UNIVERSITÄT Zentrum für Technomathematik !"#%$'&)( *,+-(/.102!#%$

Volumenanteile versus Massenanteile –der Dilatometerversuch aus der Sicht

der Kontinuumsmechanik

Michael Wolff Michael BöhmSerguei Dachkovski

3)4/576 8:9<;>=7?;@

Berichte aus der Technomathematik

Report 03-01 Januar 2003

@

"! #$%&'() +*,

-! ./*+,0! 1324'5 6(1(*+,7

89;: 6 < ==>89?@AB->C9DFEGHAI 6)JLK)INM

OPRQTSNUVU3WHXRYSZQZQNPNXN[\V]6^`_ba;c;dbed^ZfgaHh iZj;ckaRc;dmlonNf`prq fZ^Z^Zs4tTsFcvuZwgiHfZq xzy4q daa;c;d-| q h ivpnNsFc~pcdZc;d%tT^guZws4nZaRc;h h q c;dpr f`pc;daRc;fv;h q uZwvc;f]fZfgiNwTsFc;fRaTiZtgtkaNq cdnRZc;fyNwZdc;fgaaRcd"wgiZtvc;fZ^ZsyiHfgaoh ^ZfgjwgnNs4nZjRcfZcTvgjNhq uZwVLcsFZc;divpr^Zd/6^Tfga(wgiZt~cf`c;dpc;q h ^ZfjtTq fga);y0c;daRc;fViN^gtVaRc;fed^ZfgaTjNh cq uTwZ^ZfgjRcfaRc;dh q f~cviNdc;fw~c;ds4nRc;h iZtpq vq pvp _rZd t~ZcTvq c;h h cm$q tuZwZ^TfgjRc;faNq ceh cq uTwZ^ZfgjRcHfw~c;djRc;h c;q pc~p1s$q paRc;f~cfbiHfZwgiNfgaaRc;diN^`_jRcfgnNs$sFc;f~cf| q h ip/nHsFcpc;d%aTivpc;faHq c 4iZtt~c;fv^Zfgak¡nNh ^TsFc;fgiNf`pc;q h caRc;dbTcpc;q hq jvpc;fwiZt~c;fZc;dcvuZwZfvcpRy0cd%aRcf$x&¢NfZfvc;f$£q s¥¤ZiNh h c¦~nNf$q fgtgj;cvtgiNspo§y0cq~wgiTt~c;fg¨©| q cc;dwgiNh pcf~c;f¤ZnHdsFc;h ftZq fga^ZfiR;wg)fgjNq j~nNfa;c;fVc;dtguTwZq cvaRcHf~c;fZc;x§iNfZf`pc;f syiHfgaoh ^ZfgjTtZs4nZaRcHh h c;fv^ZdªLcvtuZwZdc;q ;^Zfgj«t~ZcTvq c;h h c;dwgiTt~c;fZ^TsyiNf~aHh ^ZfvjRc;f«q sz¬)piNwZhv^Zfga­x&¢NfZfvc;fFaZiHw~c;dov^ZszLct`pc;f$aRc;diNdprq jRc;d4nZaRc;h h c®c;dy0c;fga;cp$y0c;daRc;fZF¯4~ynNwZhy4q d$aNq c°]6f`y0c;fgaN^TfgjziN^`_bwgiZt~cfZ^ZsyiNfvaNh ^Zf~j;c;fzq s±¬)piNwZh vnNd]^gjRc;fVwgi;Zc;fR;jRc;h pc;f²aNq c³djRcZfZq tt~cViH^guZw$_rZdiNfga;c;d/cb4ivpc;dq iNh q cfR;tgnNh iNfgjRcaNq cVjRcprdnv__c;f~c;f²j;c;f~c;dc;h h c;f¡nNdiN^gtgtvcpv^ZfjRc;f´^`prd/c__c;fZ)|iRZc;q;q t`p_rZd4ipc;dq iNh q cf$y4q c­¬)piNwZhTc;q~a;c;f~c;f²aHq c| q uZwpc^Zf`pc;dtguZwTq cTaRc­aRc;dc;dtguTwZq caRc;f~c;fwgiZtvc;ffTq uTw`pjNdnNµ¶tZq fgaoc;q fvc¶h q f~cviNdq tTq cdpc w~cnNdq c·s4¢ZjNh q uTwRVyHwZd/cfgatgnNfgtp"s$q pfZq uTw`prh q f~cviHd/c;f4¸^gtgiHs$sFc;fZwgNfjRcNfF^$d/cuZwZf~cf$q tp³Nty4q dajRcZc;q jpaZiTtgttZq uZwZc;qaRc;d$wgiZtvc;fZ^ZsyiNfaNh ^Tfgj¹q sº¬)piNwZhaNq c®4iZtgtvc;f~»^Zfga-¡nHh ^ZsFcfgiNf`pc;q h cy0c;fZq jz~nNfvc;q fviNfga;c;d^Tf`pc;d%tuZw~cq aRcHf¥£^Zs¼y0c;fZq jRc;dViNh tc;q fdnRc;f`p¨½»aZiaHq c¾| q uTw`pc;^Zf`pc;dtguTwZq cvaRc¿aRc;dwgiZtvc;fxgh c;q ftTq fgaHÀNwZdc;fga°q faRc;dÀc;dxt`pnv__rxg^Zfga;ciHh h jRc;sFc;q fs$q p¡nNh ^ZsFc;fgiHf`pc;q h c;fjRcviHd/;c;q pc~p0y4q daowgiRZc;f4aHq c«4iZtt~c;fiNf`pc;q h caRc;f4¡nNdpc;q h ~nNf4c;sFZc;divpr^ZdN^ZfaV|c_nNds4ipq nNf$^ZfiR;wgNfjNq j²^4t~c;q fZ¤NZdxgh c;q f~cV| q uZwpcNfgaRc;d^Zfgj;c;f4_nNh jvpoiN^gt$a;c;f²c;dwiNh pc;fvc;f²eh cq uTwZ^ZfvjRc;fRTaZiZtgt²Tc;q`~nHh h t`pNfgaHq jRcdq tnvprw~c;dsFc;d syiNfgaNh ^ZfgjÁaRc;d4iZtgtvc;fgiNfpcq hNaRc;d tTq uTwb;q h aNc;fga;c;f(wiZt~cÂjNh cHq uZwa;c;sº syiHfgaNh ^ZfgjZtjNd%iTa"q t`pr6| q cvtvc;dZcvt`prq s$sptZq uZw¥iHh tÃ3^gnvprq cf`piN^gtaRc;diNxpr^~c;h h c;dÄgNfgjRcfgNfgaNc;d^Zfgj¶^TfgaaRcd(ÄNfgjNc;fgNfgaNc;d^Zfgj·fgiZuZw]+vtguTwZh ^gtgtaRc;d syiNfgaHh ^ZfjNN|iZjRcvj;c;f²x§iNfZfFa;c;d f`pc;d%tuZwZq cvaV§y4q tguZwvc;fFaRcs¥4iZtgtvc;f~£^ZfgaVaZiHs$q piN^guZwaRc;s±¡nNh ^ZsFc;fiNf`pc;q h ¨Á^Zfga¥a;c;sÅ syiNfaNh ^TfgjZtgjHd%i;aÆTc;q~nNh h t`pNfgaNq j;c;dfZq uZw`pq tnvprw~c;dsFc;d syiNfgaNh ^ZfgjZcprd%TuZw`prh q uTw4t~c;q fZ| q cvtvc]dZc;q p0c;f`ptp/iHfga-s$q ppc;q h y0c;q tvc;d f`pc;dt`pr§p^ZfgjÁaN^ZduZwaNq c-|c^`ptguZwvc"¤TnNdtuZwZ^ZfjZtgjRcsFc;q ftguZwiv_pFq sÇiNwZsFc;f4aRctF¬nNfga;c;d_nNdtguTwZ^ZfgjTt~Zc;dc;q uZw~ct4ÈRÉvÊ$Ë©| q tpnNdprq nNf²³fgjHq f~cTc;dqfjRÌr| q c«]6^`pnNd/cf²aZiNfZx`c;fÍc;ddfÍnNh jRc;d¬^Zdsk£Î À°ªdcTsFc;fg¨_rZdaNq c_rd^uZw`pviNdc;f| q tTxg^gttZq nNf~c;fv^Ts¥w~c;s4i^Zfga¤NdiN^$ªLc~pprq fi²¬^ZwZdT_TdZaNq c«l)nNddc;xÏp^Td/c;f$^Tfga$aZiZt]6f&_c;dprq j;c;f4aRc;dNe4diR;wTq x

ÐÑNQvÒÓSZÔÒÕ\ª)iZt~cvaVnNfprw~c»ugnNf`prq fZ^Z^ZssFcvuTwgiNfZq ugtprw~cVaHq h ivpnNsFcpc;dLc~ÖZc;dq sFc;f`p q t­s4nZaRc;h cvaN;]t`tT^Zs$q fgjVprwgivppwvct~ZcvuTq sFcfgt4iNdc«wgnNs4nZj;c;fgnN^gt'y4q prw²dcvtvZcvu`p;pnpc;sFZc;divpr^Zdc$iNfgaV;wgiZtvcvtZ×`aNq t`prdq ^`prq nNfaN^Zdq fj$prw~c²wgiZt~cpdiNfgt_nNds4ipq nNfy0c«aRc;dq Zc_rdnNs¹prw~cVcvØN^gipq nNfgt$n_Npwvch q f~cviNdq cva«prw~c;ds4nRc;h iZt`prq uZq p%Ù«_nNdot~TcvuZq iNh;s$q Öp^Td/cvtprw~ccvØN^givprq nNfgt«y4wZq uZwViNh h nvyÚpnugiNh uZ^Zh ivpcpwvcs4iZtt²iNfavnNh ^TsFcV_rdiZu§prq nNft²_rdn;s¥prw~c­aNq h ivpnNsFcpc;daZivpcvt£q f²prw~cugiZtvcnv_Zpyn²;wgiZtvcvtg¨©w~cb_/nHds$^Zh iZt­y0cjRc~p+iNdcq fgaRcTZc;fgaRcf`pnv_iNf`Ùxgfnvy4fs4nZaRc;h t_nNd;wgiZtvcprdiNfgtTq prq nNfgtq fbt§pcZch ¤TnNd prwZq td/cviTtgnNfVprw~cÙugiNfZcÁ^t~cva_nNdpct§pq fgjÁtZ^guTws4nZaRch tTo]h prwgnN^gjHwy0c_nZuT^gtnNft§pcZc;hHpwvcdcvtZ^Zh ptiHd/ciNh tgn~iNh q a»_nNdNnvprw~c;ds4ipc;dq iNh tRTiZtVh nNfgjViZt¦pwvc²j;c;f~c;diNhgiZttZ^ZsF~pq nNfgt$iHd/c«jNq c;fTo¤ZnNds4ipc;dq iHh tVh q x`c«t`pcZc;h`y4wvc;d/cpwvc²a;c;fgtZq pÙ²aNq _%_c;dc;fguvcvtVq f4~iHdq nN^gt²;wiZt~cvt$iHd/c«tZs4iHh hgiÂh q f~cviNdq gcvaVprw~cnNdÙq tVvnZttZq ;h c´y4wZq h cVnvprw~cdy4q t~c«y0cwgivc¦pn$unN^Zf`pNnNf$fgnNfTh q f~cviNd)d/ch ivprq nNfgtTÎ p4q ttTwgnvy4fprwgip+aN^Tdq fgj(wgiZt~cbprd%iNft`_nNds4ivprq nNfÁq ft§pcZc;hs4iZtgtiNfgaÂ~nNh ^ZsFc­_rdiZu`prq nNfgtÁwgivcÂnNfZh ÙbiÁtZs4iNh haNq __c;dc;fguvcV£h cvtt4prwgiHf´ÛgÜ«¨oiTt4prw~c$;wiZt~cvtT×`aRc;fgtTq pÙ²aNq __cd/c;fu~cvt$iNdc$tTs4iNh h ZÀwZq h cq fVs4ivpc;dq iNhtguZq c;fguvcvt$nNf~cjRc;f~c;diNh h ÙynNdx&t¦y4q prw¦~nNh ^ZsFc»_rd%iZuprq nNft²s4iZtt4_rdiZu`prq nNfgtVwgic«jZnvpNprw~c«iZavviNf`piZjRc«nv_LTc;q fjq fgaRcTZcNfaRc;f`pnv_pcsFZc;divpr^Zd/ciHfga$aRc_nNds4ivprq nHfZ¤ZnNd²tTs4iNh haRc;fgtTq pÙ¹uZwgiHfgjRcvtzq p²_nNh h nyt°_dnNsÝprw~czcvØN^gipq nNfgt¹n;piNq fvcvaÞpwivpq f¹prw~cßugiZtvcÚn_un;sF;h c~pcq tnvprw~c;ds4iNhprd%iHfgt`_nNds4ipq nNfprw~càs4iZtgt·_rd%iTu§pq nNfÝnv_Þpwvcá_nNds$q fgjâ;wgiTt~c±cvØH^giNh tÝpn¼prw~c¼tnRuiNh h cvapdiNfgt_nNds4ipq nNfaRcvjNdcZc;vw~ch iTt`ponNf~cq t'pwvc$ØN^gnpq c;f`pon_Nprw~c«iZu§pr^giNhTh c;fjvprw3uTwgiNfgj;c²iHfga«pwvch c;fgjpw$uTwgiNfgjRcivpprw~cc;fganv_ prw~cbprd%iNft`_nNds4igprq nNfZÎfugnNfpdiZtp p/nÂprwZq t­prw~cÁaNq _%_c;dc;fguvcZcpy0cZc;fprw~cs4iTtgt­_diZupq nNf-£iNfgapwvc;d/c~_/nHd/c­iNh tgn­pwvcVvnNh ^TsFc«_rd%iZuprq nNf¨iNfga­prw~c­prd%iHfgt`_nNds4ipq nNfaRcjNd/cTcugiNfZcÁdc;s4iNdx&i;;h cÁq fVprw~cugiTt~c­nv_ugnNsF;h cpc$;^`pfgnNfvrq tgnvprw~cds4iNhÏprd%iNft`_nNds4ivprq nNfZwZq t ynNdx yiZt viNdãpq iNh h Ù tZ^~ZnNdpcva ~Ù prw~c |c;^`pt`uZwvc ¤ZnNdtguTwZ^ZfgjZtjRc;sFc;q fgtuZwgiv_p±;q i prw~c¬nNfgaRcd_nNdtguTwZ^ZfgjZtvZc;dc;q uZw4ÈRÉvÊ$Ë©| q t`pnNdpq nNf²³fjNq fvcZc;dqfgjRÌw~c«iN^`prwgnHd%tprwgiNfTx²ÍnNh jRc;d¬^Zdsk£Î Àªd/c;sFcfg¨_nNd)_d^Zq p_r^ZhvaNq tguZ^tgtZq nNfgt«nNf$prw~c­s4ivppcdLiNfaªLcppq fgi¬+^ZwZd_/nHdZprw~c¦ugnNdd/cu`prq nHfgt4iNfgaprw~caHd%ivy4q fgj$nv_TpwvcjNd%i;;wZ

=

6, @96M< 4M 99 26 8 4 8 4M 9>4= 924 8½< 4M 9>4 8Á< 4MG)A 4=>8>4-D»M%< E 9-6B 4 9 4 8J 4<8rKoG)A 9 8 ?4 8 4G)A>4/86D$M G)A 9-4E < K I 9 M 6! 4>4/8"4B 5 4/8rE79>8# 96MM >4$8%&')(74

# 9-@9*24 8+(M G)A 9EH=-,M.74E/8 MK)M 4 8)>4KÁD$MG)A 9 4 ?§D4AK ?/oKNEBB 4ALEK# 99*788ÁM KNG)AK78 4 4 <v= 8>4:0&<8©B 4EoK >4+AK)I 6>4== M (M 4 9:4# 9 = 9*78-1»B 8 4G)A*>4KV:0&/8©B 4E oK2>4A*K)I 6>4== M (M 4 9 4# 9 9DFE 8rK 9 4 <<)3 4K46 <.B 4LE 9-4M< K5>8rG)A6M 4"D»M G)A 9 4# 9 # 9DFE 8rK 9 4 <<)3 4K46 <.B 4LE 9-4M< K5>8rG)A6M 47,&>4# 998 9?4 8oG)IRK)M G)A 9 M: ÂJ 6MG)A 9-B5;E>4 <:>4-8ÁM KNG)AK 74K 9E' 9-4M< 4# 99< 98 4+A 8©I 6B 5 6 4' 9 4LK>=LK 9 4B 4

?@< `A1B 9 4 8rKoG)A)>4'(DC0ALEK 4!B5;E' <. 4D*Þ8E 9-4 8©ME < J 4/8©ALE < 9-4ÁJ 6"C9EA<K)M J 6 8 6 E<4/8-EI 9 4 << 4 8¹?4>4F 9 =78GM 4H; 4+M 9 4<8 4IJ 9K;MG)Io<.L*M"4K 9N>474M: 479 4/88 6!>4 << 4I( MA>8 4 8k?4KNG)A>8 4M. 9O4M 41<3M 4K 4/8QPRLK 98 4 >4! K)M.TS 4 4I74B9 A 4 6 8 479 MKNG)A 4724 8K 9%&)MKS¥B@ <ãM G)A 4Q26 8©A 4 8rKNE>4!U>4KO24 8+( K J 6zC9rEA<:oE 9-4M< 4 4M4:0& 8©B 4 4ALE' <.*H·B@ <MG)A 4/8ÁB 4G)AHE'M KNG)A 4 8(?4 < EK 9WVrK)9AM 4 8+(YX:P²A/8-;Z >X[6=-;Z]\§9P^N= D$M%< E79 6B 479-4 8J 4 8rKoG)A 4! 74M_1»B`;E! <.>4 6A74 ?4 < EK 9 LE 4 K)M G)A

I 6B 5< 4a 4 8-4b!a>5 4 8©MB 4! 9 4kE N= >c 4Md 4740M 4Y1»B5;´E <b 9 4 8VI 6 98 61<<M 4 8:9-4 8B)7?§>D8oG)I ?Qe 6>4/8f"6 8rK)M 6K 4 < EK 9kK 6;M 4g>4>4' 4! 4BhPRI)A< J 4/8½< E N= 4/8=-6 < 79`V+1<9(9?49iX:P²A/8-; ; > ;Z+\&9?4Mj>4/8¼8 6>41<<M 4 8 >4 8kC6AEK 4!B5;E <.>4' BM 9lMG)A 9I 6LK 9rE 9 4<8m1$B5;E <.)K 9 4B 5 4 8rE79 8JM. 9 4KO 6G)A;M G)A 9©M >4'6== 474n 8rE>4!9:0&A/8 4 4MMK 679©A 4 8rB 4 8o1B5;E' <.NM 4Np 6+ALK 6! ?§8 4A< ?qP4J<8rEB(M ?< 4M G)ANM 4Ir'M. 479©MIW>4/8C6ALEK 4B5;E1<. AM.<8 4MG)A 4 >4!LE 4KNG)A>8 4M. 9>M K 9K)M 4 4MsM G)A 9 ?ÏMK 679 A74<8©B 4/81$B5;E <.LE(t)(78 4M G)A74t>4>4M: 479 V+ <9u(9?49UXD24)v<Z½>QX[MK v8Z>$X[,oE 8]v # Z>X[, 4'w>=Z>X[f ;>;Z>X[6 )w8Z½>uX8ÁM 9%v > ;Z®K 6;M 4kM 4k>6/8:97(M 9 M 4 8:9 4yx 4 << 4)\§9Gzl>4P 8 4M 9 47X8ÁM 9[v> ;Z >X[,oE/8%v # Z!;M 8%=>8M 4ÁJ 61<< K 9%&+M 74$1$B5;E <)4M.74 8/C6ALEK 4 M.uM 4E>4 8 4'4M7|1$B5;E <.K /8E!~RV%>4!<8 4>46)=98rEoK;=-6 8rBE79©M 6)\R 4B6& Ez= 6 < 4'>4/8^ 6<8©B 4 <4M. 4=A<8%9

