Universitatea Transilvania din Braşov HABILITATION THESIS

Transcription

1 Uvese Tslv d Bşov HABILITATION THESIS Coecos o Ce Ieules Reled o Cove Fucos d o Ie Poduc Sces Dom: hemcs Auho: Assoc Po PhD Tslv Uves o Bșov BRAŞOV 07

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3 CONTENT Ls o oos 4 (A) Asc 6 (B) Scec d oessol chevemes d he evoluo d develome ls o cee develome (B-) Scec d oessol chevemes 0 Che Ieules develoed om cove ucos 0 Aou he Heme-Hdmd eul Feé e eules o cove ucos 3 Two evese eules o Hmme-Bulle s eul 4 4 Youg e eules 8 5 Güss-e eules dscee om d egl om 4 5 A eeme o Güss s eul v Cuch Schwz s eul o dscee dom vles e cse 5 5 Aou he ouds o sevel sscl dcos 9 53 A geelzed om o Güss e eul d ohe egl eules 33 Che Ieules o ucols d eules o vele osve oeos 38 Ieules o ucol 38 The Jese ucol ude sueudc codos d he Jese ucol eled o sogl cove uco 39 Sevel eules o geelzed eoes 47 Ieules o vele osve oeos 57 Che 3 Ieules e oduc sce 70 3 O he Cuch - Schwz eul e oduc sce 70 3 Revese eules o he Cuch-Schwz eul e oduc sce Cosdeos ou he sevel eules e oduc sce79 34 Sevel eules d chcezo o e oduc sce 83

4 (B-) The evoluo d develome ls o cee develome 4 Fuue decos o esech 88 4 Fuue decos o esech eled o Heme-Hdmd s eul d Hmme-Bulle s eul 88 4 Fuue decos o esech eled o Youg s eul d Hd s eul Fuue decos o esech eled o eules e oduc sce 03 Coclusos0 (B-) Blogh 3

5 Ls o oos R : he se o el umes C : he se o comle umes [ ]: evl N * : he se o osve eges R : he se o el umes R + : he se o oegve el umes R : he se o ozeo el umes e(): he eoel uco log(): he loghmc uco wh he se e R : Euclde -sce (R) (C) : sces o -dmesol mces de A : deem o A : l devve k A(s ) G(s ) H(s ) : hmec geomec d hmoc mes I(s ) : dec me L(s ) : loghmc me (s ): Hölde (owe) me : R([ ]) : he sce o Rem-egle ucos o he evl [ ] C 0 ([ ]) : he sce o el-vlued couous ucos o he evl [ ] L : he sce o egle ucos o he evl [ ] wh B(H): lge o ouded le oeos o el Hle sce H : e oduc : om o A# B : us-hmec owe mes o oeos A B : weghed hmec me o oeos A! B : weghed hmoc me o oeos A# B : weghed geomec me o oeos A # B : geomec me o oeos H : he Tslls eo H : he Sho eo R : he Ré eo D : he usle elve eo R : he Ré elve eo D : he Tslls elve eo I : he Tslls usle eo (-usle eo) 4 d

6 D : ed o oo : he Tslls usle elve eo 5

7 Asc I hs hlo hess we hve desced he sgc esuls cheved he uho e og hs PhD degee hemcs om Smo Solow Isue o hemcs o he Rom Acdem 0 Ieules Theo eeses old oc o m mhemcl es whch sll ems cve esech dom wh m lcos The sud o cove ucos occued d occues cel ole Ieules Theo ecuse he cove ucos develo sees o eules The esech esuls eseed hee e coceed wh he moveme o clsscl eules esulg om cove ucos d hghlghg he lcos A uco : I R o ll I 0 whee I s evl s clled cove we hve Reled o ol heo cove uco led o he eeced vlue o dom vle s lws less h o eul o he eeced vlue o he cove uco o he dom vle Ths esul kow s Jese's eul udeles m mo eules Aohe mo esul eled o cove uco s he Heme Hdmd eul due o Heme [07] d Hdmd [99] whch sses h o eve couous cove uco : R he ollowg eules hold: 6 d Reled o he Heme Hdmd eul m mhemcs hve woked wh ge ees o geelse ee d eed o dee clsses o ucos such s: us-cove ucos log-cove -cove ucos ec d l o secl mes (loghmc me Solsk me ec) The hlo hess s ocused o he sud o mo eules om Ieules Theo d o he mc some lcos The hess cosss o ou ches I lso cludes ls o oos d logh wh eeeces I he s o hs hess we hve eseed he scec d oessol chevemes d he evoluo d develome ls o cee develome The s che sudes he eules develoed om cove ucos Ths che cos sevel ogl esuls m o hem ulshed ISI ouls These sudes e lked o sevel eules such s he Heme-Hdmd eul he Feé eul Hmme-Bulle s eul d Youg s eul I he ls o hs che we ese sevel Güss-e eules dscee om d egl om Hee we show eeme o Güss s eul v Cuch Schwz s eul o dscee dom vles e cse I he ed we hve lzed he ouds o sevel sscl

8 dcos d we hve gve geelzed om o Güss e eul d we hve oed ohe egl eules The secod che sudes he eules o ucols d eules o vele osve oeos Hee hee e eseched he Jese ucol ude sueudc codos d he Jese ucol eled o sogl cove uco We hve show sevel eules o geelzed eoes Geelzed eoes hve ee suded m eseches Ré [9] d Tslls [0] eoes e well kow s oe-mee geelzos o Sho s eo eg esvel suded o ol he eld o clsscl sscl hscs [0 04] u lso he eld o uum hscs [98] We hve lso suded he eules o vele osve oeos h hve lcos oeo euos ewok heo d uum omo heo The hd che eloes he eules e oduc sce (e- Hle sce) We emk he sud o he Cuch - Schwz eul e oduc sce d some evese eules o he Cuch-Schwz eul e oduc sce We lso mke cosdeos ou sevel eules d we meo chcezo o e oduc sce I he secod o hs hlo hess we hve eseed he evoluo d develome ls o cee develome The ls che emes sevel uue decos o esech We hve deed hee uue decos o esech mel: uue decos o esech eled o Heme-Hdmd s eul d Hmme-Bulle s eul; uue decos o esech eled o Youg s eul d Hd s eul d uue decos o esech eled o eules e oduc sce The sud s ed so s o move some esuls o clsscl eules Ogl esuls o hs hlo hess hve ee ulshed ouls such s: Aeu h I J Nume Theo J Ieul Al h Ieul J h Ieul Ge h Al h I Sc ec 7

9 Rezum Î cesă eză de le m descs ezulele semcve oțue de uo duă ce oțu lul de doco î memcă l Isuul de emcă Smo Solow l Acdeme Româe î ul 0 Teo eglățlo eeză u suec vech l mulo dome memce ce ămâe u domeu de cecee cv cu mule lcț Sudul ucțlo covee ocu ș ocuă u ol cel î eo eglățlo deoece ucțle covee dezvolă o see de eglăț Rezulele ceceălo ezee c se eeă l îmuăățe eglățlo clsce ce ezulă d ucțle covee ș evdețee lcțlo ceso O ucțe : I R î ce I ese u evl se umeșe coveă dcă vem eu oce I 0 Leg de eo olăț o ucțe coveă lcă l vloe șeă ue vle leoe ese îodeu m mcă su eglă cu vloe șeă ucțe covee vle leoe Aces ezul cuoscu su umele de egle lu Jese să l z mulo eglăț moe U l ezul mo leg de ucț coveă ese egle Heme- Hdmd doă lu Heme [07] ș Hdmd [99] ce mă că eu oce ucțe coveă couă : R vem umăoe egle: d Leg de egle Heme-Hdmd mulț memce u luc cu me ees l geelze e s edee cese eu dee clse de ucț cum : ucțle cvs-covee ucțle log-covee ucțle -covee ec ș lce lo eu med secle (med logmcă med Solsk ec) Tez de le se eză e sudee eglățlo moe d eo eglățlo ș mculu ceso î uele lcț Tez cosă d u cole De semee clude o lsă de oț ș o loge cu de eețe Î m e cese lucă m eze elzăle șțce ș oesole ș lule de evoluțe ș dezvole eu dezvole cee Pmul col sudză eglățle ezule d ucțle covee Aces col coțe m mule ezule ogle mule de ele ulce î evse ISI Acese sud su lege de câev eglăț ecum: egle Heme- Hdmd egle Feé egle lu Hmme-Bulle ș egle lu Youg Î ulm e cesu col ezeăm m mule eglăț de Güss î omă dsceă ș î omă eglă Ac vom ă o e eglăț lu Güss egle Cuch-Schwz eu vle leoe dscee î czul Î l m lz mgle m mulo dco ssc ș m d o omă geelză eglăț de Güss ș m oțu le eglăț egle 8

