Multisymplectic Integrator of the Zakharov System
Lehmer's Problem for compact abelian groups
a r X i v :m a t h /0303279v 2 [m a t h .N T ] 7 J a n 2004LEHMER’S PROBLEM FOR COMPACT ABELIAN GROUPSDOUGLAS LINDAbstract.We formulate Lehmer’s Problem about the Mahler mea-sure of polynomials for general compact abelian groups,introducing a Lehmer constant for each such group.We show that all nontrivial connected compact groups have the same Lehmer constant,and conjec-ture the value of the Lehmer constant for finite cyclic groups.We also show that if a group has infinitely many connected components then its Lehmer constant vanishes.1.IntroductionLet f ∈Z [x ±1]be a Laurent polynomial with integer coefficients.Define its logarithmic Mahler measure to bem (f )=1log |f (e 2πis )|ds.Lehmer’s Problem asks whether m (f )can be arbitrarily small but positive.Equivalently,doesinf {m (f ):f ∈Z [x ±1],m (f )>0}equal zero?The smallest positive value of m (f )known,found by Lehmer himself [6],is attained by (1.1)f L (x )=x 10+x 9−x 7−x 6−x 5−x 4−x 3+x +1,for which m (f L )≈0.16235.For accounts of this problem,see [2],[3],and for connections with dynamics see [4].Lehmer’s Problem can be formulated for arbitrary compact abelian groups.To do so,let G be a compact abelian group with normalized Haar measure µ.Let Gdenote its (multiplicative)dual group of characters,and Z [ G ]be the ring of integral combinations of characters.For f ∈Z [ G ]define its logarith-mic Mahler measure over G to bem (f )=m G (f )=Glog |f |dµ.2DOUGLAS LINDSince log 0=−∞,if f vanishes on a set of positive µ-measure then m (f )=−∞,but otherwise m (f ) 0by Lemma 2.1below.Definition 1.1.The Lehmer constant of a compact abelian group G isλ(G )=inf {m G (f ):f ∈Z [ G],m G (f )>0}.Lehmer’s Problem therefore asks whether λ(T )=0,where T =R /Z .We show in Theorem 3.2that all nontrivial connected groups have the same Lehmer constant λ(T ).On the other hand,Theorem 5.1shows that if G has infinitely many connected components,then λ(G )=0.We compute the Lehmer constant of some finite groups,and conjecture a simple formula for cyclic groups and also for products of two-element groups.Finally,we show that if λ(T )=0,then the only groups which have positive Lehmer constant are finite.2.PreliminariesLet G be a compact abelian group,and let f ∈Z [ G].If f vanishes on a set of positive µ-measure then clearly m (f )=−∞.Lemma 2.1.Let f ∈Z [ G].If µ({x :f (x )=0})=0,then m (f ) 0.Proof.The characters appearing in f generate a subgroup ∆of G.Let H =∆⊥and π:G →G/H be the quotient map.Then f is constant oncosets of H ,and so defines f∈Z [(G/H ) ]=Z [∆]such that f ◦π=f .Clearly m G/H ( f)=m G (f ),and f vanishes on a set of positive measure if and only if f does.Hence we may assume that Gis finitely generated.Let Z /n denote Z /n Z .Applying the structure theorem for finitely gener-ated abelian groups to G,we see that G is isomorphic to Z /n 1⊕···⊕Z /n r ⊕T kfor suitable integers n 1,...,n r and k .Consider f ∈Z [ G]as a function of r +1variables j 1∈Z /n 1,...,j r ∈Z /n r ,and s ∈T k .Since the set where f vanishes is null,each functionf j 1...j r (s )=f (j 1,...,j r ,s )is a nonvanishing complex combination of characters on T k ,and hence by [4,Lemma 3.7]we see that m T k (f j 1...j r )>−∞.Thusm (f )=1n 1...n rm T k (g )>−∞,whereg (s )=n 1−1 j 1=0···n r −1 j r =0f j 1...j r (s ).LEHMER’S PROBLEM3Since for each i the product over j i covers all the n i th roots of unity,the coefficients of g are algebraic integersfixed by all elements in the Galois group of thefield they generate,and so g∈Z[(T k) ].Hence m T k(g) 0by [4,Lemma3.7],showing that m(f) 0.We repeatedly use the following observation.Lemma2.2.Let H be a closed subgroup of G.Thenλ(G) λ(G/H). Proof.Letπ:G→G/H be the quotient map.If f∈Z[(G/H) ],then f◦π∈Z[ G]and m G(f◦π)=m G/H(f).The conclusion then follows from the definition ofλ.3.Connected groupsIn this section we prove that all nontrivial connected groups have the same Lehmer constant.Lemma3.1.λ(T k)=λ(T)for all k 1.Proof.Since T∼=T k/T k−1,Lemma2.2shows thatλ(T k) λ(T).To prove the reverse inequality,fixǫ>0.Choose f∈Z[(T k) ]with0<m T k(f)<λ(T k)+ǫ.For r=(r1,...,r k)∈Z k define f r(s)=f(r1s,...,r k s),so that f r∈Z[ T]. By[5]we canfind r so that|m T(f r)−m T k(f)|<min{m T k(f),ǫ}.It follows thatλ(T) λ(T k)+2ǫ.Sinceǫ>0was arbitrary,we obtain that λ(T) λ(T k). Theorem3.2.If G is a nontrivial connected compact abelian group,then λ(G)=λ(T).Proof.Since G is nontrivial and connected,it has a quotient isomorphic to T,and henceλ(G) λ(T)by Lemma2.2.To prove the reverse inequality,observe that G is torsion-free.Let f∈Z[ G]with m G(f)>0.The subgroup∆of G generated by the characters in f is therefore isomorphic to Z k for some k 1,and so ∆=G/∆⊥∼=T k. As in the proof of Lemma2.1,f induces f∈Z[∆]with m ∆( f)=m G(f). By Lemma3.1,we have thatm(f)=m ∆( f) λ(T k)=λ(T),Gproving thatλ(G) λ(T).4DOUGLAS LIND4.Finite groupsIn this section we consider the problem of computing the Lehmer constant of some finite abelian groups.Trivially λ(Z /1)=log 2.Define characters χk on Z /n by χk (j )=e 2πijk/n .Thus the character group is Z /n ={χ0,χ1,...,χn −1}.Example 4.1.Let G =Z /2.For f =aχ0+bχ1∈Z [ Z /2]we have thatm (f )=12log |a 2−b 2|.It is easy to check that the smallest value of |a 2−b 2|greater than 1is 3,sothat λ(Z /2)=13log |a 3+b 3+c 3−3abc |.Choosing a =b =1and c =0gives a value of 2for the expression inside the absolute value,which is clearly the smallest possible value greater than 1.Hence λ(Z /3)=14log (a +b +c +d )(a −b +c −d )[(a −c )2+(b −d )2] .Putting a =b =c =1and d =0gives14log 3.There is a simple upper bound for the Lehmer constant of a finite group.Lemma 4.4.Let F be a finite abelian group with cardinality |F | 3.Then (4.1)λ(F )1LEHMER’S PROBLEM5 Put f=g−χ0,so thatf(x)= |F|−1if x=0,−1if x=0.1Then m F(f)=logρ(n).nProof.Let p=ρ(n),andΦp be the p th cyclotomic polynomial.Put f p(j)=Φp(e2πij/n),so that f p∈Z[ Z/n].Then1m(f p)=log2.n1Proof.Sinceρ(n)=2we see thatλ(Z/n)logρ(n)for all n 2.n.Next,consider groups of the form Z/2⊕···⊕Z/2=Z n/2Example4.9.Let G=Z/2⊕Z/2.The charactersχij(x,y)=χi(x)χj(y) form the dual group G={χ00,χ01,χ10,χ11}.Thenm 1 i,j=0a ijχij =16DOUGLAS LINDThe expression inside the absolute value is3when a00=a01=a10=1 and a11=0.It cannot attain the value±2,since if so2would factor into a product of four integers whose pairwise differences are all even.Hence1λ(Z2/2)=log(2n−1)for all n 2.2nNote that this quantity is the upper bound forλ(F)in Lemma4.4when .F=Z n/25.Mixed groupsHere we consider groups G that have a nontrivial connected component G0of the identity.LetΓ= G,and denote the torsion subgroup ofΓby t(Γ).The totally disconnected group G/G0then has dual group t(Γ).Theorem5.1.If G/G0is infinite,thenλ(G)=0.Proof.Let∆be afinite subgroup of t(Γ),and suppose that|∆| 3.Then by Lemma2.2,(5.1)λ(G) λ(G/∆⊥)=λ( ∆) 1LEHMER’S PROBLEM7 Thenm T⊕Z/2(h)=1。
具有三个位势的广义耦合无色散可积方程的多孤子解(英文)
具有三个位势的广义耦合无色散可积方程的多孤子解（英文）2O11年5月第42巷第3期内蒙古大学(自然科学版)JournalofInnerMongoliaUniversity(NaturalScienceEdition)May2011V o1.42No.3ArticleID:1000—1638(2O11)03—0253-05 Multi—solitonSolutionsoftheGeneralized DispersionlessIntegrableEquationwithThreeCoupledPotentialsZhaqilao(C0llegeofMathematicsScience,InnerMongoliaNormalUniversity,Hohhot010022,Chin a)Abstract:AnewN—foldDarbouxtransformationofthegeneralizedcoupleddis—Dersionlessintegrableequationisderivedwiththeaidofthegaugetransforma—tionbetweencorresponding2×2matrixspectralproblemswiththreepoten—tla1s.AsanapplicationoftheDarbouxtransformation,N—solitonsolutionsof thegeneralizedcoupleddispersionlessintegrableequationareexplicitlygiven. Keywords:Darbouxtransformation;solitonsolution;coupleddispersionlessinte—grableequationCLCnumber:O175.2Documentcode:AIntr0ducti0n Thestudvofdispersionlesshierarchiesisoneofthemostprominentsubjectinthefieldofnon —linearscience,partlybecauseofdispersionlessequationsemergenceindiverseareasofmathe maticalandthe.retica1Dhysicssuchasquantumfieldtheory,conformalfieldtheory,stringtheory,sol itontheorv,etc.Inpast,thecoupleddispersionlessequationanditsgeneralizationhasalsoattracte da reatdealofinterestbecauseofitsniceintegrabilitystructureandsolitonsolutions. InRefs.[1—2],KonnoandKakuhataconsideredthegeneralformofthecoupleddispersionless integrableequation(CDIE)q+(rs)=0,,.一2r一0,s一25g一0(1) andobtainedthesingleanddoublesolitonsolutionsofCDIE(1)bytheinversescatteringtransf orm(IST).CDIE(1)hastwospecialcases:r=andr—.Theformercasewasfoundtobeequivalent tothesine—Gordone0uation[.andthelattertothePohlmeyer—gund--Reggeequation 们.Forr—s,CDIE(1)reducestothemodelwithtwopotentials.Inthepastyears,manyeffortshavebeen dedicatedtothereducedequationandrichfamiliesofexactanalyticalsolutionshavebeenobta inedeffectivemethods,suchasIST,Painlev6analysis∞,B/icklundtransformation(BT)"andDarbouxtransformation(DT)C783.However,toourknowledge,verylittleresearchhas beendoneonCDIE(1)viatheN—foldDTmethod. TheaimofthispaPeristoobtainN—solitonsolutionsofCDIE(1)withthehelpofspectral*Receiveddate:2011-05-06Foundationitem:SupportedbytheNaturalScienceFoundationofInnerMongolia(GrantNo 2009MS0108);the HighEducationScienceResearchofInnerMongoIiaAutonomousRegion(GrantNo?NJ10045)Biography:Zhaqila.(1971一),ma1e(Mongolian),anativeofOrdosofInnerMongolia,Viceprofesser,Ph?D?E_ mail:zhaqilao@.254内蒙古大学(自然科学版)2011矩problemsandtheDarbouxmatrixmethod(9-131.Tothisend,westartfromthecorresponding spec—tralproblemsofCDIE(1)andachieveanN—foldDTinsection1.ResortingtoDTinsection1.N—solitonsolutionsofCDIE(1)areobtainedinsection2.Mostimportantly,thedetailedstructure softhesinglesolitonanddoublesolitonsolutionsofCDIE(1)aregivenbothanalyticallyandgrap hical—ly.DarbouxtransformationInthissection,weshal1constructaThespectralproblemsofCDIE(1)whereU:=——iXnewN—foldDTforCDIE(1).aregivenas一U,≯===qfr5—q,V一(:--.r)十i(～0)(2)(3)Throughaa"lrectcalculation,thezerocurvatureequationUf—V+U,厂一【,『一0giverisetoCDIE(1).TheDarbouxtransformationisactuallyagau—ofspectralproblems(2).Itisrequiredthatalso一,一(+TU)T～,whereU一一iq尸sI—qlgetransformation(4)satisfiesthesamespectralproblemst—,V一(+)T-(5),V—O—+～o)Nowwediscussaconcretetransformation.LetmatrixTin(6)beintheformofT一丁c一A暑)whereN一1N～1N—lN--1A—一+∑A,B一∑B,c一∑CA一,D—一+∑D,0^=0=0k=OA,B,C^,Dkarefunctionsofzandt.A^,B,CandD^aregivenbyalinearalgebraicsystem N—l一一,∑(cI+3jD)—as117景0(6)(7)(8)怒蒋,,≤≤2N/1(9)1()+J1(J)'where一(l,2),一(1,)aretWObasicsolutionsofspectralproblems(2),and(≠J,:≠,asi≠)aresomeparameterssuitablychosensuchthatthedeterminantofcoefficientsfor (8)iSnonzero.Hence,ifwetakeDl,一--AN-一1.(10)therestA^,B^,CkandD^(1≤J≤N一1)areuniquelydeterminedby(8).Eq.(7)showsthatdetT()isa(一2N)th—orderpolynomialin,anddetT()=A()D()一B(,)C().Ontheotherhand,from(8)wehaveA()一一a,B(),CO,)一一D().,,,B+A,L∑h.代w第3期Zhaqila0Multi--solitonSolutionsoftheGeneralizedCoupled (255)Therefore,itholdsthatdet-/(,)一0whichimpliesthat,(1≤J≤2N)are2Nrootsofdet丁(),thatisdetT()===yⅡ(—)(11)whereyisindependentof.Inawaysimilartotheproofin[13],wecanverifythefollowingprop.sitions.Proposition1MatrixdeterminedbythesecondexpressionofEq.(5)hasthesameformasU.wherethetransformationformulafromtheoldpotentialsq,,-,intonewonesaregivenby ……一一邶一r+i一一瓮whereAN--1iscoefficientsof(8),i.e.AN一111…l__N÷1282),7….:LN+;;;:'.;j12N2N--N1….=L--2NN-.-2Ni.+(13)and△^…,△BareproducedfromGbyreplacingits(2N一1)th,2Nthcolumnwith(一,…, 一^.--2NN),respectively.△c1isproducedfrom△lbyreplacingits(2N一1)thcolumnwith(一,…,一2N).Accordingto(10),allJ(】≤≤2N)mustsatisfyconditions一一(一1,3,5,7,…).Proposition2MatrIxdefinedbythefourthexpressionofEq.(5)hasthesameformasV,m whchtheo1dpotentia1sg,r,saremappedintonewpotentials,r,,accordingtothesameDT (12).Propositions1and2showthatthetransformation(12)changesspectralproblems(2)intoan—otherspectra1problems(5),withU,Vand—U,一Vhavingthesameform.Thereforebothofthespec—tra1DroblemsleadtothesameCDIE(1),SO(12)istheDarbouxtransformationofCDIE(1)?F romPropositions1and2wehavethefollowingtheorem?Theorem1Let(q,r,s)beasolutionofCDIE(1).Thenfunction(,尸,)determinedbyDT (12)iSanewsolutionofCDIE(1).2SolitonsolutionsNowwecho.sethetrivialsolutionq.=fiz(|8isanarbitrarynon—zeroconstant)andro—s.一0,thecorrespondingcompatibIebasicsolutionofspectralproblems(2)canbewrittenas (.z,t,)一exp(一￡)00exp($i)(14)where￡一if1)!(一赢?Using(9),when,≠0andj≠0,wehave一exp(2岛),(:1,2)(15)Substituting(15)into(13),(12)denotesaunifiedandexplicitformulationofallk～solit.n.一lution(1≤是≤N),fr.mwhichitiseasytogetN—solitonsolutionsofCDIE(】)?Forsimplicity,wesha1】discl】sstwospectralcasesofN一1andN一2.256内蒙古大学(自然科学版)2011在Case1(N一1).Let,,,(一1,2)and2=::--A1.From(12),wehaveanexplicitsolutionof CDIE(1)asfollowing—fix-一一一一c16)'一一'where一lexp(2￡-i一蠢_1'2).If2一一1andl,2arepureimaginaryparameters,l,2arerealparameters,thesolutionde—terminedby(16)isthesinglesolitonsolutionq一—Itanhi,r一一sechi,s一一1sechi,一砉(￡一2Za12X)(17)Case2(N一2).Let,J(一1,2,3,4)and2一--;t1,4一--A3.From(12),wehaveanewso—lutionofCDIE(1)qz一卢z+i,r一i,s一i等c8whprA1一2j3 AA.△c1一一_『一一一一l一一3i一4l2iA3I23 AB.=一——.I_一where(一1,2,3,4)aredeterminedin(15).If,uz一--/zl,/14一一3and,(一1,2,3,4)arepureimaginaryparameters,J(一1,2,3,4)are realparameters,thesolutiondeterminedby(18)isthedoublesolitonsolution(seeFig.1).-4020(a】(b)(c)Fig.1Doublesolitonsolution(18)withr—s一0,一A1一2—5i,l一--p2—1,卢=0.O1,ps:一{一1,一A3=^=3iIteratingtheabovemethod,wecanobtainaseriesofmulti—solitonsolutionsofCDIE(1).3ConclusionByextendingelementsoftheDTmatrixintonegative--orderpolynomialsin,wedeveloped theDTmatrixmethod"todirectlyconstructanexplicitdeterminantformu1aforN—foldDTofCDIE(1).ThisN—foldDTformulacanbeinterpretedasanonlinearsuperpositionofasingleDT. Moreover,notonlytheconstructionofN—foldDTisverynaturalandmuchsimplerthanthatob一11111111乱1111融111l第3期ZhaqilaoMuIti—solitonSolutionsoftheGeneralizedCoupled (257)tainedintheusualwayEl-z],butalsotheN—foldDTisverysuitableforgeneratingaseriesofexplic—itsolutionsbysymboliccomputationonacomputer.References:[1][2][3][4][5]1-6][7][83[9][1o][11][12][13]KonnoK,KakuhataH.InteractionAmongGrowing,DecayingandStationarySolitonsforC oupledIntegrableDis—persionlessEquations[J].J.Phys.Soc.Jpn.,1995,64(8):2707-2709.KonnoK,KakuhataH.NovelSolitonicEvolutionsinaCoupledIntegrable,DispersionlessS ystem[J].J.Phys.Soc.Jpn.,1996,65(3):713—721.HirotaR,TsujimotoS.Noteon"NewCoupledIntegrableDispersionlessEquations"[J],J.Ph ys.Soc.Jpn.,I994,63(9):3533-3533.KotlyarovVP.OnEquationsGaugcEquivalenttOtheSine-GordonandPohlmeyer—Iund —ReggeEquations[J].J.Phys.Soc.Jpn.,1994,63:3535-3537.KonnoK,OonoH.NewCoupledIntegrableDispersionlessEquations[J].J.Phys.Soc.Jpn.,1 994,63:377—378.AlagesanT,ChungY,NakkeeranK.B~cklundtransformationandsolitonsolutionsfortheco upleddispersionlessequations[J].Chaos,SolitonsandFractals,2004,21:63—67.CbenAH,LiXM.Solitonsolutionsofthecoupleddispersionlessequation[J].Phys.Lett.A,2 007,370:281—286.HassanHJ.Darbouxtransformationofthegeneralizedcoupleddispersionlessintegrablesys tem[JⅢ.Phys.A:Math.Gen.,2009,42:l一11.NeugebaureG,MeineIR.GeneralN—so!itonsolutionoftheAKNSclassonarbitrarybackground[J].Phys.Lett.A,1984,100:467-470.MatveevVB,SalleMA.DarbouxTransformationsandSolitons[M].Berlin:Springer,1991. GuCH,HuHS,ZbouZx.DarbouxTransformationinSolitonTheoryandItsGeometricAppli cations[M].Shanghai:ShanghaiScienceandTechnologyPublishingHouse,2005.FanE(;.ComputerAlgebraandIntegrableSystems[M].Beijing:SciencePress,2004. Zhaqilao,ChenY,IiZB.Darbouxtransformationandmulti—solitonsolutionsforsomesolitonequations[J].Chaos,SolitonsandFractals.2009,41:661-670.(责任编委孙炯)具有三个位势的广义耦合无色散可积方程的多孤子解扎其劳(内蒙古师范大学数学科学学院,呼和浩特010022)摘要:利用具有三个位势的2×2矩阵谱问题的规范变换,给一个广义耦合无色散方程构造了一种新的N重达布变换.作为达布变换的应用,获得了该广义耦合无色散方程的N一孤子解.关键词:达布变换;孤子解;广义耦合无色散可积方程中图分类号:O175.2文献标志码:A收稿日期:2010—05-06基金项目:内蒙古自然科学基金项目(2009MS0108);内蒙古自治区高等学校科学研究项目(NJ10045)作者简介:扎其劳(1971一),男(蒙古族),内蒙古鄂尔多斯市人,副教授,博士.。