V @9-@ \ =V:9%\/ h £ãp/¨hh hz V @9-@2\kK)M. <>¥<:V:9%\&>z< 4 9K 5>8 4G)A 4 M 4 ,&74 >4/8 C8 674 ( ?4M >4/81$B5;E <. >; &A>8 4O>4 8/1B5;E' <.u(BW4+M 9 5I 9'9K 6;M 4QoEG)APfLKNG)A<.!ED 4 81$B5;E <. 9"D»M 4k 8@FE>4 = ;M 8+±E < K 6 <.B 4E 9-4M<`>4/8¹K)M G)A M%< >474 C6ALEK 4M. 9 4 8 5<8 4 9 M 4 8:9½9«?4MB C9E+A</ 9 4/8rKNG)A 4M 4K)M G)AjM 4·D$M G)A 9-40 4 8 4+M.)( 4 < 4 C6ALEK 4j>8; 4M: 9 1c 9-4 8 M 4K 4 8 6 8E )KNK 479[(* < 4M 9-4 ;M 8ÝEoK >4' 8)EEAB 4' 74 8r6 9©M.!BK)B 4GNALE'MIBM 9dC6AEK 4'B5;E) <>4'A 4 8>)EKNK=78J 6 << K 9%&M:>4mM K 679©A 4 8B 41$B5;E <.>4 M 4 8@FE<4 =M.oV @9-@ \B1< 4MG)A >4B 8bEKNK 4E 9 4M%< V8EKNK 4<8oG)A)\ >4/8K)M G)AM%< >4 47C6ALEK 4mM K 9©9 Kh( 4M 79K)M G)A E )G)A>0EKNK 4M¿C9E+A<UM 4 PR ; 4+M G)A (2;M KNG)A 4 8EKNK 4 ?M26 <.B 4'LE 9 4M%<ßMB ?4/8-4+M G)A J 6' 9 4 8 4+M. 4B C 8-6( 4 9 VrK9Y9¥=>8¶EoK;=A<8<MG)A 4/8-41B 9 4 8rKoG)A)>4'\°M K 9©>f;EK 4MJM 4 < 4jPR*; 4F>4 9-6 < 4 8©M 4/8LE 8MK 9½9z >4 8 "4A 4>6/8©M 4K 6 << 9-4 4!>6G)A (2;MKNG)A 4 4M >4' 8@FE<4!u 9 4<8rKNG)AM 4>4 ; 4 8%>4!9zB E << 4 4M:74 8uM G)A 9-?ÏMK 679 A74<8©B 4!l1$B5;E' <.*i; 4 8%74ÝM. >4' >4'LE 9 4 P 8 4M 9-4X8ÁM 9%v>/;Z½>JX[,)E<8%v # Z PRLK & 9[( 4'4 9K;MG)I 4 < 9>BM 9d 4 8 4U»M%< =-4$74 8B1»B`;E! <.K /8rE =kE < K I 9©M 6'LE <474 8"4+B 574<8rE7978%>4KNG)AMG)A 9 4 4KNG)A>8©M 474 ; 4/8%>4¼IRE!9:kM 4H;M 8G( 4+M >4; 4 8%>4>M K 9/M 4 8r@E<4 =·M.sV1@)9-@ \M.$M 4K 4+B_E << 4n; 4 4<8 < 4+M G)AU74BL261<.B 4! ? 76GNA>4B 8bEKNK 4E 9 4M%<9BzB ) LK 9©M 4 E <<4IRE'0M 4 M. V1@9 @ \5>4=©M.M 4 8%9 4 8@ E/4b 4E 9 M J6>4 8±K 4A>8k 8:6 E ; 4/8%>4> 6>4 8¼E oG)A E/8tM G)A 9N>4=©M.M 4 8:9 K 4M> ;M 4 =-61< 74>4 4 8< 4 4Q( 4M >4'9/zB ?'M%< @¹K)M.Þ=>84M 4ßE'74 6BÁB 4' 4D»M%< E 9-6B 479 4/8J 4 8rK)oG)AM 4>4B 4KNK 4! 4G,)&'743 4<8 C8 6 4à<²E < K I 9©M 6' 4<8!4M 9E N=%>4 9 8rE>4!9

?M Kn(B 4+M 9 5I 9 9 (M 4+A 93K)MG)A +M 4YC8 6 4kMo=-61< >4 PRI)A<.YV 6A 4Y1»B5;E! <\(oKNEBB 4>(BW4M 9 5I 99 K 4 9+( 9d+M 4o1$B5;E) <.'4M.9DE 4MM K 94KB@ <ãM G)A>EKNK(2;M KNG)A74)( 4M 9½<ãM G)A0V+(2;MKNG)A 4 9k 9 \ M 4j)B`;E <K 4+M.79 4QP^)K 4+AoEK

#

9 A 4 8 BMKNG)A0 4!M. 9-4¾C0G)A<8B 5N=-4' 4 8:9 8©M == 99 B J 40>4 8`1B5;E' <. 9 ALE79 M 4C8 6 4GE3 4LE +M 4°K 4 <: 4Q, &>4 ;M 4u(°?4+M. 4 8 1$B5;E <.) 9 9-V 6M 4¹K 6 < G)A 4C4M 9oE79©M 6 IRE' 4/8 8 4MG)A 9Q; 4/8%>4>O; 4!TM 4t"4B 5 4 8rE 9>8¥K)MG)AIM G)A 9U; 4M 9ÞJ 6L 4 81$B5;E <.K ?ÏC9E/8:9:9-4B 5 4 8rE 9 8)4 9= 4 8% 9]\§924/8rKoG)A 4O;M8 >>4o1$B5;E!1<.K <8rE= EG)AsV @9 @ \=>8/M 4oE 8[>4K 9 4 << 9 401»B5;E! <Q(7>4= MM 4 8:49:0&A< 4G;M 8$E < Kz?4!M.oJ>4$>4/88 4KNK>4 4! 9rK 5<8 4GHA 4o1B5;E' <.K 4MU ?474 9 ) 9&>

K 6°M K 9 >4 824!74 86M.7V1@9 @2\/ < 4+M G)AG<<GM 4O 8r@E<4 =ÚK 6BM 9 M G)A 9->4=©M.M 4/8:9½9 ?4!M.74;M 8(9?49 4>6G)A-BM 9Ú< U(B 4M 9 5I 9 9 ;t®AL@/8-4'(BM 9< ( 9 6>4/86BM 9z< U( 9

E N=>K 6¹MK 9-M 4u 8r@E<4 =s>4= MM 4/8:9½>6<M 4790E! 4 8/>8¦MBl(2; 4M 9 4UE << 4¹K 9%&M:u(2;M KNG)A 4' ;

@9 D»M 4K 4 4 8< 4! )>4u( 4M:>4!>EKNK>4 81B5;E' <.K /8rE°= >8mM G)A 9 ?ÏM K 679©A 4/8©B 41$B5;E <.>4 (B(M>4K 9>4M 4K 4A>8 5>8 6< 4BE 9©M KNG)A743 8@FE<4mM K 9 9zB 24 8< EFN=`LK 4 8 4/8cP 8 4M 9f( 4+M >4o;M 8>/EKNKo>4/8 E 4 84M 48EKNK 4oE 9-4M< 4YV 8bEKNK 4<8oG)A 4\F( 4Mo=EG)A 4 8-4' >6 8rB 4 <:= A<8:9½9D$M 4ÞMDM 4K 4 8JP 8 4+M 90E'>4 < 4M 9 4 9-4s76 8©B 4 < "=>8 M 4¿8bEKNK 47?B)(2;926 <.B 4'LE! 9-4+M%< 4ÞK)M.LEA&M:J 6 >4'zJ 4 8rKNG)AM 47474 1B5;E' <.KHB 6!>4 << 40(78Á?4KNG)A<8 4M¾J 6C6ALEK 4B5;E1<.74IV+(L9?49p 6A!LK 6 ?§8 4+A< ?qP4J<8rEBM \g IR@! 4±K 6B(M 9$(B "4K 9>4 8rE 8:9 M:74 8F8 6 4 << 4¹= 84K 574(M 4 << 4ßC9E+A< K 6/8:9-4'-J 4 8K; 4 4 9J; 4 8%>4!9:ÆMG)A 9©M oE! 4M MK 9M 4BÂ@! <MG)ALK 9>4oE 4"?4K 9 MBBu 6>4 8Jr4 9%M K5>4 8 D»M G)A 9 4>4/8 7479 4M<M 9-47C6AEK 49:ÆM 4 24 8% < 4M G)AK78-4G)A>4 ( 4M 74> MK 9 M 4 >4798-6)==-4' 4 4 PfLEAB 4 4/8 M 426B 6>4'M 9%&79>4 8°D$M< E79 6B 479-4/8 5<8 6! 4 ;4&A<8:4l 4<8$1»B5;E) <¼M 9 4 8&A 4 8*4 8=<< 9-X+R ;7=Z½9?4M->4 8 P oK>; 4 8:9ÞJ 6'ÆD»M%< E 9-6B 479 4 8J 4 8rKoG)A74> (9?49 (7851$B5;E' <.*¿J 6'oP^oK 9 4M 9( C 4 8<M 9>2;M 8%E << >4B 4+M.M 4Á= 61< >4>47 6 8rB 4 <J 4/8K;E79 V[ <9 (9?49X?@A7;>=Z%\9

V @9\ 5 h £θ¨h £

θh £θ¨h £

θ¨ 9

$M 4 8 74MoK)M. 5g 4 8-261<.B 4LE 9 4M<)>4KGC 4 8<M 9K>6<.Vθ\/+M 4¹EI 9 4 << 4o,)&>4G>4 8 C8 6 4G(78

"4B 5 4/8rE79 8θ> < V θ \b < V

θ\0M 44 9rK 5>8 4G)A 474$,)&)>4'>; 4!3M 4oC8 6 4D>8E )K

P^oK 9 4M 9F)(2;b9C 4 8<M 9J 4K 9>4 9D$M 4$>6 8B 41<)V1@)9\6= 61< 79 E oK5>4 8!PR ; 4O>4/80<ãM. 4E 8 48MKNG)AK78-4>41< 4( <MG)AH 4 8$26 <)B 4'LE 9 4M%< 4±E N=U+M 4±EI 9 4 << 4h,)&!>4h)âK 9 4 << 9

>4>4 4/8`>4'MB C I 9 =A 4 8%>41< 4M 9 479 4 6 8rB 4 < 4M.74&A 4 8 E<8V+ <96C I 9# 9 # 99\§9D$M 4 ?4 8 4G)Au74 88EKNK 4'LE 9-4+M%< 4mEALE!°E N=%74 6BÁB 4' 4/848 4KNK E79-4'(= 8I 6I 8 479-4C9E+A< K 6 8:9 4>(9?¦9 fw>; 6>4/8)@ ;>; 8[8 >K 6;M 4u>4/824 8% < 4M G)A3 4 84M)( 41<: 4$>6 8B 41<:G< 4M 9K 5&79 4 8 45P 8 4+M 9 4J 6<8 4+AHE1< 9 49

8

@oF' ½1` 'A1zsM 4K 4B rE/5M 9 4 < K 9 4 << 4Q;M 84M.M:74E )K 4<85r6 9 M)BK)B 4GNALE'MIsg>4 8c"4A 4>6/8©M 4>4 8¥8ÁM KNG)A)>4t 4'L@79©M 79 4 ?4 8 M == 4h)k/oKNE+BB 4A&'74h 4 8 4+M 9>"E N=U 4 4kM 4?4 8 4G)A>4=>8 4-D$M%< E79 6B 479 4 8J 4 8rKoG)A 4 8A 4' VrK9rE/5M 9 4 < =\§9

@[?!@o4 N §&6'A ÕL. 6 A1½ <4A66A A' *, A:ÆM 8J 4798EG)A 9 4-C9EA<E1< Ku>4K 9 IR@/8 5 4 8 O() < 4M G)AE < K I 6>4aM K 9©M 4/8 4 4¹8ÁM KNG)A"K 4M.74 8C6ALEK 4tV[(9?49 P^oK 9 4M 9½>$C 4/8<M 9%\gV©C4M 4A74_C I 9¹699\&9$z/B 34EAB 4!k>4 8z<M 4E 8©MK)M 4 8 9-4"4A 4>6 8 M 4 J 4 8K; 4'74 ;M 8 (78 I 6' 9 MBK)B 4G)AEM KNG)A74 8 6>4 <<M 4<8 4K8bE 9-4 8©ME < J 4/8©ALE < 9-4Kj>4 <M 4E/8©M K)M 4 8:9-4 8-4 4KNG)A 4 24 8+( 4 88 K 9-4'LK 6 8

εW74

²E oG)A*=LKNG)A 4 C)5LE)K 9-4'LK 6/8σ>k>4 8 M +M 4K 4B E <<¼BM 9_>4! E 4<8 4

C)5LEK 9 4LK 6/8 4L()KREBB 4o=%& << 9½9z?4IRE' 9½<ãM G)AT 4 < 9 4I(2;MKNG)A 4!ε

>4B24 8rKNG)AM 4'KTJ 4+I 9-6/8$ >4 8 oKNEBB 4'LAE

V69-@ \ε

Û V ∂^

∂Ö ∂^

∂Ö \ B(M 9°M%> < 4+M G)A @>N 6!>4 8 = >

V 74M 8 4M 34EBMB 4K)M 6 4'*\6K 6;M 4M 4°?4; 4 K 1< 4MG)A

V69\ρ

∂

∂9 S7M J σ

Q MΩ9

DFE 4+MHK)Mρ

M 4Þ8EKNK 4'M G)A 9-4$_ M 4ÞD$MG)A 9 4G>4 88EKNK 4I 8%&= 9 49 8ÁM 9Ω

K 4+MM 43)4= 4 8 4)(I 6N=©M 8E 9 M 6W>4KMr@ 8 5 4 8rKW 4( 4+M G)A 4 9½9u 78 oEK)M ?½K 9rE79 MKNG)A 4HP^N=%E! 4lM< 94 9rK 5<8 4G)A 4V69 =\ SO+M J

σU M

Ω9

D24 Á< 4MG)A>4'_V9\ V9 =\K)M. 3¦E) 4+M. 4>U>4 8 Á< 4+M G)A V9\(oK &79+(<M G)A PRo=rEK74M.) >4 AM)( (N=)749ozB E << 4I>4K 8 4M 4 C)5LE)K ?5<8 6< 4+BÂKÁ< E 9 479+M 4 3¦E74M.) V69 \

σ

∂Ωu EFN=

∂ΩV

∂Ω?B&!E<4/8 4 8 26/8©BE < 4HJ 4I 9 6 8E

∂Ω? 4 8=< &G)A 4o>4K$r@ 8 5 4 8rK

Ω>nS074>474 4 8

C)5LEK MG)A 9 4)J 4I 9-6 8EN=>4/8 4/8=½< &G)A 4\

@F@> 4 !°F6 %Á ".6,o *, AC4M 4M.74Ú8ÁM KNG)A¹EoK

≥ r6B 5 6 4' 9 47>4! 4' 49:ÆM8m 479 8EG)A 9-4'"AM 4 8¦C9E+A<oE < K

4M. 4 8ÁM KNG)A¥K 4M 4 8«B 4M K 9²E < KjC6ALEK 4!j 4( 4+M G)A 479 40 4o=)>4E<8:9 4oP^oK 9 4!M 9>^C 4 8<M 9>8bE 8 9 4LK)M 9 )K>;b9 1K K 4+M_E/8rE N= AM.)>4;M 4K 4> EKNK 4M.74 8 6>41<<M 4 8) >4 8C6ALEK 4B5;E1<.74>MLK 4K 6>4 8 4u>4K P^oK 9 4M 9rKM.u>4°8bE 8%9 4LK)M 9BM 9+K 4A/8' 8-6E/4 8

<

1$B5;E <.K >4KNG)A ;M.+M I 4M 9>E oG)A°B@ <ãM G)A¹M K 9>M.>4B E 4+M. 4B C I 9m78 P^)K 9-4M 96>4 88E 8 9 4LK)M 9K 4M-IRE>*; 6 4+MK)M G)A 4M. 4UC6AEK 4/8 4)( 4-EFoK M%< 479½9$M 4 8+(bJ 4 8K; 4MK 4;M 8E N=-X+, 4)J)vwZJ+M 4>6/8:9(M 479©M 4/8:9 45P 8 4M 9:49z >4 8 8ÁM KNG)AK 9rA 4>6 8 M 4h;M8% E'>4! 6BÁB 4'> ELKNKI+M 4 r6+B 5 674 9 4 M. %4>4Br@ 8 5 4 8 5I 9$K)MBu< 9E 5<8%&K 4! 9²K 4M.IR@! 4gV[ <9 (9?49BX+FE ; ;Z½> X83< <7=Z%\&9 zKUP 8 4+M 9 4(B 8bE 9 4 8©M E < J 4 8rAE < 9 4kJ 6!¶C9rEA< ; 4 8%74 +M 4Ý?4! 8©M == 4lC6ALEK 4W)Wr6B 5 6' 4' 9-4 6o=:9

K>= 6 =B 4! 9+( 9> ;4&A>8-4'M: >4 8²8ÁM KNG)AK 9 A 4 6 8©M 4 4M.74 C6AEK 4 6=:9²E < K4M.k?4/8 4M G)A4 8©Io< & 8:9M K 9>M.u>4BN>8 4M 4Ur6B 5 6! 4 9 4J 6 8©I 6BÁB 9>(9?49; 44M. 8]&FB-<M G)A74<8¦?4 8 4M G)A4M. 4K C9rEA< ; 4 8©IRK 9)G)IRKkE)(âE )K C 4/8<M 9 4K 974 VJ1<9G(9?¦9sX[,!)>;7;Z%\&9¿?4M¹C9rEA<J 4 8LEG)A< &KNK)M >4'j;M 8 4M.74 D»M ==oK)M 6' >4/8ur6B 5 6 4! 9 4 V C6ALEK 4\§>´K)M 4i;E>4 <:ºK)MG)AM. 4MLE 4 8 B ) J 4 8< 4M. 4 EB 8:9 9·D24 8·= >8 +M 4 1B5;E1<.74 ;M G)A 9©M >4r6A< 4LK 9 6==m;M 8%M.UM 4K 4B 8 6 4 <<FM G)A 9f 4 8oG)IRK)MG)A 9 M:79½9$D4K)ALE < s 4K)M 9[( 4'M M 4K 4BAM 4 8 7479 8rEG)A 9 479 4 E << +M 4 r6B 5 6' 4! 9-4' I 4M 4 J 6 << J 4 8rKNG)AM 4>4! 4D$M ==oK)M 6K >4KNG)A ;M.M:I 4M 9-4 V[ <9IX+FE ; ;Z½>IX8L< <>=Z%\ M 4 BEI 8 6K)I 6 5M KNG)A 4

8 6!>4 <<ãM 4 8ÁJ 6°C9E+A<)M K 9 4M.¹C)574(M E < =E1<<)>4 8E << >4B 4M 4!Ú8MKNG)AK 9©A 4>6/8©M 4oVrK9E oG)AX: 6 < ;Z>XDEG ;Z+\&9:ÆM 8K 9-41<< 4' 4+M.M:>4 ;M G)A 9 M: 4°?4! 8©M ==-4 4 8 4M 9 V+(9?49X?4 8%vvZ]\§9:ÆM 8/7479 8rEG)A 9 4 4MDVr6 9 8 6 << ?\261<.B 4t BM 9/74 8$8EKNK 4 B-9 4 8rK 9 4+AB 4;M 84E>EKNK5M 4°8ÁM KNG)AmA 6B 6>4'ÂK 4M9DFE!(MK 9M 4GV% 4KREB 9 ?\D$MG)A 9 4

ρ74B&E

V69 # \ρ s ¡

>4=©M.M 4 8:9½9z 4B 26 <B 4 4+AB 43+M 4°M ? 9 4$C6ALEK 4nV-6>4 8dr6B 5 6' 4 9 4!\dEK46 <.B 44M7D 4K)M 9+( 4uM 4¹8EKNK 4 B