10 Î l dole col sudem eglățle eu ucțole ș eglăț eu oeo vesl ozv Ac ese ceceă ucțol Jese î codț de sueăce ș ucțol Jese legă de o ucțe uec coveă Am ă m mule eglăț vd eole geelze Eole geelze u os sude de mulț ceceăo Eole Ré [9] ș Tslls [0] su e cuoscue c geelză cu u meu le eoe lu Sho d sude esv u um î domeul clsc l zc ssce [0-04] c ș î domeul zc cuce[98] De semee m sud eglățle eu oeo vesl ozv ce u lcț î: ecuțle oeolo eo ețelelo ș eo cucă omțlo Al ele col eloeză eglățle î-u sțu vecol îzes cu odus scl (ehle) Remcăm sudul eglăț Cuch-Schwz îu sțu vecol îzes cu odus scl ș uele eglăț vese eu egle Cuch-Schwz î-u sțu ehle De semee cem câev cosdeț cu ve l m mule eglăț ș mețoăm o cceze uu sțulu vecol îzes cu odus scl Î dou e cese eze de le m eze lule de evoluțe ș dezvole eu dezvole cee Ulmul col lzeză m mule decț voe de cecee Am dec e decț voe de cecee ș ume: voe decț de cecee lege de egle lu Heme-Hdmd ș egle lu Hmme-Bulle; voele decț de cecee eeoe l egle lu Youg ș egle lu Hd ș decțle voe de cecee eeoe l eglățle d-u sțu vecol îzes cu odus scl Sudul lo ese ț eu îmuăăț uele ezule vd eglățle clsce Rezulele ogle le cese eze de le u os ulce î evse ecum: Aeu h I J Nume Theo J Ieul Al h Ieul J h Ieul Ge h Al h I Sc ec 9

11 (B) Scec d oessol chevemes d he evoluo d develome ls o cee develome (B-) Scec d oessol chevemes Che Ieules develoed om cove ucos The sud o omzo olems s dsgushed ume o oees chcezed cove ucos These ucos l mo ole m es o mhemcs The cove ucos develo sees o eules A uco : I R whee I s evl s clled cove he le segme ewee wo os o he gh o he uco les ove o o he gh Euvlel uco s cove he se o os o o ove he gh o he uco s cove se I c we hve o ll I 0 As lcos o cove uco we hve he ollowg: eve om s cove uco he gle eul d osve homogee; he log de o he dom o osve-dee mces s cove; ohe uco emle s Eule s gmm uco 0 e d 0 ( c Eule s gmm uco s log-cove uco e we hve I 0 ); uco : I R o ll s log-cove he s lso cove; eled o ol heo cove uco led o he eeced vlue o dom vle s lws less h o eul o he eeced vlue o he cove uco o he dom vle Ths esul kow s Jese's eul whch udeles m mo eules s gve s: o el cove uco umes s dom d osve weghs w w w we hve: w w () w w Whe we hve w w w he we deduce he clsscl v o Jese's eul: 0

12 () Aou he Heme-Hdmd eul As cul cse Jese's eul o eul () we hve: () A mo esul eled o eul () s he Heme Hdmd eul due o Heme [07] d Hdmd [99] whch sses h o eve couous cove uco : R he ollowg eules hold: () d Hd Llewood d Pól eseed he ook [06] he ollowg esul whch chcese he cove ucos gve : Theoem A ecess d suce codo h couous uco e cove ( ) s h (3) d h o h < + h I c e show h hs esul s euvle o he s eul () whe s couous o [ ] Reled o he Heme Hdmd eul m mhemcs hve woked wh ge ees o geelse ee coue d eed o dee clsses o ucos such s: us-cove ucos log-cove -cove ucos ec d l o secl mes (loghmc me Solsk me ec) I he moogh [5] Dgom d Pece eseed m chcezos o he Heme-Hdmd eul Io Rș [65] mde he ollowg emk coeco wh he ove eeme o Heme-Hdmd eul: : R he (4) c c d h h s cove uco o eve c 0 d c s mml wh hs oe 4 A sees o oos d movemes o he Heme-Hdmd eul wee gve ove me (see [ ]) I [] Bessee led Heme- Hdmd eul o smlces d Bessee d Páles eslshed [] sevel eules o Heme-Hdmd e o geelzed cove ucos A eeso o he Heme-Hdmd eul hough sumoc uco ws lso gve hălescu d Nculescu [39] The Heme-Hdmd eul s he sg o o Choue s heo [66]

13 Beoe sg he esuls we ecll some useul cs om leue Dgom Ceoe d Soo ese [56 57] he ollowg esmes o he ecso he Heme-Hdmd eul: Pooso Le : R e wce deele uco such h hee es el coss m d so h m " The (5) d (6) m 4 m d 4 d These eules ollow om he Heme-Hdmd eul o he cove ucos m d Theoem 3 (culee-o [45]) Le : R e wce deele uco such h hee es el coss m d so h m " The (7) m o ll 0 Remk 4 B egg ech em o he eul (7) o [0 ] wh esec o he vle we ecove he eul (6) Cooll 5 (culee-o [45]) Pesevg he oo o Theoem 3 he ollowg eules hold: (8) m 8 8 o ll 0 Remk 6 Noce h egg ll ems o he eul (8) o [0 ] wh esec o he vle we ecove he eul (5) The ollowg esul cooes he clssc seme o he Heme-Hdmd eul Cooll 7 (culee-o [45]) Suose : R s deele d cove The (9) d g d 0 d (0) d d 0 o ll Feé e eules o cove ucos Feé [7] sudg goomec olomls oed some eules whch geelse he Heme-Hdmd eul d hus eslshed he ollowg well-kow weghed geelzo:

14 Theoem I : R s couous d cove d g : R egle d smmec wh esec o he le / g / g / The () g d g d g d s h s oved he ove esuls he e [45] we hve show ohe eules o Feé e: Theoem (culee-o [45]) Le : R e wce deele uco such h hee es el coss m d so h m " Assume g : R s egle d smmec ou The he ollowg eules hold: m () g d m 8 d g d g d g (3) g d g d g d g d Remk 3 Fo he cul cse g evls we ge: 8 d we l Theoem o he (4) m d whch eeses moveme o he Hmme-Bulle eul [66] gve : (5) d The ollowg heoem gves ew Feé-e eules Theoem 4 (culee-o [45]) Le : R uco wh " 0 d g : R e deele cove e couous The he ollowg semes hold ) I g s mooocll decesg he (6) g d g d ) I g s mooocll cesg he g d g d 0; (7) g d g d g d g d 0 o ll We ed hs seco wh weghed seme o kow esul coceg cove ucos I he lgh o Pooso he ollowg seme es s vl geelzo o esul due o Vsć d Lckovć [05] d Luș [8] (c Pećć e l [76]) d we om s oo 3

15 Pooso 5 Le d e wo osve umes d Le g : R e egle d smmec ou A The he eules A (8) g d g A A A d hold o 0 d ll couous cove uco : R d ol m Ths eul s due o Bee d Alze [5] Fom eul (6) led o he cove uco wh 0 \ we hve (see [45]) (9) A S A S whee Hee A / / A g A s he owe me d S 0 s he Solsk me Also he lm cse (o we m euvlel s he cse o he cove uco ) gves us (0) H L H L whee H s he hmoc me d L s he log log loghmc me Some o he evous esuls whee meoed he ollowg es hus: [68] Nezgod eslshed some geelzos o Feé eul o cove seueces {69} he gve sevel eules o cove seueces d odecesg cove ucos d [] Ku e l oud ew eules o Heme-Hdmd-Feé e o hmocll cove ucos v col egls d 3 Two evese eules o Hmme-Bulle s eul Fo ce coss o [5] Dgom d Pece oud moveme o Hmme-Bulle s eul gve he ollowg: Theoem 3 Le : R e wce deele uco such h hee es el coss m d so h m " The he ollowg eules hold: (3) m 4 d 4 Ths esul ws meoed Remk Ne we ovde wo evese eules o Hmme-Bulle s eul 4

16 Lemm 3 (culee-dcu-rțu [46]) Wheeve : R deele uco we hve he ollowg eul: s wce (3) whee d " d Remk 33 ) Clel o oe hs 0 eleme comuos oe os: Theeoe o eve 4 we c we 3 B some m " Iegg om o mullg / d usg elo (3) we o he eules om (3) ) Ieules (3) c lso e oed lg he Hmme-Bulle eul o he cove ucos m d I he ollowg we gve evese eul o Hmme-Bulle s eul Theoem 34 (culee-dcu-rțu [46]) Le : R e wce deele d cove uco The he ollowg eul holds (33) ' ' d 6 Alg he eul o Güss (see [98]) we o he ollowg: Theoem 35 (culee-dcu-rțu [46]) Le : R e wce deele uco d ssume hee es el coss m d such h: m " The o ll (34) ' ' m d 6 64 I we cosde Ovousl s cove uco Accodg o Theoem 34 oe hs: A A S 6 5