代数英语
(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。
0+||zero-dagger; 读作零正。
1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。
AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。
BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。
B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。
C0 类函数||function of class C0; 又称“连续函数类”。
CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。
Cp统计量||Cp-statisticC。
Empirical processes of dependent random variables
2
Preliminaries
n i=1
from R to R. The centered G -indexed empirical process is given by (P n − P )g = 1 n
n
the marginal and empirical distribution functions. Let G be a class of measurabrocesses that have been discussed include linear processes and Gaussian processes; see Dehling and Taqqu (1989) and Cs¨ org˝ o and Mielniczuk (1996) for long and short-range dependent subordinated Gaussian processes and Ho and Hsing (1996) and Wu (2003a) for long-range dependent linear processes. A collection of recent results is presented in Dehling, Mikosch and Sorensen (2002). In that collection Dedecker and Louhichi (2002) made an important generalization of Ossiander’s (1987) result. Here we investigate the empirical central limit problem for dependent random variables from another angle that avoids strong mixing conditions. In particular, we apply a martingale method and establish a weak convergence theory for stationary, causal processes. Our results are comparable with the theory for independent random variables in that the imposed moment conditions are optimal or almost optimal. We show that, if the process is short-range dependent in a certain sense, then the limiting behavior is similar to that of iid random variables in that the limiting distribution is a Gaussian process and the norming √ sequence is n. For long-range dependent linear processes, one needs to apply asymptotic √ expansions to obtain n-norming limit theorems (Section 6.2.2). The paper is structured as follows. In Section 2 we introduce some mathematical preliminaries necessary for the weak convergence theory and illustrate the essence of our approach. Two types of empirical central limit theorems are established. Empirical processes indexed by indicators of left half lines, absolutely continuous functions, and piecewise differentiable functions are discussed in Sections 3, 4 and 5 respectively. Applications to linear processes and iterated random functions are made in Section 6. Section 7 presents some integral and maximal inequalities that may be of independent interest. Some proofs are given in Sections 8 and 9.
带强奇异边界积分方程的迦辽金边界元解法_英文_
第37卷第11期中国科学技术大学学报Vol.37,No.11 2007年11月JOU RNAL OF UNIVE RSI TY OF SCIENCE AND TECHNOLOGY OF CHINA No v.2007 A rt icle ID:0253-2778(2007)11-1357-06Galerkin boundary element method for solving the boundary integral equation with hypersingularityZH U Jia-lin1,ZH A NG Shou-gui1,2(1.Sc hool of M athe matics and Ph ysics,Chongqing Univ ersity,Chongqing400044,Ch ina;2.S ch ool of M athematic s and Comp uter S cie nce,Ch ong qing Normal Univ er sity,Ch ong qing400047,China)Abstract:A Galer kin boundary elem ents metho d is applied to solve the integ ral equatio n w ith hyper sing ular ity,w hich can be deduced from the double layer solution for the N eumann pro blem of Laplace equation.T he scheme of integration by parts in the sense o f distributions is perfo rmed to reduce the hy persingularity integ ral into a w eak o ne,w hich shifts the par tial derivativ es o f hyper sing ular ker nel to the unknow n function in the variational fo rmulation.T hus,the boundary ro tation of an unknow n function is used to substitute fo r the original unknow n function in the variatio nal equatio n.When linear boundary elem ents ar e used in tw o-dimensional cases,the boundar y rotation can be discr etized into a constant vector on each element,so that the integr ations can be perfo rmed in a simple w ay.T he num erical tests illustr ate the effectiv eness and practicality o f the scheme presented.Key words:Galerkin bo undary element method;double layer po tential;hyper singular integral;Laplace equation;N eumann pro blemCLC number:O241.82Document code:AAMS Subject Classification(2000):Primary65N38;Secondary35J05带强奇异边界积分方程的迦辽金边界元解法祝家麟1,张守贵1,2(1.重庆大学数理学院,重庆400044;2.重庆师范大学数学与计算机科学学院,重庆400047)摘要:采用双层位势来表示二维Laplace方程Neumann问题的解,导致求解含超强奇异性的边界积分方程,将其转换为边界上的Galerkin变分方程求解.针对超强奇异积分的计算,运用分步积分,详细地推导了基于边界旋度的变分公式及边界旋度的表达式,最终把超强奇异的积分计算转化为弱奇异积分的数值计算.当采用线性边界单元来离散Galerkin变分公式时,在每个离散的单元上边界旋度成为常向量,因此,数值积分变得很简单.数值算例验证了方法的有效性和实用性.关键词:Galerkin边界元法;双层位势;超强奇异积分;Laplace方程;Neumann问题Received:2006-12-15;Revised:2007-03-04Foundation item:Supported by the National Natu ral S cien ce Foundation of Chin a(No.50679097)and the National Natural Science Fou ndation for Distinguish ed You ng S cholars(No.50625824).Biography:ZH U Jia-lin(corres pon ding author),male,born in1943,Prof.Res earch field:boundary elemen t method.E-m ail:jlz hu@0 IntroductionIn boundar y element methods,w e oftenencounter the calculation of hypersingular integr ations.Such as the follo wing Neum ann problem o f Laplace equation,$u(x )=0,x I 8;5u(x )5n (x )#=g(x ),x I #;(1a )or$u(x )=0,x I 8c ;5u(x )5n (x )#=g(x ),x I #.(1b )H ere 8is an open bounded dom ain w ith bo undary #in R 2,and 8c is the infinite co mplem entar y of8=8+#.When a double layer solutio n is used (w here c is a constant,U =[u ]=u +-u-is aninterm ediary unknow n function on the boundary to be deter mined),u(x )=12P Q#U (y )55n y ln 1|x -y |d s y +c,(2)from w hich the corr espo nding boundary integ ral equation has a hypersing ular ker nel.12P Q#U (y )525n x 5n yln 1|x -y |d s y =g(x ).(3)Actually,this ex pression of the norm alderivative of such a potential is not an integ ral.It is a finite part ex pr ession.Therefore,w e use a Galerkinvariational formulation.By Green p sformula,w e have [1,12]P v I (H 1(8)/p 0)@w 10(8c )and w 10(8c )=v r ln rI L 2(8c ),Dv I L 2(8c ).A variational form ulation of problem (1)is g iv en byQ8G 8cgrad u #g rad v d x =3g,L 4H -120(#)@(H 12(#)/p 0),(4)w here L =v +-v -I H 12(#)/p 0,and p 0is a constant.Further,o ne can obtain an explicit ex pressionof the bilinear fo rm in the left hand side o f Eq.(4),w hich needs to do double integration on #,b(U (x ),L (y ))=14P Q #Q#(U (x )-U (y ))(L (x )-L (y ))# 525n (x )5n (y )ln 1|x -y |d s x d s y .(5)The follow ing variational equatio n has a uniquesolution (allow ing a difference of constant )in H 12(#)/p 0,b(U ,L )=QHg (x )L (x )d s x,P L I (H 1(8)/p 0)@w 10(8c ).(6)The variational formulation (4)is the key starting point to the num er ical so lution,but since Eq.(5)is inconvenient for implem entation,w eshall use another ex pression for the bilinear fo rm b (U ,L )deduced dir ectly from Eq.(4).Accor ding to the idea of integration by parts in the sense o f distributions,one can shift the partial derivatives of hypersing ular kernel to the unknow n function in the v ar iational formulatio n.T hen,w e try using the boundary rotation o f unknow n function to substitute fo r the orig inal unknow n function in the variational equation.Finally the calculation o f coefficients of the corresponding matrix beco mes very simple in spite of the com plicated deduction.The schem e of integ ration by parts in the sense of distributions is co mmo nly used in or der to reduce hyper sing ularity,but reg ular ization can be acco mplished in different w ays [14].Som e appr oaches transform the norm al derivative to the tangential der iv ative first,then perfo rm the integratio n be parts [6,7,8,15].Som e other techniques w ere usedalso,suchasusingsing ularitysubtraction technique to reduce the hyper -sing ular operator to a w eakly singular one [18];som e calculatetheH adamardfinite -par tofahy persingular integ ral [17];so me use the Maue -typeidentities to replace the hyper -singular operator w iththe w eakly -sing ularoperato r [16].WhenGreen p s functio n instead of fundamental solution is used,one can dev elo p the hypersing ular kernel in series and separate the singular parts[9~11],but thedomain should be regularly shaped,otherw ise we can1358中国科学技术大学学报第37卷not find the explicit expression of corresponding Green p s functio n.The appro ach to shifting the par tial derivatives o f hy persingular kernel to the boundar y r otation (curl)of the unknow n function in three -dim ensional cases w as propo sed by Nedelec [2~4],Duonggave the num er icalimplementatio n later [5].The appr oach presented inthis paper used the same idea fo r 2D problemsdifferent from the appro aches cited abo ve.1 Variational formulation based onboundary rotationIn R 2,suppose #D is an open neighbo rhood of boundar y #.For a proper D ,any point x in #D has a local projection p (x )fr om #D onto #.T hen fo r any smooth functio n U (x )defined o n #,one can define a v ector field ortho gonal to the planrot #U (x )=grad U (x )@n ,P x I #,(7)w her e n (x )is the unitary exterior normal to theboundar y #,and the function U (x )is ex plicited on #D by<(x )=U (p (x )),x I #D ,p (x )I #.(8)One can define a scalar functio n rot #X (x )to a vecto r field X on #byrot #X (x )=n (x )#ro t X (x ).(9)H ere,r ot is the ordinary rotation o perato r in R 2,X (x )is a vector field defined in #D byX (x )=X (p (x )),x I #D ,p (x )I #.(10)Accor ding to the definitions abov e,one can obtain the Laplace -Beltr am i oper ator on #,$#U (x )=r ot #rot #U (x ).(11)T hen,for the double layer potential u (x )(Eq.(2)),w e hav eg rad u(x )=-12P Q#rot #U (x )@ grad y ln1|x -y |d s y ,x I R 2-#.(12)After a long deduction,the bilinear for m b (U ,L )g iv en by Eq.(4)has the expressio n [13],b(U ,L )=12P Q #Q#ln |x -y |rot #U (y )#r ot #L (x )d s y d s x .(13)T hen,w e are g oing to so lve the fo llow ingvariational equation w ith the left hand termex pr essed by Eq.(13),b(U ,L )=12P Q #Q#ln |x -y |r ot #U (y )# rot #L (x )d s y d s x =Q#g (x )L (x )d s x.(14)2 Implementation of the variationalformulationWe now g iv e the approx imation pro cess to solv e Eq.(14).The boundar y #is appr oxim ated by #h =G Ni =1#i ,w here boundar y elem ents #i =P i P i +1ar e straight lines,P i denote boundary no des (i =1,2,,,N ).For the closed boundary P N +1=P 1.We use linear elements,then,the approx imatio n of a distribution U (or L ),using linear basis functions,may be w ritten as a linear combinationU =E Ni=1U i7i(i =1,2,,,N ),(15)w here the basis 7i ar e the linear po lynomials,and the coefficients U i ar e determined at the N nodes.When i =N ,#N +1=#1,7i =W (i),x I #i ;W (i+1),x I #i+1;0,other case .(16)On each bo undary elem ent,w e establish a coo rdinate system show n as Fig.1:Suppose that the so ur ce point is Q ,and the field point P situated on the bo undary elem ent #i =P i P i +1,r =PQ .T he bases co nsist of tw o vectors (n ,m );n is the unitary normal of #i and m is the unitary vectorFig.1 The local coordinates1359第11期G aler kin boundar y element method for so lving the boundar y integr al equatio n w ith hy per sing ular ityalong the direction of #.The distances fro m P,P 1and P 2to the no rmal n are denoted by s,s 1and s 2respectively.d is the distance fro m Q to #i .We cho ose s as the integr al variable,thenW (i)=s (i)-s (i)1l i ,W (i+1)=s (i+1)2-s (i+1)l i+1.(17)T he letter w ith up -script (i )denotes the local coordinate on the i th elem ent,and l repr esents the length of the elem ent.Substitute Eq.(15)into Eq.(14),and let L =7j (j =1,2,,,N ),w e g ot the linear systemE Ni=1aijU i =b j (j =1,2,,,N ),(18)w her eb j =Q#jG #j+1g(x )7j d s x(j =1,2,,,N ;#N+1=#1);(19)a ij =12P Q #jG #j+1ro t #7j(x )#Q #iG #i+1ln |x -y |rot#7i (y )d s y d s x(i,j =1,2,,,N ).(20)Further more,the entry a ij of the co rresponding matrix can be rew ritten as the combination o f four integr als,a ij =12P Q #j Q#iln |x -y |ro t #W (i)(y)#ro t #W (j)(x )d s y d s x +12P Q #j Q#i+1ln |x -y |rot #W (i+1)(y )#rot #W (j )(x )d s y d s x +12P Q #j +1Q#iln |x -y |rot #W (i)(y)#rot #W (j+1)(x )d s y d s x +12P Q #j +1Q#i +1ln |x -y |rot #W (i+1)(y)#rot #W (j+1)(x )d s y d s x(i =1,2,,,N ;j =1,2,,,N ).