9 DFE! K)Mu>4 8F26 <.B 4LE' 9-4+M%< 5

> 4 88bEKNK 4LE 9 4M%<

χ K 6;M 4M 4°D»M G)A 9 4

ρ >4 8 M ? 9 4 C0ALEK 4 4 9rK 5>8-4G)A 4! 78G)A

V6998\ 5 ¡ ¡ > χ

s s > ρ s ¡

>4=©M.M 4 8:9½9 8M.A 6+B 6>4' 4-8ÁM KNG)A>4 ; 4 8%>4'bEAE)J 6'3V9 # \>V98\-M 4 8r@E/4ρ>

5>χ >ρ M. 4>4+B r6@ 8 5 4/8 5I 9<4 8©Io< & 8%9>M>4B 4; 4M%< K >4 8 8 4)(2; 4/8 9 M 4K 4/8' 8@FE<4=>8

4M. K)M G)A EFN= 4M. 4! C I 9o(oKNEBB 4)(M 4A 4'>4KN6 <.)B 4T 476BB 4 ;M 8% 9QVD$M 4!aM K 9 4)( M 4K 4 8 8 4(2; 4 8:9 4K 4MRE 4' 6BB 499\?4IRE 9½<ãM G)A>4 < 9-4EM 4Á= 6 < 4>4°?M< E)( 4'3V©M.E << 47C I 9 4u (bE1<< 4'`4+M 9 4)\

V699<\ ∑

5@> ∑

χ @

∀M @>999> 5

≥; >

χ ≥; 9

P^oKV69 # \§>V9 8\1=-61< >4!>4 8 oKNEB(B 4ALE'(2;M KNG)A74(8EKNK 4! ?m)6 <.B 4E 9-4M< 4

V699w\χ

ρ ρ5

w

K 6;M 43+M 4°8ÁM KNG)AK 8 4 4 <:=>8M 4°D$MG)A 9-4GM 4 8:4(M5/8-6+I 4"D»M G)A 9:47V+1<9X?4 8+vvZ%\

V699v\ρ

∑

ρ 5>

V69-@ ;!\ Ûρ ∑

Ûρ χ

9

D$M 4jC8 6FI 9-4ρ 5; 4 8%>4ÚEoG)A0C E 8:9©M E < M G)A 9:4U>4oE 99P^)K774B ?MK)A 4 8©M 74Ú= 6 < 79½>

E EUM 4or4 9%M K>4 84D$MG)A 9-4'ρ

ρ

EFoK78 4M G)A 9> B±BM 9$M< = 4mJ 6$V699w\ 26 <B 4 ?á8bEKNK 4E 9 4M%< 4ÝM 4M.oE>4 8B (78-4G)A 49D24 8 26/8:9-4M<B>4 88EKNK 4 ?u>474 4/8>4j26 <B 4LE 9 4M%< 4Y 4K 9 4A 9 4 6GNA E 8 M>RE EYM 4K 4b; 4 4<8­J 6 >4 84"4+B 5 4 8rE79>8 6G)AkJ 6W)B5;E1<.K;= 8 4M 4·B 4G)ALEMKNG)A 4' 24 8[( 4/88)>4¶EA)&+M K)M. > E M 4E 9 4M%<ãM >4 8bEKNK 4 MB r6 9 8-61<< J 61<.B 4! K)M G)A M G)A 9àJ 4/8%&74 8>¼K 6 < E>4 I 4M 4C6ALEK 4B5;E1<.74¶E N= 9 8 479 4!9DFE 474ºIR@' 4 K)MG)A M 4 6 <.)B 4LE 9-4M< 4E << 4M. >8rG)A 4 < EK 9 MKNG)A 4 4 8+( 4 8 8 4 &'>4/8>=E << K K)M G)A M 4 4 < EK 9 MKNG)A 4! 8 6< 4 >4/84M.)( 41<: 47C6AEK 4J 6! 4M.E>4/8- 9 4 8rKNG)A 4M:>4!92P^)K`>4/86?4(M 4+AOV699w\ = 6 < 79E=78J 4 8rKNG)AM 4 4 4N"4B 5 4 8E 9>8 4Lk24 8[( 4 88 >4NSÅK 6 < E'>4 I 4M. 4 1»B5;E) <>4K 9E79:9= M>4 S

V69-@>@*\χ

ρ £ θ ε ¨ρ£θ ε ¨

5Vθ > ε

\ ρ £ θ

ε ¨

ρ£θ

ε ¨ 5 V θ §> ε

\§9

V½DE 4M 4AL@/8 4 4 9K 5<8 4GHA 4θ > ε

)(2;9θ&>

εG( 4M 4B oK 9E'*199\

@F@oF !1 §AÞÏ'9A Ú#»%*,½ !1 §A 1zs>4 84r6 9©M.BKHB 4G)ALEMI¹B(M 9`C6ALEK 4')B`;E <>4!kMK 9 4K o@79©M > >4QPR 9 4M%<- 4 824 8+( 4 88)>4>)74 8 78rG)A D»M G)A 9 4&!>4 8 4A 4/8J 6/8%>4 8;= 45;M 8%> ( 4K 9©MBB 49:ÆM 847479 8rEG)A 9 4 4M V% 8:4M MB 4LK)M 6'LE < 4K \r6 98 61<< J 61<.B 4 BM 94>4 8D»M G)A 9 4

ρ&>BEK

V[(9?49¦M.o= 6 < >4ÞJ 60C6ALEK 4 ?4$e 6 4 8/"4B 5 4 8E 9 >8%&74 874\´K 4+M.$26 <B 4EN=lB(M 9U>4 8Q 4 4 D»M G)A 9:4

ρ&'>4/8:9½9 24AB 4N;M 8ÚE>GEKNKáK)M G)ALM 4Å8EKNK 4I>4K

r6 98 6 << J 6 <.B 4K5M G)A 9&74 8:9>oK 6M< 9<6== 4!LE 8

V69-@;\ ¡ ? ¡ ¡ ρ

?ρ

ρ9

C4M7EKhr6 9 8 6 << J 6 <B 4 4M : >8=-41<u>4 8YrE 9:4< &'>4 <9?4M¹E << >4B 4M.E>476BB 4 4/8Jz%K 679 8-6 5M 4>4/86 <.B 4!&>4/8M%< 9EV69-@ =!\ nV½< ∆

< \ >; 6 4+M

∆<^M 4u,)&'>4)& 4 8 %4>4/8<4+M.)( 4 < 4rE 9 4mM K 99C6BM 9<4 8©AoE < 9-45;M 8

v

V69-@ \ ¡ ? ¡ ¡ £h ∆

h ¨h

h = ∆

hh =uV ∆

hh \ V ∆ hh \

9

P^oKV69-@;\ uV9 @ \4 8 AoE < 9-4'`;M 8

V69-@ # \ ∆hh

@ ρ

?ρ

ρ?@9

?4MNIo< 4+M. 4-D»M G)A 9 4& 4<8>4= 6 < 79E oK V69-@ # \J 78G)Au,M 4E 8©M K)M 4 8)

V69-@ 8!\ ∆hh Û ρ

?ρ

ρ=>8fM 4 8 4 < E79©M J 4b,)&!>4&'* 4 8)kM %4>4/8 3MG)A 9 9?4Md 4kB 4MK 9-4'oPR ; 4' )>4E N= C6ALEK 4B5;E! <>4 MB C9EA<âKNG)A 4M 9 M 4 4Mo=EG)A 4 8-4 6<8©B 4 <IV9 @ 8\>4 8 4G)A 9=-4/8:9©M 79>)Eu+M 4 8 4 < E79©M J 4'"D$MG)A 9 4&>4/8>4|§Io< 4M.~K)MUVrK9*; 4M 9-4 8/ 9-4'\9 DEA 4/8; 4 8%>4;M 8M.>4 8/>6 < >4°B 4MK 9B(M 9->4 84+M.o=rEG)A 4 8:4U>6 8B 41<V9 @ 8\E/8 4M 9 4$Q>8¦MBC I 9 I)>8+(EN=M 44 8K; 4! 374 8- 6 8rB 4 < V9:@ # \ 4M>4A 4'9

@F@ , § 1 )5 ", ! 'A1D$M 4 Io< EKNK)M KNG)A 4 DRALEB 4 < ? 24 BÂE!7?&?4(M 4A74 >4/8 <ãM. 4E/8 4 MK 679 8 6/5 4"4A 4 8B 6>4 < EK 9 M (M 9%&79o; 4/8%>4 M. >4 8¿: 4M K 4 B 6M = M (M 4/8:9½>oEKNK K)M 4 M 4 4 8+( 4<88>4M. 9 4 8©M 4 8-4!>M 4±EoK R>4 8>4k>4/80C6ALEK 4 4 9rK 9 4A 49¹D24 8 0M.o=rEG)AA 4+M 9zALE < 4 8E 8 4M 9 4 ;M 8°BM 97>4/8Þ<M.74E/8 MK)M 4 8:9 4 6/8 B 4 <5V9 @ 8\9OzB C I 9 9 @9 4 8r@/8:9-4 8%W;M 8$M 4r6LK 4F 4)( 4>; 4!zBM 9/>4 8R 6 8rB 4 < V 9 @ # \B>4E 8 4M 9-4 9 ;M 8]9P N=>4 8/ 8) < E!>4 J 6V69-@ 8!\ K 479[( 4'`;M 8E

V69-@ <!\ε

Û ν³ σ ? ν³ 9 8V σ \ δ

Û ρ

ρ£θ¨

ρ£θ¨ δ

>

; 6 4+Mρ0M 4ÞD$MG)A 9 47>4 8 3 4= 4 8 4(I 6o=©M 78rE79©M 67VrE < K 6Q 4( 6>4"EN=

θ\f

ρVθ\jM 4

EI 9 4 << 4¶D$MG)A 9 4K)M 9V8ELG)ABE <IRE 4K 9+(<M G)AK 4M.>R>4¶< 4 9+( 9-4'0"4 8rB M V9 @ <\E N=+(oK 5LE < 9-4M 9©A 4 8rBMKNG)A 4G)71»B5;E1<.KTJ 4 8+( 4 88)>49DE!M%< 9

V69-@ w!\ε

Û ν³ σ ? ν³ 9 8V σ \ δ α

Vθ?

θ\

δ Û ρ

ρ£θ ¨

ρ£θ ¨ δ

9

1B 9 4 8ρVθ\¿M K 9 479+( 9'+M 4¹D»M G)A 9 4O>4KEI 9 4 << 4DC6ALEK 474BM KNG)ALK>E! 4 8J 4( 6>4'ÂEFN=-M 4

3)4= 4 8 4)( 9 4B 574 8E79>8θ§> (­J 4<8rK 9 4A 49

>ν

*α

K)M. 4 9K 5<8 4G)A 4 26F7?§8 6!<> C 6M KNK 67? E+A< : & 8 B 4EFoK 74AK)I 6 4==©M (M 4 9 4() <ãM G)A

θ&>

δM K 9d 4<8B6MA 4M 9K 9 4LK 6/89d>88E 9-4 8©ME <M 4

6A 4z8ÁM KNG)A)K)I 6B 5 6 4 9 4!UV C6ALEK 4)\¦=]&1<< 9 >4/8F< 479+( 9 4G"4/8©B M.UV9 @ w\m; 41>-ÞBE4 8©A& < 9 M 4 Io< EKNK)M KNG)A 4 DRAEB 4 < ? 24 )BE ?§?4(M 4A 4 >4 8 <ãM. 4E<8 4"4A 4 8B 6>4 < EK 9 M (M 9%&79½9

@ ;

:ÆM 8 4+B 4/8©I 4> EKNK M +M 4K 4B 8 674 << M 4 8GHA "4B 5 4/8rE79 8 ? C6ALEK 4&>4 8>4'(A 4 8J 6/8%>4 8;=-4' 4 24/8+( 4 88 43-C 5LE 4M K 679 8 6/5K)M9DR>8rG)A5P^N=< @K 4' 4<8< 4M G)A)>43V9 @ <\/uV9 @ w\ LEG)A

σ4 8 AE < 9 45;M 8

V69-@ v!\σ

áµ ε λ

V:9 8ε\

δ – K ρ

ρ£θ¨

ρ£θ¨ δ,

V69<;!\σ

áµ ε λ

V:9 8ε\

δ?= r

αVθS

θ\

δ – K ρ

ρ£θ ¨

ρ£θ ¨ δ

B(M 9>4u,NEB>? r6>4)== M (M 4 9-4λ

µ>NK 6;M 43 4BNr6B 5<8:4KNK)M 6K)B 6!<`rt>4B6&FE

V69@*\µ

³ N£rÛ

ν¨ > λ

ν³

£rÛν¨ £rÛ4

ν¨ > r ³

H£Û4 ν¨ 9

:ÆM 8 ALE' 4M 4m8@ <MG)AI 4M 9½> >4-< 479[( 9 4 "4 8©BÝM.uV9 @ <\ M K V69>;\4 9; 4>4 8BM 9:9 4 < K V9 v\ >8rG)A3M 426 <oB 4E 9 4M%< 4 6!>4 8 BM 9 9 4 < KV9 @ ;\/ 78rG)A M 4¹8EKNK 4LE' 9 4M< 4°E oK2()/8oG)I 49p 4>4 4+M.)( 4 < 4 C0ALEK 4 IRE MA>8 4 MM JM: 4 << 4> 9 4B 5 4 8E 978rEA)&+M >48bE 9-4 8©ME < 5LE/8rEB 479-4 8 4K)M 9+( 49/F>8>4'D$M< E79 6B 4 9-4 8/J 4 8rKoG)AzM K 94KQMG)A 9B 679K; 4'M 1>-M 4 8rE>4n(GM K)I) 9©M 4/8 4>JLEGNAO; 4 < G)A 4Ú8ÁM KNG)AK 8-4!>4 <:G; 4 < G)A 4Q 8r@E<4'D(n 4 8 4GHA 4K)M. 9zB C I 9 ; 4 8%>4! ;M 8 E oK5>4u 4+M 74D 6<8©B 4 <)V9 @2<\ )$V9 @ w\ 4M. 4o>6 < 4 8A 4 8< 4M 9-4'>kM 4 =>8 4 :0&<8©B 4E )K >4+A)K)I 6>4==©M (M 4 9 4 +M 4 PR ; 4 >4/88MKNG)AK78-4>41<B(M 9/>4$26 <B 4oE 9 4M%< 4>J 4( 6 4®E H=M 4 34=-4 8-4!)( 9 4B 5 4 8rE79 8

θ&>

LEA 4 < 4799DFEsM 4o26 <B 4LE 9 4M%< 4zJ 6' >4/8B"4+B 574<8rE79784 >4o24 8+( 4/8 8)>4'ßEA&'>40V+1<9V69-@>@*\ \§> >4! 4 ;M 8 74 8EKNK 4LE 9-4M< 4! >4 26 8+() 9¶DELEGHA =A 8 4 ;M 8WM 44 9rK 5<8 4G)A 4>4?479 8rEG)A 9)>4¿B(M 96 <.B 4E 9-4M< 4s 78rG)A9R KQM%< 96==:4LE/8R>4+B6& E>4 8d 6/8 B 4 <2V9 @ ;\

V69\ ρ

ρ£θ¨

ρ£θ¨ nV ∑

Û

ρ £ θ ¨ χ \mV ∑

Û

ρ £ θ ¨ χ

\ S"@>

; 6 4+MχM 4iPR)=EK)BEKNK 4LE! 9 4M%< 4ÝK)M.19 P^oKY74 ?4(M 4A>4!lV9<;\§> V9\

K 6;M 4YV69-@ \=-6 < 79^EK8E79 4 8 ME < 4K 479+(=>8 M 4iV©M K 679 8 6/574\<ãM. 4E 8 4 "4A 4/8©B 6 41< EK 9©M (M 9]&79BM 9C6ALEK 4B5;E1<.74 VE < K 6=>85r@ 8 5 4 8©>BM 4·8MKNG)A>4kK)M.)\ÁM 74 8fPRA&)M I 4+M 9J 6>4 24 8rKNG)AM 4>4l>)>4/8"4B 5 4/8rE7978

θK 6;M 4>4 8 8EKNK 4LE' 9 4M< 4

χ9

PRLE < 6(3V9\4/8©ALE < 9-4'`;M8 BM 9:9 4 < K V9 v\-M 4°DFE/8rK 9-4 <<.BM 9:9-41< K 26 <.B 4!LE' 9-4+M%< 4'6(

V69<=!\ ρ

ρ£θ¨

ρ£θ¨ nV ∑

ρ Vθ\5

Vθ\J\/V ∑

ρ Vθ\5

Vθ\J\ S @)>

; 6 4+M 5M 4 PRo=rEKTJ 61<.)B 4LE 9 4M< 4K)M: 9²D»M 4?4(M 4A74 V+9 <\ ; 4/8%>4¥= 8^M 4

PRo=rEK)BEKNK 4 ? )(2;9SâJ 6 <.B 4'HE' 9-4+M%< 4 J 6/8rE oK >4K 4 9+( 9½9Á:0&A>8 4tM 4 ?4(M 4A

@ @

V69\ 4() <ãM G)Aχ

<M 4E 8ÁM K 9VE'* 4K 4A 4¾J 6 4M. 4/8ÁB@ <MG)A 4jPRA&M:I 4M 9`>4 8C E 8rEB 479 4<8J 6

χ\§>)M K 9<4KV9/=\ 4()1<M G)A74 8 5

M G)A 99»M 4 8©M.-<ãM 4 9<4M ; 4+M 9 4<8 4 826 8%9 4M%<

4M >4 8P 8 4M 9 BM 9+8EKNK 4'LE 9-4+M%< 4'9

@41& 5 A"!1 %1/*, A 60! Ú#»)Ï'(Õ1½1½ *,

:ÆM 8t 4798rEG)A 9-4 4M.74 (2=<M 8©M KNG)A74 C 8-6! 4 I 8:4M K;=r@ 8©BM >4 6>4/8 I 8:4M K78©M=r@<8©BM 4x 4 8KNG)A!M 9:9K)>`+M 4ÝK)M G)Al74 < EK 9 479uB5;E>4 < K 61<<9 P^oK·BE79rA 4+BÂE 9 MKNG)A 4' 8>4LEAB 4 ;M 8VE>BEKNKo( ?4M 74Ks(Y 9 4 8rKoG)A 4>4' 1B5;E <K 5>8 6( 4KNK 4K)>E < K 6 4M 4 8BV13)4= 4/8-4')( ? \c"4B 5 4/8E 978

θ§>-M 4 C 8-6' 4 M 4j,&>4 <j >4sVKP^E<4! ?\

DR>8rG)AB 4KNK 4/8Y 4K)M 9+( 4 9:kM 8J; 4MK 4o 4K 6!>4 8rK-EFN=M 4K 4 1»BK 9E)¿AM>-EßM.U>4/8D$M%< E79 6B 479 8©M 4E1< K P^oK EK< &>4 < >4;@+A<MG)A M 4 ,&>4Y>4 8uC8:6 4 74M±<;

°

>4 6BB 4U;M 8% 96: 6 << 4U;M 8B(9?49/M 4 1B5;E < 4M 4 8J 61<< K 9%&!M:E )K 9 4M 9 MK)M 4 8:9 4C8 6 4G>4K fw>; M.DC 4 8½<ãM 9/ 9-4/8rKoG)A 4>K 6 IR@ 4;M 8 (9?¦9

θ $<)<=

° 674 8E oGHA 4M.

