; TeX output 1999.04.01:1712y?`DtGGcmr17Sample7tPqapserforthe߆TG cmtt12amsmathPackageI`File7tname:testmath.texXQ cmr12AmericanMathematicalSoScietry;29OctobSer1996 "VVersion1.027T>Nff cmbx121VLIntros3duction>K`y cmr10ThisUUpapGercontainsexamplesofvqariousfeaturesfrom !", cmsy10AU>'M S-L5ffٓRcmr7A͉TU>'ExX. >2VLEnumerationofHamiltonianpathsinagraph>Lett "V cmbx10A=( b> cmmi10a 0ercmmi7ij )bGetheadjacencymatrixofgraphG..ThecorrespondingKirchho >matrixK=(kij )isobtainedfromAbyreplacinginAeachdiagonalentryby>the}degreeofitscorrespGondingvertex;i.e.,Gtheithdiagonalentryisidenti ed>withUUthedegreeoftheithvertex.qItiswellknownthatՍadetq9K(iji)=mtheUUnumbGerofspanningtreesofG; i=1;:::;n8(1)>whereUUK(iji)istheithprincipalsubmatrixofK.!w>#toFeachspanningtreeofG.DenotebyCiVJ=u cmex10S WUj8C:i(jg) J.Itisobviousthatthe>collectionofHamiltoniancyclesisasubsetofCiTL.0Notethatthecardinalityof>CiisUUkii@detaxK(iji).qLetx䍑rbX.3=f^x1 3;:::;MS^xn Bg.>$\wh?X=\{\hatx_1,\dots,\hatx_n\}$>De neUUmultiplicationfortheelementsofx䍑rbXpbyՍq^0Ƶxiٲ^<.xj=i^xj W^ xiH; MT^ x2፴iP=0; i;jY=1;:::;n:8(2)Ǎ>Let\q^| kij_=kij^ xjt˲and\q^| kij=P 8jg O!cmsy76=i\qT^kij'Qʲ.Then| thenumbGer| ofHamiltoniancycles>Hch6isUUgivenbytherelation[8]f^δn ,Y tTjg=1x^]xj%^Hc=<$K1Kwfe (֍2\q p*^ -kij޲detqƍ)`b:'K0(iji);i=1;:::;n:8(3)1*y>%p0J cmsl10SampleUUpapGerfortheamsmathpackqagej2?>The*taskhereistoexpress(3)inaformfreeofany_^xi 6,3;i=1;:::;n.cThe*result >alsoUUleadstotheresolutionofenumerationofHamiltonianpathsinagraph.MItiswellknownthattheenumerationofHamiltoniancyclesandpathsin>acompletegraphKn andinacompletebipartitegraphKnZcmr51 n2²canonlybGe>found6from$': cmti10 rstc}'ombinatorialprinciplesW[4].jOne6wondersifthereexistsa>formulawhichcanbGeusedveryecientlytoproGduceKnandKn1 n21<.\\Recently*,>usingLagrangianmethoGds,GouldenandJacksonhaveshownthatHcݲcanbGe>expressedcintermsofthedeterminantandpGermanentoftheadjacencymatrix>[3].However,ܱtheformulaofGouldenandJacksondeterminesneitherKn 3nor>Kn1 n2"e ectively*.zIn}thispapGer, usinganalgebraicmethod, weparametrize>theFadjacencymatrix.lTheresultingformulaalsoinvolvesthedeterminantand>pGermanent,butZitcaneasilybeappliedtoKn زandKn1 n21<.dInaddition,we>eliminateвthepGermanentfromHc㓲andshowthatHc㓲canbGerepresentedby>aO*determinantalfunctionofmultivqariables,eachvqariablewithdomainf0;1g.>F*urthermore,weshowthatHccanbGewrittenbynumbGerofspanningtreesof>subgraphs.qFinally*,UUweapplytheformulastoacompletemultigraphKn1 :::  nO \cmmi5p.MTheuconditionsaij ٲ=ajgi ,|i;jl=1;:::;n,|areunotrequiredinthispapGer.All>formulasUUcanbGeextendedtoadigraphsimplybymultiplyingHch6by2. ->3VLMainffTheorem>Notation.l*F*orp;q2FPandn2!fwewrite(q[;n)(p;n)ifqpandAq@L;n2=>Ap;n в.H>\begin{notation}?For$p,q\inP$and$n\in\omega$>...>\end{notation}MLetKXB=(bij )bGeann$nmatrix.nsLetn=f1;:::;ng.nsUsingthepropGerties>ofr(2),UUitisreadilyseenthat>LemmaT3.1.&oY t3i2!f$cmbx7nr^UX t,jg2nbij^ xi`^9=^ IY t -i2n w^j̵xi#v4^,}PpGerwher}'epGerpXBisthepermanentofB.MLetx䍑ǫbUUY#=f^y1 c;:::;Z^^yn Eg.qDe neUUmultiplicationfortheelementsofx䍑ǫbYDzbyaX^yiò^ yjC+^8yj a^ VyiY.=0; i;jY=1;:::;n:8(5)>Then,UUitfollowsthat>LemmaT3.2.&Y tVi2n^xX tO jg2n bij^ yj(ß^LO=^ IY t -i2n^j̵yi"7^+Sdet;;B:8(6) ty>SampleUUpapGerfortheamsmathpackqagej3?MNotethatallbasicpropGertiesofdeterminantsaredirectconsequencesof >Lemma3.2.qW*riteaȟX tjg2n~9bij^ yj۲=X tQjg2n㉵b:()Zij -^ ~whereb:()ZiiET=iTL; b:()Zij=bij ; i6=jR:8(8)ˍ>LetޡB^()*&=(b:()Zij ~<). By.(6)and(7),itisstraightforwardޡtoshowthefollowing>result:>TheoremT3.3.detBB=>n X l `=0|sX ㉴Ilҷn'#:Y 7&1ʹi2Il4Ų(biiax8iTL)det8B() ~<(IlȸjIl);8(9)"$g>wher}'eͧIlA~=/fi1|s;:::;ilȸgandB^() ~<(IljIl)istheprincip}'alsubmatrixobtainedfrom>B^()#bydeletingitsi1|s;:::;ilr}'owsandcolumns.>R}'emark3.1.xkLet5MbGeannn5matrix.g:Theconvention5M(njn)=1hasbGeen>usedUUin(9)qandhereafter.MBeforeZproGceedingwithourdiscussion,wepausetonotethatTheorem3.3>yields*immediatelyafundamentalformulawhichcanbGeusedtocomputethe>coGecientsUUofacharacteristicpGolynomial[9]:>CorollaryT3.4. ZWritedetww(B8xI)=Pލ USn% USl `=0(1)^lȵblx^l.Thenu.bl=jџX Ilҷn*det%2B(IlȸjIl):8(10) MLet%sK(t;t1|s;:::;tnq~)=V0BBfi@d}D1t4øa12xt2\a:::t a1n mtn a21xt1;9D2t\a:::t a2n mtn :S::::::::::::::::::::: an1 mt14=an2 mt2\a:::{&Dnq~tV߫1CCfiA;8(11)'q֍>\begin{pmatrix}?D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\>-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\>\hdotsfor[2]{4}\\>-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}>whereHϵDid=X tQjg2n㉵aij tj6; i=1;:::;n:8(12)8y>SampleUUpapGerfortheamsmathpackqagej4?MSetJ"zDG(t1|s;:::;tnq~)=<$ȄKwfeo (֍`tzudet K(t;t1;:::;tnq~)jcXt=1rXY:bO>Then~DG(t1|s;:::;tnq~)=X t`i2n㉵Didet,K(t=1;t1|s;:::;tnq~;iji);8(13)h>where8K(t=1;t1|s;:::;tnq~;iji)istheithprincipalsubmatrixofK(t=1;t1|s;:::;tnq~). MTheorem3.3UUleadstoK;detZɷK(t1|s;t1;:::;tnq~)=ԟX 7IJ2nHX(1)j`IJj Vtnj`IJj2Y 7oi2I$ti3͟Y 7jg2I/IJ(Djo+8j6tj)det8A(t)(}fe.8I.8j}fe.8I): 8(14))>NoteUUthatHXdetXGK(t=1;t1|s;:::;tnq~)=ԟX 7IJ2nHX(1)j`IJj =Y 7 =i2Isti3͟Y 7jg2I/IJ(Djo+8j6tj)det8A() ~<(}fe.8I.8j}fe.8I)=0: 8(15)MLetUUtid=i^xi Ҁ;i=1;:::;n.qLemma3.1UUyieldsMH^OqܟX tPi2n_ali2`xiTL^ [hdetK(t=1;x1|s;:::;xnq~;l2`jl)`\=^ IY t -i2n w^j̵xi#v4^5u X j,}PIJnfl `gLޓ(1)j`IJj =pGerݮA() ~<(IjI)det8A()(}fe.8Ig[8fl2`gj}fe.8I[fl2`g): (16)>\begin{multline} >\biggl(\sum_{\,i\in\mathbf{n}}a_{l?_i}x_i\biggr)>\det\mathbf{K}(t=1,x_1,\dots,x_n;l?|l)\\>=\biggl(\prod_{\,i\in\mathbf{n}}\hat?x_i\biggr)>\sum_{I\subseteq\mathbf{n}-\{l?