(21)3 C alcu lation of the b ound ary rotationSuppose that the coo rdinates of P 1,P 2,P ,Q are P 1:(x 1,y 1),P 2:(x 2,y 2),P:(x ,y ),Q:(x c ,y c ),l =|P 2-P 1|,(22)n =y 2-y 1l ,x 2-x 1l,(23)r 1=(x 1-x c ,y 1-y c ),r 2=(x 2-x c ,y 2-y c ).We hav es 1=r 1#m ,s 2=r 2#m ,s =r #m ,d =r 1#n =r 2#n .Therefore,s -s 1l =(r -r 1)#m|r 2-r 1|=(x -x 1)(x 2-x 1)+(y -y 1)(y 2-y 1)l2.(24)Let i ,j denoted the coor dinate axis on the plan and k =i @j ,grad s -s 1l =x 2-x 1l 2i +y 2-y 1l 2j .(25)According to the definition (7),rot #s -s 1l =gr ad s -s 1l @n =-1lk .(26)Sim ilar lyrot #s 2-s l =grad s 2-sl @n =1l k .(27)Finally,the coefficients o f the m atrix can be calculated in a simple fo rmulaa ij =12P l i l j Q #j Q#iln |x -y |d s y d s x - 12P l i+1l j Q #j Q#i+1ln |x -y |d s y d s x - 12P l i l j+1Q #j+1Q#iln |x -y |d s yd s x+12P l i+1l j+1Q #j +1Q #i+1ln |x -y |d s yd sx(i =1,2,,,N ;j =1,2,,,N ).(28) Now that the integrations are simplified,thefirst integ ral can be calculated by an analytical for mula,and the seco nd integ ral is calculated by Gaussian quadr ature.The linear system (18)gives the values U i(i =1,2,,,N)on boundary no des,and theappr ox im ate solution of the problem (1)w ill be given by a u h =12P E Ni=1c =1360中国科学技术大学学报第37卷12P E Ni=1Q#iW (i)(y -x )#n y |y -x |2d s y +Q#i+1W (i+1)(y -x )#n y|y -x |2d s y U i +c.(29)We need so me fixed values to determ ine the constant c ,w hen x I 8,but fo r the ex terio r problem ,the solution is unique.4 Numerical resultsIn order to test the efficiency o f the proposed num er ical appro ach,w e did so me numerical tests on pro blems w ith analy tical so lutions know n.U niform mesh refinement is used.Example 411 8={(x ,y )|x 2+y 2\1}.u =1r 2cos 2H =x 2-y 2(x 2+y 2)2.T he boundary co ndition is5u 5n #=2co s 2H =2x 2-2y 2. T he r esults for this case are show n in Tab.1.We cansee that the relationship o f appro ximation error E (u )=|u -u h |w ith the element lengthh(h =2P N)can be described by ex pr ession likeFig.2 ln E(u)U 2ln h +ln A (A is a positive constant)E(u)=0(h 2).The conv er gence rates are displayed in Fig. 2.Example 412 8is a square dom ain w ith side length 8.u(x ,y )=xy.Subjected to the boundary condition g =?y at x =?4;g =?x at y =?4.The results fo r this case are show n in T ab.2.The conv erg ence rate is also abo ut O (h 2),which ag reesw iththe theor eticalconv erg ence forGalerkin boundary element metho d (12).Fig.3illustrates the visual sim ulated results.Tab.1 The comparison of the solutions with the numbers of boundary elements increasingr ap proximate solution u hN =16N =32N =64N =128an alytical solu tion 1.50.41808470.43784930.44281420.44403960.44444443.00.10444790.10946380.11070360.11101000.11111115.00.037600950.039406170.039853260.039963570.040000007.00.019184580.020105870.020333310.020389580.020408169.00.011605040.012162250.012300390.012334440.0123456812.00.0065277830.0068412130.0069189680.0069381190.006944444=Note > N is numbers of the elem ents,r is the distance to the centre of th e circleTab.2 The comparison of the solutions with the numbers of boundary elements increasingx y ap prox imate solution u hN =16N =32N =64N =128analytical solution3.0 3.00.9018947E+10.8994118E+10.8999805E+10.8999805E+19.01.0 1.00.9981121E+00.9997572E+00.1000141E+10.1000001E+1 1.00.50.20.9987735E-10.9997454E-10.1000052E+00.1000004E+00.1-0.4-0.60.2396128E+00.2399411E+00.2400023E+00.2400005E+00.24-2.0-2.00.3988905E+10.3998950E+10.3999978E+10.3999992E+14.03.0-2.0-0.598808E+1-0.5998041E+1-0.599986E+1-0.599998E+1-6.01361第11期G aler kin boundar y element method for so lving the boundar y integr al equatio n w ith hy per sing ular ityFig.3Visual simulated result5ConclusionThis paper presents an approach that overcomes the barrier of hyper singular integration,enco unter ed w hile applying the bo undary element m ethod, w hen a double layer solution for the Neumann problem is used.We introduce a Galerkin variational formulation in order to implem ent the integration by parts.T he differ ence fr om other integratio ns by par ts is the employm ent o f the bo undary ro tation instead o f the tang ential derivativ e.While linear boundar y elem ents are used in tw o-dimensional cases,the bo undary rotation can be discr etized into a constant vector on each elem ent,and the hyper singular integrals tur n into w eak o nes,so that the integr ations can be per for med smoothly. Numerical tests sho w go od co nverg ence behav io r.References[1]Giro ire J,Nedelec J C.N umer ical so lutio n o f anex terio r Neumann pro blem using a double layer potential[J].M athematics of Computat ion,1978,32 (144):973-990.[2]N edelec J C.F inite element fo r exter io r pr oblems usinginteg ral equat ions[J].Int J fo r N um M eth in Fluids,1987,7(11):1229-1234.[3]N edelec J C.Integr al equat ions with no n-integ r ablekernels[J].Integr al Equatio ns and O perator T heor y,1982,5(1):562-572.[4]N edelec J C.Equations Integr ales,Chapitr e XI,V olume6[M]//Analyse M athematique et Calcul N umer ique po ur les Sciences et les T echnique.P aris:M asson,1984-1985:666-678.[5]Duong T H.A finite element method for the double-layerpotential so lution of the Neumann ex ter ior problem[J].Math M eth in the Appl Sci,1980,2:191-208.[6]Ghosh N,Rajiyah H,Ghosh S,et al.A new boundar yelement method fo rmulat ion for linear elast icity[J].ASM E,T r ansactio ns,Jo ur nal of A pplied M echanics,1986,53:69-76.[7]Sladek V,Sladek J.Reg ularization of hypersingularand nearly singula r integr als in potential theory and elasticity[J].Int J N umer M ethods Eng,1993,36(10):1609-1628.[8]T anaka M,Sladek V,Sladek J.R egular izationtechniques applied to boundary element met ho ds[J].Appl M ech Rev,1994,47(10):457-499.[9]Han Ho u-de.Numerical solutions of integro-differentialequations for elliptic boundary value problems[J].Science in China Ser ies A,1988,2:136-145(in Chinese). 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[15]Nico lazzi L C,Barcellos C S,F ancello E A,et al.Gener alized bo undary element metho d for Ga lerkinbo undar y integ rals[J].Eng ineer ing Analysis w ithBoundar y Elements,2005,29:494-510.[16]K alliv okas L F,Juneja T,Bielak J.A sy mmet ricGaler kin BEM var iatio na l framew ork for mult-i domainint erface pr oblems[J].Co mputer M etho ds in A pplied M echanics and Eng ineering,2005,194:3607-3836. [17]Aimi A,Dilig enti M.H y per sing ular ker nel integr ationin3D Galerkin boundar y element metho d[J].J ofCo mp and Appl M ath,2002,138:51-72.[18]H ar ris P J,Chen K.On efficient preconditioners fo riterative so lutio n o f a Galerkin boundar y elementequat ion for the three-dimensional ex terior Helmholt z pr oblem[J].J o f Co mp and A ppl M ath,2003,156:303-318.1362中国科学技术大学学报第37卷。
广义循环布尔矩阵三明治半群中的完全正则元
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Multiple Integrals
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Double Integrals in Polar Coordinates Recall that the polar coordinates (r, θ) of a point in the rectangular coordinates (x, y ) are defined as: y x = r cos θ , where r = x2 + y 2 and θ = tan−1 with 0 ≤ θ < 2π . y = r sin θ x If f (x, y ) is continuous on a polar rectangle R given by 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β , where 0 ≤ β − α ≤ 2π , then
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Consensus and Cooperation in Networked Multi-Agent Systems
Consensus and Cooperation in Networked Multi-Agent SystemsAlgorithms that provide rapid agreement and teamwork between all participants allow effective task performance by self-organizing networked systems.By Reza Olfati-Saber,Member IEEE,J.Alex Fax,and Richard M.Murray,Fellow IEEEABSTRACT|This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow,robustness to changes in network topology due to link/node failures,time-delays,and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided.Our analysis frame-work is based on tools from matrix theory,algebraic graph theory,and control theory.We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators,flocking,formation control,fast consensus in small-world networks,Markov processes and gossip-based algo-rithms,load balancing in networks,rendezvous in space, distributed sensor fusion in sensor networks,and belief propagation.We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms.A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions.Simu-lation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations.KEYWORDS|Consensus algorithms;cooperative control; flocking;graph Laplacians;information fusion;multi-agent systems;networked control systems;synchronization of cou-pled oscillators I.INTRODUCTIONConsensus problems have a long history in computer science and form the foundation of the field of distributed computing[1].Formal study of consensus problems in groups of experts originated in management science and statistics in1960s(see DeGroot[2]and references therein). The ideas of statistical consensus theory by DeGroot re-appeared two decades later in aggregation of information with uncertainty obtained from multiple sensors1[3]and medical experts[4].Distributed computation over networks has a tradition in systems and control theory starting with the pioneering work of Borkar and Varaiya[5]and Tsitsiklis[6]and Tsitsiklis,Bertsekas,and Athans[7]on asynchronous asymptotic agreement problem for distributed decision-making systems and parallel computing[8].In networks of agents(or dynamic systems),B con-sensus[means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents.A B consensus algorithm[(or protocol)is an interaction rule that specifies the information exchange between an agent and all of its neighbors on the network.2 The theoretical framework for posing and solving consensus problems for networked dynamic systems was introduced by Olfati-Saber and Murray in[9]and[10] building on the earlier work of Fax and Murray[11],[12]. The study of the alignment problem involving reaching an agreement V without computing any objective functions V appeared in the work of Jadbabaie et al.[13].Further theoretical extensions of this work were presented in[14] and[15]with a look toward treatment of directed infor-mation flow in networks as shown in Fig.1(a).Manuscript received August8,2005;revised September7,2006.This work was supported in part by the Army Research Office(ARO)under Grant W911NF-04-1-0316. R.Olfati-Saber is with Dartmouth College,Thayer School of Engineering,Hanover,NH03755USA(e-mail:olfati@).J.A.Fax is with Northrop Grumman Corp.,Woodland Hills,CA91367USA(e-mail:alex.fax@).R.M.Murray is with the California Institute of Technology,Control and Dynamical Systems,Pasadena,CA91125USA(e-mail:murray@).Digital Object Identifier:10.1109/JPROC.2006.8872931This is known as sensor fusion and is an important application of modern consensus algorithms that will be discussed later.2The term B nearest neighbors[is more commonly used in physics than B neighbors[when applied to particle/spin interactions over a lattice (e.g.,Ising model).Vol.95,No.1,January2007|Proceedings of the IEEE2150018-9219/$25.00Ó2007IEEEThe common motivation behind the work in [5],[6],and [10]is the rich history of consensus protocols in com-puter science [1],whereas Jadbabaie et al.[13]attempted to provide a formal analysis of emergence of alignment in the simplified model of flocking by Vicsek et al.[16].The setup in [10]was originally created with the vision of de-signing agent-based amorphous computers [17],[18]for collaborative information processing in ter,[10]was used in development of flocking algorithms with guaranteed convergence and the capability to deal with obstacles and adversarial agents [19].Graph Laplacians and their spectral properties [20]–[23]are important graph-related matrices that play a crucial role in convergence analysis of consensus and alignment algo-rithms.Graph Laplacians are an important point of focus of this paper.It is worth mentioning that the second smallest eigenvalue of graph Laplacians called algebraic connectivity quantifies the speed of convergence of consensus algo-rithms.The notion of algebraic connectivity of graphs has appeared in a variety of other areas including low-density parity-check codes (LDPC)in information theory and com-munications [24],Ramanujan graphs [25]in number theory and quantum chaos,and combinatorial optimization prob-lems such as the max-cut problem [21].More recently,there has been a tremendous surge of interest V among researchers from various disciplines of engineering and science V in problems related to multia-gent networked systems with close ties to consensus prob-lems.This includes subjects such as consensus [26]–[32],collective behavior of flocks and swarms [19],[33]–[37],sensor fusion [38]–[40],random networks [41],[42],syn-chronization of coupled oscillators [42]–[46],algebraic connectivity 3of complex networks [47]–[49],asynchro-nous distributed algorithms [30],[50],formation control for multirobot systems [51]–[59],optimization-based co-operative control [60]–[63],dynamic graphs [64]–[67],complexity of coordinated tasks [68]–[71],and consensus-based belief propagation in Bayesian networks [72],[73].A detailed discussion of selected applications will be pre-sented shortly.In this paper,we focus on the work described in five key papers V namely,Jadbabaie,Lin,and Morse [13],Olfati-Saber and Murray [10],Fax and Murray [12],Moreau [14],and Ren and Beard [15]V that have been instrumental in paving the way for more recent advances in study of self-organizing networked systems ,or swarms .These networked systems are comprised of locally interacting mobile/static agents equipped with dedicated sensing,computing,and communication devices.As a result,we now have a better understanding of complex