8r@ E<4/8 4Kθ ; &A< 4!9D$M 4l,&)>4 < M K 9E! M 4W,)&>4 > M 4 M 4NC8 6 4H 4MR>4/8

"4B 5 4/8rE79 8θ 74K)M 9+( 9 9PV<< 4I&FE<4/8 4 r)8%&=%9 4 S EoG)ALM 4 C0G)A ; 4 8 I 8rE=:9GS K 4M 4

J 4 8LEG)A< &KNK)M 799D$M 4á= 6 < 4>4L8r@/8:9-4/8 4l 4(M 4A 4' K)MGNAl < 4M G)A74 8rBE E>4 E H=7M 4 26 << ?gNM 426A< 5<8 6 4 9?4M >4! 24 8rKoG)A 4!5;M 8%6=%9 >4 8JV]F)! 4\m26A< 5<8 6 4u>4 8!26 8+()u74 4 4>; 4M<dE' M 4gPR!LEAB 4 4M 4/8A 6B 6!>4 4!0"4B 574 8E79>85 C6ALEK 4HJ 4<8:9 4M%< 4A 4 8>4 8 4G)A 9=-4/8:9©M 79>4 8rKNG)A 4M 9 9 ­==-4LE 8 M K 9V-=19-@ \

σ;

,o@K>4/8K 9E79©M KNG)A74Á< 4MG)AV-=19\ ?-M J

σ ;

s>4') 9 4 34E!' 4M>4o 4 8²C 5LEK;= 8:4MA 4+M 99 : 4M 9-4/8$K 4M'+M 4 a ?%P$G)ALK 45LE 8rE << 4 < (>8F=<M.>4/8EG)ALK 4 9C6+B(M 9=-6 < >4E oKV 9 @ <\

V-=19 =\ε

Û ρ

ρ£θ¨

ρ£θ¨ >

V-=19 \ε

ε Û ρ

ρ£θ¨

ρ£θ¨ >

E << 4ÚE 4 8 4sr6+B 576 4 9 4°J 6εJ 4/8KNG)A*;M.>49?4ME'>4 6BB 4 4 8R6B 6!>4M 9]& 9

>4 8 "4B 574<8rE79>8 74 8 C6ALEK 4'HE' 9-4+M%< 4 4 9K 5<8©M G)A 9 >4+B @/8:9<M G)A I 6LK 9E 9-4C)5LEK2(oK 9rE) 4M.@/8:9½<ãM G)AI 6LK 9rE! 9 4 824/8+( 4 8 8K2(oK 9rE! 9zKLK* 4K 6 4<8 4M< 9

V-=19 # \ε ∆

hh > ε

ε ∆

aa >

@

E 4MM K 9 ∆aa M 4 8 4 < E 9 M J 4 R>4 8)Q>4K¹Df 8rG)AB 4KNK 4/8rK 19 C6BM 9 4/8©ALE < 9-4'u;M 8=>8 M 4

<M.74E 8 4A 6B 6>474U"¦A 4/8©B 6 4 < EK 9©M (M 9%&79VBM 94C6ALEK 4B5;E!1<. 4MB K 5LEK;= 8 4M 4E << M 4(2; 4+M < 4M G)A74

V-=1998\ ∆hh Û ρ

ρ£θ¨

ρ£θ¨ >

V-=199<\ ∆aa Û ρ

ρ£θ¨

ρ£θ¨ >

M 4jM 4Yz%K 6 9 8 6/5M 4j 4<89©A 4/8 BMKNG)A 4 i1B5;E1<.KTJ 4<8+( 4 8 8>4!ÆEFoK /8)G)I 49V?4MD$M%< E79 6B 479-4 8J 4/8rKoG)A 4!0V% 9/A9>6A 4o()K &79[(<ãM G)A 4¾?4 < EK 9 >4'*\ M 9BM 4oÁ< 4MG)AjV = 9 <\>4>4 4/8-V =998!\I 4M 4(oK & 9[(<ãM G)A 4z)=-6/8©BE79 M 6 49:ÆM 85 4798rEG)A 9-4 479+( 9 4+M.0F; 4M 5AoEK 4B 6>4 <<> (9?49BM 4i1B`;E <ÆJ 6 P^oK 9 4M 94(C 4 8<M 9½9D$M 4¥8EKNK 4E 9-4M< 4ÞK 4M 4¹BM 9

χ

χ&> M 4$261<.)B 4LE 9:4M%< 4zBM 9 5 5

4( 4MG)A 479>MA>8 4 PRo=rEK>; 4/8 9 4K 4M 4!V-=199w\

χ @>

χ ; > 5 @> 5 ;19

1B 9 4 8?4EG)A 9)ÂJ 6 G @^S"G 4 8ALE < 9-45;M 8E oKV69\ uV = 9 8\

V-=199v\ ∆hh Û V ρ

Ï£θ ¨

ρ£

θ¨ ?2@ \

Û ρ

Vθ\ ρ

£θ¨

ρ £

θ¨

ρ£

θ¨

ρ £

θ¨ χ

§>

; 6 8rE oK5E-< 4+M G)A 9= 6 < 79

V-=19-@ ;!\χ dV = ∆

hh @*\ ρ

£θ¨

ρ£

θ ¨ S"@ ρ

£θ¨

ρ£

θ¨

ρ £

θ¨ >

; 6 4+M M 4 8@E/4χ&> ∆

hh

θM%9E9 ( 4M 9rEA)&M K)M. 9

P^oKu>4 8R>6/8©B 41<V = 9-@ ;\«IRE'"E < K 6oM 4jJ 6 <. 9 M 6D>4 8$8bEKNK 4LE! 9 4M%< 4 4K C 4<8<ãM 9rK < 4MG)A 9

4 8 4G)A 4 9 ; 4 8%>4!> ; 4' M 4 ( 4M 9½<ãM G)A 4' 24 8½< &N=-4 J 6 "4B 574 8rE79 >8 ,)&>4'&74 8

∆<>4B 4KNK 4!; 4 8%>4!9P´< K

θg;M 8]$+M 4O"4B 5 4 8rE79 >8J(¹?4+M.7>4 8

8 4KNK74; &A< 991$B M 4b>6/8©B 41< V-=19-@ ;\VK 6;M 4 4 9rK 5<8 4G)A 4>4ß= 6 < 4>4¥E!)( ; 4 4>4MK 9 M 4jr4! 9%MK>4 8D$M G)A 9-4s>4 85 4M 9-4+M%<M:79 4 C6AEK 4E < K I 9©M 674 74 8f"4B 5 4/8rE79785 679K; 4)M: 9 zBC I 9 4A 4 ;M84>4K 6'* 4 8:9^E 8rE ;= 4+M.> ;M 4K)M G)AkE oKg>4kE N=%>4 6BB 474 V; 4M 9-4 8 4¥DE79-4!\4M.740>6/8©B 4 <=78BM 4Q"4+B 5 4 8rE7978rEA)&M:I 4M 9 74 8VD$MG)A 9 4U>4;M 4< &KNK 99P^oKU 4 6/8 B 4 < 0V = 9 8\bV = 9 @ ;\= 6 < 9 >EKNK 4<88EKNK 4LE 9:4M%<- 4 8´K)MG)A0M%< 4>4C6ALEK 4 K 6 < E'>4h<<< 4M. 9>"K 6 < E74 I 4M.74 C6ALEK 4B5;E <) K 9E79:9=©M. 479½9ÞD$M 4K 4"0E79rKNEG)A 4 IE'774M >4 84D$MG)A 9-4! 4K 9 MBBuG>4/8 ( 4/8rK 9J 6 << K 9]&'*+M J 6 8©AHE'>4 4!UC6ALEK 4 4 9+( 9; 4/8%>49DEKl!a>5 4/8©MB 4 9°BuoKNKE oG)ANMG)A 9G 4M`>4/8 C 4 8<M 9 ?ÏC9rE 8%9:9-4+B 5 4 8rE7978

@ =

4>6! 4o; 4/8%>4>4K 6>4/8 4KMK 9´E oG)A 4M.74 AL@A 4/8 4j"4B 5 4/8rE7978BÂ@! <MG)A>FK 6 < E>4I 4M. 4E>4/8-4'3M G)A 9>4; LKNG)A 9 4!7== 4I 9 4EH= 98 4 9 49PRLE < 6o(sV = 9 @ ;\­IR@ 4$;M 824+M. 4 4' 9K 5>8 4G)A 4! 4k?4(M 4AÞ=>8B>4o26 <B 4E 9 4M%<>4KC 4 8<M 9<4 8©AoE < 9 492P oKV9<=\ OV = 998\1=-6 < >4

V-=19-@>@*\ ∆hh Û V ρ

£θ ¨

ρÏ£

訣

ρ£

θ¨

ρ £

訨L £

θ¨ ?@ \§>

V-=19-@;\ 5 dV = ∆hh @*\

ρ Vθ\ ?

ρ Vθ\ -V = ∆

hh @*\/V

ρ Vθ\ ?

ρ V

θ\ \ 9

z V = 9 @;\°A&! 9>4/8 26 <B 4oE 9 4M%<«M%9E 9«J 6! >4 854M 93J 6' >4 85"4+B 5 4 8rE79>8E>

4 8%M 4KmM K 9->4 8J)KNEBB 4AE$(2;MKNG)A 4 5 ∆hh M G)A 9¦B 4+A<8VrE==©M. ?\V<M.74E/8J;M 4

>4 8 (2;MKNG)A 4χ ∆

hh MUV = 9-@ ;\§978 24 8-4+M.o=EG)A$>4 8C0G)A>8-4+M. ; 4M K 4ÞK 479[( 4!G;M 8

E 479+( 9

V-=19-@ =!\γV:9]\ $V ∆ hh \ V:9%\

h £p¨Lhh

=>8M 4 8 4 < E79©M J 4G,)&! 4&)>4 8)(B 4M 9 5I 9 9½9:ÆM 8 B@G)A 9 4 4M.M:>4mC0G)A<oKNK 4E)K`>4'(A 4 8%>41< 4M 9 479 4 6 8rB 4 < 6(M 4A 49?4MN`, §(1$ b1+!AoVrE < K 6BM 9

θ

θ\ 4 8©ALE1< 9 45;M 8EFoK V = 9 @ ;!\/GV =9 @ =\

V-=19-@ \χ V:9%\ =

γV:9%\ ρ

£θ ¨

ρÏ£

θ ¨L

ρ £

θ ¨ >

K 6;M 4(EFoKV = 9:@;\/GV =9-@ =\

V-=19-@ # \ 5 Vθ§> 9%\ =

γV:9[\ ρ

£θ ¨

£ρ£

θ ¨L

ρ £

θ ¨¨£Û γ

V%9%\ ¨V;M 4 4 8oB BM 9M G)A 9½<ãM. 4E 8 4<8JPfA&+M I 4M 9 J 6

γV 9[\\§9-V: 4>4U74 8$8 6>4 << E!LEAB 4UM< 9

LE79>8<M G)AγV:9[\ @9 \

P^oK5>4 oKNEBB 4A& 43V9 w\&>V9 v\= 6 < 79M 4G>4A< 4/8E'LKNG)A&79+(

V-=19-@ 8!\ 5 Vθ&> 9%\?

χ V:9%\

≤ ρÏ£

θ ¨

ρ £

θ ¨

ρ £

θ ¨ V1@ S'5 V

θ&> 9%\ \

χ V:9%\§9

D24 8? 8)G)AM0V = 9-@*8\bM K 9$= 8C9E+A<6Io< 4M>J;M 40 < 4M G)AzK)M G)A 9LE 8 ;M 8% 9RF>8R>4kC9EA< Rw>;< E 9 4$M 4¿D$M G)A 9-47 4KP^oK 9 4M 9KGU>4K$C 4 8½<ãM 9rK=>8

θ s8>;>;

°MVK9mX-3 6Av8Z>-E 4M

K 9EBB 4 +M 4 PRE74=>8 >4lC 4 8<M 9E)K >4 8E>474 474 "E 4 << 4 >4+M 4= 8>4P^oK 9 4M 9;6>8%>4 >8rG)A-<M 4E 8 4 3 4 8 4KNK)M 6 4 8©ALE < 9-4!9 \V-=19-@ <!\

ρ Vθ\ u<8w!w >

ρ V

θ\ u<8 ## X+r-5eB Z>

K 6;M 4 4' 9K 5/8-4GNA 4'Á=78 4 @ ;>; 8%8 =78θ O8>;>;

°

V-=19-@ w!\ρ Vθ\ u<<@ # >

ρ V

θ\ u< # vv X+r-5eB Z9

@

C6BM 9= 6 < >4E oK V = 9 @28\/*GV = 9 @*w\= 874 fw>; V-=19-@ v!\ 5 V

θ&> 9%\?

χ V:9%\

≤; > ;>; V1@ S'5 V

θ§> 9]\\

χ V:9%\

≤;1> ;>;@ @

K 6;M 4(EFoKV = 9:@ 8\/GV =9-@*v\= 874 @ ; ; 8%8 V-=19<;!\ 5 V

θ&> 9%\?

χ V:9%\

≤; > ;@ V@ S 5 V

θ§> 9[\\

χ V:9[\

≤; > ;>;>=!w

E < K BÂEaMBE < 4/8: 4 8:9 479K;Ei(>8ALE < 4l1B5;E' <.94Ý?4M.l ( J>4W>4/8J 6 << K 9]&M >4'71$B5;E'*1<.®M K 9 M 4K 4 8 >4A< 4 81< 4MG)Au<<9F>8JM 4 < 4¹?4 < E)>4°M K 9 E < K 63 4 81B 9 4 8rKNG)AM 4! (2;M KNG)A74 8EKNK 4' ?o 26 <B 4LE 9 4M%< 4MC9rEA< M G)A 9 4 8©A 4!<MG)A9?4M4M. 4 8¼I 6LK)MK 9-4! 9 4 8 6!>4 <<ãM 4 8) K 6 << 9 4 4/8k 4EG)A 9-479H; 4/8%>49á?4ME'<M 474>4C)5LE>4IR@! 9 4>4/8-1c 9-4/8rKNG)AM 4 4 9K;EK` 8r@E<4 8; 4 8%>49:ÆM 8f 4B 4/8 I 4'>dEKNK-=>8 M 4jLKÞM. 9:4 8 4KNK)M 4/8 4>4 "4B 5 4 8rE79 >8 4>0E < K 6 9 4 8 @ ; ; ;

°

>4 8iP^oK 9 4M 9 4; 4M%< K M 4 8r@E<4 8 4 D»M G)A 9 4 E1< K 4<8tC 4 8<M 9N 4K)M 9+( 9kV 74M < 4M G)A74 8"4B 5 4/8rE79 8%\§>K 6 EKNK$M 4¾D$M == 4 8 4( 4¿M. >4' 4 4 8%s>4 8mP^oK 8oG)I 4¾M.gV-=19-@ ;\ =-6 < >4 4uM G)A 9)J 4 8rKNG)A ;M.749

>8l+M 4 ' !9A1 %+&,1& $ b1+! 'A 4 8%M 9 K)MG)A 4+M. 4 ; 4K 4 9½<ãM G)A 424 8 4M.)=EG)AÂJ 6' V =9 @ \§9 FEG)A | 4 <ãM G)A-< E'* 4 84M 9]~ +M%< 9

V-=19@*\ @ χ V

∞\ =

γV∞\ ρ

£θ ¨

ρÏ£

θ ¨L

ρ £

θ ¨

VγV∞\ M K 94M 4 8 4 < E 9©M J 4H,)&>4!&'>4/8 oEG)A 4 8VJ 6 << K 9%&M:>4! 1$B5;E <)\§9BP^)K

V-=19@*\IRE' 4+M. 40 4 8f 4M 4s 479 4M<M 9-4kD$MG)A 9 4j74<8 4G)A! 479B; 4 8] 49 P^oKgV = 9@ \V-=19-@ \= 6 < 79 E +M 4 95(AoEAE 4>4l 6 8©B 4 <=>8 >4º8EKNK 4'LE 9-4+M%<B>4 8K)MG)AM%< >4>4'OC0ALEK 4

V-=19\χ V:9]\

γV:9[\mV

γV∞\J\ h £p/¨h

h £∞¨oh 9

:ÆM 4Q 4 8 4M 9KmM.Ur6E/5M 9 4 <<@ÁE oK 4= A<8:9/VrK9d 6 8rB 4 <V1@9 @ \\Ï>=©M.>4794K)M G)A7M 4K 4s>6 8rB 4 <9%E19M>4jP 8 4+M 9 4 X8ÁM 9:9 4B 4 %4 8>4,oE 8rKNK 6!0e°8EE/8]Z½9D26/8:9bE1< Kg| 4 8 474 6= 9 8rELK;= 6BE79 M 6)~>4=©M.M 4 8:9-E < K | =8rEG 9 M 6 98rELKT=-6 8rB 4~ 6 <.B 4!LE 9-4M<;M. 9 4/8 5<8 479©M 4 8 99z>4 8 C8rEaM K^;M 8%K 9rE79%9| 41<M G)A 8:6 E<4/8!4M 9[~4M 4 4M 9 9

∞K 6u/8 6E7>4;4&A< 9½>)EKNK M 4

1$B5;E <.¥E < K¹E! 4KNG)A< 6KNK 4U>4 < 9 4¥IZE9zKU>4 4 9K 5<8 4GHA 4>4 6<8©B 4 < U;M 8[E

∞ >8rG)A 9

∞4 8rK 479+( 9 9

z(E'LE < 6!>4/8: 4MK 4= 6 < 79E oKV =19-@ # \ M 4U76 8©B 4 <T= 8>426 <.B 4!LE' 9-4+M%< 74M;J 6 << K 9%&M:>4 8M K 679©A 4 8©B 4/8-1$B5;E) <.