17 / whee A s he hmec me d S 0 s he Solsk me Fo log 0 we hve h s cove uco Alg Theoem 34 o we d he eul A 6 G e I / whee G s he geomec me d I e s he dec me I [culee-floe-fuuch 47] ou uose ws o eslsh sevel eules eled o Heme-Hdmd eul We lso oved geelzo o he Hmme-Bulle eul d Le H H : R e wo ucos deed : H H Lemm 36 Le : R " 0 o ll The we hve h he ucos H d oegve d cove Sce he ucos H d d d e wce deele uco such h H e H Hdmd eul we o he ollowg: e cove he lg he Heme- Theoem 37 (culee-floe-fuuch [47]) Le : R deele uco such h " 0 o ll (35) d H d d (36) d d 0 H d 0 e wce The we hve d Remk 38 B ddg elos (35) d (36) we deduce he ollowg eul: 6

18 (37) d H H d 0 d : e wce The we hve Theoem 39 (culee-floe-fuuch [47]) Le R deele uco such h " 0 o ll (38) d (39) whee d d 4 H d 3 H s H d 0 4 H d 3 H s H d 0 s To geelze he ove esuls we c eed he ucos H H : R o he ucos H gh g : R whch e deed : H g g d g d d H g d g d I we ke he ollowg ucos: H gh g : R deed : H g g d g d d H g d g d he we deduce he ollowg: Theoem 30 (culee-floe-fuuch [47]) Le : R e wce deele uco d g : R s deele uco smmec ou The he ollowg eules hold: (30) g d g d g d 4 g d g d 7

19 We lso eslshed esmo o Fée eules o dee kds o ucos I hs coe we show leve oo d geelzo o Theoem 4 [4] cosdeg he egl Rem-Seles Fuue decos o esech eled o Hmme-Bulle s eul wll e lzed he e [culee-nezgod-o 4] 4 Youg e eules The Youg egl eul s he souce o m sc eules Youg [08] oved he ollowg: Theoem 4 Suose h : 0 0 s cesg couous uco lm The such h 0 0 d 0 0 (4) d d Y ; Thee hs ee much wok o dee oos d geelsos o (4) (Bulle [7] d ovć e l [55]) I s es o see h elo (4) s lowe oud o he Youg ucol Y (;) I 974 ekle [38] showed h hee co e ue oud o Y (;) whch s deede o He oves he ollowg heoem whch ovdes evese eul Suose he codos o Theoem 4 hold The (4) Y ; m Lemm 4 I sses he ssumos o Theoem 4 he 0 0 (43) d d Y ; We emk he elo: (44) Y ; Y ; Wkowsk hs e [06] showed ohe evese Youg s egl eul hus: ude he ssumos o Theoem 4 he eul (45) Y ; holds wh eul d ol I [40] guzz geelzes hs eul Usg coveel eul (45) o we d he ollowg eul: (46) Y ; Comg elos (44) d (46) we o g eul (45) Ag Wkowsk [06] gve ohe esul eled o Youg s egl eul hus ude he ssumos o Theoem 4 he eul d m 0 0 (47) m d holds wh eul d ol 8

20 Ceoe [33] oved h he ue oud oed Wkowsk gve (45) s lws ee h h o ekle (4) Fo Theoem 48 we deduce he eul: (48) Fo m m Theoem 4 we deduce he Youg eul: (49) o ll 0 d wh guzz [40] oved evese Youg s eul he ollowg w: (40) 0 o ll 0 d wh u u Ths eul s euvle o he ollowg eul o : u I 0 d 0 9 u we chge d d he he Youg eul ecomes: (4) Bu hs s ue whe 0 d 0 Esecll whe we lk ou Youg's eul we wll ee o he ls om Ne we ese some eemes d some evese eules o Youg s eul whch we hve used ou esech Oe o evese eules o Youg eul ws gve Tomg [00] usg he Sech o he ollowg w (4) S o osve el umes d 0 whee he Sech o [78 93] ws deed o osve el ume h lms h Noe h h h h S h h elog h h d S(h) = S(/h) > o h h 0 We cll he eul (4) o-e evese eul o Youg s eul Tomg lso showed [00] he ollowg eul: (43) Llog S o osve el umes d 0 whee he loghmc me [6] L() s deed

21 L L log log We cll he eul (43) deece-e evese eul o Youg s eul Bsed o he scl eules (4) d (43) Tomg showed wo evese eules o vele osve oeos I [Fuuch-culee 76] we eseed wo eules whch gve wo dee evese eules o Youg s eul mel: (44) 0 e m d (45) 0 log whee 0 m m m o ll 0 The ove esuls e he cul cses o he ollowg heoem om [Fuuch-culee 76]: Theoem 43 Le : R e wce deele uco such h hee es el cos so h 0 " o The he ollowg eules hold: 0 (6) o ll 0 Fo Cwgh-Feld s eul (see e g [30]) m e we s ollows: (47) m whee 0 m m m o ll 0 Ths eul s moveme o Youg s eul d he sme me gves evese eul o Youg eul Remk 43 The s eul o (47) clel gves moveme o he s eul (45) d (46) Fo 0 < < we d he gh hd sde o he secod eul o (47) gves ghe ue oud h h o (46) om he eul o > 0 Fo > we d he gh log log hd sde o he secod eul o (45) gves ghe ue oud h h o (47) om he eul o > 0 I ddo we d he log log gh hd sde o he secod eul o (47) gves ghe ue oud h h o (45) o 0 om e Ne we ocus o wo mmede cul cses o Theoem 3 (culee- o [45]) h hel us o gve movemes o he well kow hmecgeomec me eul (lso kow s Youg s eul) ) We l elo 47 o he uco : R ( > 0) deed log whch leds o (48) e 0 e

22 Sce e 0 we o eeme o Youg s eul whee We lso oed evese eul o Youg s eul : loglog e d we ve he ollowg eul: ) Ne we l elo 47 o he uco R (49) log log deed whee 0 d 0 Youg s eul ws eed Keh d sh [6] hus whee 0 d m The use hs eul o he sud o m om eules I [78] Fuuch moves eul (4) hus (4) S whee 0 d m d he uco S ws gve ove Koe oved [9] geel esul eled o moveme o he eul ewee hmec d geomec mes whch o = mles he eul: (40) (4) whee 0 d m Ths eul ws edscoveed Keh d sh [6] A geelzo o eul (4) c e oud e o Aldz [] I [culee 5] we ese ohe moveme o Youg s eul d evese eul s ollows (43) 0 d m Ths eul c e eseed wh Koovch cos: K h K h o he osve el umes d (44) h whee >0 0 m Kh d h Noce h he 4h s eul (44) ws oed Zou e l [] whle he secod ws oed Lo e l [4] Fll we gve [culee 5] ohe moveme o Youg s eul d evese eul gve s: (45) Alog Blog whee 0 m

23 d B A 4 4 Remk 44 ) Sce A 0 d B 0 we 4 4 o eeme o he Keh-sh eul d eeme o Youg s eul ) Ieules (48) d (49) gve wo movemes o Youg s eul c) Ieul (49) c e oud [culee-o 45] d m ohe e o Dgom (see e g [53]) d) Fo (49) we o (46) whee log log 0 B he sum o elos (49) d (45) we deduce 0 d (47) log log The Hez me [83] s deed s H whee 0 d 0 I s es o see h H Bu usg elo (47) we hve log log so we deduce log A H log Fom eul (45) we deduce ohe eul o he Hez me hus: (48) Alog A H Blog whee 0 m A A d B B 4 4 Ne we mke lle shess o some ece esuls ou Youg s eul I he ece e [09] Zho d Wu ovded wo eg ems o Youg s eul hus: Le 0 () I d 0 0 he 4 (440) () I he 0

24 4 (44) whee m d m 0 I he sme e we d he evese vesos o ove eules: Le 0 0 () I d 0 he 4 (44) () I he 4 (443) whee m d m 0 Que ecel [94] Sheh d osleh gve ull desco o ll ohe eemes o he evese Youg s eul hus: Le 0 0 () I d 0 he (444) S () I he (445) S whee s he gees ege less h o eul o d S k k k k k k k k s k k k k k k s k k k k k Fuuch Ghem d Ghkhlu gve [83] evese Youg s eul o R mel: Le 0 N such h d R The () I he (446) () I he (447) k k k k k k