phenomena such as flocking [19],or design of novel information fusion algorithms for sensor networks that are robust to node and link failures [38],[72]–[76].Gossip-based algorithms such as the push-sum protocol [77]are important alternatives in computer science to Laplacian-based consensus algorithms in this paper.Markov processes establish an interesting connection between the information propagation speed in these two categories of algorithms proposed by computer scientists and control theorists [78].The contribution of this paper is to present a cohesive overview of the key results on theory and applications of consensus problems in networked systems in a unified framework.This includes basic notions in information consensus and control theoretic methods for convergence and performance analysis of consensus protocols that heavily rely on matrix theory and spectral graph theory.A byproduct of this framework is to demonstrate that seem-ingly different consensus algorithms in the literature [10],[12]–[15]are closely related.Applications of consensus problems in areas of interest to researchers in computer science,physics,biology,mathematics,robotics,and con-trol theory are discussed in this introduction.A.Consensus in NetworksThe interaction topology of a network of agents is rep-resented using a directed graph G ¼ðV ;E Þwith the set of nodes V ¼f 1;2;...;n g and edges E V ÂV .TheFig.1.Two equivalent forms of consensus algorithms:(a)a networkof integrator agents in which agent i receives the state x j of its neighbor,agent j ,if there is a link ði ;j Þconnecting the two nodes;and (b)the block diagram for a network of interconnecteddynamic systems all with identical transfer functions P ðs Þ¼1=s .The collective networked system has a diagonal transfer function and is a multiple-input multiple-output (MIMO)linear system.3To be defined in Section II-A.Olfati-Saber et al.:Consensus and Cooperation in Networked Multi-Agent Systems216Proceedings of the IEEE |Vol.95,No.1,January 2007neighbors of agent i are denoted by N i ¼f j 2V :ði ;j Þ2E g .According to [10],a simple consensus algorithm to reach an agreement regarding the state of n integrator agents with dynamics _x i ¼u i can be expressed as an n th-order linear system on a graph_x i ðt Þ¼X j 2N ix j ðt ÞÀx i ðt ÞÀÁþb i ðt Þ;x i ð0Þ¼z i2R ;b i ðt Þ¼0:(1)The collective dynamics of the group of agents following protocol (1)can be written as_x ¼ÀLx(2)where L ¼½l ij is the graph Laplacian of the network and itselements are defined as follows:l ij ¼À1;j 2N i j N i j ;j ¼i :&(3)Here,j N i j denotes the number of neighbors of node i (or out-degree of node i ).Fig.1shows two equivalent forms of the consensus algorithm in (1)and (2)for agents with a scalar state.The role of the input bias b in Fig.1(b)is defined later.According to the definition of graph Laplacian in (3),all row-sums of L are zero because of P j l ij ¼0.Therefore,L always has a zero eigenvalue 1¼0.This zero eigenvalues corresponds to the eigenvector 1¼ð1;...;1ÞT because 1belongs to the null-space of L ðL 1¼0Þ.In other words,an equilibrium of system (2)is a state in the form x üð ;...; ÞT ¼ 1where all nodes agree.Based on ana-lytical tools from algebraic graph theory [23],we later show that x Ãis a unique equilibrium of (2)(up to a constant multiplicative factor)for connected graphs.One can show that for a connected network,the equilibrium x üð ;...; ÞT is globally exponentially stable.Moreover,the consensus value is ¼1=n P i z i that is equal to the average of the initial values.This im-plies that irrespective of the initial value of the state of each agent,all agents reach an asymptotic consensus regarding the value of the function f ðz Þ¼1=n P i z i .While the calculation of f ðz Þis simple for small net-works,its implications for very large networks is more interesting.For example,if a network has n ¼106nodes and each node can only talk to log 10ðn Þ¼6neighbors,finding the average value of the initial conditions of the nodes is more complicated.The role of protocol (1)is to provide a systematic consensus mechanism in such a largenetwork to compute the average.There are a variety of functions that can be computed in a similar fashion using synchronous or asynchronous distributed algorithms (see [10],[28],[30],[73],and [76]).B.The f -Consensus Problem and Meaning of CooperationTo understand the role of cooperation in performing coordinated tasks,we need to distinguish between un-constrained and constrained consensus problems.An unconstrained consensus problem is simply the alignment problem in which it suffices that the state of all agents asymptotically be the same.In contrast,in distributed computation of a function f ðz Þ,the state of all agents has to asymptotically become equal to f ðz Þ,meaning that the consensus problem is constrained.We refer to this con-strained consensus problem as the f -consensus problem .Solving the f -consensus problem is a cooperative task and requires willing participation of all the agents.To demonstrate this fact,suppose a single agent decides not to cooperate with the rest of the agents and keep its state unchanged.Then,the overall task cannot be performed despite the fact that the rest of the agents reach an agree-ment.Furthermore,there could be scenarios in which multiple agents that form a coalition do not cooperate with the rest and removal of this coalition of agents and their links might render the network disconnected.In a dis-connected network,it is impossible for all nodes to reach an agreement (unless all nodes initially agree which is a trivial case).From the above discussion,cooperation can be infor-mally interpreted as B giving consent to providing one’s state and following a common protocol that serves the group objective.[One might think that solving the alignment problem is not a cooperative task.The justification is that if a single agent (called a leader)leaves its value unchanged,all others will asymptotically agree with the leader according to the consensus protocol and an alignment is reached.However,if there are multiple leaders where two of whom are in disagreement,then no consensus can be asymptot-ically reached.Therefore,alignment is in general a coop-erative task as well.Formal analysis of the behavior of systems that involve more than one type of agent is more complicated,partic-ularly,in presence of adversarial agents in noncooperative games [79],[80].The focus of this paper is on cooperative multi-agent systems.C.Iterative Consensus and Markov ChainsIn Section II,we show how an iterative consensus algorithm that corresponds to the discrete-time version of system (1)is a Markov chainðk þ1Þ¼ ðk ÞP(4)Olfati-Saber et al.:Consensus and Cooperation in Networked Multi-Agent SystemsVol.95,No.1,January 2007|Proceedings of the IEEE217with P ¼I À L and a small 90.Here,the i th element of the row vector ðk Þdenotes the probability of being in state i at iteration k .It turns out that for any arbitrary graph G with Laplacian L and a sufficiently small ,the matrix P satisfies the property Pj p ij ¼1with p ij !0;8i ;j .Hence,P is a valid transition probability matrix for the Markov chain in (4).The reason matrix theory [81]is so widely used in analysis of consensus algorithms [10],[12]–[15],[64]is primarily due to the structure of P in (4)and its connection to graphs.4There are interesting connections between this Markov chain and the speed of information diffusion in gossip-based averaging algorithms [77],[78].One of the early applications of consensus problems was dynamic load balancing [82]for parallel processors with the same structure as system (4).To this date,load balancing in networks proves to be an active area of research in computer science.D.ApplicationsMany seemingly different problems that involve inter-connection of dynamic systems in various areas of science and engineering happen to be closely related to consensus problems for multi-agent systems.In this section,we pro-vide an account of the existing connections.1)Synchronization of Coupled Oscillators:The problem of synchronization of coupled oscillators has attracted numer-ous scientists from diverse fields including physics,biology,neuroscience,and mathematics [83]–[86].This is partly due to the emergence of synchronous oscillations in coupled neural oscillators.Let us consider the generalized Kuramoto model of coupled oscillators on a graph with dynamics_i ¼ Xj 2N isin ð j À i Þþ!i (5)where i and !i are the phase and frequency of the i thoscillator.This model is the natural nonlinear extension of the consensus algorithm in (1)and its linearization around the aligned state 1¼...¼ n is identical to system (2)plus a nonzero input bias b i ¼ð!i À"!Þ= with "!¼1=n P i !i after a change of variables x i ¼ð i À"!t Þ= .In [43],Sepulchre et al.show that if is sufficiently large,then for a network with all-to-all links,synchroni-zation to the aligned state is globally achieved for all ini-tial states.Recently,synchronization of networked oscillators under variable time-delays was studied in [45].We believe that the use of convergence analysis methods that utilize the spectral properties of graph Laplacians willshed light on performance and convergence analysis of self-synchrony in oscillator networks [42].2)Flocking Theory:Flocks of mobile agents equipped with sensing and communication devices can serve as mobile sensor networks for massive distributed sensing in an environment [87].A theoretical framework for design and analysis of flocking algorithms for mobile agents with obstacle-avoidance capabilities is developed by Olfati-Saber [19].The role of consensus algorithms in particle-based flocking is for an agent to achieve velocity matching with respect to its neighbors.In [19],it is demonstrated that flocks are networks of dynamic systems with a dynamic topology.This topology is a proximity graph that depends on the state of all agents and is determined locally for each agent,i.e.,the topology of flocks is a state-dependent graph.The notion of state-dependent graphs was introduced by Mesbahi [64]in a context that is independent of flocking.3)Fast Consensus in Small-Worlds:In recent years,network design problems for achieving faster consensus algorithms has attracted considerable attention from a number of researchers.In Xiao and Boyd [88],design of the weights of a network is considered and solved using semi-definite convex programming.This leads to a slight increase in algebraic connectivity of a network that is a measure of speed of convergence of consensus algorithms.An alternative approach is to keep the weights fixed and design the topology of the network to achieve a relatively high algebraic connectivity.A randomized algorithm for network design is proposed by Olfati-Saber [47]based on random rewiring idea of Watts and Strogatz [89]that led to creation of their celebrated small-world model .The random rewiring of existing links of a network gives rise to considerably faster consensus algorithms.This is due to multiple orders of magnitude increase in algebraic connectivity of the network in comparison to a lattice-type nearest-neighbort graph.4)Rendezvous in Space:Another common form of consensus problems is rendezvous in space [90],[91].This is equivalent to reaching a consensus in position by a num-ber of agents with an interaction topology that is position induced (i.e.,a proximity graph).We refer the reader to [92]and references therein for a detailed discussion.This type of rendezvous is an unconstrained consensus problem that becomes challenging under variations in the network topology.Flocking is somewhat more challenging than rendezvous in space because it requires both interagent and agent-to-obstacle collision avoidance.5)Distributed Sensor Fusion in Sensor Networks:The most recent application of consensus problems is distrib-uted sensor fusion in sensor networks.This is done by posing various distributed averaging problems require to4In honor of the pioneering contributions of Oscar Perron (1907)to the theory of nonnegative matrices,were refer to P as the Perron Matrix of graph G (See Section II-C for details).Olfati-Saber et al.:Consensus and Cooperation in Networked Multi-Agent Systems218Proceedings of the IEEE |Vol.95,No.1,January 2007implement a Kalman filter [38],[39],approximate Kalman filter [74],or linear least-squares estimator [75]as average-consensus problems .Novel low-pass and high-pass consensus filters are also developed that dynamically calculate the average of their inputs in sensor networks [39],[93].6)Distributed Formation Control:Multivehicle systems are an important category of networked systems due to their commercial and military applications.There are two broad approaches to distributed formation control:i)rep-resentation of formations as rigid structures [53],[94]and the use of gradient-based controls obtained from their structural potentials [52]and ii)representation of form-ations using the vectors of relative positions of neighboring vehicles and the use of consensus-based controllers with input bias.We discuss the later approach here.A theoretical framework for design and analysis of distributed controllers for multivehicle formations of type ii)was developed by Fax and Murray [12].Moving in formation is a cooperative task and requires consent and collaboration of every agent in the formation.In [12],graph Laplacians and matrix theory were extensively used which makes one wonder whether relative-position-based formation control is a consensus problem.The answer is yes.To see this,consider a network of self-interested agents whose individual desire is to minimize their local cost U i ðx Þ¼Pj 2N i k x j Àx i Àr ij k 2via a distributed algorithm (x i is the position of vehicle i with dynamics _x i ¼u i and r ij is a desired intervehicle relative-position vector).Instead,if the agents use gradient-descent algorithm on the collective cost P n i ¼1U i ðx Þusing the following protocol:_x i ¼Xj 2N iðx j Àx i Àr ij Þ¼Xj 2N iðx j Àx i Þþb i (6)with input bias b i ¼Pj 2N i r ji [see Fig.1(b)],the objective of every agent will be achieved.This is the same as the consensus algorithm in (1)up to the nonzero bias terms b i .This nonzero bias plays no role in stability analysis of sys-tem (6).Thus,distributed formation control for integrator agents is a consensus problem.The main contribution of the work by Fax and Murray is to extend this scenario to the case where all agents are multiinput multioutput linear systems _x i ¼Ax i þBu i .Stability analysis of relative-position-based formation control for multivehicle systems is extensively covered in Section IV.E.OutlineThe outline of the paper is as follows.Basic concepts and theoretical results in information consensus are presented in Section II.Convergence and performance analysis of consensus on networks with switching topology are given in Section III.A theoretical framework for cooperative control of formations of networked multi-vehicle systems is provided in Section IV.Some simulationresults related to consensus in complex networks including small-worlds are presented in Section V.Finally,some concluding remarks are stated in Section VI.RMATION CONSENSUSConsider a network of decision-making agents with dynamics _x i ¼u i interested in reaching a consensus via local communication with their neighbors on a graph G ¼ðV ;E Þ.By reaching a consensus,we mean asymptot-ically converging to a one-dimensional agreement space characterized by the following equation:x 1¼x 2¼...