V-=19<=!\ 5 Vθ§> 9%\

γV:9%\ V

γV:9

∞\-\ V =

γV:9

∞\ @ \ V =

γV:9[\ @ \ h £p¨Lh

h £p∞¨h

h £p∞¨ ;h

Vh £p¨L ;h 9

@ #

zB E << 44+M: 4 8J 6 << K 9%&!M 4ºMK 679rA 4 8rB 4W1$B5;E) < IR@ 4b;M 8EoK V = 9\UV-=19<=!\ M 4 PR ; 4M G)A(2;MKNG)A 4! 8EKNK 4! ?m)26 <.B 4'LE! 9-4+M%< 4'ÂK 4+A 8 9 4K 9©MBÁB 49:ÆM 8<4 8©ALE1< 9 4bE )KV:= 9\mOV-=19<=\¦< 4MG)A 9

V-=19 \χ V:9]\'S 5 V

θ§> 9[\ h £p/¨h

h £p∞¨h V1@fS

­h £p∞¨o Nh

h £p¨o Nh \ =h £p%¨hh £p

∞¨L)h

h £p∞¨L)h £p¨

Vh £p¨ Nh 9

?4M >4 81M K 679©A 4/8©B 4u1B`;E <74J 65P^oK 9 4M 9!( C 4 8<M 9M%< 9V-=19 # \ <

≤<.V:9%\

≤<V:9

∞\>

K 6BM 9 K)M.G74M 74°C4+M 9 4-M.u>4/8-Á< 4MG)A)QV-=19 \mMG)A 9 74E79©M J>GM 4 26 <.B 4'LE 9-4+M%< 4K)M.°M3M 4K 4B E << 4GMG)A 9Io< 4+M. 4 8 E < K`+M 4¹8EKNK 4!LE 9-4M< 4 9P oKV = 9 \ 4 8rALE1< 9 4;M 8 E oG)AM 4mBEaMBE < 4 PR ; 4+M G)A >M.>4Bi;M 8M 4G I 9 M 6

V-=198!\ϕVrK \m = t4h

h £p∞¨Lh

h £p∞¨oNt

$tF Nh

E N=$>4B zK 9 4/8JLE1<< X<>°<.V:9∞\Z 9 4 8rKoG)A 49Â:kM 4 < 4M G)A 9oLEG)A)(78-4G)A 4!>GMBB 9UM 4

I 9©M 6'ϕMA<8 8EaMBuoBáMBkC I 9

V-=19<!\ K / h V

h £p∞¨L Hhh \

E9DFEBM 9f 4798]&! 9BM 4BbEaMBE < 4QPf ; 4MG)AQ(2;M KNG)A 4o26 <B 4 ? ¾8bEKNK 4E 9 4M%< 4MvJ 6 << K 9%&M:>4 8M K 6 9 A 4 8rB 4 8 1$B5;E <)V-=19w!\

ϕVrK \Ï9

:ÆM 8 4+B 4/8©I 4>EKNK4M 4 8r@ E<4mM.V = 9w\ 78J 6 >4u,)&!>4'6(Á?4MuUd74u>4 81$B5;E <."EA&'799 D4E$M 4 8:4 < E79©M J 4'U,)&!>4&'74 8)>4"K 4A>8¦Io< 4M"K)M. >IRE'7M 4: 8[( 41<;M.uV = 9<\m 9-4 8?4 9+(ÂJ 6

V-=19v!\ Vh £p∞¨L Hhh ≈

@ h £p

∞¨Lhh

&A 4 8)K>; 4M K 4GB74= 6 8rB 9; 4 8[ 4>oK 6EKNK ;M 8E!>4&+A 4 8 9

V-=19 =>;!\ K ≈Û V<.V:9 ∞

\ < \§>

V-=19 =@*\ϕVrK \

≈ h £p

∞¨Lh

­h £ãp∞¨Lh ≈

h £p∞¨ohh 9

4 8©ALE < 9 49«8ÁM 9E)>4 8 4¿: 6 8 9 4>RM 4·BbEaMBÂE1< 4 PR ; 4M G)A*b(2;M KNG)A 4j261<.B 4! ?u*8bEKNK 4LE 9:4M%<J 4798%&79 8j 8:4M-$M 4 8:9-4 <J>4/8 8 4 < E79 M J 4'0 4KNEB 9< & 4&>4/8 9«D$M 4K 4PR ; 4+M G)AQ;M 8] 4 8 8 4MG)A 9½>F; 4!oM 4j, &>4s74 84C8 6 4 479K;EMA>8-4¾ALE < 4QP^)K >4A4 8=]&A/8%99 KK 4M²E>4B 4/8 I 9>`EKNK M 4¼< 479+( 9 4 3 4K< 9E79 4MV-4!E <6!ÝBM 9 6 4 8 6A74 3B\LEA&M:J 6 >4/8C9rEA< K 6 8:9 4K)M >m;M 8AE 4b>8`+M 4¶?4(M 4A V = 9 # \74 9[( 9>

@*8

E < K 6u+M 4U6M:>4LKNG)AE= 9>)EKNK >4/8P oK 9-4'M 9 4M. 4u 8@E/4 8 4¹D»M G)A 9 4-E < K 74 8/C 4 8<M 9AoE799'F>8-M 4B>4I 4A>8:9-4 1»B`;E! <.K)M. >8M V = 9 # \M 4 3)41< E 9 M 6 4b(s; 4G)ALK 4 < ) EK oE <ãM 9rE79 M J 4 3)4K< 9E 97< 4M 9½> 787EKNK 4 9+( 97>4/8Ú8EKNK 4LE' 9-4+M%<5MG)A 9¹Io< 4+M. 4 8°E < KW 4 826 <.B 4'LE 9 4M%<;MK 9½9 4 8%M 4KK)MGM 4K 4 PRLKNG)A&79+(74-E oG)A=>80E! 4 8 4o1»B`;E <>4"E*; 4LE/8>K 6< E 4 >8 (2; 4M0C6ALEK 4 EFN=%98-4 9-4!>sM 4 1»B5;E! < J 6 << K 9]&!M: MK 9½>I 6!LK 9rE 9 4/8"4B 5 4/8rE79 84K 9rE79:9=©M.>4 9 *QM 4jC6AEK 4zI 4+M. 4o/8 6 E<4zD$MG)A 9-4F 9 4<8rKNG)AM 4>4ÞE N=K; 4MK 49C6LK 9 BuoKNKE N=>4 8 81< E>4ÂJ 6 6 8rB 4 < V9 @ # \ >4E 8% 4M 9-4 9; 4/8%>49

P^oKV-=19-@ ;!\ 4 8©ALE < 9:45;M 8=78M 4Á%) !' A°*,&`, §( » +!' A

V-=19 =\ @ χ V:9

∞\ -V =

γV:9

∞\ @ \ ρ

Ï£θ£p

∞¨¨

ρ£

θ ¨ S"@ ρ

£θ£p

∞¨¨

ρÏ£

θ£p

∞¨¨L

ρ £

θ£p

∞¨¨ >

; 6 8rE oKáK)M G)Aρ Vθ\ 4 <ãMBM.M 4 8 4â< &KNK 9>°B(M 9

θoS C9E 8:9:9 4B 5 4/8rE79 8g

θV:9

∞\QS

J79 4B 5 4 8E79 8½9)C6+B(M 9 IR@745;M 8-V:= 9 @ ;\ BKNG)A>8 4M. 4'6(

V-=19 =>=!\χ V:9]\ γ

£p¨ Û

γ£p

∞¨ Û ρ

£θ£p¨¨

ρ £

θ£p

∞¨¨ S"@ ρ

£θ£p¨¨

ρ£

θ£p¨¨L

ρ £

θ£p%¨¨ 9

PRLE < 6 4 8©ALE < 9 4 ;M 8E oK V = 9 @;\ =78>4'66 <.)B 4LE 9 4M%<

V-=19 = \ 5 VθV:9%\§> 9%\ £ γ

£p¨ Û`¨ρ£

θ£p%¨¨L£

γ£p

∞¨ Û`¨

ρ £

θ£p

∞¨¨

£ γ£p¨ Û`¨£

ρ£

θ£p¨¨

ρ £

θ£p%¨¨¨ 9

D$M 4$ 6 8©B 4 < V = 9 =>=\ GV:= 9 = \K)M$LE!A&M:ÁJ 6!3>4/8!P 8:9'>4/8"4B 5 4/8rE79>8[&>4 8 >4KBuoKNK 4'`;4&A<8 4 4 8-1B5;E' <. 8BÂEaMBE < (2; 4+MC6ALEK 4 5<8[&K 4 9K 4M9:ÆM 8$; 6 << 4âE oG)A AM 4/8 +M 4 PR ; 4M G)A (2;M KNG)A74Å8ELKNK 4' ?jt6 <.B 4!LE 9-4M< 4ELKNG)A&79+( 49 PRLE < 6(V =19-@ 8!\ 4 8©ALE < 9:45;M 8

V-=19 = # \ 5 VθV 9%\§> 9]\ ?

χ V:9]\

≤ ρ£

θ£p¨¨o

ρ £

θ£p¨¨

ρ £

θ£p¨¨ V @ S 5 V

θV 9[\§> 9[\\

χ V 9%\>

;EKt( I 4+M. 4 8E>4 8 4 FoE <M 9E 9©M J 4hP^oKNKNE!>4 E1< KhLEG)A V =19-@ 8!\=A>8:9QP oG)A =78M G)A 9©M K 6 9 A 4 8rB 4 1$B5;E) <.>4 K)M* M 4 1c 9 4<8rKNG)AM 4>4 (2;MKNG)A 4! 8EKNK 4! ?I*26 <.B 4'LE 9 4M%< 6= 9)J 4 8LEGHA< &KNK)M E 8 9D$M 4 D$M == 4 8 4)( (2;MKNG)A 4-8EKNK 4LE! 9 4M<

χ Q1$B5;E <.K /8E! =ÚE oKV1@9 @ \ 4M)>4/8

J 6 << K 9]&M >4'3M G)A 9 ?ÏMK 679 A74<8©B 4u1B5;E <. 4 8 4GHA 4 9K)M G)A (

V-=19 =8!\χ V:9]\'S =V:9[\ γ

£p¨ Û

γ£p

∞¨ ÂÛ ρ

Ï£θ£p/¨¨

ρ £

θ£p

∞¨¨ S"@ ρ

£θ£p%¨¨

ρÏ£

θ£p%¨¨

ρ £

θ£p¨¨ S γ

£p¨

γ£p

∞¨ 9

8M 9^>4'jHEG)Ao=-61< >4 4' 4/8< 4F>4o( 4M:>4o;M 8 > EKNKoM 4b1B 9-4 8rKNG)AM 4>4j(2;M KNG)A 48bEKNK 4LE 9:4M%<

χk1»B`;E) <.K /8rE = =>8uM G)A 9 ?ÏM K 679©A 4 8©B 4k1»B5;E1<.>4

>8rG)ALE oK6M G)A 9J 4 8%LEG)A< &KNK)M:LE/8K 4+M.IR@ 49

@*<

26 8©A 4 8F 4B 4 8 I 4!5;M 8C6 < E>4 I 4M. 4G1»B`;E! <.K 9E79:9=©M. 479 >< 4M. 9'74 88EKNK 4LE 9:4M%<>4 8K)MG)A$M%< >4)>4 C6AEK 4Q<<>() < 4M G)A¹IRE'"K)MG)A¹M.o= 6 < >49©A 4 8BM KNG)A 4 8¦D24A°E 4 8 4 8 4M 9rK5>4/8-1$B5;E) <.K 8E!3&'74 89P^oG)Ag 4M |K 6=-6/8:9%~474M.! 4)>4j1$B5;E'*1<. 4ßK)M) 9 4M%< Ko 4 9 8%&G)A<MG)A 4Y1B 9 4 8rKNG)AM 4!>4BÂ@! <MG)A>;M 4LM 4±=:6 < >4'>4WP oK;=A<8)>4N( 4M >4!9¹D»M 4 DFE79 4W; >8%>4M;M 4>4 8)BX-3 6Av8Z/4 9% 6+BB 4B-9:ÆM 8-K 9-41<< 4'yLK 4M. 4M 4 <M 4M:>4! PRI)A< J 4 8< E ;=J 6'H< # ;

° ( # ;

° = 84M. 4

D$M%< E79 6B 479-4 8 5>8 6 4O>4K 4F 9 4I 9 6M 4"CE79 A< K Bw>;mJ 6 8©9D$M 4K 4K"4B 5 4/8rE79>8©M. 9-4 8JLE1<<2; 4<8%>4M G)A 9J 4 8< EKNK 4>LK 6uEKNK EG)A-AM 8 4MG)A 4 8 6 E>4 84+M 9 4M. 4ÂJ 6 << K 9%&M:>4$1B5;E' <.M.UC 4 8<M 9K 9E79:9= M>49 D$M 4¹BEaMBE < 48 4 < E 9©M J 4Q,)&>4!&>4/8D 4 8 4G)A 479K)MG)A E$EGNA>4 8d 6/8 B 4 <2V = 9 8\F(

V-=19 =<!\γV:9

∞\ Û ρ

£ÉvÈZÊZ¨ρ £ ZÈZÊZ¨

£ ZÈZÊZ¨ >

;EKÂBM 9>4' DE 9-4E )K X-3 6Av8Z(V-=19 =w!\

γV:9

∞\ ?;> ;>;<!<>;w

=A/8 99 : 4+M 9 4<8b 4AB 4 ;M8kE>oEKNKIM 4_PRI)A< K 6 J 4/8< E H=-4 >oEKNKh 4M$>4/8"4B 5 4/8rE79 8$<>; ;

° M 48 4 < E79©M J 4Q,)&)>4&'74 8( 4M.74<8 479©MBB 9 44M 9'9 >4E

4M.7 o= 9:4 <J 6γV:9

∞\Q 4 98E!>49FEG)A>4/8 76 8©B 41< V = 9 =8\4/8©ALE < 9-4'4;M 8E'

V-=19 =v!\χ V:9 \S-=V:9 \ ?; > # @9

1B 9 4 8²?4EG)A 9)o>4KNK 4>-EKNK7M 4U 8r@E/4χl(2;MKNG)A 4 ; @ <M 4!79RV= IRE!

K 9 M:>4/8; 4M K 4 (2;MKNG)A 4)( 4+M 9<M G)A±E oGHAt 4E 9©M Jl; 4 8%>4)\§>°M K 9UM 4MPR ; 4M G)A M.V-=19 =v!\(·A 6G)A9 zKLK 4K 6! 4 8 4·= 8(< EKNE+B 4 PRI)A< J 4 8< & ;= 4áIREºK)MGNA4M /8-6FE<4/8>4A< 4 8<4/8%>4 4> 4MRKNG)A 4 << 4 8 4 PRI)A< J 4 8< &FN=-4!5;M 8%3+M 4K 4 8 >4/8©M.>4/89 4A 4 ;M 8(9?49K 4A<8KNG)A 4 <<FEN= # ;

° AE < 9 4 M 4K 4i"4B 5 4 8E 9>8>VK 6 <ãM 4 9 4M. 4 M K 679©A 4/8©B 4

1$B5;E <.J 6/8>M 43 8@FE<4χoÝ=K 9 MBB 4!5S;M 4 6' 4' 4( 4M 9SU 4 8 4M9

@46241½*+, 'A°! 1F#$*, % 6 7N`%"!1 1½`6

C4M.NM 4 D$M G)A 9-4l>4 8"E N=:9 8 479 474TC0ALEK 4kM G)A 9®EFoKÝE!>4 8 4!yx 4 << 4N 4IRE! 9½>B )KNK 4âK)M 4 S 9 4/8U AM%< =-4!LE+AB 4 ; 4M 9 4 8 4/8DE79-4'lS E )Ky 4 D»M%< E 9-6B 479 4 8%LE 9-44 88-4G)A 479 ; 4 8]749:kM 8 < 4M. 4' 74M74 8$C4M 9)E79 M 6U>4 8¦J 6 << K 9]&M >4'01B5;E1<.o>4KP^oK 9 4M 9rKàM.L>4' C 4 8<ãM 9s74MB C9rEA< Rw ; > 6*; 6A<EKIC8©M)(M5 E oG)AâE N=¥E)>4 8-4C4M 9oE79©M 674u 4/8:9 8rE!LE 81M K 99

@*w

:ÆM 84 8@ 8%9 4 8u(2; 4M8@ <ãM G)AI 4M 9-4!>; 674MM 4 4 8rK 9 4J 6'7>4/8d 4LEM:I 4M 9 >4/8d 4>6B 4798 M 4EA&79½> E=>8-E' 4 8°I 4M 4 D$M G)A 9-4 4K 9©MBB LEGHAl>4B P 8rG)AMB 4!M KNG)A 4!TC 8 M)(M54 8=-6 8%74 8:99P^oK5>4B oK;EBB 4AEuV = 9 8\4 8+M. 9K)M G)A 78rG)Au1$BK 9 4 <<.

V 9-@ \ρVθ\

ρVθ\/V1@ = ∆

hh \ >

; 6 4+M ∆hh M 4 8 41< E79©M J 4j,&)>4& 4 8¿M K 9>dM 4s74 8/"4B 5 4/8rE7978

θ4! 9rK 5<8©M G)A 9 9V:kM 4

6 4(EFoK >4)=A<8:9 >74(M 4A 9+K)MG)A M 4P oK EK< &)>4 <E N=-+M 4"4B 5 4 8E 9 >8θ&9C6 < E 4

>8JP^oK 9 4M 96J 6<8©ALE'74ÚMK 9½> K)M.Q 4M 4°: 4/8 9 4U 4<8$D»M G)A 9 4zM.oV 9-@2\fM 4Q>4K P^oK 9 4M 9rK9zLK 6=-4 8 M K 9 4Ks LK 9©M >4M 4oPRI)A< J 4 8< E N=R( ;4&A< 4'>f 4M>4B +M 4 1$B5;E'*1<.M G)A 9o( KNG)A 4 << 4/8=-61< 79> 6 4 8s74M 4B +M 4tPfo=E'K 9-4B 5 4/8rE7978j 4 8s>4/8YC 4 8<M 9-?C9E 8 9:9 4B 5 4/8E 978 <ãM 4 9½9 : 4/8%>4 4M 5LEE/8 "4B 5 4 8E79 8 4 74LK 9 () 4AL@/8©M >4,)&>4'&74 874I>4B 4KNK 4>¹K 6 IR@74 4M.M >4 : 4/8 9 4 >4/8D»M G)A 9 4 HEG)A V 9 @ \ 4 8 4G)A 4 9; 4 8%>4!>oK 6M 4"D»M G)A 9 4

ρVθ\n 4IRE' 96M K 9 >*; 6 8rE ;=!;M 8'1< 4MG)A( 8)G)INI 6BB 49

DFELEG)AIRE! (9?49= >8f 4'UP^)K 9-4M 9^>8rG)A<ãM. 4E 8 4 !a 98rE>576 < E79©M 6' 4M. 4 >6/8©B 4 < = 8^M 4D$M G)A 9-4>4; 6 4'`; 4/8] 491$BlM 4 D$M G)A 9-4

ρVθ\$(74<8 4G)A! 4> ;M 8%uM 4¹8EKNK 4±B 74 8/C8 6 4G 4'L@79©M 79½9?4M 4M 4 8

(2=<M. 8©M KNG)A 4 6 << 5<8 6 4 ; 4 8%>4! >4 8:4 ,)& 4 P^E/4 78rG)AB 4KNK 4 8 4M 4 8"4B 5 4/8rE79 8

θk 4'H@ 9 M:79½9?4M6Io< EKNK)MKNG)A 4D»M%< E79-6B 479 4/8 < &KNK 9$K)M G)A6=:9R 8BM 4g,)&>4

B 4KNK 49/6M.OP^oK>; 4o 4K 9 4A 9E<8©M>-M 4 C8 6 4 4+M¦MBB 4 8:9 4B 574<8rE79>8 (mJ 4 8B 4KNK 4V,&>4 > P^E<4 ? z 4 78G)AB 4KNK 4 8 4M 6A< 5<8 6 4\§9 P oK 4B²A<ãM G)AI 4M 9KTJ 4 8 A& < 9M KV 9\ <.V

θ\/) V θ \ ¹<:V θ

\/) V θ \

4 8©A& < 9 BbE >4 4L@ 9 M:79 4 P E<4>8rG)AB 4KNK 4 8 V θ \§> K 6;M 4 ELE1< 6 >4z74 78rG)AB 4KNK 4/8u

Vθ\§9V[(MRK 9 4A 9=>84MBB 4 8:9 4B 5 4/8rE7978½>*(9?49H<;

°\9

C6BM 9<4 8 AE < 9 4!`;M 8EK 26 <.)B 4 RVθ\G>4 8-C 8-6' 4( 8!"4B 5 4/8rE79 8

θ EEG)A

V 9 =\ρVθ\

ρ Vθ\ s

¡ £θ ¨ 9

>8?4K 9©MBÁBuY>4 8D$MG)A 9 4 >4K C 4 8<M 9KÆJ 4<8=rEA<8:4 ;M 8ELE1< 6 > >8 (oB J>4 >4/81$B5;E <.AM> ; 4I 4M.gP^)K 9-4!M 9B 4A>8VJ 6 8ALE>4!M K 9>4K)M G)A >4/8C 4 8<M 9 4M; 4M 9-4 8 4 8PRI)A<.O78 6G)A6(oKNEBÁB 4)(M 4+A 9½96M. 4M>4A< 4 8 4 << 4IREYLE 8 Mº<M 4>4> EKNKjM 4WC8 6 4Y(kK 9rE 8©IzJ 6 4M 4B M >4E < 4 =<M>4 8QV26A< (2=<M>4 8+\¥E ; 4M G)A 99°DFEA 4 8$( 4+M >4M;M 8'4M. 4 (2; 4+M 9 4 8@ <MG)AI 4M 9½>uM 4

@*v

D$M G)A 9-4ρVθ\G( 4/8 4G)A 498ÁM 9'»M%< =-4>4K P 8G)AMB 4!M KNG)A 4'DC8©M.)(M 5LK 4K 9©MBB 4 ;M 8M 4

D$M G)A 9-4G74MR<;°>E < K 6

ρVθ \§>-E E<4 8]74BNM 4$,&>4u 4 8C8 6! 4 <.V

θ \>>4' 4!o=E1<< K 4M

<;°92P^)K`>4' 4/8< 4 74>4KC I 9 4K9 = 9;= 6 < 79E!M 4G>6 8B 41<

V 9 \ρVθ\

ρ Vθ\

ρVθ \mV h £ θ ¨h £

θ ¨ \

9

PRLE < 6 < &KNK 9K)M G)A M 4 D»M G)A 9 4 >4K C 4 8<M 9K ( 4M 4 8g"4B 5 4 8rE7978EB J>4 >4/81$B5;E <.u74K 9©MBB 4!9

@<A A A @[?!@o4 /*+, §% § 'A°! ¹#»/*,`#4 , 6A1 °( 6,'A1?M K)A 4 84ALE' 4;M 8d+M 4¥<ãM. 4E 8©M K)M 4/8:9 4s>6 8B 41<V9 @*8\¦J 4 8; 4>4 9> B 4uoKNE+BB 4ALE!(2;M KNG)A74 4 88-41< E79©M J 4$, &>4'&!>4 8QG>4 8D$M G)A 9[&>4 8 G(O 4KNG)A/8-4+M. 49:ÆM 8>4 4¼E>5;M 4±K)MG)A 4M.M >4I>6 8rB 4 <:N&74 8> ; 4LV9-@*8\o >8rG)AyM 4IMG)A 9<M.74E/8-4?4(M 4+AGV9 @ # \4/8rK 479[( 9;M 8+ 92PRK 9-4 << 4ÂJ 6V69-@ <\'M K 9E