\}}>(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)>\det\mathbf{A}^{(\lambda)}>(\overline?I\cup\{l\}|\overlineI\cup\{l\}).>\label{sum-ali}>\end{multline}CMByx(3)@,UU(6)t,UUand(7),wehave>PropQositionT3.5. ׍,Hc=<$1Kwfe (֍2nn X l `=0 J(1)lȵDl;8(17)e>wher}'eFֵDl=jџX Ilҷn*DG(t1|s;:::;tnq~)2jGti*=J9#cmex7n#00;U&t}\cmti7ifzi2Il⍍01;Totherwise61;ni=1;::: ;nhJ:8(18))Jy>SampleUUpapGerfortheamsmathpackqagej5?>4VLApplication>W*eFconsiderheretheapplicationsofTheorems5.1and.5.2toacompletemul- >tipartitergraphKn1 :::  np./ItcanbGeshownthatthenumbGerofspanningtreesof>Kn1 :::  npEmayUUbGewrittenuWbT*=np2p 獍Y tni=1z(n8niTL)ni*18(19);>whereϵn=n1S+8g+8npR:8(20)MItUUfollowsfromTheorems5.1and5.2that/鍍koHczh=<$1Kwfe (֍2nn X l `=0 J(1)lȲ(n8l2`)p2(X nl1 + +lp2Ա=l?p 獍;Y t;vRi=1JI^<$Qqni 卑RliZƣ^#w618[(nl2`)(ni,liTL)]ni*liǷ^(nl)2Sbp 獍X tmjg=1(ni,liTL)2|s^ Ñ:8(21)1>...?\binom{n_i}{l_i}\\>and,e5c|Hcs$u=<$K1Kwfe (֍2 'n1  6<X  ;l `=0(1)lȲ(n8l2`)p2(X nl1 + +lp2Ա=l?p 獍;Y t;vRi=1JI^<$Qqni 卑RliZƣ^! |>8[(nl2`)(ni,liTL)]ni*li9^1<$slplwfe (֍np?2^D@[(nl)(np2lpR)]:8(22).uMThenenumerationofHcinaKn1   np ̲graphcanalsobGecarriedoutbyThe- >orem Ll7.26or7.3togetherwiththealgebraicmethoGdofmS(2)4r.djSomeelegant>representations`HmaybGeobtained.F*orexample,cHcs)inaKn1 n2n3j"graphmaybGe>written)&E=೅hHc=<$?n1|s!n2!n3!dwfe7 (֍n1S+8n2+n3"X tqi˿^^<$c%n1 卒貵i2^<^<$yn2 卒n3S8n1+i"v^)X^<$D`+n3 卒1/̵n3S8n2+ib ^L{+l^<$ ȇn1S81 卑ꃵi&~s^-^<$?Jn2S81 卑57[n3S8n1+if%^mq^<$an3S81 卑t n3S8n2+iן^K^:8(23)38>5VLSecretffKeyExchanges>MoGderncryptographyisfundamentallyconcernedwiththeproblemofsecure>privqatecommunication.$ASecretKeyExchangeisaprotoGcolwhereAliceand>Bob,havingtnosecretinformationincommontostart,areabletoagreeon6y>SampleUUpapGerfortheamsmathpackqagej6?>aMcommonsecretkey*,conversingMoverapublicchannel.[^ThenotionofaSe- >cret"KeyExchangeprotoGcolwas rstintroGducedintheseminalpaperofDie>andHellman[1].VQ[1]presentedaconcreteimplementationofaSecretKeyEx->changemprotoGcol,t!dependentmonaspGeci cassumption(avqariantonthediscrete>log),6spGeciallyǣtailoredtoyieldSecretKeyExchange.ȰSecretKeyExchangeis>ofIcoursetrivialiftrapGdoorIpermutationsexist.NRHowever,thereisnoknown>implementationUUbasedonaweakergeneralassumption.MTheconceptofaninformationallyone-wayfunctionwasintroGducedin[5].>W*eUUgiveonlyaninformalde nitionhere:>De nitionT5.1.A pGolynomialftimecomputablefunctionf=1ffk됸gisinfor->mationallyޏone-wayifthereisnoprobabilisticpGolynomialtimealgorithmwhich>(withqprobabilityoftheform1r4kP^e Mforqsomee>0)qreturnsoninputy"2f0;1g^k>aUUrandomelementoff^1 (y[ٲ).MIn2[thenon-uniformsetting[5]showthatthesearenotweakerthanone-way>functions:>TheoremT5.1([5](non-uniform)).UThe8existenc}'eofinformationallyone-way>functionsimpliestheexistenc}'eofone-wayfunctions.MW*e willsticktotheconventionintroGducedabove ofsaying\non-uniform">bGeforethetheoremstatementwhenthetheoremmakesuseofnon-uniformity*.>ItRshouldbGeunderstoodthatifnothingissaidthentheresultholdsforboth>theUUuniformandthenon-uniformmoGdels.MItUUnowfollowsfromTheorem5.1that>TheoremT5.2(non-uniform).j|We}'ak]SKE]impliestheexistenceofaone-way>function.MMorenrecently*,'thepGolynomial-time,interiorpGointalgorithmsforlinearpro->gramminghavebGeenextendedtothecaseofconvexquadraticprograms[11 ,13 ],>certain7jlinearcomplementarity7jproblems[7,10 7l],=fandthenonlinearcomplemen->tarityproblem[6].hTheconnectionbGetweenthesealgorithmsandtheclassical>NewtonUUmethoGdfornonlinearequationsiswellexplainedin[7].!č>6VLReview>W*eUUbGeginourdiscussionwiththefollowingde nition:>De nitionT6.1.AfunctionHn:&<^n !\<^n issaidtobGeB-di er}'entiableo\atthe>pGointz7if(i)HxisLipschitzcontinuousinaneighbGorhoodofzp,+and(ii)Q@thereex->ists)apGositivehomogeneousfunctionBqH(zp)p:8ݸ<^n8!<^nq~,2calledtheB-derivative>ofUUH%Satzp,suchthat5dlim-;v@L!0<$OtH(zw+8v[ٲ)H(zp)BqH(zp)vOtwfey$ (֍4k9v[ٸk)n=0:EĠy>SampleUUpapGerfortheamsmathpackqagej7?>TheNfunctionHLisB-di er}'entiable$insetSo۲ifNitisB-di erentiableateverypGoint >inUUS.qTheB-derivqativeBqH(zp)issaidtobGestr}'ong7Ҳif]Elimꪍs(v@L;v 0ncmsy50)!(0;0)<$HH(zw+8v[ٲ)H(zw+v[ٟ^0*)BqH(zp)(vv[ٟ^0*)Hwfec (֍BkGv8v[ٟr0*kO=0:>LemmaT6.1.}Ther}'eqexistsasmoothfunction 0|s(zp)de nedforjwzpj W>ʎ12a>satisfyingthefollowingpr}'operties!:Fr(i)W 0|s(zp)isb}'oundedaboveandbelowbypositiveconstantsc1C 0|s(zp)c2.`Cb(ii)WIfj[zpjW>1,then 0|s(zp)=1.@R(iii)WF;orallz~inthedomainof 0|s,0'ln' 0C0.B'(iv)WIf182a0.>Pr}'oof.]UIW*eiwchoGose 0|s(zp)tobearadialfunctiondependingonlyonr/²=襸jµzpj .,Let >h(rG)30$beasuitablesmoothfunctionsatisfyingh(r)3c3for$1d2a318a,UUandh(rG)=0UUforjrzpjŵ>18 lal&feVp,2.qTheradialLaplacianLqa\0'ln' 0|s(rG)=^<$ ɵd^2 Vwfez (֍drr2=˲+<$l1lwfe (֍r<$ 7d zwfe (֍dr3^D8lnN7 0|s(r)c>has;smoGothcoecientsforr>;1gy2a.CyTherefore,we;mayapplytheexistence>and uniquenesstheoryforordinarydi erentialequations.Simplyletln 0|s(rG)>bGeUUthesolutionofthedi erentialequationm>^<$2̵d^2Ÿwfez (֍drGr2ôβ+<$l1lwfe (֍r<$ 7d zwfe (֍dr3^;ln: 0|s(rG)=h(r)>withUUinitialconditionsgivenbyln UT 0|s(1)=0UUandln ^[ٷ0l0|s(1)=0.MNext,Ȑlet~QD xbGea nitecollectionofpairwisedisjointdisks,allofwhich>arepcontainedintheunitdiskcenteredattheorigininC.ZW*eassumethat>D = fz|zjjzw8zɸj(L<`g.jSuppGose6thatDɲ(a)denotesthesmallerconcentric>diskȵDɲ(a)=fzjjlzw8zj'Ɉ(1׸2a)`g.W*ede neasmoGothweightfunction>0|s(zp)forz72CϟS !&ŗDɲ(a)bysetting0(zp)=1whenzT!