¼x n :This agreement space can be expressed as x ¼ 1where 1¼ð1;...;1ÞT and 2R is the collective decision of the group of agents.Let A ¼½a ij be the adjacency matrix of graph G .The set of neighbors of a agent i is N i and defined byN i ¼f j 2V :a ij ¼0g ;V ¼f 1;...;n g :Agent i communicates with agent j if j is a neighbor of i (or a ij ¼0).The set of all nodes and their neighbors defines the edge set of the graph as E ¼fði ;j Þ2V ÂV :a ij ¼0g .A dynamic graph G ðt Þ¼ðV ;E ðt ÞÞis a graph in which the set of edges E ðt Þand the adjacency matrix A ðt Þare time-varying.Clearly,the set of neighbors N i ðt Þof every agent in a dynamic graph is a time-varying set as well.Dynamic graphs are useful for describing the network topology of mobile sensor networks and flocks [19].It is shown in [10]that the linear system_x i ðt Þ¼Xj 2N ia ij x j ðt ÞÀx i ðt ÞÀÁ(7)is a distributed consensus algorithm ,i.e.,guarantees con-vergence to a collective decision via local interagent interactions.Assuming that the graph is undirected (a ij ¼a ji for all i ;j ),it follows that the sum of the state of all nodes is an invariant quantity,or P i _xi ¼0.In particular,applying this condition twice at times t ¼0and t ¼1gives the following result¼1n Xix i ð0Þ:In other words,if a consensus is asymptotically reached,then necessarily the collective decision is equal to theOlfati-Saber et al.:Consensus and Cooperation in Networked Multi-Agent SystemsVol.95,No.1,January 2007|Proceedings of the IEEE219average of the initial state of all nodes.A consensus algo-rithm with this specific invariance property is called an average-consensus algorithm [9]and has broad applications in distributed computing on networks (e.g.,sensor fusion in sensor networks).The dynamics of system (7)can be expressed in a compact form as_x ¼ÀLx(8)where L is known as the graph Laplacian of G .The graph Laplacian is defined asL ¼D ÀA(9)where D ¼diag ðd 1;...;d n Þis the degree matrix of G with elements d i ¼Pj ¼i a ij and zero off-diagonal elements.By definition,L has a right eigenvector of 1associated with the zero eigenvalue 5because of the identity L 1¼0.For the case of undirected graphs,graph Laplacian satisfies the following sum-of-squares (SOS)property:x T Lx ¼12Xði ;j Þ2Ea ij ðx j Àx i Þ2:(10)By defining a quadratic disagreement function as’ðx Þ¼12x T Lx(11)it becomes apparent that algorithm (7)is the same as_x ¼Àr ’ðx Þor the gradient-descent algorithm.This algorithm globallyasymptotically converges to the agreement space provided that two conditions hold:1)L is a positive semidefinite matrix;2)the only equilibrium of (7)is 1for some .Both of these conditions hold for a connected graph and follow from the SOS property of graph Laplacian in (10).Therefore,an average-consensus is asymptotically reached for all initial states.This fact is summarized in the following lemma.Lemma 1:Let G be a connected undirected graph.Then,the algorithm in (7)asymptotically solves an average-consensus problem for all initial states.A.Algebraic Connectivity and Spectral Propertiesof GraphsSpectral properties of Laplacian matrix are instrumen-tal in analysis of convergence of the class of linear consensus algorithms in (7).According to Gershgorin theorem [81],all eigenvalues of L in the complex plane are located in a closed disk centered at Áþ0j with a radius of Á¼max i d i ,i.e.,the maximum degree of a graph.For undirected graphs,L is a symmetric matrix with real eigenvalues and,therefore,the set of eigenvalues of L can be ordered sequentially in an ascending order as0¼ 1 2 ÁÁÁ n 2Á:(12)The zero eigenvalue is known as the trivial eigenvalue of L .For a connected graph G , 290(i.e.,the zero eigenvalue is isolated).The second smallest eigenvalue of Laplacian 2is called algebraic connectivity of a graph [20].Algebraic connectivity of the network topology is a measure of performance/speed of consensus algorithms [10].Example 1:Fig.2shows two examples of networks of integrator agents with different topologies.Both graphs are undirected and have 0–1weights.Every node of the graph in Fig.2(a)is connected to its 4nearest neighbors on a ring.The other graph is a proximity graph of points that are distributed uniformly at random in a square.Every node is connected to all of its spatial neighbors within a closed ball of radius r 90.Here are the important degree information and Laplacian eigenvalues of these graphsa Þ 1¼0; 2¼0:48; n ¼6:24;Á¼4b Þ 1¼0; 2¼0:25; n ¼9:37;Á¼8:(13)In both cases, i G 2Áfor all i .B.Convergence Analysis for Directed Networks The convergence analysis of the consensus algorithm in (7)is equivalent to proving that the agreement space characterized by x ¼ 1; 2R is an asymptotically stable equilibrium of system (7).The stability properties of system (7)is completely determined by the location of the Laplacian eigenvalues of the network.The eigenvalues of the adjacency matrix are irrelevant to the stability analysis of system (7),unless the network is k -regular (all of its nodes have the same degree k ).The following lemma combines a well-known rank property of graph Laplacians with Gershgorin theorem to provide spectral characterization of Laplacian of a fixed directed network G .Before stating the lemma,we need to define the notion of strong connectivity of graphs.A graph5These properties were discussed earlier in the introduction for graphs with 0–1weights.Olfati-Saber et al.:Consensus and Cooperation in Networked Multi-Agent Systems220Proceedings of the IEEE |Vol.95,No.1,January 2007。
Explicit symplectic integrators for solving non-separable Hamiltonians
HˆA = eT Tˆ + eV Vˆ + ε2eTT V [TˆTˆVˆ ] +ε2eV TV [Vˆ TˆVˆ ] + O(ε4)
(2.9)
where eT , eTV , eTTV , etc., are functions of {ti} and {vi} and where condensed commutator brackets, [TˆTˆVˆ ] = [Tˆ, [Tˆ, Vˆ ]], [TˆVˆ TˆVˆ ] = [Tˆ, [Vˆ , [Tˆ, Vˆ ]]], etc., are used. From the way Lie operators are defined via (2.1), one can convert operators back to functions1,7 via [T, V ] → {V, T } = −{T, V }, yielding
HA = eT T + eV V + ε2eT TV {T T V } +ε2eV T V {V T V } + O(ε4),
(2.10)
where again, condensed Poisson brackets, {T T V } = {T, {T, V }}, etc., are used. For a separable Hamiltonian of the form (2.3), we have
The product of operators in (2.7) can be combined by use of the Baker-Campbell-Hausdorff (BCH) formula to give
N
etiεTˆeviεVˆ = eεHˆA ,
The Quantized Knizhnik-Zamolodchikov Equation in Tensor Products of Irreducible sl(2)-Modul
1. Introduction. 1.1. In this paper we solve the rational quantized Knizhnik-Zamolodchikov difference equation ( qKZ ) with values in a tensor product of irreducible highest weight sl2 -modules. The rational qKZ equation is a system of difference equations for a function Ψ(z1 , ..., zn ) with values in a tensor product M1 ⊗ ... ⊗ Mn of sl2 modules. The system of equations has the form Ψ(z1 , . . . , zm + p, . . . , zn ) = Rm,m−1 (zm − zm−1 + p) . . . Rm,1 (zm − z1 + p)e−µhm × ×Rm,n (zm − zn ) . . . Rm,m+1 (zm − zm+1 )Ψ(z1 , . . . , zn ), m = 1, . . . , n, where p, µ are complex parameters of the qKZ equation, h is a generator of the Cartan subalgebra of sl2 , hm is the operator h acting in the m-th factor, Ri,j (x) is the rational R-matrix RMi Mj (x) ∈ End(Mi ⊗ Mj ) acting in the i-th and j -th factors of the tensor product. In this paper we consider only steps p with negative real part. The qKZ equation is an important system of difference equations. The qKZ equation was introduced in [FR] as an equation for matrix elements of vertex operators of a quantum affine algebra. An important special case of the qKZ equation had been introduced earlier in [S] as equations for form factors in integrable quantum field theory. Later, the qKZ equation was derived as an equation for correlation functions in lattice integrable models, cf. [JM] and references therein. Solutions of the rational qKZ equation with values in a tensor product of sl2 Verma modules Vλ1 ⊗ ... ⊗ Vλn with generic highest weights λ1 , . . . , λn were constructed in [TV1]. The solutions have the form Ψ(z ) =
Endosperm - an integrator of seed growth and development
Endosperm:an integrator of seed growth and development Fre´de´ric Berger1,Paul E Grini2and Arp Schnittger3Plant reproduction relies on interactions between parental and zygotic components.Elaborate reciprocal signaling pathways enable coordination of the genetic programs between these components.Afirst and important step in this communication is the tight control of cell cycle events in the gametes prior to fertilization.This prepares for coordinated fertilization and the initiation of seed development.The dialog between the various actors of reproduction extends after fertilization,with the endosperm taking a central role.Importantly,the endosperm mediates a maternal input that is based on memory of the transcriptional states of imprinted genes,which is crucial for harmonious seed growth.Our current knowledge suggests that the endosperm is an integrator of the different components and genetic programs that are involved in seed development.Addresses1Temasek Lifescience Laboratory,1Research Link,National University of Singapore,117604,Singapore2Department of Molecular Biosciences,University of Oslo,Blindernveien 31,0316Oslo,Norway3University group at the Max-Planck-Institute for Plant Breeding Research,University of Cologne,Department of Botany III,Max-Delbru¨ck-Laboratorium,Carl von Linne´Weg10,50829Ko¨ln,GermanyCorresponding author:Berger,Fre´de´ric(fred@.sg),Grini,Paul E (p.e.grini@bio.uio.no)and Schnittger,Arp(schnitt@mpiz-koeln.mpg.de) Current Opinion in Plant Biology2006,9:664–670This review comes from a themed issue onCell biologyEdited by Laurie G Smith and Ulrike MayerAvailable online29th September20061369-5266/$–see front matter#2006Elsevier Ltd.All rights reserved.DOI10.1016/j.pbi.2006.09.015IntroductionIn plants,the diploid generation produces a special lineage leading to meiosis.The haploid meiotic products(spores) then develop as morphologically distinct life forms,the male and female gametophytes,which will differentiate a germ-line and produce the male and female gametes (Figure1).Inflowering plants,double-fertilization has evolved parallel to specialization of the haploid gameto-phyte towards reproduction.The gametophytic life form has been drastically reduced to be nearly equivalent to the germ line.In Arabidopsis,as in the majority offlowering plants,the pollen grain develops from an initial asymmetric division of the haploid meiotic product,Pollen Mitosis I (PMI)(Figure1).The smaller generative cell undergoes PMII and divides again,giving rise to two sperm cells.The female embryo sac develops from only one of the meiotic products and typically undergoes three successive syncy-tial divisions followed by cellularization into seven cells, including the two female gametes,namely the egg cell and the central cell(Figure1;[1]).At the time of fertilization,the pollen grain grows a pollen tube that delivers the two sperm cells to the embryo sac. Despite significant advances in our understanding of guidance mechanisms for the pollen tube and the tran-scriptional activity of the gametes[2–4],our knowledge on the actual mechanism of fertilization remains extre-mely limited.A remarkable recent study has shown, however,that the fusion between the sperm cells and the female gametes involves the conserved protein GENERATIVE CELL SPECIFIC1(GCS1)[5 ]. One sperm cell fertilizes the egg cell,giving rise to a zygote from which the embryo develops.The other sperm cell fuses with the central cell generating the endosperm,which nurtures embryo development in the seed(Figure1).In this review,the signaling events that follow double-fertilization inflowering plants are outlined with parti-cular emphasis on the following points.What signals control the cell cycle at fertilization?What mechanisms co-ordinate the relative growth of the three main seed components,the embryo,the endosperm and the mater-nal seed integuments?Finally,what roles do gametes play in embryo and endosperm development?Control of cell cycle