh `ubld±k bkgcr kb b heohtkrdh.

DDD.

@DF.. frhaks Loesdus, oe lko﬈ldobto ao vdslkldaha ao 8.>> phrt±ĸĸluehs luehs ao pkeob, ob lhsk ao quo eeofhrhb h lkedsdkbhr, h phrtdr ao oso gkgobtk so vkev±ĸĸhh h lheluehr ubh buovh trhyoltkrdh ob oe rofdgob crkwbdhbk. ,

lkedsd±kb obtro oeehs, so lkgpkrtob ob oe dbsthbto lkgk lkedsdkbos oehstdlhs o dbgoadhthgobto aospu±ooss buovhgobto ostob rostrdbfdahs pkr oe gkvdgdobtk crhwbdhbk, utdedzhbak oe hefkrdtgk rdtgk ao Oue Oueor or b-a b-adgo dgobsd bsdkbh kbhee kct kctobo obogks gks ehs sdfudo sdfudobto btoss trhyol trhyol-tkrdhslkbadldkbos phrh ehs phrt± ĸĸluehs luehs gkstrhahs ob eh @DF.1 chjk ehs gdsghs phrh oe hfuh.

H.

Phrd hrdhbt hbtos os

Ob osth pr±hltdlh hltdlh quo so mdzk vhrdhr ob tkahs ehs sdguehldkbos quo so rohedzhrkb, `uo oe pktobldhe, yh `uorh quo ehs phrt±ĸluehs ĸluehs bk osthchb skgotdahs chjk bdbfub pktobldhe, yh `uorh oe pktobldhe frhvdthldkbhe lorlh ao eh supor﬈ldo ao eh Qdorrh, oe pktobldhe oe oeoltrdlk k gojkr adlmk oe pktobldhe ao Lkuekgc, puos phrh tkahs ehs sdguehldkbos tkghgks lkbadldkbos bkrgheos phrh oe hfuh, akba±o sus phrhgotrks yh ost±hb hb aotorgdbhaks, pkr ojogpek eh togporhturh, oe lko﬈ldobto ao vdslkldaha, otl. T phrh eh `ubld±kb kb heohtkrdh ao tdpk Fhussdhbh su prkpdk

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DP DP..

SO\_E SO\_EQ QHAK\

^rdgorh sdgu ^rdgorh sdguehldk ehldkb, b, ubh phrt± phrt±ĸlueh ĸlueh ao pkeob, pkeob, chjk bdbfu bdbfub b pktobldhe, aojhah lhor ub hfuh ob lkbadldkbos bkrgheos. \ofubah sdguehld±k kb, b, ubh phrt±ĸĸlueh lueh chjk ub pktobldhe frhvdthldkbhe, ob hfuh chjk lkbadldkbos bkrgheos. Qorlorh orlorh sdguehld sdguehld±k± b, ubh ubh phrt phrt±±ĸĸlue luehh ch chjk jk ub pkto pktobl bldh dhee Lkuekgcdhbk, ob hfuh chjk lkbadldkbos bkrgheos. _etdgh sdguehld± sdguehldkb, k± b, 8> phrt± phrt±ĸluehs ĸluehs ao pkeob chjk ub pktobldhe frhvdhthldkbhe, ob hfuh ob lkbadldkbos bkrgheos.

 

1

@DF. ?: ^hrt @DF. ^hrt±±ĸĸlueh lueh quo so aojh lhor ob hfuh, chjk bdbfub pktobldhe, guostrh guost rh poquob bhs ˕ hs vdcrhldkbos heroaoakr ao su pksdld±k kb b krdfdbhe.

@DF. =: \o guostrh eh trhyoltkrdh ao 8> phrt± phrt ±ĸluehs ĸluehs ao pkeob, chjk ub pktobldhe frhvdthldkbhe

P.

LKBL LKBLE_\D E_\DKBO\ KBO\

Eh sdg sdgueh uehldk ldkbos bos quo roh rohedz edzhgk hgkss skb chs chsthb thbto to pro prolds ldshs, hs, hubquo bk so utdedzk ob bdbfub gkgobtk tokroghs prkchcdedstdlks, yh quo phrh ub ostuadk ostrdltk ao eh adb±hgdlh hgdlh crkwbdhbh y phrh ubh ghykr oxhltdtua so roqudoro ao ehs eoyos quo rdfob eh prkchcdedaha, pkaogks lkbleudr quo ob foborhe buostrhs sdguehldkbos chjk ad`orobtos pktobldheos, ob hfuh h lkbadldkboss bkrgh ldkbo bkrgheos eos rosuoevo rosuoevo oe prkceo prkceogh gh ao ehs phrt phrt±±ĸluehs ĸluehs ao pkeob lhyobak ob luhequdor ffiudak, puos eks ubdlks phrhgotrks quo lhgcdob skb eh togporhturh y eh vdslkldaha eh luhe osth aotorgdbhah pkr eh togporhturh, phrh eh adb± adbhgdlh h± gdlh crkwbdhbh, akbao buostrk gkaoek os ubh guy cuobh hprkxdghld kb k± b aoe lkgpkrthgdobtk ao eh adb±hhgdlh gdlh ostuadhah.

@DF. 0: \o guostrh eh trhyol trhyoltkrdh tkrdh ao ubh phrt phrt±±ĸluh ĸluh ao pkeob chjk ub pktobldhe frhvdthldkbhe.

PD.

HFSHA FSHAOLDG OLDGDOBQ DOBQK\ K\

Xuorogks mhlor bkthr eh cuobh lkekckrhld±k kb b y fu±ĸĸhh ao aoee prkpr k`oskr @orbhbak Cust Custhgobt hgobtoo Lhst Lhsth˕ h˕boah, boah, qudob dgphrto lehsos ob eh `hluetha ao Ldobldhs, pkr crdbahrbks su hyuah h obtobaor eks `obkgobks bhturheos, lkgk ob osto lhsk.

PDD. PDD.

@DF.. 6: Ob eh frh @DF frh﬈lh ﬈lh so guo guostr strhh eh trhyo trhyoltk ltkrdh rdh ob lkekr lkekr hzue ao eh phrt±ĸĸlueh lueh ao pkeob, chjk eh hlld k kb ± b ao ub pktobldhe Lkuekgcdhbk,

± CDC CDCEDK EDKFSH FSH@ @DH

So`.8 Qmkrbtkb, \topmob Q.9 Ghrdkb, Jorry C. (3>>-?=40-0. So`.3 @oybghb, Sdlmhra (8444). Qmo @oybghb Eolturos kb ^mysdls. ^orsous ^ucedsmdbf. D\CB >-6>43-8 So`.< Fkeastodb, Morcort9 Lmhreos ^. ^kkeo9 Jkmb E. \h`ik (3>>3) (3> >3).. Leh Lehssd ssdlhe lhe Golmhb Golmhbdls dls (8-0?6>3--048->?4?)9 G. \gkeulmkwsid, Hbb. ^mys. (Eodpzdf) 6?0 (84>0).

 

? So`.0 J. ^orrdb, L.S. Hlha. \ld.,810, \ld.,810, 406 (84>=)9 J. ^orrdb ^orrdb,, Hbbheos ao Lmdgdo y ao ^mysdquo, 8=,8 (84>4)9 J. ^orrdb, Eos htkgos. Oa. Helhb, ^hrds 848=).