=72 sSʟ;Dand0(zp)=> 0|s((zRzɲ)=`)@CwhenzڲisanelementofD.jItfollowsfromLemma6.1that0>satis esUUthepropGerties:Gq(i)W0|s(zp)$DisbGoundedabove$DandbelowbypGositiveconstantsc1C0|s(zp)c2.`D(ii)W0'ln'0C04Uforallz72CS L8Dɲ(a),:thedomainwherethefunction0WisUUde ned.A(iii)W0'ln'0Cc3|s`^2rwhenUU(182a)'Let-dA '-denotetheannulusA (=/0f(12a)A=S oAɲ.CfThe0propGerties(2)and(3)of0Fmaybesummarizedas0'ln'0C>c3|s`^2 WA,UUwhereA isthecharacteristicfunctionofA.cffdffYffffSty>SampleUUpapGerfortheamsmathpackqagej8?MSuppGose!that + isanonnegativerealconstant.uW*eapplyPropGosition3.5 >with:(zp)H=0|s(z)e^ j`zI{jۑr2u.uIf:u2C^1l0 0(Rǟ^2ɱ9wS Ο3?Dɲ(a)),6rassumethatDWisa>bGoundeddomaincontainingthesupportofuandAPDXmRǟ^2ʸVS PXDɲ(a).`A>calculationUUgives@YcZ_ɟ yDgr gr kǟ feU$@pu  y"2 0|s(zp)e j`zI{jۑr2<c4 "cZB\ yDj"uj%2!Lز0e j`zI{jۑr2s+8c5`2 cZV9 yAqjKuj%)2+D0e j`zI{jۑr2u:yMTheUUbGoundedness,property(1)of0|s,thenyieldsyqcZw yD   feU$@UUisconstantonthebloGcksofX$.iZRPX a<=f2M3j=X$g;QX=f2M3jX$g:8(24)>IfUUX ythen=Y AforsomeYX7sothat\tL[QX a<=[ YX8,PYG: 5->ThusUUbyMobiusinversionqˍThusUUthereisabijectionfromQX ytoWc^BW=(XJ).qInparticularjrQX$j,=wD^b(XJ)M.MNextnotethatb(X)vB=dim˘X. FW*eseethisbychoGosingabasisforX>consistingUUofvectorsv[ٟ^kde nedbyqЍv[ٴki=\(. S1UifUUi2k>(;fc S0Uotherwise.qэ>\[v^{k}_{i}=>\begin{cases}?1&\text{if$i\in\Lambda_{k}$},\\>0?&\text{otherwise.}\end{cases}>\]>LemmaT6.2.}L}'etAbeanarrangement.ThenY(A;t)=mX B+A勲(1)j`B+j tdimT(B+) w: MIn?ordertocomputeRǟ^0b^0 Exrecallthede nitionofS(X:;Y8)fromLemma3.1.>Sinceh}H2BM۲,mGAH |B.?Thush}ifTc(B)=Yathenh}B42S(HA;Y8).?LetL^0N9^0w=L(A^0N9^0r). fy>SampleUUpapGerfortheamsmathpackqagej9?>ThenL596Rǟ0b0= X H2B+A!^˲(1)j`B+j tdimT(B+)j=̟X 7Y2L04s0&XDX jB+2Sa(H;Y)@{a(1)j`B+j tdimY=\X 7Y2L04s0%;ԟX jB+2Sa(H;Y)?^(1)j`B+AmHlujtdimY=\X 7Y2L04s0(HA;Y8)tdimY7=(A0N90r;t):8(25)Ovč>CorollaryT6.3. ZL}'et(A;A^09;A^0N9^0r)b}'eatripleofarrangements.Thenhb[ٲ(A;t)=(A09;t)8+t(A0N90r;t):>De nitionT6.2.Let(A;A^09;A^0N9^0r)bGeatriplewithrespecttothehyperplaneH2 >A.qCallUUH%Sasep}'aratorhifUUTc(A)62L(A^09).>CorollaryT6.4. ZL}'et(A;A^09;A^0N9^0r)b}'eatriplewithrespecttoH2A.Fr(i)WIfHcisasep}'aratorthenhbص(A)=(A0N90r)Wandhenc}'e6Kjh(A)jUW=j5(A0N90r)j$&:JCb(ii)WIfHcisnotasep}'aratorthenŔ(A)=(A09)8(A0N90r)WandŸj(A)j%Ѳ=j5(A09)j#&%+8j(A0N90r)j$W:>Pr}'oof.]UIItUUfollowsfromTheorem5.1that[ٲ(A;t)hasleadingtermA(1)r7(A)(A)tr7(A):>The7conclusionfollowsbycomparingcoGecientsoftheleadingtermsonbGoth >sidesoftheequationinCorollary6.3.6%IfHisaseparatorthenrG(A^09)handUUthereisnocontributionfrom[ٲ(A^09;t).ffdffYffff᭍MThe0VPoincarGepGolynomialofanarrangementwillappGearrepeatedlyinthese>notes.T-ItwillbGeshowntoequalthePoincarGepGolynomialofthegradedalgebras>whichwearegoingtoassoGciatewithA.'(ItisalsothePoincarGepGolynomial ucy>SampleUUpapGerfortheamsmathpackqagej10?|Cz+FigureUU1:qǵQ(A1|s)=xy[zp(x8z)(x+z)(yz)(y+z)CWA[FigureUU2:qǵQ(A2|s)=xy[zp(x8+y+z)(x+yz)(xy+z)(xyz)>ofEthecomplementM(A)foracomplexarrangement.Hereweprovethatthe >PoincarGepGolynomialisthechambercountingfunctionforarealarrangement.>TheUUcomplementM(A)isadisjointunionofchambGers2=M(A)=ߟ[ jC}2Cham?(A)1iµC(:!u>TheUUnumbGerofchambGersisdeterminedbythePoincarGepGolynomialasfollows.>TheoremT6.5.L}'etAR bearealarrangement.Then࠸jCham3(AR8-)jMɲ=[ٲ(AR8-;1):>Pr}'oof.]UIW*encheckthepropGertiesrequiredinCorollary6.4:0(i)followsfrom >[ٲ(lȵ;t)=1,UUand(ii)isaconsequenceofCorollary3.4.bffdffYffff Cy>SampleUUpapGerfortheamsmathpackqagej11?>TheoremT6.6.L}'etIbeaprotocolforarandompair(X:;Y8).bIfoneofN(x^09;y[ٲ) >andN(x;y[ٟ^0*)isapr}'e xoftheotherand(x;y[ٲ)2SX;Y,thenhj6(x09;y[ٲ)i1፴jg=1Dz=hj(x;y[ٲ)i1፴jg=1Dz=hj(x;y[ٟ0*)i1፴jg=1V:>Pr}'oof.]UIW*eUUshowbyinductiononithathj6(x09;y[ٲ)iijg=1Dz=hj(x;y[ٲ)iijg=1Dz=hj(x;y[ٟ0*)iijg=1V:>The@inductionhypGothesisholdsvqacuouslyfori`=0. 4Assume@itholdsfor>i1,inparticular[j6(x^09;y[ٲ)]䍴i1jg=1;=[j(x;y[ٟ^0*)]䍴i1jg=1V.AGThenoneof[j6(x^09;y[ٲ)]^1;Zjg=i>and.[j6(x;y[ٟ^0*)]^1;Zjg=i]Visapre xoftheotherwhichimpliesthatoneofiTL(x^09;y[ٲ)and>iTL(x;y[ٟ^0*)r"isapre xoftheother..IftheithmessageistransmittedbyPX then,>byĚtheseparate-transmissionspropGertyandtheinductionhypGothesis,፵iTL(x;y[ٲ)=>iTL(x;y[ٟ^0*),hencetoneofi(x;y[ٲ)andi(x^09;y[ٲ)isapre xoftheother. Bythe>implicit-termination$propGerty*,neitheriTL(x;y[ٲ)nori(x^09;y[ٲ)canbGeaproperpre-> x*oftheother,3hencetheymustbGethesameandiTL(x^09;y[ٲ)=i(x;y[ٲ)=i(x;y[ٟ^0*).>If;theithmessageistransmittedbyPYDzthen,symmetrically*,iTL(x;y[ٲ)=i(x^09;y[ٲ)>bytheinductionhypGothesisandtheseparate-transmissionsproperty*,and,then,>iTL(x;y[ٲ)=i(x;y[ٟ^0*)yLbytheimplicit-terminationpropGerty*,Iprovingtheinduction>step.=ffdffYffffMIfisaprotoGcolfor(X:;Y8),and(x;y[ٲ),(x^09;y)aredistinctinputsinSX;Yģ,>then,UUbythecorrect-decisionpropGerty*,hj6(x;y[ٲ)i^1;Zjg=1Ǹ6=hj(x^09;y[ٲ)i^1;Zjg=1V.MEquation/(25)de nedPYn'sambiguity/setS:XJjYfk(y[ٲ)(tobGethesetofpossibleX>vqaluesQwhenY=y[ٲ.QThelastcorollaryimpliesthatforally"2SY&,themultiset^1>ofUUcoGdewordsfN(x;y[ٲ):x2S:XJjYfk(y)&Y`gUUispre xfree.!č>7VLOne-WfayffComplexity,ʍx䍑@C^>C1I#(XjY8)eIA,Ctheone-waycomplexityofarandompair(X:;Y8) א,CisthenumbGerof>bitsSPX ymusttransmitintheworstcasewhenPYisnotpGermittedtotransmit>any*feedbackmessages.8StartingwithSX;Y#,fthesuppGortsetof(X:;Y8)m,wede ne>G(XjY8)a,UUthechar}'acteristichypergraphofUU(X:;Y8) ,UUandshowthatx䍒^TC1lw(XjY8)ڭ=dlog?(G(XjY8)#)eUU:MLetǏ(X:;Y8)&RbGeǏarandompair.vF*oreachy#hinSY՝,$thesupportsetofY8,>Equation(25)de nedS:XJjYfk(y[ٲ)*XtobGethesetofpossiblexvqalueswhenY|=Ry[ٲ.