progression beforeand after fertilizationIn contrast to cell cycle control in animal gametes,cell cycle control in plant gametes has remained largely unknown until recently.In Arabidopsis,direct measure-ments of DNA content suggest that sperm cells enter a new S-phase after PMII,which is presumably completed either before or during pollen tube growth[6,7].Thus, when delivered to the embryo sac,sperm cells are likely to be in G2phase[8].Further evidence of a fusion at G2 phase comes from a mutant in the Arabidopsis cyclin-dependent kinase gene CDKA;1,which encodes a homo-log of the yeast cdc2a kinase.In cdka;1mutant pollen,the generative cell fails to enter PMII and remains in the G2 phase with a DNA content of about2C[9 ,10].Impor-tantly,this mutant pollen is able to complete fertilization [9 ].To prevent aneuploidy and allow proper develop-ment of the zygote,the two nuclei of the gametes need to fuse in the same cell cycle phase.The simplest assump-tion derived from the cell cycle phase of the male gamete is that female gametes also reach the G2phase,matchingthe cell cycle stage of sperm cells.This assumption was recently supported by measurements of DNA content in tobacco gametes [8].Conserved chromatin remodeling Polycomb group (Pc-G)complexes control the arrest of the Arabidopsis female gametophyte.The Pc-G complex FERTILIZATION INDEPENDENT SEED (FIS)contains the SET domain protein MEDEA (MEA)[11],the VEFS domain protein FIS2[12],and the WD40domain protein FERTILIZA-TION INDEPENDENT ENDOSPERM (FIE)[13].Loss-of-function mutations of FIS genes cause autono-mous onset of cell division in the central cell in the absence of fertilization [14,15].The FIS complex also includes the WD40protein MULTICOPY SUPPRESSOR OF IRA 1(MSI1)[16,17].Unexpectedly,when compared to the fis mutants,msi mutants show additional autonomous division in the egg cell that leads to a non-viable parthenogeneticembryo [18 ].This suggests that partially distinct mechan-isms control the arrest of the egg cell and the central cell (Figure 2a).MSI1interacts with plant homologs of the mammalian tumor suppressor protein Retinoblastoma (Rb)[19].Autonomous cell proliferation in the embryo sac has been reported in plants that carry null alleles of the Arabidopsis Rb homolog RBR1[20].However,the pleiotropy of the phenotype has prevented unambiguous identification of the origin of proliferating cells in rbr1.The rbr1mutant phenotype is likely to be compound.RBR1can interact with components of the FIS-complex [21],but could also act upstream of the FIS Pc-G complex and together with MSI1to directly control cell cycle progression in the female gametophyte.In analogy to its function in root stem cells,RBR1might also be involved in specifying gamete identity [22].Endosperm in seed growth and development Berger,Grini and Schnittger 665Figure 1Life cycle of flowering plants.In flowering plants,the predominant life form is the sporophyte (the green plant).In the flowers,the sporophyte will asexually generate female and male spores.The male spore will undergo two rounds of cell divisions,PMI and PMII.The mature pollencomprises a large vegetative cell and embedded in this,two sperm cells,the actual gametes (shown with red cytoplasm).Female gametogenesis includes three rounds of nuclear divisions followed by cellularization,resulting in a seven-cell embryo sac.The embryo sac contains the egg cell (again shown with red cytoplasm)flanked by two synergid cells and the central cell (shown in yellow),which contains a large nucleus.During the double fertilization process,sperm cells are delivered by the pollen tube to the embryo sac.One sperm cell fuses with the eggcell and generates the zygote,whereas the second sperm cell fuses with the central cell,giving rise to the endosperm.The endosperm supports the developing embryo.Double-fertilization represents the end of the gametophytic life stage and the beginning of the sporophytic phase.In conclusion,the cell cycle arrest of the central cell is controlled by the FIS pathway,whereas the egg cell arrest probably depends on a MSI1-dependent action that is independent of the Polycomb Group FIS pathway. Proper differentiation of both female gametes might involve the function of RBR1.A recent report[9 ]on a functional knock-out of the Arabidopsis CDKA1;1has provided evidence for mechanisms that might be involved in cell-cycle control prior to fertilization.Co-ordination of growth between the different seed componentsDuring seed growth,co-ordination is necessary between the embryo,the endosperm,the maternally contributed seed integuments and the nucellus,which is also of maternal origin.No experimental data have yet been obtained for the role of the nucellus,but recent advances have demonstrated that interplay between endosperm and seed integument growth is essential for seed size determination in Arabidopsis(Figure2).Mutations in the HAIKU(IKU)genes decrease endosperm size and even-tually embryo and seed size[23].The HAIKU genes IKU2 and MINISEED3(MINI3)encode a leucine-rich-repeat kinase and a WRKY transcription factor,respectively[24]. These genes are expressed in endosperm immediately after fertilization.The decreased endosperm size of iku mutants is accompanied by a decrease in cell elongation in the seed integuments,indicating communication between these two genetically distinct seed components (Figure3c–f;[25 ]).Similarly,reducing the degree of cell elongation in the seed integuments reduces endosperm growth.Conversely,increasing the number of seed inte-gument cells causes a symmetrical increase in endosperm growth[26,27].In Arabidopsis,thefinal size of the seed is determined before the embryo initiates the major phase of cell proliferation after the heart embryo stage (Figure2).These results indicate the capacity of seed integuments to regulate endosperm growth by a maternal sporophytic effect(Figure2b).In cereal species,the embryo remains confined to a small volume in the mature seed and the endosperm extends its development until seed maturity and stores reserves.Hence,the model of seed size control established in Arabidopsis is likely to be applicable to seed development in cereals.The mechan-istic nature of the communication between the endo-sperm and the integuments remains unknown.A likely factor could be biophysical forces;a growing endosperm might exert mechanical tensions on the integument cells [25 ,28].How the embryo adjusts its degree of cell proliferation to the space available(as seen in iku seeds)is not yet understood.However,this observation indicates some kind of communication from the endosperm to the munication from the embryo to the endo-sperm has been revealed by analysis of the mutant for the Arabidopsis homolog of the cyclin dependent kinase cdc2a, CDKA;1[9 ].In heterozygous cdka;1null mutants,50% of the pollen contain a single sperm cell[9 ,10]that exclusively fertilizes the egg cell.The development of the zygotic embryo has a dramatic effect because it triggers nuclear divisions of the central cell and initiates endosperm development(Figures3a,b;[9 ]).Thus, either the fertilization process or the developing embryo provides a signal that counteracts cell cycle arrest in the central cell and stimulates endosperm development666Cell biologyFigure2Interactions between seed components.(a)Before fertilization,a negative signal embodied by the action of the chromatin remodelingcomplex FIS restricts the premature proliferation of the central cell.An equivalent repression depends on MSI1and targets the egg cell.After fertilization,a positive signal is derived from the fertilization of the egg cell that triggers endosperm development if the central cell is not fertilized.The origin and nature of this positive signal is currently not known.(b)During later stages,reciprocal signaling between the endosperm(yellow with black nuclei),the embryo(blue)and the seed integuments(green)is required to achieve proper seed growth.The endosperm takes a central role as an integrator and mediator of these signals.The endosperm controls cell elongation(black double arrows)in the seed integuments(dark red arrows). Reciprocally,the seed integuments restrict the potential growth of the endosperm(red arrows).As a result,the final seed size is determined.(c)Finally,endosperm trophic inputs and potential signals to the embryo sustain cell proliferation in the embryo.(d)At the mature stage,the embryo occupies most of the space initially created by the interactions between the endosperm and the seed integuments.(Figure 2a).The developing endosperm in cdka;1-ferti-lized seeds arrests after a few rounds of the cell cycle,whereas the embryo develops until the globular stage.Such seeds remain small because they do not expand the seed integument.These observations support the essen-tial role played by endosperm and the comparatively limited role played by the embryo in controlling seed size.Coordination of maternal and zygotic cues during seed developmentAlthough extensive screens have been performed,only a handful of mutations that cause maternal effects have been isolated over the past few years [29,30].However,using a transposon-based mutagenesis approach,a recent study discovered a larger number of such mutants [31],and suggests that there might be a larger diversity of maternal effects than previously thought.The maternal influence might be inherited by the endo-sperm by two routes.First,the central cell as progenitor of the endosperm contributes much cytoplasm to the endo-sperm.The endosperm could inherit maternally derived mRNA and proteins that are located at specific sub-domains in the central cell.Indeed,the central cell shows cytological evidence of polarity,which could prefigure the future endosperm antero-posterior organization [32].Direct maternal effects that originate from the central cell constitute an attractive hypothesis,but one that is supported by little,if any,direct evidence to date.The origin of the maternal effect of DICER-LIKE 1remains mysterious [33],and the maternal effects that are asso-ciated with PROLIFERA (a homolog of MINICHRO-MOSOME MAINTENANCE 7)are complex and might involve several mechanisms [34].The second origin of maternal influence on seed develop-ment appears to be a specific activation of maternal alleles and silencing of paternal alleles,resulting in maternal genomic imprinting.Two of the components of the FIS complex are subject to maternal genomic imprinting [35,36].The paternal allele of MEA and FIS2is silenced.As a consequence,the maternal inheritance of a null allele,mea or fis2,cannot be rescued by a paternal wildtype MEA or FIS2allele and is sufficient to cause maternal effects on endosperm development [11,14,15,17].The mechanisms that lead to MEA and FIS2imprinting are in part distinct ([35,36];Figure 4).FIS2is silenced throughout most of the plant life cycle by DNA methyla-tion on a specific region in its promoter [36].DNA methylation is maintained by the methyltransferase MET1during vegetative development and male game-togenesis.At the end of female gametogenesis,however,FIS2silencing is relieved in the central cell by the DNA glycosylase DEMETER (DME).Hence,a transcription-ally active maternal FIS2allele is provided to the endo-sperm whereas the paternal FIS2allele remains silenced by MET1.By contrast,MEA silencing depends on the repressive methylation marks on lysine residue 27of HISTONE 3(H3K27)[37 ,38 ].H3K27methylation of the MEA locus depends on the activity of Pc-G com-plexes that are active during the vegetative phase and male gametogenesis.In endosperm,a loop involving negative feedback by the Pc-G complex,including MEA and FIS2maintains the silencing of the paternal MEA allele ([37 ,38 ];Figure 4).The activation of the maternal allele of MEA in the central cell depends on several mechanisms.Real-time quantitative PCR has shown that MEA expression in the central cell is inhibited by MEA itself [39].DME is required for MEA activation [40]and is able to remove methylated cytosine from domains linked to the MEA locus [38 ].However,the degree of DNA methylation of MEA varies depending on natural Arabidopsis accessionsEndosperm in seed growth and development Berger,Grini and Schnittger 667Figure 3Phenotypes of mutants in which seed development is affected.(a,b)The effects of pollination of wildtype ovules by cdka;1/+plants.(a)One and a half days after self-fertilization,the seeds contain an octant stageembryo (e)and stage VI (26–30nuclei)endosperm (ed)surrounded by seed integuments (int).(b)By contrast,wildtype ovules that are fertilized with cdka;1mutant pollen produce seeds that have a two-or four-cell-stage embryo one day after pollination.(c,d)In comparison to (c)wildtype seeds,(d)iku seeds contain an endosperm (ed)that is reduced in size and whose seed integuments (int)do not elongate,whereas embryo growth is not altered.