V # 9-@ \ε

Û ν³ σ ? ν³ 9 8V σ \ δ

@ ρ

?ρ

ρ?2@

δ E < KOP^)K EK 5I 9 (n; &A< 49B 8 4ÆD»M%< E79-6+B 479 4 8J 4 8K)G)AÚ=-61< 79RE!¥ELK 9:4 << 4zJ 6V-=1998\ +M 4°?4(M 4A

V # 9\ ∆hh

@ ρ

?ρ

ρ?@9

oKNEBÁB 4kBM 94V9L\YV =99w!\ 4/8©ALE < 9-4Q;M 8²=>84LK 4/8-4 C¦M 9oE79©M 6074 851$B5;E!1<.J 6 P^oK 9 4M 9 ( C 4 8<M 9 4M. 4 24 8rE << >4B 4M. 4 8 >4/8 6/8 B 4 <WV = 9 @ ;\ =78I>48bEKNK 4LE 9:4M%< >4K3C 4 8½<ãM 9rK

V # 9 =\χ V:9]\ ρ

Ï£θ¨

ρ£

θ ¨ V1@ γ

V 9+\\ S@ ρ £

θ¨

ρ£

θ¨

ρ £

θ¨ 9

D$M 4i,M.74E 8©M K)M 4 8) 4<8Á< 479+( 9 4 6 8rB 4 < 4()1<M G)Aγ

4 8%M 9K 6=-6 8%9V =9-@ ;\§9/P oK V # 9 =\4 8%M. 9ÚK)MG)A±BM 9U>4' 6M:>4 74<8< 4 74L+M 4t24 8E << >4B 4M 4/8t(IV = 9 =>=\jV=>8J 6 << K 9]&M >4G1»B`;E <>4\Ï>&B-<MG)A

V # 9 \χ V:9]\ £rÛ γ

£p¨¨

£rÛ γ£p

∞¨¨ ρ

Ï£θ£p/¨¨

ρ £

θ£p

∞¨¨ S"@ ρ

£θ£p¨¨

ρÏ£

θ£p%¨¨

ρ £

θ£p¨¨ 9

>8J 6 << K 9]&!M:>4mM K 679©A 4/8©B 4u1$B5;E <)>4=-61< 79E oKV # 9 =\< 4M G)A 9

V # 9 # \χ V:9]\ £Û

γ£p¨¨

Û

£Ûγ£p

∞¨¨Û 9

/;

&A 4 8K>; 4+M K 4 4 8]+M. 9 K)M G)A =>8 |§Io< 4+M. 4~ 8 4 < E79©M J 4 ,)&'>4)& 4<8>4 M 4 4 8 4MLK 9©MBBuzBM 9 4BNPRM< K 8E =¶E oKoV@9 @ \§94D$M 4QPR ; 4M G)Ag 479 8]&!79$M.M 4K 4+B E <<

V # 998\χ V:9]\'S =V:9[\ £rÛ

γ£p¨¨

Û

£Ûγ£p

∞¨¨Û ? γ

£p¨

γ£p

∞¨ 9

6M. 4 8rKNG)A< E( 4M:79½>EKNK M 4K 4PR ; 4M G)A)°M.3 4<8 8@ E/4 6 8% J 6 ; > ; ; # M K 9

ÚK 6+B(M 99 6 < 4 8 M 4LE/84KNG)A 4M 9 9 V 8=EA<8)K 4B&Eg 4/8rK 9 4M 79γV 9

∞\Y>4BÅ?4 9 8rE>4QLEG)A

M G)A 9 ;1> ;@9D24 8JP^oK 8oG)I-M.oV # 9 8\4 88 4M G)A 924M¥8EaMB B M.U>4/8 &A 4 J 6γV 9]\ ;> #

γV:9

∞\&9 \

?4M M G)A 9©M K 679©A 4 8©B 4 8 1$B5;E! < IR@!74 K)MG)A S 4K 674 8rK>h; 4 8 6E<4"4B 5 4/8rE79 8 9-4/8rKNG)AM 4!>4 J 6/8<M 4>4 S M 4 +== 4I 9 4 >4 8 1B5;E' <. >4/89 A 4 8 BMKNG)A 4!ßD4A* J 4 8rK 9[& 8 I 4'9z:kM 8»IR@ 4UM 4kD$M == 4/8-4')(D 4 8fHEG)A >40>6 8rB 4 <:V-=19-@ ;!\ uV # 9 =\J 4 8 4G)A7479-4'(8EKNK 4oE 9 4M< 4OM%< >4 4/8 AE < 9 4

V # 99<\ M == ρÏ£

θ¨

ρ£

θ ¨ ρ

£θ£p¨¨

ρ£

θ£p¨¨L

ρ £

θ£p¨¨ V = γ

V:9]\ γ V 9]\\§9

6M. 74 8rKNG)A< EmBM 9>4' DE79 4=>8>4 fw>;V # 99w\

θ u< # ;

°­>

θV:9[\ u< ;>;

°>

θV:9

∞\ # ;

°­>

γV 9%\ ; >

γV%9

∞\Ï>

γV%9

∞\ ? ; > ; ;<<7;w

=A/8 9J 9-4 86?4EG)A 9ÂJ 6!uX-3 6Av8Z(V # 99v\ M == ; > ; # 69V½DE 4M ; >8%>4!uV-=9 =w\§>V = 9 =v\ 4! 9+( 9½9\4D$M 4K 4PR*; 4MG)A 4 8rKNG)A 4M 9KNG)A 6' 8 4G)A 9'/8 6 E 9zB 6M)( 4 < =rE << M K 94M 4/8 5<8N= 4' >4K B@ <MG)A 40>4+A< 4/8rK°= 824M 4kI 6'I 8 479 4¾C9EA< K 6 8:9-4

Á= 8>4M.I 6I 8 479 4K "4+B 574<8rE7978©M. 9 4 8JHE1<<2L@79©M 19

@F@)*, A A 0!1 1& ! , 6A 7 5 )% :ÆM 8 I 6BB 4!I)>8+(E N=+M 47 6 8rB 4 < uV9 @ <\§>V+9 @ w\F( 8)G)Io>M. >4 4 4M. oKRE+B(B 4ALE'(2;M KNG)A743>4°D$M G)A 9-4uu>4BkVrBM 9:9½< 4 8 4'"<M 4E 8:4\1:0&<8©B 4EoK >4+AK)I 6>4==©M (M 4 9-44 9©ALE < 9 4M K 99 78 4-D$M%< E79 6B 479 4/8J 4 8rKoGNAuV©C)5LE' < 4+M G)A<< \ = 6 < 79 < 4M G)A 9

V # 9-@ ;!\ Û ρ

ρ£θ¨

ρ£θ¨

αVθ?

θ\ Û ρ

ρ£θ ¨

ρ£θ ¨ 9

K-K 4ME 8rE'4<8©M 4 8 9½> EKNKρ +M 4¥D»M G)A 9:4U>4 83 4= 4 8 4(I 6o=©M 78rE79©M 6oVE1< K 6 4( 6>4

E N=θ\­MK 9½>F; &A<8 4

ρVθ\t

ρVθ\lM 4¥EI 9 4 << 4¥D$MG)A 9-4'ßK)M >m 4( 6>4!¥E N=RM 4

EI 9 4 << 4"4B 574 8E79>8θ(2;b9oE ;=+M 4'3)4= 4/8 4)( 9 4B 5 4/8rE79>8

θ&9

D$M 4G1»BÂK 9 4 <<.OLEG)A6>4B·:0&<8©B 4E oK >4+AK)I 6>4==©M (M 4 9-4α

4/8%M 9< 4+M G)A 9

@

V # 9-@>@*\α

ρ

V£θ

θ ¨ V

Ûρ£θ¨ ?

Ûρ£θ ¨ \§9

D$M 4°8ÁM KNG)AK78 4>4 <v=78M 4 8:4(M5/8-6+I 4"D»M G)A 9:47V 9 @ ;\=A<8:9(

V # 9-@;\α

ρ

V£θ

θ ¨ ∑

V Û

ρ £ θ ¨? Û

ρ £ θ ¨ \ χ

Û= £

θ

θ ¨ V ∑

χ

ρ £ θ ¨ \ ∑

V Û

ρ £ θ ¨? Û

ρ £ θ ¨ \ χ

∑

χ V ∑

ρ £ θ

¨χ

ρ £ θ ¨ \

α

; 6 4+M M 4 8@E/4α

>4+B6& E

V # 9-@ =!\α ρ £ θ

¨V£

θ

θ ¨ V

Ûρ £ θ ¨

? Ûρ £ θ

¨ \

M 4zE N=θ 4( 6>474°: & 8 B 4E oK 4AK)I 6 4==©M (M 4' 9-4' 74 8«M ? 9 4 C0ALEK 4¥K)M9 P^)K

>4B oK;EBB 4AEuV9 w\ (2;M KNG)A 48EKNK 4 ?/6 <.)B 4LE 9 4M%< 4=-61< >4

V # 9-@ \χ

ρ £ θ ¨

ρ£θ ¨ 5

Vθ\§>

χ ρ £ θ

¨

ρ£θ ¨ 5

9

1B 9 4 8?4' 9+()zJ 6gV # 9-@ \ 4 8rALE < 9-4'U;M 8$E oKgV # 9-@;\4M 4<ãM. 4E 8 48ÁM KNG)A)K78 4>4 <=>8>4 : & 8 B 4E oK 4AK)I 6 4o==©M (M 4' 9-4' BM 9 »M%< =-4 >4/8 E N=

θ 74( 6>4! 4

26 <.B 4'LE 9 4M%< 4

V # 9-@ # \α

∑

5Vθ\

α 9

zPR! 4798EG)A 9 >4 8- 4 8©M.74PR*; 4MG)A>4!u(2;M KNG)A 4z8EKNK 4' ?R$261<.B 4!LE 9 4M< 4GS(BM.>4K 9 4MB C9rEA<*SÆ< &KNK 90K)M G)AQ&A 4/8K>; 4MK 4¹E oG)AD+M 4¥<ãM. 4E 8 4z8ÁM KNG)AK 8 4>4 <B(M 9>48EKNK 4LE 9 4M%< 4J 4 8K; 4!>4'9

@F@ »"*,6Aà! 1& ! , 6A 7o1 ` %¼ Ú +!1 ½

4 A `

z0>4' <MG)A 4!s"0E! 4 << 4YVrK9/(9?49 X-3 6Av8Z©>^X 3MG <>=Z%\ ; 4 8%>40M 4 V<M 4E 8 4kBM 9:9½< 4 8 4'\: & 8 B 4EFoK 74AK)I 6 4==©M (M 4 9 4!M >4/8 3)4!>4 <4E ;=

θ /;

° 4( 6 49DE M 4

3)4= 4 8 4)( 9 4B 574 8E79>8 4MJPRLKNG)A<8 4G)IvJ 4/8rKoG)A 4! 4 8©A 4'<MG)A E)J 6'E ; 4+M G)A 4IZE>M K 94K L@79©M >B>4!¥:0& 8B 4E oK >4+AK)I 6>4)== M (M 4 9-4EFN= 4M. 4kE'>4/8 4¶?4(K 9-4B 5 4 8E79>8

θ B (>8 4G)A 49:kM 8 >4A 4'-J 6D>4'ÚD24=©M.M 9 M 6 4-=>8/M 4OP^oK >4+AK)I 6>4==©M (M 4 9 4E oK>oE < K 6ÂJ 6

V # 9-@ 8!\ h £θ¨Lh £

θ ¨

h £θ ¨

αθ V θ \/V θ ?

θ\§>

V # 9-@ <!\ h £θ¨Lh £

θ ¨h £θ ¨

αθ

Vθ\/V

θ?

θ \§9

P^oKV # 9-@ 8!\ 4 8%M. 9K)MG)AV # 9-@ w!\ <.V

θ \ $V1@ αθ V θ \mV θ ? θ

\-\4<:Vθ\§9

P^oKV # 9-@ 8!\ uV # 9 @ <\ = 6 < 79- 9 4 8?4EGNA 9J 6!uV # 9-@ w!\ M 4>4K)G)A 9 4$>6/8©B 4 <

V # 9-@ v!\αθ

Vθ\ αθ

£ θ ¨Û

αθ £ θ ¨£ θ θ

¨ 9

@F@#4 A¹!1( %!*,-!¹#/*, D$M 4Q>6 8©B 41<Z=>8>4'26 <B 4E 9 4M%<2>4KC 4 8<M 9KOV-=19 @;\IRE!-K 6QoB>4KNG)A 8©M 4 4; 4/8%>4>EKNK >8 M 4·D$MG)A 9-4' >4Ko 4BM KNG)ALKzK 6;M 4b>4 8 4M 4 C6AEK 4!sP^oK 9 4M 9 ) C 4 8<ãM 9E N=:9 8 479 4!9V:kM 8F 4798EG)A 9-4 ;M 4! 4 8)BlM 4$1$B5;E'*1<. 4M. 4/8C 8-6' 41>M 4O("?4M.G>8E oK P^oK 9 4M 9H 4K 9 4A 9½>j( C 4/8<M 9½9 \k1c 9 4<8á?4EG)A 9 >4/8ÝK 9-4 9K E>4' 6B(B 4 426B 6>4'M 9%&79 >4/8d76 8 B 4 < V = 9 8\4/8 AE < 9 45;M 8E oK V = 9 @;\

V # 9<;!\ 5 Vθ\ ρ

£θ¨

ρ£θ¨

ρ£

θ¨

ρ £

θ¨ 9

D$M 4K 4D 6 8rB 4 <v= 6 < 79E oG)AK 6= 6<8:9EoK5>4 8 8ÁMKNG)AK 8 4>4 <2V9 v\1= 8M 4°D$MG)A 9 4 9

@ @#4 A¹!1( %!*,-! A :ÆM 4mM.>4 8 6M.< 4M%< 4/8K; &A 9½>2; 4/8] 4 4+M >4 8P^oK>; 4 8:9J 6-D»M%< E 9-6B 479 4 8J 4 8rKoG)A74B 4+M K 9d 6<8©B 4 < 6>4 8P 8:9

V @9\ 5 Vθ\ h £

θ¨h £

θ¨

h £θ¨h Ï£

θ¨

J 4 8K; 4 479½9mz0LK 4/8-4 8$C9rEE 8[K)M 9oE 9 M 6K)M. 5 Vθ\k>4 86 <.B 4E 9-4M< >4K0C 4 8<M 9rK>

<.Vθ\bM 4 EI 9 4 << 4 ,&74 >4 8 C8 6 4h(78s"4+B 574<8rE7978

θ>¥< V θ \ < V

θ\ M 4

4 9rK 5<8 4G)A 4>4$, &>4'>; 4!7M 4sC8:6 4Q78E oK P^oK 9 4!M 9m)(2;b9 C 4 8<M 9/ 4K 9>419 D$M 4>6 8©B 41<V1@9\=-6 < 79´EFoKs>4/8 Pf ; 4 )Y>4 8<M 4E 8 48ÁM KNG)A)K78 4 4 </74() <ãM G)A >4/826 <.B 4'LE 9 4M%< 4E ;=4M 4EI 9 4 << 4 ,)& 4b>4 8 C8 6 4 9¦:kM 8f; 6 << 4AM 4 8I)>8+( 74 8 8E!>4LEG)A>4+A 4> 6 +M 4K 4N>6/8©B 4 < V @9\mK)M G)A M 4 8 4+M.LK 9©MBB3¼BM 9>4áM. C I 9 =A 4 8%>4 < 4M 9 479 4 4= M7479½9:ÆM 8` 8oG)I 4' M 4¶M V @9\J 6 8©I 6BÁB 4'74 ,)&>4!BM 9 $M< = 4Y>4 84(YV # 9 @ 8\E!LE < 6>4>6 8©B 41<:V # 9@*\ <.V

θ\ ¹< α

Vθ?

θ\¦< §>

V # 9\ < Vθ\ ¹< α

/Vθ?

θ\4< &>

V # 9<=!\ < V θ \ ¹< α Vθ?

θ\4<

/=

E oK9DFE 4+M3K)M.¶E << 4 :0&/8 B 4E oK 4AK)I 6 4==©M (M 4' 9-4'E ;=6+M 4C9rE 8 9 9 4B 5 4 8rE7978θ

4( 6>4'9P^oK$ 4<8 24 8K)GNALKNE' 6 8[' 4 8%M 93K)MG)A < < 9?4 9[( 4n;M 8=78α

M 48MKNG)AK78-4>41< V # 9 @ # \§>NK 6Á=-6 < 79E oKV1@o9\ 7V # 9@ \!SUV # 9<=\

V # 9 \ 5 Vθ\ 5 V

θ\ £

θ

θ ¨£

α

α ¨

£θ

θ ¨£

α

α ¨ h

h h

α £

θ

θ ¨ h )h h

9

:ÆM 8%3+M 4G1$B8 4G)AKZ= 6 8©B 4 < V9:@>@ \ 4 9[( 9>oK 6 4 8©AHE1< 9 45;M 8EFoKV # 9 \4< 4M G)A 9

V # 9 # \ 5 Vθ\ 5 V

θ\ ρ £

θ¨

ρ

ρ£θ¨

ρ

£θ

θ ¨£

α

ᩨ

£θ

θ ¨£

α

α ¨ h

)h h

α £

θ

θ ¨ h )h h

>

E 4MRK)Mρ$M 4°D»M G)A 9 4>4 8dC8 6! 4Á=78

θ/V]1< 4MG)A6>4 86D$MG)A 9 4>4K P^oK 9 4M 9KV=>8

θ\&>

ρ

M 4¿D$M G)A 9-4Q>4KUC 4 8½<ãM 9rK= >8θ&>

ρ V

θ\M

ρVθ\ 4 9rK 5<8 4G)A 4oM 4D»M G)A 9 4'n 4K C 4<8<ãM 9rK

$>4K 4B(MKNG)A 4KÁ= 8θ9: &/8 4$M 4 6 8rB 4 <V@9\/>4E °MB 3¦EAB 47>4/84MB C I 9=

>4798-6== 4! 4jPR!LEAB 4>VK 6BuoKNK 9:4 >4 8ÁM.WV # 9 # \-EN=>4/88 4GHA 9-4'ºC4+M 9 4W 4M 5 Vθ\

K 9-4A74>4 EI 9 6/8 1< 4MG)A 4M.LK K 4M.9oPRAE I 6I 8 479 4 8ÝD»M%< E79-6B 479 4 8[E79-4' M K 9 4M.24 8% < 4+M G)A>4 8- 6 8rB 4 < 3V1@9\ OV-=19-@;\'BÂ@! <MG)A9