>Theۅchar}'acteristic[hypergraphG(XjY8)+Wof(X:;Y8)$zhasSXrasitsvertexsetand>theUUhypGeredgeS:XJjYfk(y[ٲ)*ffffv J= "5-:Aacmr61L|{Ycmr8ASampleUUpapGerfortheamsmathpackqagej12?>TheoremT7.1.L}'et R^n ebeanopenset,letu2BqV8( ;R^m),andletRbQTcux e =^ Gy"2Rm _:y=j~u(x)8+^<$ 1DGu wfeL (֍jDGuj(x);zp^C@forsomemBJz72Rnq~^8(26)ō>foreveryxH2 nSu:b.kL}'etٵf/:UvR^m !R^kibeaLipschitzcontinuousfunctionsuch >thatf(0)=0,andletv"=f(u)p:8 !R^k.Thenv"2BqV8( ;R^k됲)andKJ9v"=K(f(u+)8f(u)) usBHn1  wSu-:8(27) T>In9addition,Kfor  x䍑e!Du  ,-almosteveryx2 9ther}'estrictionofthefunctionfM[toT^cux>isdi er}'entiableat7ò~u Me(x)and'x䍒۫etD;v"=r(33fj 6Tux)(~u~)<$x䍑eD u33wfe 5  x䍑ceUUD u  8 8 8 x䍑e5Du  D:8(28)!dMBeforeCprovingthetheorem,zywestatewithoutproGofthreeelementaryremarks>whichUUwillbGeusefulinthesequel.kQ>R}'emark7.1.xkLet!xI: ] 0;+1[/!b])0;+1[+bGeacontinuousfunctionsuchthat>![ٲ(t)!0UUast!0.qThenlimh!0+/g[ٲ(!(h))=L, limh!0+3g[ٲ(h)=L K>forUUanyfunctiongxI: ?^] {0;+1[,͕!R.kQ>R}'emark7.2.xkLet5+gxI:RR^n !<%RbGeaLipschitzcontinuousfunctionandassume>that'cL(zp)= limh!0+<$fg[ٲ(hz)8g[ٲ(0)fwfe5 (֍h~>existsVforeveryz:h2Q^nuandthatLisalinearfunctionofzp.vThengвisdi eren->tiableUUat0.>R}'emark7.3.xkLetJAp:JR^nٸ![R^m "bGealinearfunction,andletf/:^R^m !RbGea>function.WThentherestrictionofftotherangeofAisdi erentiableat0ifand>onlyUUiff(A)p:8R^n8!Risdi erentiableat0andK#r(33fj 6Imh(A) ,!)(0)A=r(f(A))(0):ɼ>Pr}'oof.]UIW*eUUbGeginbyshowingthatv"2BqV8( ;R^k됲)andɸjQDGv[ٸj(Bq)Kaĸj(DGuj⸲(B)8BG2B( );8(29)>whereJK~4>0istheLipschitzconstantoff.nBym(13)~andbytheapproximation >resultD?quotedinx3,zitispGossibleto ndasequence(uh.)C^13( ;R^m)D?converging>toUUuinL^1|s( ;R^m)andsuchthatč,limh!+1+>cZɹx y љwj`ruh.j]dx=j5DGujH ( ): y>SampleUUpapGerfortheamsmathpackqagej13?>TheTfunctionsvh=f(uh.)areloGcallyLipschitzcontinuousin ,Tandthede ni- >tionofdi erentialimpliesthatjWrvh.j{)Kaĸj(ruh.jalmosteverywherein .#The>lowerUUsemicontinuityofthetotalvqariationand(13)qyield(I yGQjS DGv[ٸjf6( )liminfAh!+1 *j"ĵDGvh.j9( )]=liminfAh!+1 *cZ% y -j0_rvh.jH,dx򍍍]KaIJliminfGh!+1ScZ%S y -3j/ruh.jITrdx=Kaĸj(DGuj⸲( ):8(30)>SinceUUf(0)=0,wehavealso0M^cZۘ y ûjƂv[ٸj޵dxKağcZ y jujmdx;*>thereforepu\2BqV8( ;R^k됲).RepGeatingthesameargumentforeveryopGenset >A ,weget(29)TforeveryBG2B( ),bGecausejH7Dv[ٸjҥ,jrDujareRadonmeasures.>T*oUUproveLemma6.1, rstweobservethat&SvfSu:b; %~vg(x)=f(~u~(x))8x2 nSu:8(31)>InUUfact,forevery">0UUwehaveMlfy"2B$(x):j5v[ٲ(y)8f(~u~(x))jJ׵>"gfy2B$(x):j5u(y[ٲ)8ܼ~u^(x)j=P>"=Kg;>henceԓOlimb+!0+<$j3fy"2B$(x):j5v[ٲ(y)8f(~u~(x))jJ׵>"gjwfe> (֍Grn>=0m(>wheneverx2 nSu:b.OByasimilarargument,ifx2Su)|isapGointsuchthatthere >existsUUatriplet(u^+;u^;u:b)UUsatisfyingx(14)@,(15)u,then`(v[ٟ+i(x)8v[ٟڲ(x)) vf=(f(u+(x))f(u(x))) u:cifex2Sv>andUUf(u^(x))=f(u^+(x))UUifx2Su:bnSvN.qHence,UUby(1.8)weget)I yGD钵J9v[ٲ(Bq)=cZUR yBW=\Sv`(v+ 8I8vڲ) vjdHn1V=cZUR yBW=\Sv`(f(u+)8f(u)) u dHn1򍍍V=cZUR yBW=\Su>(f(u+)8f(u)) u dHn1 V5p>andUULemma6.1isproved.n܄ffdffYffff龍MT*oprove(31)~,itisnotrestrictivetoassumethatke=f1.YMoreover,to>simplify4ournotation,'fromnowonweshallassumethat =R^nq~.^The4proGofof>(31)Sisdividedintotwosteps.بInthe rststepweprovethestatementinthe>one-dimensionalcase(n=1),usingTheorem5.2.CInthesecondstepweachieve>theUUgeneralresultusingTheorem7.1.Ϡy>SampleUUpapGerfortheamsmathpackqagej14?>*N cmbx12Step1J>Assume8thatn=1.Since8SusEisatmostcountable,q(7)qyieldsthat  x䍑e8Dv[٫ [ [ Q(Su:bnSvN)=@>0,soRthat(19)and(21)implythatDGv=x䍑RyeDD ҦvƷ+j޵J9v+istheRadon-Nikod#pymde- >compGositionofDvFزinabsolutelycontinuousandsingularpartwithrespecttoq͍> > > x䍑CceAUUDIu  R.qByUUTheorem5.2,wehave"⍍<$x䍑kbeiDrdvf>Ewfe 5  x䍑ceUUD u  |d4(t)= Tlims!t+<$zDGv[ٲ([t;s[Lq) wfe0 5  x䍑ceUUD u  d([t;s[Lq)K";<$x䍑Ae32D"uݟwfe 5  x䍑ceUUD u  -̲(t)= Tlims!t+<$7DGu([t;s[Lq) wfe0 5  x䍑ceUUD u  d([t;s[Lq)'ft> > > x䍑CceAUUDIu  R-almost5everywhereinR.|hItiswellknown(see,mforinstance,[12 ,2.5.16])>thatBeveryone-dimensionalfunctionofbGoundedvqariationw3%hasauniqueleft>continuousrepresentative,\i.e.,afunction^w ޲suchthat^w X=walmosteverywhere>andUUlim8㐴s!t*ڲ^(w/T(s)=@^w 4(t)UUforeveryt2R.qTheseUUconditionsimplyr^quw|(t)=DGu(]1;t[c); %^vg(t)=Dv[ٲ(]1;t[c)8t2R8(32)>and^Wv(t)=f(^u~(t))8t2R:8(33)>Let8t<2RbGesuchthat  x䍑 eeWD!u  ([t;s[Lq)>0foreverys>tandassumethatthe>limitsUUin(22)qexist.qByx(23)Wand(24)wegetY?6eэ̍<$uA^u7Ƶvzl(s)8[^vm(t)tϟwfe0 5  x䍑ceUUD u  d([t;s[Lq) =<$Kf(^u~(s))8f(^u~(t))KwfeK 5 *ϫ  * * x䍑e$Du  #3([t;s[Lq):7 =30Kf(^u~(s))8f(^u~(t)+<$x䍑ϫehDOulwfe 5  x䍑ceUUD u  (t)   x䍑deD u  H ([t;s[Lq))KfeW 5=d =d =d x䍑C7 eA(DIMu  TpȲ([t;s[Lq):+30lf(^u~(t)8+<$x䍑ϫehDOuwfe 5  x䍑ceUUD u  (t)   x䍑deD u  H ([t;s[Lq))f(^u~(t))lfeD- 5=I =I =I x䍑BYe@DI-u  S([t;s[Lq)y>SampleUUpapGerfortheamsmathpackqagej15?>forUUeverys>t.qUsingUUtheLipschitzconditiononfhwe nd1獍ꍑ? ? ? ? ? ? ? ? ? ? ? <$E^EM'vJ(s)8[^vm(t)D30wfe0 5  x䍑ceUUD u  d([t;s[Lq)xP630lf(^u~(t)8+<$x䍑ϫehDOuwfe 5  x䍑ceUUD u  (t)   x䍑deD u  H ([t;s[Lq))f(^u~(t))lfeD- 5=I =I =I x䍑BYe@DI-u  S([t;s[Lq)ꍒs s s s s s s s s s s 7"huKaī a a a a a <$#^^u 9(s)8ܼ^u^(t)Lwfe0 5  x䍑ceUUD u  d([t;s[Lq);R<$x䍑ϫehDOulwfe 5  x䍑ceUUD u  (t)  l:#荑>ByI (29)*,/the%}functions!  x䍑*emDu  d|([t;s[Lq)%}iscontinuous%}andconverges%}to0ass#t.q͍>ThereforeUURemark7.1andthepreviousinequalityimply.