25 5 Güss-e eules dscee om d egl om I hs seco we ove eul whch wll hels us d ew eeme o he dscee veso o Güss eul We hve lso coued he esech hs eld d we show some eules h hve ee oed ([culee-rțu- Pečć43] [culee-cudu 49]) The dscee veso o Güss eul [3 0 3] hs he ollowg om: 4 whee e el umes so h d o ll I 935 Güss (see [98]) oved he ollowg egl eul whch gves omo o he egl o oduc o wo ucos ems o he oduc o egls o he wo ucos: Le d g e wo ouded ucos deed o [] wh d g whee e ou coss The we hve: g d d g d d he eul s sh he sese h he cos /4 c e elced smlle oe Ae he ume o es ulshed hee c e oced ge ees o hs eul I s well kow h mo esouce o sudg eules s [ ] I [8] Peg d o eslshed om o eul o Guss e o ucos whose s d secod devves e soluel couous d he hd devve s oud Also [59] Dgom eseed sevel egl eules o Guss e d [60] he showed some Guss e eules e oduc sces d lcos o he egl Aohe moveme o Guss eul ws oed ece [36] oeove [5] Guss e eul ws used ode o o some sh Osowsk-Guss e eules Lu Kechos d Delss showed [3] sevel eemes o Guss eul e oduc sces usg Kue s esuls o Gms New geelzos o he eul o Guss wee eseed [47] usg Rem- Louvlle col egls Ceoe d Dgom suded [3] some eemes o Guss eul As lcle we o some oees o ouds o he vce he sdd devo he coece o vo d o he covce eled o sevel sscl dcos o dscee dom vles e cse ([culee-rțu-pečć43] [culee-cudu 49]) 4 4

26 5 A eeme o Güss s eul v Cuch Schwz s eul o dscee dom vles e cse The vce o dom vle wh oles P o s s secod cel mome he eeced vlue o he sued devo om me E : V E 5 The eesso o he vce c e hus eded: V E E We oe RV he se o dom vles wh oles P o The covce s mesue o how much wo dom vles chge ogehe he sme me d s deed s CovY E E Y EY d s euvle o he om CovY EY E EY Usg he eul o Cuch-Schwz o dscee dom vles we d he eul gve o he om Cov Y V V Y Y V V Y Cov Ne we show eeme o hs eul Lemm 5 I d Y e dscee dom vles e cse he hee s he ollowg eul (5) V( Y ) V( ) V(Y ) Cov( Y ) whee d e el umes Cooll 5 I d Y e dscee dom vles e cse he hee e he ollowg eules: (5) V( Y ) V( ) V(Y ) Cov( Y ) d (53) V( Y ) V( ) V(Y ) Cov( Y ) Remk 53 Fom elos (5) d (5) we d he llelogm lw ems o vce mel: (54) V( Y ) V( Y ) V( ) V(Y ) Lemm 54 I Y Z d T e dscee dom vles e cse he hee s he ollowg eul (55) Cov( YcZ dt ) ccov( Z ) dcov( T ) ccov(y Z ) dcov(y T )

27 whee c d d e el umes Theoem 55 (culee-rțu-pečć[43]) I Y d Z e dscee dom vles e cse wh kz he we hve he eul (56) Y Cov Z CovY ZV V V Z Cov Z Cov 0 V Y V Y Cov Poo Fo he dscee dom vles Y d Z gve e cse wh V 0 we ke he ollowg dom vle: Cov Y Cov( Z ) W Y Z We clcule he vce o dom V Z Z Cov Y Cov vle W hus: V W V Y d V V lg elo (5) we hve Cov Y Cov Z V W V Y V Z V V Cov V Cov V Y Cov Y Cov Z Cov Y Z V V Usg Lemm 54 we deduce he ollowg eul Cov Y Cov Z Cov Y Cov( Z ) Cov Y Z Cov( ) V V V V( ) Cov Y Cov( Z ) Cov( Z )Cov Y Cov(Y Z ) V V Cov Y Cov( Z ) Cov(Y Z ) V Reug o clcule he vce o dom vle W we hve Cov Y Cov Z V(W ) V Y V Z V V Y Cov Z Y Z V Cov Y Cov Z V Z V V Cov Y Cov( Z ) Cov(Y Z ) V Theeoe we deduce he eul V( )V(W ) V( )V Y Cov Y V( )V Z V( )Cov(Y Z ) Cov Y Cov( Z ) Sce V ( )V W 0 ollows h Cov Z V( )VZ Cov Z V( )Cov(Y Z ) Cov Y )VY Cov Y 0 V( o eve R Ths mles h Cov( Z ) 6

28 V( )VZ Cov Z V( )VY Cov Y (57) V( )Cov(Y Z ) Cov Y Cov( Z ) Tkg o ccou h ( )V Z Cov Z 0 dvdg ( )V Z Cov Z 7 V ecuse kz d V we o he eul o he seme Remk 56 Le Y d Z e dscee dom vles e cse wh V Y 0 d V Z 0 we ke he ollowg dom vle: Cov Y W Y Z he we hve he eul V Y (58) vege Y Cov Y Z Cov ZV Y VY V Z Cov 0 V Y Le e el umes ssume V Y Cov o ll d he I 935 Poovcu (see eg [0 84]) oved he ollowg eul V 4 (59) The dscee veso o Güss eul hs he ollowg om: (50) whee 4 e el umes so h d o ll Fom he elo Cov Y EY E E Y d usg he eul o Cuch-Schwz o dscee dom vles gve Cov Y V V Y we o oo o Güss s eul Bh d Dvs show [0] h he ollowg eul V (5) The eul o Bh d Dvs eeses moveme o Poovcu s eul ecuse 4 Theeoe we wll s hve moveme o Güss s eul gve he ollowg elo: (5) Y Y 4 I Y d Z e dscee dom vles e cse wh kz he we hve om eul (58) he ollowg elo: (53) Cov Y Cov Z CovY ZV Cov Y V V Y V V Z Cov Z Le z z z e el umes ssume o ll d o el ume k The lg eul (53) kz we deduce secod eeme o Güss s eul gve

29 8 (54) S whee C B A S wh z z A z z B d z z z z C Remk 57 I [3] Kechos d Delss demosed ohe eemes o he dscee veso o Güss eul Cooll 58 (culee-rțu-pečć[43]) I d Y e dscee dom vles e cse he hee s he ollowg eul (55) V(Y ) ) V( Y ) ( V Remk 59 Ths eul ems o sums ecomes Y Y Dvdg d mkg he ollowg susuos: d Y we o he eul whch s c he kowsk eul he cse 0 d 0 Cooll 50 I d Y e dscee dom vles e cse he hee s he ollowg eul (56) V(Y ) ) V( Y ) ( V Poo Fom elo (53) we hve Y ) Cov( V(Y ) ) V( Y ) ( V Y ) Cov( Y V V V(Y ) ) ( V Alg he eul o Cuch-Schwz o dscee dom vles we o V(Y ) ) V( Y V whch mles he eul o he seme I [6 7] he Lukszk Kmowsk mec s uco deg dsce ewee wo dom vles o wo dom vecos I cse he dom

30 vles d Y e chcezed dscee ol dsuo he Lukszk Kmowsk mec D s deed s: Y P P Y D Ne we wll gve ohe mec o he se RV We c look he se RV s veco sce The ul w s oducg d usg he sdd e oduc o RV The e oduc o wo dom vles d Y s deed Y Cov( Y ) The e oduc o wh sel s lws o-egve Ths oduc llows us o dee he "legh" o dom vle hough sue oo: Cov( ) V Ths legh uco sses he eued oees o semom d s clled he Euclde semom o RV A semom llowed ssgg zeo legh o some o-zeo vecos The se RV wh hs semom s clled semomed veco sce Fll oe c use he om o dee mec o RV d( Y ) Y V Y Ths dsce uco s clled he Euclde mec o RV Coseuel he se o dom vles RV om Hle sce d semomed veco sce Some o he evous esuls wee meoed he e [33] whee sed-jme d Ome eloe he oees o he covce ledg o ew clsses o eules cludg he Osowsk d Osowsk-Güss eules 5 Aou he ouds o sevel sscl dcos Sscl dcos l ve mo ole he chcezo o he vous ocesses: ecoomc socl d echologcl I sscs he geel oo o sceg (vce o dseso) we ee o he dvdul vlues o mesule devos om he cel vlue Ne we wll o some oees o ouds o he vce he sdd devo he coece o vo d o he covce eled o sevel sscl dcos o dscee dom vles e cse The esuls e develomes o he esech eseed (culee-rțu-pečć[43]) The weghed hmec me (me vlue) o dom vle wh oles P o d s gve E 9