[38 ,41].Moreover,DNA methylation does not play a major role in silencing MEA in vegetative tissues or in pollen[37 ],and loss of DNA methylation on MEA paternal allele does not alter their silenced state in endosperm[36,38 ].Hence,DNA methylation probably plays a limited regulatory role in MEA imprinting.What mechanism is responsible for the removal of H3K27 methylation that is essential for MEA imprinting is unclear.DME DNA glycosylase activity causes single-strand breaks in DNA[38 ].It is possible,therefore,that DNA repair mechanisms that follow DME action involve the deposition of new chromatin and that methylated H3K27are removed,causing loss of repression of MEA transcriptional activity(Figure4).Although accumulating evidence on epigenetic regula-tion mechanisms has contributed to our understanding of how differential genetic programs are set up in the different components of the seed,little is known about the downstream targets that are controlled by,for exam-ple,the Pc-G FIS complex.Small-scale microarray analyses using combinations offis mutants have to date discovered two genes that are repressed by the FIS-complex,and these genes encode the MADS-box protein PHERES1(PHE1)and the SKP1-like protein MEIDOS[16].What functions these proteins have in seed development and growth is not known,but PHE1 has been shown to be expressed exclusively from the paternal genome after fertilization[42 ].In contrast to the gametophytic maternal effects discussed previously, genes such as PHE1might thus be responsible for acti-vating paternal-specific gene sets in endosperm. ConclusionsOur current knowledge on seed development underlines the importance of the endosperm not only as a source of nutrients but also as an integrator of seed growth and development.Different signaling routes are now emer-ging between the embryo,the endosperm and seed integuments.A reciprocal signaling process between seed integuments and endosperm appears to coordinate the proper course of seed growth.Signals between embryo and endosperm are probably also important for proper seed development but remain to be characterized.668Cell biologyFigure4A model for dual control of parental genomic imprinting of MEA in Arabidopsis.The gene MEA is silenced during vegetative developmentand male gametogenesis by the continuous action of Pc-G complexes that are responsible for maintenance of H3K27methylation(pink circles) at the MEA locus.MEA expression is activated in the central cell at the end of female gametogenesis by the DNA glycosylase(DME)and potentially by other activities that lead to the removal of H3K27methylation.After fertilization,the MEA maternal allele(m)is expressed in endosperm and produces the MEA subunit of the Pc-G complex,which contains FIS2,FIE and MSI1and is responsible for maintaining the silencing of the paternal MEA allele(p).Significant progress has been made in the understanding of maternal effects and the role of the endosperm as a relay between the mother and the embryo.Here,a possible role of gametophytic maternal factors that are already present in the unfertilized egg and central cell calls for attention.A major constraint to isolate these signaling components is the small size of reproductive structures in Arabidopsis. 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AR,Grossniklaus U,Meinke DW,Ray A:SHORT INTEGUMENTS1/ SUSPENSOR1/CARPEL FACTORY,a dicer homolog,is amaternal effect gene required for embryo development inArabidopsis.Plant Physiol2002,130:808-822.34.Springer PS,Holding DR,Groover A,Yordan C,Martienssen RA:The essential Mcm7protein prolifera is localized to thenucleus of dividing cells during the G1phase and is required maternally for early Arabidopsis development.Development 2000,127:1815-1822.35.Kinoshita T,Yadegari R,Harada JJ,Goldberg RB,Fischer RL:Imprinting of the MEDEA polycomb gene in the Arabidopsis endosperm.Plant Cell1999,11:1945-1952.36.Jullien PE,Kinoshita T,Ohad N,Berger F:Maintenance of DNAmethylation during the Arabidopsis life cycle is essential for parental imprinting.Plant Cell2006,18:1360-1372.37. Jullien PE,Katz A,Oliva M,Ohad N,Berger F:Polycomb group complexes self-regulate imprinting of the polycomb group gene MEDEA in Arabidopsis.Curr Biol2006,16:486-492.The authors demonstrate that imprinting of the polycomb group geneMEDEA depends on regulations by Polycomb group complexes.Together with the work described in[36],this study provides evidence for silencing mechanisms during male gametogenesis and endospermdevelopment that are essential for the maintenance of imprinting.38. Gehring M,Huh JH,Hsieh TF,Penterman J,Choi Y,Harada JJ, Goldberg RB,Fischer RL:Demeter DNA glycosylase establishes MEDEA polycomb gene self-imprinting by allele-specific demethylation.Cell2006,124:495-506.This study dissects the biochemical mechanism of DME,which isessential for the activation of the maternal allele of imprinted genes. The authors also provide evidence for the self-regulation of imprinting bythe Polycomb group gene MEA in endosperm.39.Baroux C,Gagliardini V,Page DR,Grossniklaus U:Dynamicregulatory interactions of polycomb group genes:MEDEAautoregulation is required for imprinted gene expression inArabidopsis.Genes Dev2006,20:1081-1086.40.Choi Y,Gehring M,Johnson L,Hannon M,Harada JJ,Goldberg RB,Jacobsen SE,Fischer RL:DEMETER,a DNAglycosylase domain protein,is required for endospermgene imprinting and seed viability in Arabidopsis.Cell2002,110:33-42.41.Spillane C,Baroux C,Escobar-Restrepo JM,Page DR,Laoueille S,Grossniklaus U:Transposons and tandem repeats are notinvolved in the control of genomic imprinting at the MEDEAlocus in Arabidopsis.Cold Spring Harb Symp Quant Biol2004, 69:465-475.42.Ko¨hler C,Page DR,Gagliardini V,Grossniklaus U:TheArabidopsis thaliana MEDEA polycomb group proteincontrols expression of PHERES1by parental imprinting.Nat Genet2005,37:28-30.This study provides an elegant demonstration of how the expression of a paternally expressed imprinted gene is controlled in plants.43.Roy PJ,Stuart JM,Lund J,Kim SK:Chromosomal clustering ofmuscle-expressed genes in Caenorhabditis elegans.Nature2002,418:975-979.44.Day RC,Grossniklaus U,Macknight RC:Be more specific!Laser-assisted microdissection of plant cells.Trends Plant Sci2005, 10:397-406.45.Honys D,Twell D:Transcriptome analysis of haploid malegametophyte development in Arabidopsis.Genome Biol2004, 5:R85.46.Hennig L,Gruissem W,Grossniklaus U,Ko¨hler C:Transcriptionalprograms of early reproductive stages in Arabidopsis.Plant Physiol2004,135:1765-1775.47.Wang D,Tyson MD,Jackson SS,Yadegari R:Partiallyredundant functions of two SET-domain polycomb-groupproteins in controlling initiation of seed developmentin Arabidopsis.Proc Natl Acad Sci USA2006,103:13244-13249.This study demonstrates redundant functions of the polycomb-group proteins SWINGER and MEDEA in complex with FERTILIZATION-INDE-PENDENT SEED2in controlling the initiation of endosperm development.48.Makarevich G,Leroy O,Akinci U,Schubert D,Clarenz O,Goodrich J,Grossniklaus U,Ko¨hler C:Different Polycomb group complexes regulate common target genes in Arabidopsis.EMBO reports2006,7:947-952.This careful study reports the location of repressive epigenetic mark on PHERES1by several Pc-G complexes that act during vegetative and reproductive development.49.Gutierrez-Marcos JF,Costa LM,Evans MMS:Maternalgametophytic baseless1is required for development ofthe central cell and early endosperm patterning in maize(Zea mays).Genetics2006,174:317-329.This study provides the exciting report of a direct maternal influence from the female gamete on polarity of zygotic products in plants.670Cell biology。
FastDirectSoluti...
Fast Direct Solution of the Combined Field IntegralEquationAlex Heldring #1, José Maria Tamayo#2 and Juan Manuel Rius #3# Department of Telecommunications and Signal Processing, Polytechnic University of CataloniaC. Jordi Girona, 31, 08034, Barcelona, Spain1****************.edu2*************************.edu3************.eduAbstract—The Method of Moments for electromagnetic scattering and radiation problems is often used in conjunction with the EFIE because the EFIE allows the analysis of open surfaces. For electrically large closed surfaces, the CFIE is often much more efficient because it yields a well conditioned impedance matrix. This is particularly important when an iterative solution method is used. This paper compares the EFIE and the CFIE for a novel fast direct (non-iterative) solution method, the Multiscale Compressed Block Decomposition method.I.I NTRODUCTIONThe Method of Moments [1] is a popular tool for the simulation of electromagnetic radiation and scattering. Often it is applied to a surface integral equation (SIE) which is discretized using local basis functions defined on a triangulation of the structure under investigation. If this structure consists exclusively of perfectly conducting surfaces then the typical SIE that are used are the Electric Field Integral Equation (EFIE), the Magnetic Field Integral Equation (MFIE) or a combination of the two (CFIE).The MoM leads to a dense linear system, with a direct solution time that scales with the cube of the number of basis functions N, or, for a discretization that is fixed with respect to the wave length, with the sixth power of the frequency. This proportionality is referred to as the computational complexity of the method. Evidently this poses a strict limit to the maximum electrical size of the problem under investigation.Many methods to accelerate the solution have been proposed over the past decades, most of them using an approximate representation of the MoM impedance matrix which can be used in an iterative solution algorithm. The most famous among these methods is the Multilevel Fast Multipole Method (MLFMM) [2] which reduces the complexity to KN log N where K is the required number of iteration steps. Although these methods have brought an enormous gain in efficiency for electrically large problems, they typically have a number of drawbacks. To start with, many problems lead to a badly conditioned impedance matrix. Iterative methods then need to be preconditioned to converge in a reasonable number of steps. But the choice, parameter adjustment and construction of a good preconditioner can be a tedious and unpredictable process. Another drawback is that for every single independent vector, the iteration must be restarted. Furthermore, some of the most efficient iterative methods, such as MLFMM, are kernel dependent. In short, a direct (non-iterative) solution method with a reduced complexity remains an interesting alternative, even if it is asymptotically less efficient than the above mentioned iterative methods.In [3] we have presented such a method, the Compressed Block Decomposition (CBD). In [4] we showed that the complexity of this method is N2log2N. In [5] we presented a multiscale version of the CBD applied to static or low-frequency problems which was shown to yield an N log3N complexity. Due to the oscillatory kernel this is not true in the high frequency regime, and in fact the multiscale method from [5] needs several adaptations to be efficient at high frequencies. In [6] we present a multiscale extension to the CBD for the high frequency regime (MS-CBD) and demonstrate with a numerical example that it is superior to the single-scale CBD. Its primary feature is that, just like single-level CBD, it is entirely algebraic, since it exclusively uses ACA [7], truncated SVD and QR decompositions to achieve block-wise compression of the impedance matrix and maintain it after factorization. In the paper we also show that the MS-CBD has N2 complexity and is therefore asymptotically (with respect to frequency) superior to the CBD.Although by avoiding an iterative algorithm we much alleviate the problems presented by badly conditioned matrices, the MS-CBD is not insensitive to the impedance matrix condition number. A consequence of a high condition number is that small perturbations in any of the matrix elements may cause large errors in the solution. Since the block-compression through ACA and truncated SVD involves an approximation (governed by a threshold W on the singular values), a badly conditioned matrix requires a lower threshold and therefore less compression.The EFIE is the most commonly used formulation for perfect conductors, since it is applicable to both closed and open geometries (open geometries have edges). The MFIE and therefore the CFIE are restricted to closed geometries. However, the EFIE, besides suffering from a well known low-frequency breakdown [8], is also often badly conditioned. This is in contrast with the MFIE and with the CFIE to an even greater extent, which are well conditioned independently of the electrical size of the problem.2010 URSI International Symposium on Electromagnetic TheoryWe expect that, although the compression rate will not bebetter for CFIE than for EFIE as a function of the threshold value, CFIE will allow higher threshold values for a given accuracy. It may therefore be expedient to use CFIE when the problem geometry allows it (closed problems), even taking into account that the MS-CBD fully profits from the symmetry of the EFIE impedance matrix, yielding a gain of almost a factor two in computation time.II. M ULTI S CALE -C OMPRESSED B LOCK D ECOMPOSITION . The MS-CBD is essentially a nested implementation of the well known Partitioned Matrix Inverse formulas [9]. For a matrix Z partitioned into four blocks: 111122111Z Z Z Z Z ªº «»¬¼, (1)with11211121222211121222122121111211121221211111111)(~~~~~~~ Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z , (2)The first step is the construction of the impedance matrix incompressed form. The basis functions are subdividedrecursively into groups based on geometrical proximity. Allmatrix blocks, at any scale, that are not on the diagonal, are compressed into truncated SVD form. This means that the SVD is computed and all columns and rows in the three SVD factors that relate to singular values smaller than W times the largest singular value are discarded. The computation of the SVD, which is an expensive operation, is accelerated by using the ACA algorithm. Blocks on the diagonal are either subdivided again, or, but only at the smallest scale, inverted directly by LU decomposition.The second step is the inversion of the compressed matrixaccording to (1). As equation (2) shows, the only operations involved are multiplication and summation of matrices (and inversion, but this is recursively done by the same MS-CBD algorithm, except at the smallest scale). To maintain the compression, every multiplication and summation requires a restoration