獍<$x䍑dӫeblDk>v_wfe 5  x䍑ceUUD u  u=(t)= limh!0+30ff(^u~(t)8+h<$x䍑eD u33wfe 5  x䍑ceUUD u  Y"(t))f(^u~(t))ffe~ (֍By8(22)X,^@u (x)~="~u 8z(x)ëforeveryx~2RnSu:b;moreover,@applyingëthesameargu- >mentUUtothefunctionsu^09(t)=u(t),UUv[ٟ^0*(t)=f(u^0(t))=v[ٲ(t),UUweget/[ <$x䍑c+eaĵDjsXv^Mwfe 5  x䍑ceUUD u  tr(t)='?limh!030Hҵf(~u~(t)8+h<$x䍑eD u33wfe 5  x䍑ceUUD u  Y"(t))f(~u~(t))Hҟfe~ (֍andUUourstatementisproved.e>Step2uT>LetKusconsidernowthegeneralcasen;>1.CLetKި2R^n ɲbGesuchthatjbhjŲ=1,>andnlet=TfyM-2R^nbҲ:hy[;i=0g.Innthefollowing,uwenshallidentifyR^n*with>{R,Band=weshalldenotebyythevqariableranginginandbytthevqariable>rangingfinR.eBythejustprovenfone-dimensionalresult,jandbyTheorem3.3,>weUUget@4lim>Ԭh!030RVff(~u~(y+8t)+h<$x䍑eD uy33wfeC 5  x䍑ceUUD uy· · · '(t))f(~u~(y+t))RVffe؟ (֍Uh鉲=<$x䍑 TeDNvyKwfeC 5  x䍑ceUUD uy· · · (t)   x䍑feD!uy· · · 1-a.e.qinS_Rb>forUUHn1-almosteveryy"2ɲ.qW*eclaimthatc<$Ӹhx䍑geDu;i}b~wfe&- 5  UUhx䍑geDu;i  T޲(y+8t)=<$x䍑 ^eOD4uyKwfeC 5  x䍑ceUUD uy· · · (t)   x䍑feD!uy· · · 1-a.e.qinS_R8(34)ʣy>SampleUUpapGerfortheamsmathpackqagej16?>forUUHn1-almosteveryy"2ɲ.qInfact,byx(16)Wand(18)qweget!܍IcZO8 y<$x䍑afe_Dguy[wfeC 5  x䍑ceUUD uy· · · x8 8 8 x䍑e5Duy· · · OsdHn1(y[ٲ)=cZUR yx䍑NeD%{uyy/dHn1(y[ٲ)%jJFT,=hx䍑geDu;i=<$Ohx䍑geDu;iKwfe&- 5  UUhx䍑geDu;i  .%8 8 8 5hx䍑geDu;i  +%=cZUR y<$ohx䍑geDu;iwfe&- 5  UUhx䍑geDu;i  andUU(24)qfollowsUUfrom(13)u.qBythesameargumentitispGossibletoprovethat܍<$4hx䍑geDv[;i}b~wfe&- 5  UUhx䍑geDu;i  T޲(y+8t)=<$x䍑 TeDNvyKwfeC 5  x䍑ceUUD uy· · · (t)   x䍑feD!uy· · · 1-a.e.qinS_R8(35)>forUUHn1-almosteveryy"2ɲ.qByx(24)Wand(25)qweget2ЬJ;limHیh!030\]Ff(~u~(y+8t)+h<$hx䍑geDu;i33wfe&- 5  UUhx䍑geDu;i  )%(y+t))f(~u~(y+t))\]Ffeզ& (֍gh5=<$hx䍑geDv[;iKwfe&- 5  UUhx䍑geDu;i  +쫲(y+8t)>forUUHn1-almosteveryy"2ɲ,andusingagain(14)u,(15)qwegett%'limrh!030Ff(~u~(x)8+h<$hx䍑geDu;i33wfe&- 5  UUhx䍑geDu;i  )%(x))f(~u~(x))Ffe (֍Hh!=<$hx䍑geDv[;iKwfe&- 5  UUhx䍑geDu;i  +쫲(x)'ft> > > AUUhx䍑geDu;i  d--a.e.qinUUR^nq~. MSinceBthefunction  hx䍑geDu;i  ,=   x䍑deD u  "isstrictlypGositive  hx䍑geDu;i  +-almostev-q͍>erywhere,UUweobtainalso:K limIh!030],bf(~u~(x)8+h33 33 33 hx䍑geDu;i  33fe&- 59 9 9 x䍑Ie ;D"u  )%(x)<$hx䍑geDu;i33wfe&- 5  UUhx䍑geDu;i  (x))8f(~u~(x))],bfekş (֍cgh!m*0=K K K Ohx䍑geDu;i  Kfe&- 59 9 9 x䍑Ie ;D"u  +쫲(x)<$hx䍑geDv[;i33wfe&- 5  UUhx䍑geDu;i  )%(x)> > > AUUhx䍑geDu;i  d--almostUUeverywhereinR^nq~.߱y>SampleUUpapGerfortheamsmathpackqagej17?MFinally*,UUsince! \׾ \׾ \׾ `-hx䍑geDu;i  \׾fe&- 59 9 9 x䍑Ie ;D"u  <$Rhx䍑geDu;iQwfe&- 5  UUhx䍑geDu;i  ɲ=<$Khx䍑geDu;iKwfe  5   x䍑 e9Dtu  ( =\*<$x䍑lBe]۵Dou wfe 5  x䍑ceUUD u  ".u;\+I I I x䍑OPieMBDUЖu  `-a.e.qinRn*܍\׾ \׾ \׾ `-hx䍑geDu;i  \׾fe&- 59 9 9 x䍑Ie ;D"u  <$hx䍑geDv[;iQwfe&- 5  UUhx䍑geDu;i  ɲ=<$Khx䍑geDv[;iKwfeŸ 5N N N x䍑 ?eصD2lu  'X=\*<$x䍑eandsincebGothsidesof(33)uarezero  x䍑 WneIDכu  n-almosteverywhereon  Ihx䍑geDu;i  +߲-q͍>negligibleUUsets,weconcludethat5vclimah!030u1ffZ70 7@ "~ ~8u7(x)8+h\*<$x䍑OeAkDu wfe 5  x䍑ceUUD u  !(x);\+Z@1 @A|Ns8f(~u~(x))u1ffe+ (֍S4Gh%W5=\*<$x䍑e > > x䍑CceAUUDIu  R-a.e.fin4 R^nq~.Sinceisarbitrary*,:byRemarks7.2and7.3therestrictionof >ftosytheanespaceT^cux jisdi erentiableatU~u ,(x)for  x䍑5eεDWbu  f5-almosteveryxS2R^n>andUU(26)qholds.RffdffYffff MItUUfollowsfrom(13)u,(14),and(15)qthat؍RDG(t1|s;:::;tnq~)=ԟX 7IJ2nHX(1)j`IJj֕1=jεIj!˟Y 7!"i2I0ti3͟Y 7jg2I/IJ(Djo+8j6tj)det8A() ~<(}fe.8I.8j}fe.8I):8(36)C>LetUUtid=i^xi Ҁ,i=1;:::;n.qLemmaUU1leadstoERɑDG(^x1 3;:::;MS^xn B)=Y ti2nSb^xinX 7fǴIJ2n*(1)j`IJj֕1=jεIj!˲per1Byx(3)@,UU(13)u,UUand(37),wehavethefollowingresult:ݍ>TheoremT7.2.WeρHc=<$1Kwfe (֍2nn X l `=1!2l2`(1)l `1 Nwher}'eԍ}fA:()6lET=jџX Ilҷn*pGer&kA() ~<(IlȸjIl)det8A(()=(}fe.8I.8l@j}fe.8I.8l);jqŵIlȸjw=l2`:8(39)"MIt5isworthnotingthatA:()6lof(39)issimilartothecoGecientsblGvofthe>characteristicdpGolynomialof+(10).uItiswellknowningraphtheorythatthe>coGecients blѲcanbeexpressedasasumover certainsubgraphs.toUUseewhetherAlȲ,=0,UUstructuralpropGertiesofagraph.y>SampleUUpapGerfortheamsmathpackqagej18?MW*e maycall(38)4aparametricrepresentationofHc.qIncomputation,*the >parameter8i playsveryimpGortantroles.qThechoiceoftheparameterusually>depGends2|onthepropertiesofthegivengraph. =F*oracompletegraphKnq~,ilet>id=1,UUi=1;:::;n.qItfollowsfrom(39)qthatw}4A:(1)6l =\(. Sn!; d(if)Եlx=1fc S0; d(otherwiseI:8(40)x>Byx(38)ǩLHc=<$K1Kwfe (֍2 -(n81)!:8(41)Fw>F*orUUacompletebipartitegraphKn1 n21<,letid=0,i=1;:::;n.qByx(39)@,Al=\(. Sn1|s!n2!n1 n21<;PifYlx=2fc S0;Potherwise}:8(42)\?>Theorem7.2UUleadsto Hc=<$1Kwfe!2 (֍n1S+8n2&`vn1|s!n2!n1 n21<:8(43)8=MNow,UUweconsideranasymmetricalapproach.qTheorem3.3leadstonIdetY8K(t=1;t1|s;:::;tnq~;l2`jl)ln= ՟X jIJnfl `g#([(1)j`IJj =Y 7 =i2Isti3͟Y 7jg2I/IJ(Djo+8j6tj)det8A() ~<(}fe.8Ig[fl2`gj}fe.8I[fl2`g): (44)MByx(3)VandUU(16)qweUUhavethefollowingasymmetricalresult:>TheoremT7.3.g3Hc=<$K1Kwfe (֍2X j 'IJnfl `g,9j(1)j`IJj =pGerݮA() ~<(IjI)det8A()(}fe.8Ig[8fl2`gj}fe.8I[fl2`g)8(45)>whichr}'educestoGoulden{JacksonP'sformulawhenid=0;i=1;:::;n[9p]. ]>8VLVfariousfffontfeaturesofthe+߆Tff cmtt12amsmathpackage>8.1\Boldversionsofsp`ecialsymb`olsuT>Invtheamsmathpackqage\boldsymbolisusedforgettingindividualbGoldmath >symbGols andboldGreekletters|everythinginmathexceptforlettersofthe>LatinUUalphabGet,whereyou'duse\mathbf.qF*orexample,>A_\infty?+\piA_0\sim>\mathbf{A}_{\boldsymbol{\infty}}?\boldsymbol{+}>\boldsymbol{\pi}?\mathbf{A}_{\boldsymbol{0}}>loGoksUUlikethis:nnA1 Ų+8[A0CA0ђcmbsy71 Ȇ+*,DF cmmib10$A0 *y>SampleUUpapGerfortheamsmathpackqagej19?