31 P The vce o dom vle o d eeced vlue o he sued devo om me : wh oles s s secod cel mome he E V Sdd devo ( ) hs sml ole wh vege le devo u keeg he dseso chcescs; sscs used hs dco whch s clculed s me o dvdul devos sued om he cel edec d he evl ( ) s he medum evl o vo whee we V Coece o vo (CV()) s elve mesue o hve sceg whch desces he o ewee he sdd devo d he hmec me d s gve he omul: V CV E Two vles hve sog sscl elosh wh oe ohe he e o move ogehe Accodg o [69] coelo s mesue o le elosh ewee wo vles d Y d s mesued he coelo coece gve : Y I s es o see h Y Cov Y V VY Thee s he ollowg eul: 4 m d m m Fo wh (5) m whee d we deduce Poovcu s eul: m 4 Ths eul suggess ue oud o dcos o he vce he sdd devo he coece o vo d o he covce hus: m m m CV 4 d Cov Y m Q 4 (5) whee m d m Q The dscee veso o Güss eul he weghed om hs he ollowg om: 30

32 whee 4 mq e el umes so h m d Q o ll The egl v o eul o Güss [98] esdes lcos mhemcl lss hs some sscl d cul lcos We kow h he dscee veso o Güss eul hs he ollowg om: whee 4 mq e el umes so h m d Q o ll Thee e m cles whch eed hs eul egl v (see eg [4] [59] [60] [0] [36]) We wll ocus eo o he dscee veso o Güss eul eg movded useuless o hs eul we sud he eul o Güss he coe o elemes o sscs usg he coces o vce d covce o he dom vles Bh d Dvs show [0] o wh h: m Bu he eul o Bh d Dvs ems vld o Thus we deduce ue ouds ee h elo (5) hus: d m m (53) CovY m Q Y Y C V m wh I hs ee show [36] A cd ece h o dscee dom vles e cse we hve: (54) whee h h s he hmoc me o dscee dom vles e cse Fom [culee 44] elceme wh he coelo coece eul (55) we deduce he eul: Y Z Y Z Y Z Ne we wll ese sevel movemes o he ove eules eled o vce Pooso 5 Fo dscee dom vle e cse hee s he ollowg eul m g g (55) 3

33 3 whee he geomec me g s h vlue whch shows h we elce ech dvdul vlue he oduc would o chge d we hve he omul: g wh Poo I he e [30] Cwgh d Feld oved he ollowg eul: m whee 0 d Bu whch mles o he eul o he seme Remk 5 ) Fo hs eul we o g m whee g ) I s es o see h eul (55) s eeme o eul (54) ecuse he geomec me s hghe h he hmoc me Ieul (55) ovdes ohe oud o he vce u s ve dcul o come he ems g d m o see whch s ee Comg he ove eules d kg o ccou eul (55) we oud ohe ouds o he sdd devo he coece o vo d o he covce hus: (56) m g g (57) C m g V g d (58) g Y g Y Q Y Cov Now we w o d ue oud ee h he Bh d Dvs o he ove dcos Theoem 53 Fo dscee dom vle e cse hee s he ollowg eul (59) m m Poo We evlue he sum m d we deduce he ollowg:

34 so we hve m whch s euvle o he eul m m m m m m m m m 53 A geelzed om o Güss e eul d ohe egl eules I 935 Güss (see [98]) oved he ollowg egl eul: Le d g e wo ouded ucos deed o [] wh g whee e ou coss The we hve: 33 d (53) g d d g d d he eul s sh he sese h he cos /4 c e elced smlle oe I he ollowg esech o eg he Güss eul we used he sme wok mehods s he oes used he dscee veso The ollowg esuls wee eced om ou e [culee-cudu 49] Floe d Nculescu [70] eed he olem o esmg he devo o he vlues o uco om s me vlue The esmo o he devo o uco om s me vlue s chcezed ems o dom vles We deoe R([ ]) he sce o Rem-egle ucos o he evl [ ] d C 0 ([ ]) he sce o el-vlued couous ucos o he evl [ ] The egl hmec me o Rem-egle uco : R s he ume d I d h e wo egle uos o d h d 0 geelzo o he egl hmec me s he ume h h clled he h-egl hmec me o Rem-egle uco h d d 4 he

35 We d he ollowg oe o he h-egl hmec me o Rem-egle uco : k h k whee k s el cos I he uco s Rem-egle uco we deoe v( ) he vce o The eesso o he vce o c e eded hs w: v h d d I he sme w we deed he h-vce o Rem-egle uco vh ( ) h h The eesso o he h-vce c e hus eded: h d vh h d h d h d I s es o see ohe om o he h-vce gve he ollowg: vh ( ) h h d we hve v k v h whee k s cos I [9] Aldz showed eeme o he A-G eul d used he oo h / d / s mesue o he dseso o ou s me vlue whch s c comle o he vce / v whee / d The covce s mesue o how much wo Rem-egle ucos chge ogehe he sme me d s deed s cov g g g d s euvle o he om cov g g g g d d g h d / d 34

36 I c he covce s he Cheshev ucol ched o ucos d g I [3] s we s T( g) The oees o he Cheshev ucol hve ee suded Elezovć guć d Pečć he e [66] Fo ohe geelzos o Güss eul see [56 75] The h-covce s mesue o how much wo dom vles chge ogehe d s deed s cov g g g d s euvle o he om cov h g g g h h h h h 35 h g h h d h h h d g h I [74] Pečć used he geelzo o he Cheshev ucol oo ched o ucos d g o he Cheshev h-ucol ched o ucos d g deed T g; h Hee Pečć showed some geelzos o he eul o Güss he Cheshev h-ucol I s es o see h ems o T g; h cov g he covce hs c e we s I ems o covce he eul o Guss ecomes (53) cov g 4 Ad ems o Cheshev ucol he eul o Guss ecomes T g 4 I hee s ddol omo ou he me vlues o he wo ucos he eul o Güss he Zks gued hs e [0] h he eul c e sheed d he lso gve olsc eeo o Lemm 53 ([culee-cudu 49]) Le e Rem-egle uco deed o [] wh whee e wo coss The we hve: (533) v h 4 whee h : 0 s Rem- egle uco wh d 0 h h h Lemm 53 ([culee-cudu 49]) Le e Rem-egle uco deed o [] wh whee e wo coss d Remegle uco h : [0 ) wh h d 0 d The we hve he ollowg elos: h d h d (534) vh h d hd We c ove eul o egle ucos sml o he eul o Cuch-Schwz o dom vles gve he ollowg

37 Theoem 533 ([culee-cudu 49]) I g h R([ ]) he we hve he eul (535) cov g v v g h Pooso 534 ([culee-cudu 49]) Le d g e wo Remegle ucos deed o [] wh d g whee e ou coss d we hve Rem-egle uco h : [0 ) wh h d 0 h The we hve (536) cov g T g g g h h h h h h 4 Theoem 535 ([culee-cudu 49]) I g R([]) wh v h 0 he we hve he eul cov g cov cov (537) h h gvh v v cov 0 v k d h h h h h h h v g cov g Lemm 536 ([culee-cudu 49]) Le d g e wo Rem-egle ucos deed o [] The we hve (538) g h h Alg he eul ewee he hmec me d he geomec me d Lemm 536 we deduce he ollowg elo: Theoem 537 ([culee-cudu 49]) Le d g e wo Remegle ucos deed o [] The we hve 0 v v g cov g g (539) h h h h Ne we show eeme o Güss eul o omlzed sooc le ucol Thee e m decos whch he eul o Guss [98] hs ee geelzed Usg he oo o omlzed sooc le ucol whch es he e [5] we wll gve geelzo o he eul o Guss whch s eled o heoem o Adc d Bde [3] Le E e oem se L le clss o el-vlued ucos d g : E R hvg he oees: (L) g L ml (α + βg) L o ll αβ R (L) L e 0 () = ( ) E he 0 L A sooc le ucol ( [3] s clled osve dee ucol) A : L R s ucol ssg: (A) A(α + βg) = αa( ) + βa(g) o ll g L d αβ R (A) I L d 0 he A( ) 0 (A3) The mg A s sd o e omlzed A() = Theoem 537 ([culee-cudu 49]) Le L e such h L d ssume h hee es el umes d so h The o omlzed sooc le ucol A : L R oe hs he eul (530) A A A A h h h 36