@'@<4 +*+7o %*,&A A-6¹6/*,4 b !1 ! '%*,6 'A )Õ1! ?M K)A 4 8°ALE 4i;M 8-E>4 6BB 4!>EKNK M 4tB5;E)>4 <:>4IC8 674 EFoK >4E (2; 4Mr6B 5 6 4' 9-4' V C6ALEK 4)\$ 4K 9-4A 4 >J 6 >4 4 M 4 4M. 4áM +M 4ÝE 4<8 4 oB5;E>4 < 4 924 8rKNG)AM 4!>4 9<MG)A IRE4K J 6/8 I 6+B(B 4> EKNKiMG)A 9-E 4 8U1B5;E' <.N 4 9-4M<M:79-48MKNG)AK)I 6B 5 6' 4 9 4'J 6 8rAE>4K)M >*;M 4 (9?49 C8©MB & 8©IRE/8 M G 4+MB C9EA< @ ;>; 8[8 M.PRA&M I 4+M 9­J 6 74jP^)K 9-4M 9©M K)M 4 8 K 4M>49:kM 8 4AB 4E!>4EKNK M 4K 4r6B 5 6 4' 9-4 A 6B 674J 4/8:9-4M< 9-mMA>88EKNK 4LE 9-4M<²GV-; (G @ \ 74IE' 9K 4M96M. 4àB@ <ãM G)A 4 8 6>4 <<M 4 8¼E < K¼D8-4+MI 6B 5 6 4 9 4BM KNG)A) BM 9 4M 4 8ÞI 6!LK 9rE 9 4r6B 5 6 4' 9-4 4 9K 5<8:4G)A 4 4DFE 8½< 4 >4'MB C I 9= 4 8=-6/8%>4/8:9 M 4 r4 9%M KU 4 8D$M G)A 9-4Q>4 8 MG)A 9B5;E'74 <: 4 C6AEK 4¹E < KQ I 9©M 6$>4/8 "4B 5 4/8E 9>8©9DFEK°IR@' 9 4ÞE N= 8r@ E<4/8 4$C8 6< 4B 4K 9-6E/496M.E 4 8 4 8): 4K)M 4A 9;M 4Á= 61< 79EFoK:ÆM 8 4AE>4 < M 48ÁM KNG)A)E < KG ; 4MI 6B 5 6' 4! 9-4'LK>=LK 9 4B-96:kM 8f 4798EG)A 9-4'ßE < K 6E < K4 8rK 9-4 C0ALEK 4o(9?49/EKU 4BM KNGHA¥E oKGP oK 9-4'M 9 YC8©MB & 8©IRE<8M: ¥E < K$(2; 4+M 9 4 479K;EEK­J 6DC 4 8<M 9 )UC8©MB &/8©IRE 8M: 9;:kM 8J 4 8=EA/8-4'`;M 4°MBLC I 9<=GE 8+>4 < 4799 D$M 4ÞD$MG)A 9 4B )KNK 4 E!ºM >4/8MB C I 9 4KNG)A 8©M 4 4 4': 4MK 4W>4; 6' 4Y; 4 8+>49 FEG)A

8©ALE < 9 >4KÞ8EKNK 4E 9 4M%< K-= 8EKG 4BMKNG)A C 4 8<ãM 9-e`C 8 MB&<8©IRE 8qM 1> ;M 8%sM 4K 4 8$BM 9B>4 80EA<5V1@fS"G \ B < 9©M5<ãM (M 4 8%99

@F@>,6&7+61Õ1 (

C4M. r6B 5 6 4 9 4IV C0ALEK 4\ÆB(M 9K)MG)A¼J 4<8%& 4 8 >4!NPR 9 4M%< 4 EL 4 81$B5;E <.o 4 9 4M%<ãM 79BV+(9?¦9P oK 9-4'M 9 (sC 4 8<M 9R)8E<8:9 4LK)M 9[\§>K 6Þ=)A<8:9EKO26 8%>4A 4LEG)AL>4B rE/5M 9-4 < =t( 4M 4+B 9-4 8 4K 9©MBB 9 4I< 4M G)A*KNK>=LK 9-4B =>8 kS @

FE 8©M E< 4 91C4 <:K 94MÚ8 4KNK 47 4<8 x 4/8] 4AD;678%>4G; 4>4!D>4 86B 6>4!M 9%&79I 4M 4(oK &79+(<M G)A74 z)=-6/8©BE79 M 6! 4 8>8 M>4 V[ <9jV =9 # \\Ï9o"4A 4>6 8 479©M KNG)A >4I LE/8g;4& 8 41> 4M. 4; 4M 9-4 8 4 5A =LK)MIRE <MKNG)A 4u 8@E/4 >(9?¦9>4! 4 < 4I 98 MKNG)A 4:kM:>4 8rK 9rE1>K)MB < 9rE3(mB 4KNK 49?4MN = IR@ 9 4jE0+M 4 J 6 <. 9©M 6' >4/8C6ALEK 4LE 9 4M%< 4Y 4K 9©MBÁB 9 ; 4/8%>49­D24BK 9-4A740 8 6FE<4 C8:6< 4+B 4 9 4G)AMKNG)A 4/8BP 8:94>4>4 4/8>/;M 4 (9?49¦:kM 4<8rK 9rEKHB 4KNK 4MÚA76A 4I"4+B 5 4 8rE7978 4>¥K 9E 8©I 4/8 6Mo=<.oKNKh>4/8G)A 4BM KNG)A 4 /oKNEBÁB 4LK 479+() > 4;M. 4<8á:kM:>4 8rK 9[& 4 = >8k+M 4 4M)( 4 <:74 C6ALEK 4 E < K I 9©M 6' 4 4 8"4B 5 4/8rE79 8>>4/8 M>4o1c 9-4/8rKNG)AM 4>4G(2;MKNG)A 47>4 C6AEK 4'¹MBN24 8[1< 4MG)A7(°B@ <ãM G)A 4C9@ 8=rEI 9 6 8 4! 8 4KNK;= 4A< 4/8 9^z >4/8«: 4 8©IRK 9-6)== I)>4g; 4/8] 40EA 4/8E>4/8 4¾: 4!>4>4E>43V+1<9 (9?49X8EG v Z]\Ï9

Õ1½§ !"#$%'& ()+*,-/.&01&23"14 1$2"&4 5( !6$0

7 &8596 5: !6"6; 7=< &>01+?8@&> 59A "A @ 1B> 14 5A-C5D&4>EF)HGC%JIIKLM NO6P6LMQR9PPS

6L T$8 !6U#$V$%'& (T)=*,W-X Y&40Z < 34 1\[]."8@ 7 5@_^0`!aM4 20"1b8Ucd&0 7=< &4>0"14e?f4g-h534> Af!ab^ 591 7 3ijMk)HGh%IlmKLM 6LNLQQRon+ I

`8QQ `85(> f5(X*ER$%p,rqXf 7 .f 59 7 &58@ &> ts 7 !a& !a&>r2&"? f0MU&u5v@0!w5(0M&4>&&> x "6-y.M@ 1R@z8@> &1|` <~ f8@nQQQ

` P `4 7 W%)r04>EW%TV-y!6 7 65vkFef4> ZZvW%,$[?&>0&5: f4fZW?"&@ f06B.&6R5v&4 5( f 7 f> bZfn+ q=O;Z9f8&..> !&5: f4 u!f 77 8!6 &>r[C%.f1& 7 H5@f&..&8F cHf!a 1efZk5(;LM@598@qkfZ(fbGh8 7 &>Hcyf!6aA%'f>> 1&qrf 7 .05-X 7 0> &5( fM|`&4"!wx&"!8+L P

&!a 6L &!6fa?6 E-rmh 7 %p, |Hf5Df 59Gh 7 f.> &+5( !6 5x < 5@cd"&qF&1@ !"5&06U8G!6f 7 &5@" 7 &65( = 6LMR( 6X ?"8 59365X 7 +L 6L

Ql 4 Y-k,Yqrf4 8@ 1 -d5v6Rcd"& GH&6Zf 7 &5( f 5&!w5( f6 5(qr&4> !60> &5( f4efZ=)H&65rGH&5 7 5Uh 0&4>-d5vX A%'!& !+tfZ/-yf> < 5@ch&Bqr&"16=[4 5Df8,=8?" >> 4 E(W%'4 !a"($WF-y.M@ "18=; $|H < f8@nQQl

9 QO !""(8M4-y0"cHG&&&H,Gy&Z9f8 7 &5( fR "0"!a.> &a5( !6 5xM4KGHrc8N..>E6%'!aMh"?M+MQ..MPnwlWsPOMn+QQO

#

)&0 )&40.5vpcH,$qrf45( 00 7 %'!a"& !+_& Ghf@ fZ%'&58@ &> -/.M@ 18R@zH@> &1`8@> M+L

)fZ@ 6L )fZZ 7 &$i8> (22WGdM%U&6xM(cy,r a59f5( f [d1 "@ 1$sz8^012(6!601p '85( 101)HGh%IlmKL LNP+L`n+PR@Ln+l

)f0SO )f40"1&"xM)cH & 7 &^&4 EH,6rm 7 .59f6?!&> !60> &5( ffZ859p5v&aZDf8 7 &5: ffZ59>E+5>6&!aUIl KvnQSONQ+OOR@ln

)r0 P )r04>E6%,cH@ ?&5% 55 >01)r0QQ )r04>Eh%22dGHX)f6ZZ 7 &r8y%'&6xM(Xcy,/%'f4>> @0"1bk2& 5( !64

0p.@> 5: !6"p 7=< &">01p2 +-h53> 4M6)HGh%BIK n+QQQNO&8QI &8f-r4%'&"1&88%TV,H5@ 7 &5( ffZH.&m5v&6Z9f 7 &5( ft 5( !+ < "

59 7 .@ 1 7 &859 5: ! "&8 > f < &4>> f6xb54> " GC)RhcR[4]"RonwLMP6dr65: 5(05f6ZG!6"f4> f16xMC4.&5 7 5fZ8%U!6&4 !+&>[d1 @ 1 . 1- < nQQI

2S6 42> f8*yHyH6?"&0a]Md*, " < "5: ! 7 f>Zf8r& f59 7 &4> 7 5&>>01 !&4>5v&4Zf 7 &5( f6e t54> t !6>0 1e6ZZ@!w5rf6Z&065 5D 1&4 ^8/! 5D& %'65vhPL..n+PlUs n+OnaQS

"?aQS6 "? 5&zV,Gh"8 7 f 7 !& !&4>5@"f8x f6Z 7 &56 5( ! ."&5&4Zf 7 &5( f6 "4> &5: ! 7 &5@ &4> 65 a*-Hf4> '-d5v@0!+5@0MP6I6..MSSQs Q 8nQQS

!68q=!68,GC 7 fax& 7 6-y.M@ 1R@zX@> &1 @> M+L %'&!aQ6L %'&!6"8&40"!aMW[H,WcH& 5( a0 7 Ab@5fZZ(+0Tn+ 8XF0Z(> &18rzr < 1 -hfM

&40!" < 1 naQQL% 59 L % 55@ 7 4 ([y=*U-yf 77 8(,$-hf> +59&5 .&6Y5&Z9f8 7 &65( f4 a 5( !+,'&

7 f0> &5v&Zf8 7 &5( f 7 f>EH%59&>>+0QPpKLM LNIPILMR(POn% 59QL % 55@ 7 4 (X[Hd*,XH"? < s;F&> x fZX5@"ua 5( !+tfZ.&b5&4+Z9f 7 &5( f6"

*f40M@&>+f6Z%'&6598@ &> W!6 "!+$LlPQllMR(PQSlMn+QQ6L%> lP6 %>> ( V,Gh 7 f6x& 7 ~s #@0"4> &1 4 " %&59@ &> 5:f@ 8p594> 7 &

r ?" 5@35+?"8@> &16> f8Zvn+QlP9X !+lP X !6659(B, r < !"5( 15@ .ax &> 6!6" [h 16!6&Z5@?f IL [h R

< 8@59fZZ@M+z8@> &1-d5@&> 6M6> f8Zvn+QlP9HfQO Hf4> fZZv)&5@&,pch+x &> !" [h 16!6&ZV59Y12M30!6> !"-d593> 8

z8@> &1-C5&> 6'# 7 2)r> f8ZvnQQOgf> L gf4> ZZv$%$4 7 $%V,'cC&0 7=< &4>01Y0 7=< &4>0"1.> &5: ^ 5352

-h5@3> 7 f^4.5 Gh 7 f> &5@fRch> &a5( ^ 535v `@ !a659 &06 8G!6"f 7 &65( 7 &5( 4P6r ?" 5D35 7 M`4.f5 LMR( 6I+L L

8

"! +$#%#'&)(+*),-/.0213

http://www.math.uni-bremen.de/zetem/berichte.html? 24 8:98 M 4 78rG)A 45P^ 9-6 8S

4 ©Õ .6 !/54'@76 58+8

vw>?;@9 C 479 4 86?4 4 8½>4MI 4$EEF 474 8

9;:=<?>A@CBEDGFHD?IJB KMLONQPRIJSUTVIWNQXZYCKQ>A@CBGN+F[I\DGF^]+SQ:5F_Q`QPAaZN+IJb5`cXMdW`+T2I\b+NMYeKQ>f@eBGN+F[I\DGFgD heNQ:jikS+B?lQNZmTn`Qo+BGNp>Mqp<ãM @ vvw19

vw>?;9 24+MI 4$E EF 4>4 8

YBrDJXQDr:QhtsZDE:5X+`CutYFHbUNQ>=NQPfd`+TCvwDrP[FxTnN5IWNy]+NQScPRI$zY|5lRSpTnNQPO9@+@ T"`xQDE>MSCIJDJ`Q:yo[K~+TD he`Q:5`>=N5I?TDJFm`pB KQ:5`>ZDrS+BEPxq)p<ãM @ vvw 9

vw>?;>= 9 C 479 4 86?4 4 8½>8E/8©M. 4 < «EK 9©M%<< 6 >J/8 M 4 CF9)xM. 9rELE 8:9n

mS5TS+BEBGNQB[mSUTVI\DrScB[YIJSco+DrBED _Dr:QhA9OB hC`+TD?IJb+>ZPd`+Te]RSUTheNA]+DE:5NQS5Tjw`Q:IWT`pB[YCKQPRI?N>ZPo> p<ãM@ vvw19

vw>?; 9 C 479 4 86?4 4 8

`Q>A@2lcIJSI\DG`p:RScB[aMNUIJb+`+XQPAd`RTe]+Dr:UNQS+TVzn%lS5X+TSI\DGFA@2I\Dr>ZD _SI\DG`p:>>P^) oK 9@ vvw 9vw>?; # 9

C 479 4 86?4 4 8½>>3¦E < 5A-?= 4 8K>J>8 M 4mCF9)xM 9ELE 8:9n> 8 4>6/8©M 6xM. 9rE!LE 8 9n+Y`QBikDr:Rh9;BhCNQo5TS+DJFLwDGFxFHSI\D[g|5lSCIJDJ`Q:+PA`Q:MmSUT"ScBrBGNpBR`Q>A@2lQIWN+TPMPxDr:Rh;NCuI?`Q: PaZN5IJbU`+XuD\I\bMgeQS5F[I]+DE:5NZYNQS5TnFb>2P^)FoK 9@2vvw 9

vw>?;8 9 ,oE 8rK5 8 4 >EM E':kM 8%9©A

:I\b5NZTSIWNZ`+dQFx`Q:jiN+ThCNQ:5FxNZ`QdQDE:5d?DE:+D\IWN'b5`QTD _Q`Q:=XQDEP[Fx`plR:I?NcXM`H@2I\Dr>ZS+B+ikScB\lNdlR:5F[IJDJ`Q:+Po>6)J 4B 4 8@ vv!w 9

vw>?;< 9 C 479 4 86?4 4 8½>2261<I 4 868 4A<8©BE>6) 6^!

9;`5IWNZ`Q:IJb5NZlx>=N+TDJFHS+B[Y`QB?l+IJDJ`Q:M`cd5`Q>%@CBGNMwS+>ZDrB\IW`Q:+DESQ:MSc:5XZY[NCuzS+>ZDEB\IW`Q:+DrSc:gD hCNQ: ikS+B\lQNZmT`Qo+BJNQ>ZPo>26)J 4B 4/8@*vvw 9

vw>?;w 9 J 4 8 AE 8%"?J&LKNG)A>?/>8©INALE/8%GF@A!

l>=N+TDJFHS+BxPxDr>lBEScI\DG`p:`cd[SZPxDEBrDGFx`p:dWBG`pSIJDE:Rhy_Q`p:5NuD\I\bZSMdTnNRNZFHS[@eDrBrBES5TKyPRlTndWS5FxN>D24( 4+B374 8@ vvw19

vv>?;@9 24+MI 4$E EF 4>4 8~Qb5NZmSUT"SQ>=N5IWNQT"D_QN+XZYL9;BhC`+TD\I\b+>d`+TjYCKQ>A@CBJN+F[I\DGFl+IIWNRTndWB Ka'SIWTDGFxNQPo>>4' 8oE 8@ vvv19

vv>?;9 24+MI 4$E EF 4>4 8

gTnTn`+TQ9;:cS+B KQPxDEPf`+d+I\b5N'PKQ>@CBJN+F[I\DGF^]+S+:5F_Q`QP%aMNUIJb5`QXZd`RTIJbUNyPK>A@CBGNcF[IJDJFgD heNQ:jikS+B?lQNZmTn`Qo+BGNp> >8 & 8+( @ v!vv 9

vv>?;>= 9 J 4 8 AE 8%"?J&LKNG)A>2P´< = 8 4mC0G)ABM: 9

YDr>lBESI\DG`Q:=`+d+XcNQ:+X+TD\I\DGF%FxT$KQP[I\S+BhCTn`CuIGbuD\I\bIJb5NQT>ZS+B[FH`Q:jiN+F[I\DG`Q:>L8 & 8+( @*vvv 9vv>?; 9

J 4 8 AE 8%"?J&LKNG)A

CDE:+D\IWNZNQBGNQ>ZNQ:IQXpDrP[FxT"NUIJD_SIJDJ`Q:=`Qd5IJbUNMwSjikDGNQT¡zYIW`Q[NQPN+|UlSI\DG`Q:+PZuD?IJb'S=d$TN+NFHS[@CDEBrBrSUT$K'PRlQTndWS5FN>8`& 8+(@ vvv 9

vv>?; # 9 C 479 4 86?4 4 8

a'SIJbUNQ>ySI\Dr%Dr:=X+NQT N+T¡lQdWP@TS[QDEPo> p<ãM @ vvv 9vv>?;8 9

PR 8-4;ºD²9?49C EMG 4 >EM E' 3$9:ÆM 8:9 AL;`QoClPRIJ:UNQPxPf`cd[:U`Q:+BrDE:5NQS5TUPKQPRIWNQ>ZP¢S+:UXI\b+NQDGT X+`Q>'S+Dr:+PA`+dSII?T£S5F[I\DG`Q:> P^) oK 9@ vvv19

<

vv>?;< 9 C 479 4 86?4 4 8½>J/8 M 4mCF9)xM 9rELE 8:9n> 8 4>6/8©M 6xM. 9E'LE 8:9+

S+BES+:5FxN+X ~+T¡l:5FHSI\DG`p:MaZ`+X+NQB[LON+X5lQFxI\DG`Q:=`+d]+S5ThCN+zYFHS+BJNZv;NQ:+P[NZYCKQPRIWN>ZPf`Q:mS5TS+BEBGNQBRw`Q>A@2l+IWNRTPo>C4/5 9 4B 4 8@ vvv 9

vv>?;w 9 3)6'LE < mC9r@)J 4 8

`QBEBG`+FHSCIJDJ`Q:Z>NUIJb+`+XQPAd`RTeP[`QB$ikDE:RhBrDE:5NQS+TjXQDJd$dNQTnNQ:I\DrS+B?zScB hCNQo5TS+DJFo5`5l:UXQS+T$KikS+B?lQN@jTn`Qo+BJNQ>ZPo>oC4/5 9 4B 4 8@ vvv 9

vv>?;v 9 R)K 4=M.5P²IRGNE=

aZ`+X+NQBEBrDr:Qh uD\I\bT¡IJb5`p:5`QT">ZS+B[S+PxDrP l:5F[I\DG`Q:cPo>oC4/5 9 4B 4 8@ vvv 9vv>?@ ; 9