>8.2\\Po`orman'sbold"uT>IfabGoldversionofaparticularsymbGoldoesn'texistintheavqailablefonts, then >\boldsymbolϵcan'tbGeusedtomakethatsymbGolbold.Emeansathat\boldsymbolcan'tbGeusedwithsymbolsfromthemsamandmsbm>fonts,amongothers.AOInsomecases,pGoorman'sbGold(\pmb)canbeusedinstead>ofUU\boldsymbol:<$Gŵ@8xGşwfe q (֍:@8y蝣 蝣 蝣 蝣 8܍օ օ օ օ g g g g <$@8ywfe M (֍@8zqύ>\[\frac{\partial?x}{\partialy}>\pmb{\bigg\vert}>\frac{\partial?y}{\partialz}\]>So-called\largeopGerator"symbolssuchasPcandQGrequireanadditional>command,\mathop,to0proGduceproperspacingandlimitswhen\pmbisused.>F*orUUfurtherdetailsseeTheTKß'E-Xb}'ook.<àXoAXi\[\sum_{\substack{i\prod_\kappa?\kappaF(r_i)\qquad>\mathop{\pmb{\sum}}_{\substack{i\mathop{\pmb{\prod}}_\kappa?\kappa(r_i)>\]!č>9VLComps3oundffsymbolsandotherfeatures>9.1\Multipleintegralsigns>\iint,\iiint,andѢ\iiiintgivemultipleintegralsignswiththespacingbGe->tween'-themnicelyadjusted,[inbGothtextanddisplaystyle.N\idotsintgives>twoUUintegralsignswithdotsbGetweenthem.@\FcZcZ/۴Af(x;y[ٲ)dxdycZ4cZ#[cZ/qA/f(x;y[;zp)dxdydz8(46)!&nPcZtpcZ{!cZPҟcZwA{f(wD;x;y[;zp)dwdxdydzAcZ!/9cZ"zA:f(x1|s;:::;xk됲)8(47)ly>SampleUUpapGerfortheamsmathpackqagej20?>9.2\OverandunderarrowsuT>SomeextraoverandunderarrowopGerationsareprovidedintheamsmathpack- >age.q(BasicUUL5ffA͉TU>'ExXprovides\overrightarrowand\overleftarrow).3୸Z>Ae!୵ cx(t)EtVhq= cx(t)EtVh3Af>Ae!f 0>Ae୵ cx(t)EtVhq= cx(t)EtVh3 0>Ae ]ލ>!୵ cx(t)EtVhq= cx(t)EtVh3 ]ލ>!>\begin{align*}>\overrightarrow{\psi_\delta(t)?E_th}&>=\underrightarrow{\psi_\delta(t)?E_th}\\>\overleftarrow{\psi_\delta(t)?E_th}&>=\underleftarrow{\psi_\delta(t)?E_th}\\>\overleftrightarrow{\psi_\delta(t)?E_th}&>=\underleftrightarrow{\psi_\delta(t)?E_th}>\end{align*}>TheseUUallscalepropGerlyinsubscriptsizes:@ԥcZS3ޠq!ȉ3ABaxdx@>\[\int_{\overrightarrow{AB}}?ax\,dx\]6>9.3\DotsuT>NormallyyouneedonlytypGe\dotsforellipsisdotsinamathformula.gThe>main2exceptioniswhenthedotsfallattheendoftheformula;ԡthenyouneed>toYxspGecifyoneof\dotsc(seriesdots,Zafteracomma),\dotsb(binarydots,for>binaryJrelationsoropGerators),;\dotsm(multiplicationdots),or\dotsi(dots>afterUUanintegral).qF*orexample,theinput>Then?wehavetheseries$A_1,A_2,\dotsc$,>the?regionalsum$A_1+A_2+\dotsb$,>the?orthogonalproduct$A_1A_2\dotsm$,>and?theinfiniteintegral>\[\int_{A_1}\int_{A_2}\dotsi\].>proGducesfThen(*wehavetheseriesA1|s;A2;:::,13the(*regionalsumA1Z+ފA2+WdUQ,UUtheorthogonalproGductA1|sA2'|j,andthein niteintegral@ocZj yA1cZ; yA2x!1y>SampleUUpapGerfortheamsmathpackqagej21?>9.4\AccentsinmathuT>DoubleUUaccents:؍U^*x䍒^ H~8m8ȍx䍒m;)~Cdf~服x䍒i~KTUg*x䍒lA۵AU*x䍒ݵGd_服x䍒_Dd服x䍒-?DUD*x䍒GԵB(2~fx䍒(2%B"8~]ލx䍒8~7HV>\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad >\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad>\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad>\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad>\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]>ThisxdoubleaccentopGerationiscomplicatedandtendstoslowdownthepro->cessingUUofaL5ffA͉TU>'ExX le.6>9.5\Dotaccents>\dddotYܲand\ddddotareavqailabletoproGducetripleandquadrupledotaccents>inUUadditiontothe\dotand\ddotaccentsalreadyavqailableinL5ffA͉TU>'ExX:tt...㌍ݞǵQWt&W....ꩍꪵR>\[\dddot{Q}\qquad\ddddot{R}\]6>9.6\Ro`ots>Inatheamsmathpackqage\leftrootand\uprootallowyoutoadjustthepGosition >ofUUtheroGotindexofaradical:>\sqrt[\leftroot{-2}\uproot{2}\beta]{k}>givesUUgoGodpositioningofthe :|ٟ[. JVpZVfeV k6>9.7\Boxedformulas>Theotcommand\boxedputsabGoxarounditsargument,ulike\fboxexceptthat >theUUcontentsareinmathmoGde:>\boxed{W_t-F\subseteq?V(P_i)\subseteqW_t} JffiyffWt68F*V8(PiTL)Wth8ffffffiy.:-y>SampleUUpapGerfortheamsmathpackqagej22?>9.8\ExtensiblearrowsuT>\xleftarrow̲and\xrightarrowproGducearrowsthatextendautomaticallyto >accommoGdateunusuallywidesubscriptsorsuperscripts.hThetextofthesub->scriptorsupGerscriptaregivenasanoptionalresp.mandatoryargument:Ex->ample:Yqa,0T΍T 2 Fo84[n1]x[1+@0 (b)Q$>Q"!#۵E@0 buX>\[0?\xleftarrow[\zeta]{\alpha}F\times\triangle[n-1]H\xrightarrow{\partial_0\alpha(b)}?E^{\partial_0b}\]>9.9\2߆T cmtt12\overset,\underset,and\sideset>Examples:捍*aƷSUXXs*faSXϦb獑>\[\overset{*}{X}\qquad\underset{*}{X}\qquad>\overset{a}{\underset{b}{X}}\]MThe(command\sidesetisforaratherspGecialpurpose:[mputtingsymbolsat>thesubscriptandsupGerscriptcornersofalargeoperatorsymbolsuchasPor>QGqɲ,UUwithouta ectingtheplacementoflimits.qExamples: ލ-%-VYލ/%/ Ńشk8X޷0 =,0imQ0EiTL x2>\[\sideset{_*^*}{_*^*}\prod_k\qquad>\sideset{}{'}\sum_{0\le?i\lem}E_i\betax>\]>9.10cThe\textcommand>TheUUmainuseofthecommand\textisforwordsorphrasesinadisplay:v*y=y(0 ifUUandonlyifOZy[ٷ0፴k=k됵y:(k+B)>\[\mathbf{y}=\mathbf{y}'\quad\text{if?andonlyif}\quad >y'_k=\delta_k?y_{\tau(k)}\]>9.11cOp`eratornames>The0morecommonmathfunctionssuchaslog,˞sin,˞andlim5haveprede nedcon->trol8sequences:\log,q\sin,\lim.Theamsmathpackqageprovides\DeclareMathOperator8 *>and;F\DeclareMathOperator*forproGducingnewfunctionnamesthatwillhave>theUUsametypGographicaltreatment.qExamples: k fk1D=esssupPqƴx2R ,n69j9Vf(x)j5y>SampleUUpapGerfortheamsmathpackqagej23?>\[\norm{f}_\infty= >\esssup_{x\in?R^n}\abs{f(x)}\]dUmeaskU1ofu2RDZ1+: mfs(u)> zg=measn|fx2Rǟn:hʸj/f(x)j*5< g 8 В>0:4ƍ>\[\meas_1\{u\in?R_+^1\colonf^*(u)>\alpha\}>=\meas_n\{x\in?R^n\colon\abs{f(x)}\geq\alpha\}>\quad?\forall\alpha>0.\]>\esssupUUand\measwouldbGede nedinthedocumentpreambleas>\DeclareMathOperator*{\esssup}{ess\,sup} >\DeclareMathOperator{\meas}{meas}MThefollowingspGecialoperatornamesareprede nedintheamsmathpackqage:>\varlimsup,\varliminf,\varinjlim,anda\varprojlim.Here'swhatthey>loGokUUlikeinuse:4ƍ fe 㑟$limaNn!1ĖQ(unq~;un^8u#ɲ)08(48)limljfe 㑎aNn!1o>j6[an+1jѵ=jqŵanq~j=08(49)lim3!