38 Fom he eul o Cuch-Schwz o omlzed sooc le ucol [5] we hve o g g L whee g :E R d A : L R s omlzed sooc le ucol: (53) Ag A Ag Reled o coue o he Cuch-Schwz eul we hve he ollowg: Theoem 538 Le g g L such h g L d d g whee e gve el umes The o omlzed le sooc ucol A : L R oe hs he eul (53) Ag A Ag A A Ag Ag Fll we d sevel lcos Tkg o ccou he egl hmec me d h-egl hmec me o Rem-egle uco : [ ] R we c ewe he ollowg eules: ) I he cse whe 0 he egl om o he eul om Theoem 4 (see [7]) ws gve Theoem 5 Ude he codos o Theoem 5 he eul ecomes m m (533) g g () I [64] oc gve ew eeme o Rdo s eul Usg he egl om o he evese o eul om Theoem 5 (see [7]) we o o ( 0) m ( 0) m d g : [ ] R e wo egle ucos o [ ] wh g() > 0 ( ) [ ] couous uco o [ ] he eul m m (534) g g I ou e [Rțu-culee 89] we hve show sevel eemes d coues o Rdo s eul We eslsh h he eul o Rdo s cul cse o Jese s eul Sg om sevel eemes d coues o Jese s eul Dgom d Ioescu we o coue o Rdo s eul I hs w usg esul o Smć we d ohe coue o Rdo s eul We o sevel lcos usg oc s eul o move Hölde s eul d Luov s eul To deeme he es ouds o some eules we used l ogm o dee cses 37

39 Che Ieules o ucols d eules o vele osve oeos I ucol lss d he clculus o vos ucol s uco om veco sce o s udelg eld o scls Amog he mos suded ucols he heo o eules we emk he Jese ucol d Chechev ucol Ne we sud he Jese ucol ude sueudc codos d he Jese ucol eled o sogl cove uco Reled o oeos oeo mes ouded le oeo o comle Hle sce H whou seced We sud sevel oees whch ml he eslshme o eules ewee dee es o oeos Ieules o ucols I s el vlued uco deed o evl I 0 such h I d he he Jese ucol s deed J d he Chechev ucol s deed T Ude he codos om Deo 8 we hve deed he geelzed Jese ucol k k k Jk ( k k) := k k k d he geelzed Chechev ucol : k k k Tk ( k k) := k k k I [79] Pečć d Beesck dscuss ou he moooc oe o dscee Jese s ucol Dgom (see [58]) vesged oudedess o omlzed Jese s ucol h s ucol J ssg oed he ollowg lowe d ue oud o omlzed ucol: 0 m J J m J He 38

40 whee : K s cove uco o cove suse K o le sce K e osve el - d ules wh The Jese s eul c e egded moe geel me cludg osve le ucols cg o le clss o el vlued ucos The Jese ucol ude sueudc codos d he Jese ucol eled o sogl cove uco I hs seco he s we gve ece whch desces ue d lowe ouds o he Jese ucol ude sueudc codos Some esuls volve he Chechev ucol We gve moe geel deo o hese ucols d eslsh logous esuls These esuls wee show ou e [o-smeods-culee 58] Fo he ede s coveece le us el se kow cs egdg he cl ools sueudc d he Jese ucol See Amovch d Dgom [] o dels d oos Deo ([]) A uco deed o evl I 0 o sueudc o ech I hee ess el ume C() such h C () 39 0 s o ll I We s h s suudc uco s sueudc The se o sueudc ucos s closed ude ddo d osve scl mullco Emle ([3]) The uco s sueudc wh ' g > 0 s sueudc wh C() = 0 Also h() = log wh C() = h () = ( log + ) s sueudc uco (u o moooe d o cove) Some eleme ucos e o sueudc such s () = d () = e Lemm ([]) Le e sueudc uco wh C() deed s ove () The (0) 0 () I (0) = (0) = 0 he C() = () wheeve s deele > 0 () I 0 he s cove d (0) = (0) = 0 Deo 3 ([]) Le e el vlued uco deed o evl I le / C Smll I d le 0 e such h Jese ucol s deed (3) J d he Chechev ucol s deed (4) T The

41 Pooso 4 ([]) Le 0 d 0 wh I s sueudc he (5) J Theoem 5 ([o-smeods-culee 58]) Le e sueudc I 0 0 I 0 uco deed o evl o such h d el ume 0 d The we hve (6) Poo Le e sueudc uco wh C() deed s ove d el ume 0 The elcg whee 0 we deduce he eul C (7) Now eul (7) we mke he ollowg susuos: Theeoe we hve C ullg 0 hs eul d summg om we deduce he seme Remk 6 Fo we o eul om Pooso 4 Cooll 7 ([o-smeods-culee 58]) Le 0 e sueudc uco deed o evl I 0 o 0 I d 0 such h The we hve (8) J Poo Fo Theoem 5 we hve he eul (9) Fom Lemm we kow h s cove Theeoe lg Jese s eul we hve d 40

42 Usg hs eul d eul (9) we o eul whch mles he eul (8) oved he ove esuls we oduce ul w ohe ucols Deo 8 Assume h we hve el vlued uco deed o evl I he el umes o ll k k (we u = ( d e such h 0 )) = I o ll k d = ( k) > 0 e such h We dee he geelzed Jese ucol (0) k Jk ( k k) := k k k k d he geelzed Chechev ucol : () k k k k Tk ( k k) := k k k We lso esl oce h o k = hs deo educes o Deo 3 I [60] he ollowg esmo s oed: s cove uco he we hve kk () m Jk ( k k) kk k k Jk ( k k) kk m Jk ( k k) kk k k I hs seco we vesge ue d lowe ouds h we hve he uco s sueudc Now we eed he ele esuls The ollowg lemm desces he ehvo o he ucol ude he sueudc codo: Lemm 9 Le e s Deo 8 I s sueudc he we hve k 4

43 k (3) Jk ( k k) k k whee k 4 k k Usg he sme ece s he oo o Cooll 7 we ge: Cooll 0 ([o-smeods-culee 58]) Le e s Deo 8 Le 0 e sueudc uco deed o evl o I 0 0 I d 0 such h The we hve k k (4) Jk ( k k) k k k The e esul c e eessed s: Theoem ([o-smeods-culee 58]) Le e s Deo 8 d he osve el umes k d e such h o ll k m k k We u = kk kk o ll k m m k k I s sueudc uco he: (5) Jk ( k k) mjk ( k k) k k k m k m k k k k d (6) Jk ( k k) Jk ( k k) k k k k k k k k whee k kk kk d Remk Le = = k = d = = k = I hs cse we see h Lemm 9 ecove Pooso 4 oe esuls c e oud e [o-smeods-culee 58] I [8] Kluz d Nezgod uoed he ove esuls o he oduco d sud o Jees Csszá d Jese Csszá -dvegeces Some ouds o Cooks d L es o such dvegeces e ovded To hs ed he cocv o he comoso o moooe ucos s dscussed Ne we desce some esuls coceg ue d lowe ouds o he Jese ucol eled o he coce o sogl cove uco Deo 3 A uco deed o evl I s sogl cove wh modulus c 0 [o c-sogl cove]

44 (7) c o ll I 0 We cll sogl cove hee ess c 0 such h s sogl cove wh modulus c Sogl cove ucos wee oduced Polk [8] A uco s clled sogl cocve wh modulus c (o omel cove o ode [70]) s sogl cove wh modulus c Ovousl eve sogl cove uco s cove Ae ucos e o sogl cove The uco () = c + + s sogl cove wh modulus c d he eul (7) holds wh eul sg Accodg o H Uu d Leméchl [09] we hve: Pooso 4 The uco s sogl cove wh modulus c d ol he uco g () = () c s cove I [37] he ollowg esul s oved: Pooso 5 Cosdeg 0 wh d he uco sogl cove wh modulus c we hve (8) J c Ths s e-oved usg he olsc och e o R d Wsowcz [87 Cooll3] Noce h he se o sogl cove ucos s closed ude ddo d osve scl mullco I wh ollows we shll lso e eesed moe geel Jese ucol d s ehvou he coe o sog cove Theoem 6 ([o-smeods-culee 59]) Le e sogl cove uco wh modulus c deed o evl I I 0 such h The (9) o 0 d c oeove om (9) o we ge doule eul whch ees he eees-nkodem eul (8): Pooso 7 ([o-smeods-culee 59]) Le e sogl cove uco wh modulus c deed o evl I I d 0 such h The (0) c J c We se he ollowg lemm ou he ehvou o he geelzed Jese ucol ude he sog cove codo: 43