24+MI 4$E EF 4>4 8©>D²9C9-4LJ 4(8EG)I 4=> $M%< 6 N= 4 88EG)I 4=

S+>ZDEB\IW`Q:ZS+:5XxSUFx`Qo+D[Fx`Q>ZNMdlBEB[FHDGTnFHBGNxS5Fx`po+DxS+B he`RTD?IJb+>ZPd`+TePRIJT¡lQF[I$lTnN+XS+>ZDEB\IW`Q:+DrSc:NpD hCNQ:x@jTn`QocBGN>ZPo> I 9 6! 4 8@ v!vv 9

vv>?@>@9 C 479 4 86?4 4 8½>2$MLG 4 9 4G24 8q 4(>2PR 9:6M 6$C EK 9-6 8

:I\b5NBGNpDr:+>ZS+:Z<IWN+TSCIJDJ`Q:=d$`QTjO`Q:+PRI\SQo+DrBED _S+o+BJNZYCKQPRI?Np> > I 9-6' 4 8@ vv!v 9vv>?@;9

C 479 4 86?4 4 8½>4MI 4$EEF 474 8

9KQo5TDGXZaZN5I\b5`QXZd$`+TjI\b5NZl>=N+TDGFHScB[Y`QB\lcIJDJ`Q:M`+dcvDrP[FxTnN5IWN+z~QD>=N^9;B hCNpo5TS+DGF%LDJFxFHSIJDg|5lSCIJDJ`Q:+Po>6)J 4B 4 8@ vvv19

vv>?@ = 9 C 479 4 86?4 4 8½>J/8 M 4mCF9)xM 9rELE 8:9n> 8 4>6/8©M 6xM. 9E'LE 8:9+

l>=N+TDJFHS+B[Y`QB?l+IJDJ`Q:=`Qd+YFHblQT YI\S+o+BGN]+Dr:UNQS5TCa'SIJTD Zg|5lRSIJDJ`Q:+PA`Q:a=lB\I\DGFx`Q>A@2lcIJN+TPo>26LJ 4B 4/8@ vvv9

vv>?@ 9 J 4 8 AE 8%"?J&LKNG)A>rE 8 6 <N8ÁMI < E

9XQS[@IJDikD\I KyDE:Uv<?>ZSRhUNZmTn`+FxNQPxPxDE:Rh >D24( 4+B374<8@ vvv 9;>;>?;@9

C 479 4 86?4 4 8½>2261<I 4 868 4A<8©BE>6) 6^!

mN+T¡IlQTo+SI\DG`:%9;:+S+B KQPxDrPAd$`+TIEb5NZgDhCNQ:jikS+B?lQNZmT"`po+BGNQ> `QdQS= C`+T>yS+B[mTn`+XUlQF[I`+da'SI?TDJFxNQPxq)pNEoE 8/;>;>; 9

;>;>?;9 4M 5M.7foEF CDE:+D\IWNZgBGNQ>=NQ:CI[aZN5I\b5`QXZd$`cTCayD +N+XZmT"`po+BGNQ>ZPZuD?IJb=mNQ:QS+B?IK>*poEoE/8/;>;7; 9

;>;>?;>= 9 ÂM Eo=8rELG 4KNG 6°8E<8:9©M.MG 6 L;N+F[lpT"PxDiN'>NpPxbMTnN+dWDr:+NQ>=NQ:IDr:5v«>74<8NE 8<;>;7; 9

;>;>?; 9 d 4 8©ALE 8%°?-&LKNG)A>A<8©M K 9 6/5A7 4 8rK> Á<M J 4 8 8 4+M.LG)I 4 > $M G 61< 479rEC0G 8 9~QS[KQBJ`+T¡zn`+lQN5II?NZYCKQPRIWN>)uD\I\b9OPKQ>M>ZN5IWT"DJF`5l:5XS5TK=`Q:5XpD\I\DG`:+Po>74/8)E<8<;>;7; 9

;>;>?; # 9 C 479-4 8?4! 4 8YCKQ>A@CBGNcF[IJDJFScBrS+:5FHDE:Rh`+d+wSR>ZDEB\IW`Q:+DrSc:aSIWT"DJFxNQPxq'74>8oE 8/;>;>; 9

;>;>?;8 9 E'M 6 ²E+B(M<<ãM%>,oE/8rK5 8 4 >E'M E:kM 8 9 A

9TNhjlBrSUT"D_SI\DG`Q:Z`+d lo5`ei PAN+|5lSI\DG`Q:=d`RT Tn`[oClPRIQX+`>ZS+Dr:cPf`+dSIIWTS5F[I\DG`Q:cq>8 & 8+(<;>;>;19

;>;>?;< 9Ý8ÁM G)AE>4 <: 6 < ==½>d 4 8©ALE/8%"?J&LKNG)A>8ÁM G)ALE 4 <N?@AB->D26+B(MM GÁDFELJM KaZ`+X+NQBEBrDGNcTVlR:Rh=X+NcT9;o+b+B?l:Rhi`Q:ZYIJScb+Bro5TSQ>Z>=NQ:>8 &<8+(<;>;>;9

;>;>?;w 9C9 4/5AE-DFEA<I 41>C 479 4 8 8EEE>) 4/8%"4KNG)AI 4 <?:CI?N+T@j`pBrSI\D:RhYFHS+BEDr:Rh 2l:5F[I\DG`p:+PMuD\I\b=vlS+BEPo>>P5<8©M%<v<; ;>; 9

;>;>?;v 9Yp 6G)A74-?4A>8 4LK>EM E:kM8:9 A9 hUBG`QocS+BrD_SIJDJ`Q:A@jTn`+FxNQX5lQTnNyd`+TeBG`+FS+BrB KyPRI\S+o+DEBrD_Dr:RhFx`Q:I\T`pBrBGNcT"Po>8EMZ<; ; ;19

;>;>?@ ; 9lC 4 9-4 8 8EEFE >) 4 8["4KNGHAI 4 >: 4 8% 4 8:kM%<<ãBE>) 9 4 8: 6 <<ãBÂE!v;N5IWN+F[I\DG`Q:ZS+:UXBrS+PxPxDJd?DJFHSI\Dr`Q:=`+dQaySCI?NQTDrS+Bk9¢I$IWTDrol+IWNQPfznzR9 mT"SUF7I\DGFHS+Bk9@+@CBrDJFHSI\DG`Q:=`+dsZSjiNQBGNUI9;:+S+B KQPxDrPo>8EMZ/;>;7; 9

;>;>?@>@9C9 4=rE-?6KNG)A 4/8:9>2P´< =8 4CG)ABM 79>r/M. 4/8:9Á9C4M 4' 4/8 9>J74<8©ALE 8]m?-&LKNG)A>r4< E )K ?r: 4 8 4 8?4(L>) 4 8ALE 8%°D/(M9Io> "¦A 6BEKr6EM K 4 8

YDr>lBESI\DG`Q:=`+d+<W:5X+lPRIWTDrS+BRwT$KQP[I\S+B¢Tn`CuIEbZo[K=IJb+N+N+T¡I\DGFHS+B[TDJX[hc>MSc:MaZN5I\b5`+X>8EM<;>;>;19

w

;>;>?@;9Y261<I 4 8J, 4ABE> 4 8% "4KNG)AI 4FsZSjiNQBGNUIQS+P[N+XZaZN5IEbU`+XQPAd$`QTj<W>@ T`eiNRXsZDr:UXMmT`cd?DEBGN+TCYDhU:+S+B[mT`cFxNQPxPxDr:Rh1>L8EM<;>;>;19

;>;>?@ = 9C9 4/5AE-DFEA<I 41>C 479 4 8 8EEE-9;`5IWNZ`Q:=<?:IWNQT@j`QBrSCIJDE:Rh=YFHScBrDr:Qh l:5F[I\DG`:+Po>2P ) oK 9>;>; ; 9

;>;>?@ 9 3 6 = 34EB-< E ><3)6 < = < EG)I >6=< 4>/8 8©M G 2676 >M 8 MK)A(?E1<.9FxF[lQTSIWN^9I$IWNQ:lRSIJDJ`:w`+T"TnN+F[I\DG`p:MDE:=Ymg^~Z<?>ZS[hUDr:QhlRPxDr:Rh@2I\Dr>ZD_SIJDJ`Q:M`cdDrBEDr:5NpS5Tj l:5F[I\DG`Q:+P^S+:5X^9;PxPRl>ZDr:QhZS+: :cx:5`Cu:=YC@eScI\DrScBrB K5z QSUT$KQDE:Rh9II?NQ:5lSCIJDJ`Q:vDEPRI?TDroCl+IJDJ`Q:>C4/5 9 4B 4 8<;>;>; 9

;>;>?@ # 9lC 4 9-4 8Jr/I 41<><36E < mC9r@)J 4 8YCKQ>Z>=N5I?TDJFFx`QBEBG`+FHSCIJDr`Q:Z>=N+IJbU`+XQPAd$`QTCBEDr:5NpSpT XQDGddN+T"NQ:CIJDES+B\z£S+B hCNpo5TS+DGFoU`5l:5XQS+TKikS+B\lQN@jTn`Qo+BJNQ>ZPo>oC4/5 9 4B 4 8<;7;>; 9

;>;>?@ 8 9lEM E!:kM 8:9©A~Qb5NAhCNp:pNcT"ScBrD _QNcXyP@CN+F[IWTS+BxT"SUXQD\lP%SQ:5XZNx5IJTnNQ>ZS+Bx:+`+T>ZPo> I 9 6 4 8/; ; ; 9

;>;>?@ < 9l 8EIC9 4>4/8>2P²ABÂE'3)4(oE EALKHA ? $M< G)AM> p 4*= »M 4LKNGHA><3 6 = 34EB-< EF9 l:+DGdWDGNcXyS@+@jTn`S5FHbI?`ZIJb5N'S[@c@Tn`xQDE>ZScIWN'P[`QB\lcIJDJ`Q:M`+dcmv;g4> 26)J 4B 4/8<;>;>;9

;>;>?@ w 9lC 4 9-4 86?474<8>d<8©M 4mCF9)xM. 9E!LE>? 8%9n> 8 4>6/8©M 6xM. 9rE!LE>? 8:9+mS5TS+BEBGNQBxS+BhC`+TD\I\bQ>ZPfd`+TU>=`+X+NQBxTnN+X5lQF[I\DG`:=`+d+XpDrP[FxT"NUI?N+znIJDE>=NyPKQPRI\NQ>ZPo>D24( 4B 4/8<;>;>;19

;>;>?@ v 9 3 6 = 34EB-< E !9PRI?N+N@jNQPRIQX+NpP[FxNQ:5IS+B he`RTD?IJb+> d?`+TIJb+NAhUBG`po+S+BH>MDE:+Dr>ZD_SIJDJ`Q:M`cdj~QDExb5`:5`Cizmb+DEBrBrD @CPdlR:5F[IJDJ`Q:+S+B>D24( 4B 4 8<;>;>; 9

;@ ?;@9Y"6<8rK 9 4ur@A< 4/8½>C 479 4 8 8EEE>C 4 9-4 8): oK 9>8E 8:9 M(C4>4oEKNKgdd?DJFHDGNQ:IR>NUIJb5`cXQP¢Dr:Zb[Kx@ NRT¡I\b5N+T£>MDESIJTnNQSI\>MNQ:I7@CBrS+:Q:+DE:Rh > poE')E 8>;>;@9

;@ ?;9C 479-4/86?4 4/8><3¦E < 5A-?= 4 8K)><3¦E=E 4 <R8E= 61>J<8©M 4mCF9xM 9rEE>? 8:9> $MG 4 9 424 8>4(

Parallel Algorithms for LQ Optimal Control of Discrete-Time Periodic LinearSystems, Februar 2001.;@ ?;>= 9lC 4 9-4 86?474<8>d<8©M 4mCF9)xM. 9E!LE>? 8%9n> 8 4>6/8©M 6xM. 9rE!LE>? 8:9+gdd?DJFHDGNQ:Ipl>=N+TDGFHScBk9;B hC`cT"D?IEb+>ZPAd$`QTjS+BrS+:UFxN+XZYI?`cFHbRScPRIJDJFM~QT¡l:5FHSCIJDJ`Q:>8 &<8+(<;>; @9

;@ ?; 9lC 4 9-4 86?474<8>8E 8©M. 41< ²EK 9 M<< 6 >J<8©M 4mCF9)xM. 9rELE>? 8:9n[mS5T¡I\DrS+B[YI\S+o+DEBrD_SIJDJ`Q:=`+dQ]+S+ThCN+zYFHS+BJNMvwDrP[FxTnN5IWN5z¡~pDr>=NA]cDr:5NQSUTj`Q:IWTn`QB[YCKQPRIWN>MPo>8 & 8+(<;7;@9

;@ ?; # 9C9 4/5AE-DFEA<I 4FNQP[`CiL;NhjlBrSUT"D?IKZd`RT gX[hCNZYDE:RhjlBES5TD\I\DGNQP^Dr:=m`QB KQb5N+X+TS+B[vO`[>ZS+DE:+Po>8EMZ<;>; @9

;@ ?;8 9lEM E!:kM 8:9©A9BrDr:UNQS5TD _SI\DG`p:f@ T"DE:5FHD @eBGNyd`+TCTn`QolRPRIE:5NQPxPZuD?IJbZTNpP@jN+F[I+IJ`=I\Dr>=N+z ikS5TKQDr:Rh@jN+T¡IlQTo+SCIJDJ`Q:+Po>8EMZ<;>; @9

;@ ?;< 9C9 4/5AE-DFEA<I 41>;: 6 < =%EmDFEAB 4!>!rE 8rK 9-4'318LE'9XQS[@IJDiN sZSjiNpBGN5IQaZN+I\b5`+XQPAd?`+T YS5X+XQBJNZm`QDr:CIQmT"`po+BGNQ>ZPfz @IJD>ZS+BRw`Q:jiN+ThCNQ:5FxNLSIWNQPo> p<MZ<;>;@9

;@ ?;w 9 3 6 = 34EB-< E !aZ`+Tn`[_Q`Ci PAvDEP[FxT"Nx@eS+:5FkKZmTDr:5FHD @CBGN'd$`+T~pDrxb5`p:5`Ci;TnN[hjlRBrS+TD _SI\DG`Q:=`+d:5`p:+BrDE:5NQS+T`x@jN+TSIW`QTPo> p<MZ/; ;@9

;@ ?;v 9Ý8ÁM G)AE>4 <: 6 < ==gDr:Ud bUTVlR:RhX+NQPvOTVlpFHxPfd TjXQDJNyDE:+P[IJSI\DG`p:+TNp:YI?`Q[NpPz BGNQDJFHbl:RhCN:Z>ZD\II?NpBrPX+NcTaZN5IJbU`+X+N=i`p: S[@CBrSc:> p<MZ<; ;@9

;@ ?@ ; 9C9 4/5AE-DFEA<I 41>C 479 4 8 8EEE>) 4/8%"4KNG)AI 4 L;N+Fx`p:+PRI?T¡lQF[I\DG`p:M`+d+LON+dWBGN+F[I\D$ikD?IKZv;NQPxD?IJDJNQP¢o[KsZSjiNQBJN5IU~+TSQ:+P[d$`cT>ZPo>2P^)FoK 9>;>;@9

v

;@ ?@>@9C9 4/5AE-DFEA<I 4FNQP[`CiL;NhjlBrSUT"D?IKZd`RTIJbUNMON+l>ZS+:Q:mT`po+BGNQ>>2P^)FoK 9<;7;@9

;@ ?@;9Ý?4 8 E<8%GFEEK >6IN>8E/8©M 6 A<: 4/8%>4/8>8E<8:9©M. 3B)B 5o=>2P´< = 8 4 C0G)ABM 79>r/M. 4/8:99C¦M 4 4 8 9

bzW@2za=lB?IJDJTnNQP[`QB\lcIJDJ`[: QDEPRlS+BED _SI\DG`Q:=`+d79XQS[@jI\D$iNZ CDE:+D\IWNygBJNQ>=NQ:IQYDr>lBrSCIJDJ`[:cPo>I 9 6 4 8>;>;@9

;@ ?@ = 9C9 4/5AE-DFEA<I 41>)E' 8©M 4 < 4mC9 4M 1<>) 4/8]"4KNG)AI 4 `+`cT"ocD\IQYC@eS5FxNQP%S+:+XMS+:+S5FHbZ jTSQ>NpPf`Q:=O`[>=`[heNQ:+N+`5lPAYC@CS5FxNpPMuD\I\b9@+@CBEDGFHSI\DG`p:+PIW`^9;:+S+B K[_Dr:Rh lx:UF[IJDJ`Q:+PA`Q:MYC@eb5N+TnNQPo>>P^) )K 9<;>; @9

;<?;@9Ý8ÁM G)AE>4 <: 6 < ==½>8ÁMG)ALE74 <N?@ABulQTCaZ`+XcNQBrBEDGN+T¡l:[hX+NQT~pb5NQT">=`QNQBES+PRI?`Uz"mBrS+PRI\D _D?I I>ZD\Ipmb+S+P[NQ:Clx> uS+:5XQB?l:RhUNQ:Zo5NQDYI +b+BGNp:MP[`euDGNZX+NcT> uS+:UXQB\l:QhUP@eBrS+PRI\D _D?I I>>4/8 oE 8>;>;9

;<?;9 C9-4 5ALE-DFE+A<I 4 >C 479 4 8 8EEFE-9O:%l+I\BrDE:5NZ`+dH9XQS@2IJDiNsMS iNQBJN5IS+BGNQTxDr:=aZN+IJbU`+XQPAd$`QT~pDrxb+`Q:5`eiLON[hjlBESpTD_SIJDJ`Q:`cd+<?: iN+TP[NymS5TS+oU`QBrDJFmTn`QocBGN>ZPxq P5<8©M%<v>;>;

;<?;>= 9 P´< = 8:4mC0G)ABM 799a=lB?IJD?z"aZNQPxbZ CDE:+D\IWNZgBGNQ>=N:IQaZN+I\b5`+XZd?`+T mb+S+P[NZ CDJNQBGXZYDE>lBrSI\DG`p:+Po>>P5<8©M%<v<; ;9

;<?; 9ÁC4/8%>4= ²9DEG)AI 6)JLK)INM%>8MG)ALE74 <N?@AB 9O`5I?NZ`Q:= eDr:+D?I?N~Qb5NQT>=`x@CBrScPI\DGFHD?IKuD?IJb=mb+ScP[Nb+S+:RheNQPo> p< =b<;7;69

;<?; # 9Á8MG)ALE>4 <: 6 < ==©>8ÁM G)ALE>41<N?@ABuPhasenumwandlungen und Umwandlungsplastizität bei Stählen im Konzeptder Thermoelasto-Plastizität, July 2002.

02-06. Gerd Teschke:Construction of Generalized Uncertainty Principles and Wavelets inAnisotropic Sobolev Spaces, August 2002.

02-07. Ronny Ramlau:TIGRA - an iterative algorithm for regularizing nonlinear ill-posed problems,August 2002.

02-08. Michael Lukaschewitsch, Peter Maaß, Michael Pidcock:Tikhonov regularization for Electrical Impedance Tomography on unboundeddomains, October 2002.

02-09. Volker Dicken, Peter Maaß, Ingo Menz, Jenny Niebsch, Ronny Ramlau:Inverse Unwuchtidentifikation an Flugtriebwerken mit Quetschöldämpfern,October 2002.

02-10. Torsten Köhler, Peter Maaß, Jan Kalden:Time-series forecasting for total volume data an charge back data, November2002

02-11. Angelika Bunse-Gerstner:A Short Introduction to Iterative Methods for Large Linear Systems, November2002.

02-12. Peter Kunkel, Volker Mehrmann, Ronald Stöver:Symmetric Collocation for Unstructured Nonlinear Differential-AlgebraicEquations of Arbitrary Index, November 2002.

02-13. Michael Wolff:Ringvorlesung:Distortion Engineering 2Kontinuumsmechanische Modellierung des Materialverhaltens von Stahl unterBerücksichtigung von Phasenumwandlungen, Dezember 2002.

03-01. Michael Wolff, Michael Böhm, Serguei Dachkovski:Volumenanteile versus Massenanteile – der Dilatometerversuch aus der Sichtder Kontinuumsmechanik, Januar 2003.