(mi>:)_08(50)lim3 raN:p2Sa(A)ȵApfj08(51)>\begin{align}>&\varlimsup_{n\rightarrow\infty}b\mathcal{Q}(u_n,u_n-u^{\#})\le0\\>&\varliminf_{n\rightarrow\infty}H\left\lvert?a_{n+1}\right\rvert/\left\lverta_n\right\rvert=0\\>&\varinjlim?(m_i^\lambda\cdot)^*\le0\\>&\varprojlim_{p\in?S(A)}A_p\le0>\end{align} >9.12c\modanditsrelativesuT>TheLcommands\modand\podarevqariantsof\pmodpreferredbysomeauthors;>\modomitstheparentheses,iwhereas\podomitsthe`moGd'andretainsthe>parentheses.qExamples:Dxtxy+81 (moGdm2|s)8(52)=uDxtxy+81 moGd m28(53)Dxtxy+81 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\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds -t_\varepsilon\int?L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds?-\mathbf{E}_{x,y}\int_0^{t_\varepsilon}L_{x,y_\varepsilon(\varepsilon?s)}\varphi(x)\,ds\biggr)\biggr]\\&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),\end{split}\end{equation}Y&vy>SampleUUpapGerfortheamsmathpackqagej30?>&UnnumbGeredUUversion:LD8訍䍍Hfh;" U(x;y[ٲ)1~="Ex;y bcZibt"@M0 qL:x;y"ߏ("u)F'(x)du򍍍1~=hcZ UQLx;z u'(x)x(dzp)J:+8h^<$$1zQwfe㦟 (֍t"*^Eyy/cZiy0t"@ i0L:x;y@Lx(s)2'(x)dst"ܟcZLx;z u'(x)x(dzp)^؍Oڲ+<$1lwfe㦟 (֍t" ^`Eyy/cZiy0t"@ i0L:x;y@Lx(s)2'(x)ds8Ex;y bcZibt"@M0 qL:x;y"ߏ("s)$'(x)ds^\t^@1~=hx䍑bLx '(x)8+h"G4(x;y[ٲ);E討[t SomeUUtextaftertotestthebGelow-displayUUspacing.\begin{equation*} \begin{split}f_{h,\varepsilon}(x,y)&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}L_{x,y_\varepsilon(\varepsilon?u)}\varphi(x)\,du\\&=?h\intL_{x,z}\varphi(x)\rho_x(dz)\\&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y} \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds -t_\varepsilon\int?L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds?-\mathbf{E}_{x,y}\int_0^{t_\varepsilon}L_{x,y_\varepsilon(\varepsilon?s)}\varphi(x)\,ds\biggr)\biggr]\\&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),\end{split}\end{equation*}Y&cy>SampleUUpapGerfortheamsmathpackqagej31?>&If"theoptioncentertagsisincludedintheoptionslistoftheamsmathpack- age,3TtheequationnumbGersforsplitenvironmentswillbecenteredverticallyonUUtheheightofthesplit:, 'IPj*mI2|sj8k=     mcZinT@ 0̵ [ٲ(t)\( u(a;t)8cZi 8a@ n9(t)<$!d ۟wfeS (֍kP(G;t)6=EcZi@=F@;aFn%c(uDz)utV(;t)d\)Gdt  8kC6' ' ' '  |p  |p |p |p ŵf7cZLq y ,p ,p ,p x䍑eŵS1ɍ1;0K[a;+PW2|s( ;lȲ)  YW YW YW YW p8 p8 p8 scjv*ujT΍e2r]!9vW;5ZecA፱2( ;rm;Tc)  ̴o ̴o ̴o ̴o Ѵl:G8(65),qٍSomeUUtextaftertotestthebGelow-displayUUspacing.Y& hy>SampleUUpapGerfortheamsmathpackqagej32?>&UseUUofsplitwithinalign:G.+]'׊j*I1|sj9=    mcZ y g[Rud   #9C3'^ n9cZs y r^ 8cZi*8x@% a0sg[ٲ(u;t)d^]ɟ۱2cd ^uX1=2Bn8^cZ 8 y 7^n8u2፴xAIJ+<$1lwfeV (֍k D^+cZi+x@a#Ecut/duǟ^DJ۱2H^Zc ^.۟1=29C4' ' '  |p  |p ŵf7 7 x䍑|peS1ɍ1;0K[a;ǵW2|s( ;lȲ)  K K b b b eTjhqujT΍v2t$!=W;5ZecA፱2( ;rm;Tc)  F6 F6 F6 F3:G8(66)W?zn@'׊j*I2|sj9=    mcZinT@ 0̵ [ٲ(t)^ *u(a;t)8cZi 8a@ n9(t)<$!d ۟wfeS (֍kP(G;t)6=EcZi@=F@;aFn%c(uDz)utV(;t)d^+ dt  \9C6' ' ' '  |p  |p |p |p ŵf7cZLq y ,p ,p x䍑eŵS1ɍ1;0K[a;+PW2|s( ;lȲ)  YW YW YW YW p8 p8 p8 scjv*ujT΍e2r]!9vW;5ZecA፱2( ;rm;Tc)  ̴o ̴o ̴o ̴o Ѵl:G8(67)*SomeUUtextaftertotestthebGelow-displayUUspacing.\begin{align} \begin{split}\abs{I_1} &=\left\lvert?\int_\OmegagRu\,d\Omega\right\rvert\\&\le?C_3\left[\int_\Omega\left(\int_{a}^x g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\&\quad\times?\left[\int_\Omega\left\{u^2_x+\frac{1}{k} \left(\int_{a}^x?cu_t\,d\xi\right)^2\right\} c\Omega\right]^{1/2}\\&\le?C_4\left\lvert\left\lvertf\left\lvert\wt{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right\rvert\right\rvert \left\lvert?\abs{u}\overset{\circ}\toW_2^{\wt{A}} (\Omega;\Gamma_r,T)\right\rvert\right\rvert.\end{split}\label{eq:A}\\\begin{split}\abs{I_2}&=\left\lvert?\int_{0}^T\psi(t)\left\{u(a,t) -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} \int_{a}^\theta?c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\&\le?C_6\left\lvert\left\lvertf\int_\Omega \left\lvert?\wt{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right\rvert\right\rvert \left\lvert?\abs{u}\overset{\circ}\toW_2^{\wt{A}} (\Omega;\Gamma_r,T)\right\rvert\right\rvert.\end{split}\end{align}Y&!Sy>SampleUUpapGerfortheamsmathpackqagej33?>&UnnumbGeredUUalign,withanumberUUonthesecondsplit:K#29qjI1|sj.5=    mcZ y g[Rud   !/.5C3'\" rcZ y j^ cZi* x@&UYa1zg[ٲ(u;t)d^^X۱2dd \#vʟٱ1=2$15n8\"7cZ q y |p\(u2፴xAIJ+<$1lwfeV (֍k D^+cZi+x@a#Ecut/duǟ^DJ۱2H\)c \#gٱ1=2.5C4' ' '  |p  |p |p ŵf7 7 7 x䍑|peS1ɍ1;0K[a;ǵW2|s( ;lȲ)  K K K b b b eTjhqujT΍v2t$!=W;5ZecA፱2( ;rm;Tc)  F6 F6 F6 F3:\Q jI2|sj.5=     mcZinT@ 0̵ [ٲ(t)\( u(a;t)8cZi 8a@ n9(t)<$!d ۟wfeS (֍kP(G;t)6=EcZi@=F@;aFn%c(uDz)utV(;t)d\)Gdt  .5C6' ' ' '  |p  |p |p |p ŵf7cZLq y ,p ,p ,p x䍑eŵS1ɍ1;0K[a;+PW2|s( ;lȲ)  YW YW YW YW p8 p8 p8 scjv*ujT΍e2r]!9vW;5ZecA፱2( ;rm;Tc)  ̴o ̴o ̴o ̴o Ѵl:Dj(67^09),qٍSomeUUtextaftertotestthebGelow-displayUUspacing.\begin{align*} \begin{split}\abs{I_1}&=\left\lvert?\int_\OmegagRu\,d\Omega\right\rvert\\8 * &\le?C_3\left[\int_\Omega\left(\int_{a}^x g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\&\phantom{=}\times?\left[\int_\Omega\left\{u^2_x+\frac{1}{k} \left(\int_{a}^x?cu_t\,d\xi\right)^2\right\} c\Omega\right]^{1/2}\\&\le?C_4\left\lvert\left\lvertf\left\lvert\wt{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right\rvert\right\rvert \left\lvert?\abs{u}\overset{\circ}\toW_2^{\wt{A}} (\Omega;\Gamma_r,T)\right\rvert\right\rvert.\end{split}\\\begin{split}\abs{I_2}&=\left\lvert?