of the SVD form of the result. This is achievedwith a number of QR and truncated SVD decompositions. Thedetails of the algorithm will be published in [6].The last step is a back substitution (similar to LU backsubstitution) to obtain the solution for one or several independent (excitation) vectors. This last step is typically orders of magnitude cheaper than the previous ones, even for thousands of excitation vectors. III. N UMERICAL R ESULTS .In order to investigate the respective performance of the MS-CBD with the EFIE and the CFIE formulations, we computed the monostatic RCS of a perfectly conducting cube for a range of threshold values in both formulations. We chose to compute the monostatic RCS since this represents the typical usage of the MS-CBD (one of the major advantages of MS-CBD in comparison with iterative methods is the very small overhead for multiple excitation vectors). The setup of the experiment is shown in Fig. 1. We did the computations attwo frequency scales, a small cube (rib-size 1O and a larger one (rib-size 8O ) under otherwise identical conditions in order to see whether the result is dependent on the overall electrical size of the problem. In both cases, RWG basis functions with a mean edge size of 0.07O were used. For the small problem, this meant N = 4,608 basis functions, the large problemneeded N = 294,912 basis functions. The small problem was done with four-level MS-SBD, the large problem with ten-level MS-CBD. The incident field is polarized along the z-axis and propagates in the XY plane at angles between I =0º and I =45º (50 incidence angles for the small problem, 1,000 incidence angles for the large problem). Fig. 1. PEC cube in free space. In Figs 2 and 3 the monostatic RCS is shown for the small problem, for EFIE and CFIE respectively. Fig. 4 and 5 show the same for the large problem. The overall conclusion is that the EFIE result is accurate for a threshold value W = 10-3, while the CFIE result is already accurate with W = 10-2. This is true in both cases indicating that the size of the problem does notFig. 2. EFIE MS-CBD computation of monostatic RCS, PEC cube, 4,608basis functions, 50 incidence angles.Fig. 3. CFIE MS-CBD computation of monostatic RCS, PEC cube, 4,608 basis functions, 50 incidence angles.Fig. 4. EFIE MS-CBD computation of monostatic RCS, PEC cube, 294,912basis functions, 1,000 incidence angles.Fig. 5. CFIE MS-CBD computation of monostatic RCS, PEC cube, 294,912 basis functions, 1,000 incidence angles.Tables I to IV show the computation times and storage requirements for the different computations. All the computations were performed on a PC with a Quad Intel(R) Xeon(R) X5482 processor at 3.20GHz and 64 GB of memory. The code was implemented in Matlab, in single precision.The EFIE formulation fully exploited the symmetry of the EFIE impedance matrix, resulting in a gain of a factor two in the matrix build time, inverse build time, matrix size and inverse size (not in the back substitution time). The CFIE does not have this property.TABLE IEFIE,N=4,608 AND 50 INCIDENCE ANGLESW = 10-2 W = 10-3 W = 10-4Z build (sec) 6.8 11.6 15.9Z-1 build (sec) 1.6 2.3 3.6Backsub(sec)0.3 0.3 0.4Z Size (MB) 25 33 38Z-1 Size (MB) 20 27 34TABLE IIICFIE,N=4,608 AND 50 INCIDENCE ANGLESW = 10-1 W = 10-2 W = 10-3Z build (sec) 7.7 14.6 24.5Z-1 build (sec) 1.8 2.7 4.2Backsub(sec)0.3 0.3 0.3Z Size (MB) 23 40 55Z-1 Size (MB) 25 41 63TABLE IIIIIEFIE,N=294,912 AND 1,000 INCIDENCE ANGLESW = 10-2 W = 10-3 W = 10-4 W = 10-5 Z build(sec) 1,816 3,059 4,297 5,991 Z-1build(sec) 1,249 2,140 3,379 6,283 Backsub(sec)57 76 100 167Z Size(MB) 4,937 6,416 7,838 9,479 Z-1Size(MB) 4,909 7,186 9,743 15,401TABLE IVVCFIE,N=294,912 AND 1,000 INCIDENCE ANGLESW = 10-1 W = 10-2 W = 10-3Z build (sec) 2,197 3,890 6,266Z-1 build (sec) 929 1,942 3,601Backsub(sec)35 47 61Z Size (MB) 5,736 8,910 11,621Z-1 Size (MB) 5,611 9,551 14,184The Tables show that both the computation times and the matrix sizes grow very significantly for lower threshold values. In terms of efficiency it is therefore quite expedient to know the maximum threshold for a given desired accuracy. It is however difficult to establish a general rule for this.Another conclusion from the Tables is that, accounting for the factor two gain in the EFIE due to symmetry, the matrix sizes of EFIE and CFIE are approximately equal for equal threshold value. This is to be expected since the principal limitation to the compression rate is the oscillatory Greens function, which is present in both cases.Finally, with respect to the suggestion in the introduction that CFIE may be an interesting option for MS-CBD due tothe superior conditioning; these results indicate that in fact, the advantage does not outweigh the gain from the symmetry of the EFIE, as a comparison of the performance for converged results in both cases shows.IV.C ONCLUSIONA comparison of the performance of the Multiscale Compressed Block Decomposition (MS-CBD) in combination with the EFIE and with the CFIE has been carried out. The results indicate that the advantage due to the better conditioning of the impedance matrix for the CFIE is important but does not outweigh the gain of a factor two due to the symmetry of the EFIE impedance matrix. This conclusion is valid for the MS-CBD since it is a direct (non-iterative) method and therefore not critically dependent on the matrix condition number. If the same block-compressed impedance matrix is used within an iterative method, then the CFIE will often be much more efficient because the convergence of any iterative method is highly dependent on the matrix condition number.A CKNOWLEDGMENTThis work was supported by the Spanish Interministerial Commission on Science and Technology (CICYT) under projects TEC2009-13897-C03-01 and TEC2007-66698-C04-01/TCM and CONSOLIDER CSD2008-00068 and by the "Ministerio de Educación y Ciencia" through the FPU fellowship program.R EFERENCES[1]R. F. Harrington, Field Computation by Moment Methods, MacMillanNew York, 1968.[2]J. Song, C.C. Lu and W.C. Chew, .Multilevel Fast Multipole Algorithmfor Electromagnetic Scattering by Large Complex Objects., IEEE Trans.on Antennas and Propag., Vol. 45, No. 10, pp. 1488-1493, October 1997[3] A. Heldring, J. M. Rius,J. M. Tamayo, J. Parrón, E. Ubeda, .Fast DirectSoluton of Method of Moments Linear System,. IEEE Trans. on Antennas and Propag., Vol 55, No 11, pp. 3220-3228, Nov 2007[4] A. Heldring, J. M. Rius, J. M. Tamayo, ‘Comments on Fast DirectSoluton of Method of Moments Linear System,’ accepted for publication in IEEE Trans. on Antennas and Propag.[5] A. Heldring, J.M. Rius, J.M. Tamayo and J. Parron, ’CompressedBlock Decomposition Algorithm for Fast Capacitance Extraction,’.IEEE Trans. on Computer Aided Design, Vol. 27, No. 2, pp 265-271, Feb 2008[6] A. Heldring, J. M. Rius, J. M. Tamayo, ‘Multiscale Compressed BlockDecomposition for Fast Direct Solution of Method of Moments Linear System,’ submitted for publication in IEEE Trans. on Antennas and Propag.[7]M. Bebendorf, .Approximation of boundary element matrices,., Numer.Math. (2000) 86: 565.589[8]Y. Chu, W. C. Chew, J. Zhao and S. Chen, ‘Surface Integral EquationFormulation for Low-Frequency Scattering From a Composite Object,’ IEEE Trans. on Antennas and Propag., Vol. 51, No. 10, pp 2837-2844, Oct 2003[9]T. Banachiewicz, .Zur Berechnung der Determinanten, wie auch derInversen und zur darauf basierten Aufosung der Systeme linearer Gleichungen,. Acta Astronom. Ser. C 3, pp. 41.67, 1937。
意义、指称与实在——后期库恩对不可通约性问题的语言学辩护
第27卷，第4期科学技术哲学研究V01．27 N o．4 2010年8月S tu d i es in Ph i l o s o p h y of S c i e n c e a nd Techno logy A ug．。
2010意义、指称与实在——后期库恩对不可通约性问题的语言学辩护何兵，殷维(重庆大学贸易与行政学院，重庆400044)摘要：在库恩看来，无论采用指称的稳定性还是指称的连续性策略均不能回避不可通约性问题。
除了实指行为外，隐喻、模型在确定语言与世界(术语与其指称)的关联上发挥着重要的作用。
基于分类词典所建构的多世界理论，库恩在维护不可通约性的同时，避开了对实在本身的讨论。
关键词：指称的稳定性；指称的连续性；实指；分类词典；多世界理论中图分类号：B15文献标识码：A文章编号：1674～7062(2010)04—0055—05库恩《科学革命的结构》发表后，其不可通约性是非常罕见的。
概念隐含的反实在论立场遭到实在论者激烈的批对于这两方面的批评，库恩并不完全认同。
在评。
1970年代以来的新实在论者认为，不可通约性同这些批评者论战的过程中，后期库恩发展出基于问题来自语言与世界的表象关系，即把描述性的意分类词典的多现象世界理论，旨在不涉及实在的前义表达看成是表象的基础。
根据弗雷格的指称／意提下给出不可通约性问题更为精致的说明。
义区分和罗素的摹状词理论，大多数语言哲学家假一指称的稳定性定，一个语言表达应指涉一个对象(包括对象的类或集合)。
新实在论者相信，意义由其指称决定。
意义与指称问题是弗雷格和罗素以来分析哲学如果能够证明指称独立于意义，那么就可以克服不家们争论不休的话题。
对于这个问题，有两个传统可通约性问题。
的对立观点：一种是弗雷格的，认为意义和所指不同因此，反对语义不可通约性有两种策略：第一个的；一种是罗素的看法，认为意义和所指是同一的，是指称的策略，即坚持指称的稳定性或连续性，主张意义即所指。
群体多目标决策联合有效解类的不变凸充分条件
群体多目标决策联合有效解类的不变凸充分条件王中兴;杨雷【期刊名称】《运筹学学报》【年(卷),期】2002(006)003【摘要】In [1], the concept of a class joint efficient solution to group multiobjective decisionmaking problem is introduced, as well as some optimality necessary conditions for thesesolutions. In [2], some optimality sufficient conditions for efficient solution are studiedwhile the objective and constraint functions are convex. In this paper, several optimalitysufficient conditions for class of joint efficient solution are obtained under the conditionthat the objective and constraint function are invex and generalized invex.%对于群体多目标决策问题,文[1]引进它的联合有效解类的概念,并给出这类解的最优性必要条件.在对于问题的目标函数和约束函数附加凸性的条件下,文[2]又给出了联合有效解类的最优性充分条件.本文进一步在目标函数和约束函数具不变凸和不变广义凸的情况下,分别给出了联合有效解类的若干最优性充分条件.【总页数】8页(P27-34)【作者】王中兴;杨雷【作者单位】广西大学数学与信息科学系,南宁,530004;上海交通大学应用数学系,上海,200030【正文语种】中文【中图分类】O22【相关文献】1.一类广义(h,φ)-不变凸半无限规划解的充分条件 [J], 张庆祥;赵伏有;刘兴祥2.群体多目标决策α-较多联合有效解类的最优性条件 [J], 杨雷;于丽英;王中兴3.群体多目标决策联合有效解类的几何特性 [J], 胡毓达4.群体多目标决策联合有效解类的几个最优性充分条件 [J], 于丽英;胡毓达5.群体多目标决策的联合有效解类及其最优性条件 [J], 胡毓达因版权原因，仅展示原文概要，查看原文内容请购买。
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PACS: 02. 60. Cb, 02. 70. Bf, 45. 10. Na, 45. 20. Df, 45. 20. Dh The Zakharov system[1] iφt + φxx + 2φψ = 0, ψtt − ψxx + (|φ|2 )xx = 0 (1) is a coupled system of dispersive equations originally derived to study the interaction of an acoustic wave with a plasma envelope. Here the complex unknown function φ(x, t) represents the slowly varying envelope of the highly oscillatory electric field, and the unknown real function ψ (x, t) denotes the fluctuation of the ion density about its equilibrium value. The theoretical properties of the Zakharov system, such as the existence and uniqueness of smooth solutions, and the collision of solitons, have been studied by several authors, e.g., Refs. [2,3]. Up to now, some reliable efficient numerical methods, including the time-splitting spectral methods,[4] conservative finite difference schemes,[5,6] and Fourier spectral methods,[7] etc., have been constructed to solve the Zakharov system (1). In recent years, the multisymplectic integrator have been proposed and developed for Hamiltonian partial differential equations, because of perfectly preserving the geometry structure of the equations.[8−10] It is known that the multisymplectic Preissman scheme has many nice properties and has been discussed by researchers.[11−14] The main weakness of the Preissman scheme is its implicit nature that causes practical problems for the solutions of large nonlinear systems. The multisymplectic Euler-box scheme has been applied successfully to many Hamiltonian PDEs such as the nonlinear wave equation,[11] the Kortewegodinger de Vries equation,[15] and the nonlinear Schr¨ equation.[16] As far as we know, there have been no study on the multisymplecticity of the Zakharov system. In this Letter, we show that the Zakharov system (1) is a multisymplectic Hamiltonian system. As an application of this result, we construct the semi-explicit multisymplectic integrator based on the Euler-box scheme.
−fx = −g,
or the standard Hamiltonian PDEs (3) M ∂t z + K ∂x z = ∇z S (z ), (3)
where z = (u, v, p, q, ψ, f, g )T , M and K are the skewsymmetric matrices, 0 −1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 M= 0 0 0 0 0 0 0, 0 0 0 0 0 1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 −1 0 0 0 0 0 0 K= 0 −1 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 −1 0 1 and the smooth function S (z ) = −ψ (u2 + v 2 ) − (p2 + 2 1 1 q2 ) + ψ2 − g2 . 2 2 According to Refs. [9,10], the system (2) has a multisymplectic conservation law ∂t ω + ∂x κ = 0, (4)
3532
WANG Jian
Vol. 25
where ω = − du ∧ dv + dψ ∧ df, κ = du ∧ dp + dv ∧ dq + df ∧ dg. (5)
and time stepsize, respectively. Define the following finite difference operators
(Received 1 June 2008)
A multisymplectic formulation for the Zakharov system is presented. The semi-explicit multisymplectic integrator of the formulation is constructed by means of the Euler-box scheme. Numerical results on simulating the propagation of one soliton and the collision of two solitons are reported to illustrate the efficiency of the multisymplectic scheme.
+ n Dt zj = n+1 n zj − zj , ∆t n n zj +1 − zj + n Dx zj = , ∆x n−1 n zj − zj , ∆t n n zj − zj −1 − n Dx zj = . ∆x − n Dt zj =
For the Hamiltonian PDEs (3), when S is independent of x and t, it has local energy conservation law and local momentum conservation law ∂t E + ∂x F = 0, 1 E = S (z ) − z T Kz x , 2 (6)
(12)
1 F = z T Kz t , 2 1 ∂t I + ∂x G = 0, I = z T M z x , 2 1 G = S (z ) − z T M z t . 2
Following[11] one may obtain the multisymplectic Euler-box scheme for PDEs (3) by introducing a splitting of the two matrices M and K . For example, splitting the matrices M and K as M = M + + M −, K = K + + K −, MT + = −M − ; KT + = −K − . (13)
by E-Institutes of Shanghai Municipal Education Commission under Grant No E03004. wj3156@ c 2008 Chinese Physical Society and IOP Publishing Ltd
(7)
The corresponding scheme reads
+ n − n + n M + Dt z j + M − Dt z j + K + Dx zj − n + K − Dx z j = ∇z S (z n j ).
Implement Eqs. (6) and (7) for the Zakharov system (2) yields the local energy conservation law ∂t E + ∂x F = 0, 1 1 E = −ψ (u2 + v 2 ) + (p2 + q 2 ) + (ψ 2 + g 2 ), 2 2 F = −(ut p + vt q + ft g )Байду номын сангаас (8) and the local momentum conservation law I = ux v − ψx f, 1 G = −ψ (u2 + v 2 ) − (p2 + q 2 ) 2 1 2 2 + (ψ − g ) − ut v + ψt f. 2 ∂t I + ∂x G = 0,