45 Lemm 8 ([o-smeods-culee 59]) Le e s Deo 8 I s sogl cove wh modulus c he we hve k () Jk ( k k) c kk whee k k k Fo sogl cove ucos we hve he ollowg ouds: Theoem 9 ([o-smeods-culee 59]) Le e s Deo 8 d he osve el umes k d e such h o ll k m k k We u = kk kk o ll k m m k k I s sogl cove uco wh modulus c he we hve: () Jk ( k k) mjk ( k k) k k k c k m k k k mc k d (3) Jk ( k k) Jk ( k k) whee k k k c k k k k c k k kk kk d We show [o-smeods-culee 59] some lcos o uco gmm o Eule The uco gmm s deed v covege moe egl s e 0 d o ll 0 s kow s Eule egl o he secod kd The ollowg e oduc deo o he gmm uco s due o Weesss e e whee s he Eule-scheo cos Ths elo c e we s (4) log log log whee he se o he loghm s e 44

46 Pooso 0 ([o-smeods-culee 59]) The uco deed : 0 R log c s sogl cove wh modulus o 0 Poo Fom elo (4) we ge (5) log log log We cosde he uco g deed o [ 0 ) I es o see h d g" g' log c c 4 0 The eul elds g " 0 heeoe g s cove so s sogl cove wh modulus o 0 I s sghowd h: Cooll The uco : 0 R log c s sogl cove wh modulus 0 o Ne we gve eules eled o sogl cove uco A mo eul s gve F C o [57] s cul cse o he Dgom eul [58] o cove uco o [] we hve he ollowg eul: (6) m m o ll 0 Lemm I s uco egle d cove o we hve he ollowg eul: (7) F F whee F o eve Poo Fo 0 whe we hve 45

47 46 m d m I we elce hese eul (6) we ove he eul o he seme Ne we o evese eul o Jese s eul Pooso 3 I s uco egle d cove o we hve he ollowg eul: (8) o eve Poo I o ll he usg eul (7) we hve F F whee F B summg om o we d he ollowg eul: (9) I o ll he d usg eul (7) we hve (30)

48 47 Theeoe comg eules (9) d (30) we o he eul om seme Fom Pooso 4 we hve h: he uco s sogl cove wh modulus c he he uco g () = () c s cove We l he ove esuls o he uco g hus: Cooll 4 I s sogl cove uco wh modulus c he we hve: (7) F c F whee c F o eve Pooso 5 I s sogl cove uco wh modulus c we hve he ollowg eul: (8) F c o eve whee c F Sevel eules o geelzed eoes Geelzed eoes hve ee suded m eseches (we ee he eesed ede o [6]) Ré [9] d Tslls [0] eoes e well kow s oe-mee geelzos o Sho s eo eg esvel suded o ol he eld o clsscl sscl hscs [0 04] u lso he eld o uum hscs elo o he egleme [98] The Tslls eo s ul oe-mee eeded om o he Sho eo hece c e led o kow models whch desce ssems o ge ees omc hscs [84] Howeve o ou es kowledge he hscl elevce o mee o he Tslls eo ws hghl deed d hs o ee comleel cled e he mee eg cosdeed s mesue o he o-eesv o he ssem ude cosdeo Oe o he uhos o he ese e suded he Tslls eo d he Tslls elve eo om mhemcl o o vew Fsl udmel oees o he Tslls elve eowee wee dscussed [8] The uueess heoem o he Tslls eo d Tslls elve eo ws suded [85] Followg hs esul omc chcezo o wo-mee eeded elve eo ws gve [86] I [74] omo heoecl oees o he Tslls eo d some eules o codol d o Tslls eoes wee deved I [87] m ce eules o he Tslls eo wee suded Ad [88] he mmum

49 eo cle o he Tslls eo d he mmzo o he Fshe omo Tslls sscs wee suded Que ecel we ovded mhemcl eules o some dvegeces [89] cosdeg h s mo o sud he mhemcl eules o he develome o ew eoes We show sevel esuls om ou e [Fuuch-culee-o 75] hee we dee uhe geelzed eo sed o Tslls d Ré eoes d sud mhemcl oees he use o scl eules o develo he heo o eoes We s om he weghed usle me o some couous d scl moooc uco : I R deed () whee 0 I o d I we ke he cocdes wh he weghed hmec me A I we ke log he cocdes wh he weghed geomec me I G Tslls eo [0]: d () H l whee s ol dsuo wh 0 l he s eul o l 0 o ll d he -loghmc uco o 0 s deed l whch uoml coveges o he usul loghmc uco log() he lm Theeoe he Tslls eo coveges o Sho eo he lm : (3) lmh H log Thus we d h Tslls eo s oe o he geelzos o Sho eo I s kow h Re eo [9] s lso geelzo o Sho eo Hee we evew usle eo [6] s ohe geelzo o Sho eo Fo couous d scl moooc uco φ o (0 ] he usle eo s gve (4) I log log (4) he we hve I we ke H log I We m edee he usle eo (5) I log o couous d scl moooc uco o (0 ) I we ke log 48

50 log (5) we hve I H The cse s lso useul cce sce we ecue Ré eo mel I R whee Ré eo [9] s deed (6) R log Fom vewo o lco o souce codg he elo ewee he weghed usle me d Re eo hs ee suded Che 5 o [9] he ollowg w: Theoem A ([9]) Fo ll el umes > 0 d eges D > hee ess code ( ) such h (7) R log D R log D D whee he eoel uco D B smle clculos we d h d lm R lm s deed o [ ) D log D oved he ove esuls d ece dvces o he Tslls eo heo we vesge he mhemcl esuls o geelzed eoes volvg Tslls eoes d usle eoes usg some eules oed movemes o Youg s eul Deo Fo couous d scl moooc uco o (0 ) d wo ol dsuos { } d { } wh > 0 > 0 o ll he usle elve eo s deed (7) D log log D The usle elve eo cocdes wh he Sho elve eo log e log D log D We deoe R he Ré elve eo [3] deed (6) R log Ths s ohe cul cse o he usle elve eo mel o we hve 49

51 50 log D R log I we use he eul (43) he we o whee 0 d m I ollows h R log log We deoe (7) l l l D he Tslls elve eo whch coveges o he usul elve eo (dvegece Kullck-Lele omo) he lm : log D lmd O he ohe hd he sudes o eemes o Youg s eul hve gve ge ogess he es [ ] I he ese e we gve some eules o Tslls eoes lg wo es o eules oed [77 57] As log wh (5) we m dee ou e [75] he ollowg eo: Deo Fo couous d scl moooc uco o (0 ) d > 0 wh he Tslls usle eo (-usle eo) s deed (8) l I whee { } s ol dsuo wh > 0 o ll We oce h does o deed o he I I lm Fo > 0 d > 0 wh we dee he -eoel uco s he vese uco o he -loghmc uco e + ( ) >0 ohewse s udeed I we ke l he we hve

52 5 l H I Fuhemoe we hve l l I H Pooso 3 ([Fuuch-culee-o 75]) The Tslls usle eo s oegve: 0 I We oe hee h he -eoel uco gves us he ollowg coeco ewee Re eo d Tslls eo [0]: (9) H e R e We should oe hee H e s lws deed sce we hve (0) H 0 Deo 4 Fo couous d scl moooc uco o (0 ) d wo ol dsuos { } d { } wh > 0 > 0 o ll he Tslls usle elve eo s deed () l D Fo l he Tslls usle elve eo ecomes Tslls elve eo l D l D d o we hve l l D D Pooso 5 ([Fuuch-culee-o 75]) I s cocve cesg uco o cove decesg uco he we hve oegv o he Tslls usle elve eo: 0 D Poo We sl ssume h s cocve cesg uco The cocv o shows h we hve

53 whch s euvle o Fom he ssumo s lso cesg so h we hve Theeoe we hve l 0 sce l s cesg d l() = 0 Fo he cse h s cove decesg uco we c smll ove he oegv o he Tslls usle elve eo Remk 6 The ollowg wo ucos ss he suce codo he ove ooso: l o 0 () () o 0 I s ole h he ollowg de holds: () e R e D Ne we gve eules o he Tslls usle eo d - dvegece Fo hs uose we evew he esuls oed [57] s oe o he geelzos o eed Youg s eul d Pooso 7 ([57]) Fo wo ol vecos such h 0 0 h 0 we hve 5 d such (3) 0 m T T m T whee T o couous cesg uco : I I d uco : I J such h I d 0 o We hve he ollowg eules o he Tslls usle eo d Tslls eo: Theoem 8 ([Fuuch-culee-o 75]) Fo 0 couous d scl moooc uco ψ o (0 ) d ol dsuo wh 0 o ll d we hve (4) m 0 l l