\int_{0}^T\psi(t)\left\{u(a,t) -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} \int_{a}^\theta?c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\&\le?C_6\left\lvert\left\lvertf\int_\Omega \left\lvert?\wt{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right\rvert\right\rvert \left\lvert?\abs{u}\overset{\circ}\toW_2^{\wt{A}} (\Omega;\Gamma_r,T)\right\rvert\right\rvert.\end{split}\tag{\theequation$'$}\end{align*}Y&"1y>SampleUUpapGerfortheamsmathpackqagej34?>&A.2"1MultlineuTNumbGeredUUversion: cZib@8a^!.cZi+.b@&(a/33[f(x)2|sg[ٲ(y)2S+8f(y)2|sg(x)2]82f(x)g[ٲ(x)f(y)g(y)dx^ *dyE=cZi b@URa]^K^g[ٲ(y)2'cZi'b@ Uaf2+8f(y)2'cZi'b@ Uag2,82f(y)g(y)cZi b@8aYfg^ dy [ٲ(68)qύT*otesttheuseof\labeland\ref,DwerefertothenumbGerofthisequation here:q(68).\begin{multline}\label{eq:E}\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]?-2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy\\?=\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 \int_a^b?g^2-2f(y)g(y)\int_a^bfg\biggr\}\,dy\end{multline}UnnumbGeredUUversion: cZib@8a^!.cZi+.b@&(a/33[f(x)2|sg[ٲ(y)2S+8f(y)2|sg(x)2]82f(x)g[ٲ(x)f(y)g(y)dx^ *dyWK=cZi b@URa]^K^g[ٲ(y)2'cZi'b@ Uaf2+8f(y)2'cZi'b@ Uag2,82f(y)g(y)cZi b@8aYfg^ dyqύSomeUUtextaftertotestthebGelow-displayUUspacing.\begin{multline*}\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]?-2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy\\?=\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 \int_a^b?g^2-2f(y)g(y)\int_a^bfg\biggr\}\,dy\end{multline*}Y&#y>SampleUUpapGerfortheamsmathpackqagej35?>&A.3"1GatheruTNumbGeredUUversionwith\notagonthesecondline:fٵDG(a;r)fz72Cp:ㅸj zw8aj'SampleUUpapGerfortheamsmathpackqagej36?>&A.4"1AlignuTNumbGeredUUversion:Ev x(t)]p=(cos7tu8+sin*tx;v[ٲ);G8(72)Eݵ y·(t)]p=(u;cosߵtv+8sin*ty[ٲ);G8(73)Eӵ z(t)]p=^ #cos1õtu8+<$l lwfeo (֍  sinѵtv[;<$S4 33wfeo (֍ +sinqtu+cosGtv^#յ:G8(74)!bSomeUUtextaftertotestthebGelow-displayUUspacing.\begin{align} \gamma_x(t)&=(\cos?tu+\sintx,v),\\\gamma_y(t)&=(u,\cos?tv+\sinty),\\\gamma_z(t)&=\left(\cos?tu+\frac\alpha\beta\sintv, -\frac\beta\alpha\sin?tu+\costv\right).\end{align}UnnumbGeredUUversion:Ev x(t)]p=(cos7tu8+sin*tx;v[ٲ);E y·(t)]p=(u;cosߵtv+8sin*ty[ٲ);E z(t)]p=^ #cos1õtu8+<$l lwfeo (֍  sinѵtv[;<$S4 33wfeo (֍ +sinqtu+cosGtv^#յ:!bSomeUUtextaftertotestthebGelow-displayUUspacing.\begin{align*} \gamma_x(t)&=(\cos?tu+\sintx,v),\\\gamma_y(t)&=(u,\cos?tv+\sinty),\\\gamma_z(t)&=\left(\cos?tu+\frac\alpha\beta\sintv, -\frac\beta\alpha\sin?tu+\costv\right).\end{align*}AUUvqariation:_ʵxh}=yҲbyUU(84)G8(75)]1x0h}=y[ٟ0ҲbyUU(85)G8(76)KAx8+x0h}=y+8y[ٟ0ҲbyUUAxiom1.G8(77)SomeUUtextaftertotestthebGelow-displayUUspacing.\begin{align} x&?=y&&\text{by(\ref{eq:C})}\\x'&?=y'&&\text{by(\ref{eq:D})}\\x+x'?&=y+y'&&\text{byAxiom1.}\end{align}Y&%y>SampleUUpapGerfortheamsmathpackqagej37?>&A.5"1AlignandsplitwithingatheruTWhenpusingthealignenvironmentpwithinthegatherenvironment,#oneporthe other,r orlKbGoth,shouldbGeunnumbGered(usingthe*form);wnumbGeringboththeouterUUandinnerenvironmentUUwouldcauseacon ict.AutomicallyUUnumbGeredgatherwithsplitandalign*:ccT'(x;zp)uOb=zw8 10xx mn xmzpnuOb=zw8MrG1 xMrG(m+n)tɵxmzpnG8(78).~ǩW0gƲ=(uǟ0:)2|s;=uW1gƲ=uǟ0:uǟ1;W2gƲ=(uǟ1:)2|s;%}ߍHere1thesplitenvironment1getsanumbGer1fromtheoutergatherenvironment;numbGersUUforindividuallinesofthealign*aresuppressedbecauseofthestar.\begin{gather}\begin{split}?\varphi(x,z)&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n\end{split}\\[6pt]\begin{align*}\zeta^0?&=(\xi^0)^2,\\\zeta^1?&=\xi^0\xi^1,\\\zeta^2?&=(\xi^1)^2,\end{align*}\end{gather}TheUU*-edformofgatherwiththenon-*-edformofalign.ccT'(x;zp)uOb=zw8 10xx mn xmzpnuOb=zw8MrG1 xMrG(m+n)tɵxmzpn.~ǩ0eT=(uǟ0:)2|s;G8(79)=u1eT=uǟ0:uǟ1;G8(80)2eT=(uǟ1:)2|s;G8(81)%}ߍSomeUUtextaftertotestthebGelow-displayUUspacing.\begin{gather*}\begin{split}?\varphi(x,z)&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n\end{split}\\[6pt]\begin{align}?\zeta^0&=(\xi^0)^2,\\Y&&y>SampleUUpapGerfortheamsmathpackqagej38?>&\zeta^1?&=\xi^0\xi^1,\\ \zeta^2?&=(\xi^1)^2,\end{align}\end{gather*}Y&'κy>SampleUUpapGerfortheamsmathpackqagej39?>&A.6"1AlignatuTNumbGeredUUversion:'*εVi3=vi,8qiTLvj6;zXi?=xi,8qiTLxj6;-Uil=uiTL;forUUi6=j;G8(82)&HnVj3=vj6;yXj?=xj6;JUjUTujo+8X i6=jUQqiTLui:G8(83) UWSomeUUtextaftertotestthebGelow-displayUUspacing.\begin{alignat}{3} V_i?&=v_i-q_iv_j,&\qquadX_i&=x_i-q_ix_j,?&\qquadU_i&=u_i,?\qquad\text{for$i\nej$;}\label{eq:B}\\V_j?&=v_j,&\qquadX_j&=x_j, &?\qquadU_j&u_j+\sum_{i\nej}q_iu_i.\end{alignat}UnnumbGeredUUversion:'*εVi3=vi,8qiTLvj6;zXi?=xi,8qiTLxj6;-Uil=uiTL;forUUi6=j;&HnVj3=vj6;yXj?=xj6;JUjUTujo+8X i6=jUQqiTLui: UWSomeUUtextaftertotestthebGelow-displayUUspacing.\begin{alignat*}3 V_i?&=v_i-q_iv_j,&\qquadX_i&=x_i-q_ix_j,?&\qquadU_i&=u_i,?\qquad\text{for$i\nej$;}\\V_j?&=v_j,&\qquadX_j&=x_j, &?\qquadU_j&u_j+\sum_{i\nej}q_iu_i.\end{alignat*}Y&(Ϲy>SampleUUpapGerfortheamsmathpackqagej40?>&TheUUmostcommonuseforalignatisforthingslike{xȲ=y:byUU(66)G8(84)x[x0Ȳ=y[ٟ0:byUU(82)G8(85)fax8+x0Ȳ=y+8y[ٟ0:byUUAxiom1.G8(86)SomeUUtextaftertotestthebGelow-displayUUspacing.\begin{alignat}{2} x&?=y&&\qquad\text{by(\ref{eq:A})}\label{eq:C}\\x'&?=y'&&\qquad\text{by(\ref{eq:B})}\label{eq:D}\\x+x'?&=y+y'&&\qquad\text{byAxiom1.}\end{alignat}Y&);y>REFERENCES ?41?>ReferencesC[1]R,G.ChoGquet,M.Rogalski,J.SaintRaymond,>,attheR[10]R[11]R[12]R[13]R cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Zcmr5u cmex10