54 m H I l Poo I we ke he uom dsuo he we hve l u Pooso 7 (5) m T u T m T u 0 (whch cocdes wh Theoem 33 [57]) I he eules (5) we u l d o he we o he seme Cooll 9 ([Fuuch-culee-o 75]) Fo 0 d ol dsuo wh 0 o ll d we hve (6) m Poo Pu 0 l l l H m l l Theoem 8 l Remk 0 Cooll 9 moves he well-kow eules 0 H l I we ke he lm he eules (6) ecove Pooso [58] Cooll ([Fuuch-culee-o 75]) Fo wo vecos d o ll we hve (7) Theoem ([Fuuch-culee-o 75]) Le : I R e wce deele uco such h hee ess el cos m d so h 0 m " o I The we hve m (8) whee 0 d I o ll Cooll 3 ([Fuuch-culee-o 75]) Fo wo vecos d o ll we hve 53

55 54 (9) Cooll 4 Ude he ssumos o Theoem we hve (0) m whee 0 d I o ll We lso hve he ollowg eules o Tslls eo: Theoem 5 ([Fuuch-culee-o 75]) Fo wo ol dsuo d such h 0 0 we hve () m l l l l m l l whee m d e osve umes deedg o he mee 0 d ssg m d m o ll Cooll 6 ([Fuuch-culee-o 75]) Fo wo ol dsuo d such h 0 0 we hve () m log log log log m log log whee m d e osve umes ssg m d m o ll Poo Tke he lm Theoem 5 Remk 7 The secod o he eules () gves he evese eul o he so-clled omo eul

56 (3) 0 log log whch s euvle o he o-egv o he elve eo D 0 Usg he eul (3) we deve he ollowg esul Pooso 8 ([Fuuch-culee-o 75]) Fo wo ol such h 0 dsuo d we hve (4) log log 0 Poo I he eul (3) we ke he susuos d whch ss The we hve he ese ooso such h Aove we cosde d 0 0 o e ol dsuos Tslls elve eo (dvegece) s gve D D l I coveges o he clssc Kullck-Lele omo: lmd D log The Jees dvegece s deed J D (5) D d he Jese-Sho dvegece s deed s (6) JS D D (see eg [6]) Beoe sg he esuls we eslsh he oo The wo-mee eeded loghmc uco (see eg [6]) o he -loghmc uco o 0 s deed e l l el whch uoml coveges o he usul loghmc uco log() he lm d Ths s decesg uco wh esec o dces Coesodgl he vese uco o l s deoed e e log e 55

57 We s om he Tslls ()-usle eoes d Tslls ()- usle dvegeces s he wee deed [89] Deo 9 Fo couous d scl moooc uco o (0 ) d > 0 wh he Tslls usle eo (()-usle eo) s deed (7) I l l we hve he ollowg eoc ucol: Fo (8) H l Ths lso gves se o ohe cse o ees I l el l e whch cul cse cocdes wh Amoo s eo Deo 0 Fo couous d scl moooc uco o (0 ) 0 wh wh > 0 > 0 o ll d wo ol dsuos d he ()-usle dvegece s deed (9) D l Fo l we he ollowg: (30) D l B log o he eo comuo we d he ollowg Amoo e dvegece: D l e Pooso ([o-culee 6]) Le e el ume Assume 0 0 ss I o he we hve (3) D H e l l Theoem ([o-culee 6]) Assume h el umes ss I o he we hve (3) J J e l e l E E 56

58 whee E e l e l m As we hve see ll hese emles m cses he use o he () geelzed loghmc uco cel comlees he cue oed wh he - loghm d c e useul led es (sgl d mge ocessg omo heo) Ieules o vele osve oeos I Theo o Oeos we oud vous chcezos d he elosh ewee oeo moooc d oeo cove gve Hse d Pedese [04]Chsgm [34] I [] Kuo-Ado hs suded he coecos ewee oeo moooe ucos d oeo mes The oeo moooe uco ls mo ole he Kuo-Ado heo o oeo coecos d oeo mes Ohe omo ou lcos o oeo moooe ucos o heo o oeos me c e oud [80] Theo o oeo me ls cel ole oeo eules oeo euos ewok heo d uum omo heo Le H e el Hle sce Deoe B(H) he lge o ouded le oeos o H We we A 0 o mes h A s scl osve oeo o euvlel A 0 d A s vele We oe h I s he de oeo I [9] we oud he us-hmec owe me wegh gve A# / B A B A B 0 # wh eoe d Sevel secl cses o he ml o us-hmec owe mes e he ollowg: o we hve he weghed hmec me s ollows A B : A# B A B A B 0 ; o we o he weghed hmoc me gve s o 0 A B A! B : A# B A B 0 ; we hve he weghed geomec me gve A# B lm A# B A B A B 0 d A B commues 0 The geomec me ws deed Pusz d Wooowcz [86]: / / A# B m T 0 : T A B H A B 0 I c hs deo s he omul gve Ado [5]: / / / A BA A / / A# B A A B 0 Aohe deo o he geomec me (see eg [4] [6]) s gve A A # B su 0 d 0 A B 0 B A mo emk [4] s h he geomec me A # B s he uue osve soluo o he Rcc euo 57

59 A B The -weghed geomec me s deed [6] whee 0 A# d A B 0 B / / A BA A / / A Fuu-Ygd oved [9] he ollowg eul A! B A# B A B Fom he kow eul whch mles / / 0 0 we deduce eul o he us-hmec owe me A# B A# B A# B # Theoem B ([00]) Fo vele osve oeos A d B wh 0 mi AB I we hve () (Ro-e evese eul) () A B ShA# B () (Deece-e evese eul) A B A# B L h S h () B whee 0 Ne we show wo evese eules whch e dee om () d () gve ou e [Fuuch-culee 76] We s show he ollowg emkle scl eul: Theoem ([Fuuch-culee 76]) Le : R e wce deele uco such h hee es el cos so h 0 " o The he ollowg eules hold: 0 0 I we ke eul om ove Theoem log d ewds log he we o o ll d 0 e 0 log Fom hee we cosde ouded le oeos cg o comle Hle sce H I ouded le oeo A sses A A he A s clled seldo oeo I sel-do oeo A sses A 0 o H he A s clled osve oeo I ddo A B mes A B 0 m 58

60 Theoem ([Fuuch-culee 76]) Fo 0 oeos A d B ssg he odeg hve () (Ro-e evese eul) wo vele osve 0 mi AB I I wh h we m h () (Deece-e evese eul) A# B A B A# B log h (3) A# B A B e A# B (4) B Remk 3 I s ul o cosde h ou eules e ee h Tomg s eules ude he ssumo A B The eul h udeles he oo o eul () s oe o evese eules o Youg eul h ws gve Tomg [00] S Theeoe we come hs eul wh he eul 0 e m used he oo o eul (3) hus [Fuuch-culee 76]: () Tke h d he we hve 0 e Sh h () Tke h d he we hve 0 e Sh h Thus we c coclude h hee s o odeg ewee (3) d () I [0] Tslls deed he oe-mee eeded eo o he lss o hscl model sscl hscs The oees o Tslls elve eo ws suded [8] d [8] Fuuch Yg d Kum The elve oeo eo / / / / SA B : A loga BA A o wo vele osve oeos A d B o Hle sce ws oduced Fu d Kme [73] The mec eeso o he elve oeo eo ws oduced Fuu [9] s S / / / / / / A B : A A BA loga BA A o R d wo vele osve oeos A d B o Hle sce Noe h S0 A B SA B I [07] Yg Kum d Fuuch oduced mec eeso o elve oeo eo he coce o Tslls elve eo o oeos hus 59

61 / / / A A BA A B : 60 / A A T 0 whee A d B e wo osve vele oeos o Hle sce The elo ewee elve oeo eo S (A B) d Tslls elve T A B ws cosdeed [8] s ollows: oeo eo (5) A AB A T A B SA B T A B B A The ollowg kow oe o he Tslls elve oeo eo s gve [08]: Pooso 4 Fo scl osve oeos A d B d 0 0 wh we hve (6) TA B T A B Ths ooso c e oved he moooe cesg o R o > 0 d mles he ollowg eules (whch clude he eules (5)): A AB A = T (A B) T (A B) S(A B) T(A B) T(A B) = B A o scl osve oeos A d B d (0 ] The m esul om ou e [od-fuuch-culee63] s se o ouds h e comleme o (5) Some o ou eules move wellkow oes Amog ohe eules s show h A B e vele osve oeos d (0] he A BA I A A BA I A T A B A# B A# B B A whch s cosdele eeme o (5) whee I s he de oeo We T A B lso ove evese eul volvg Tslls elve oeo eo Theoem 5 ([od-fuuch-culee63]) Fo vele osve oeo A d B such h A B d (0] we hve (7) A A BA I A BA I A T A B A# B A# B B A Poo Cosde he uco (0] I s es o check h () s cove o Beg md he c d d ulzg he le-hd sde o Heme-Hdmd eul oe c see h whee d (0] O he ohe hd ollows om he gh-hd sde o Heme-Hdmd eul h