THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS

51.IntroductionandNotationsLetD=(V,E)beadigraphandL(D)denotethesetofcyclelengthsofD.ForuEVandintegeri21,letfo(u):={vEVIthereedestsadirectedwalkoflengthifromutov}.WedelveRo(u):={u}.Letu,vEV.IfN (v)=N (v)andN--(v)=N--(v),thenwecanvacopyofu.LotDbeaprimitivedigraphand7(D)denotetheexponentofD.In1950,H.WielandtI61foundthat7(D)5(n--1)' 1andshowedthatthereisapiquedigraphthatattainsthisbound.In1964,A.L.DulmageandN.S.Mendelsohn[2]ObservedthattherearegapsintheexponentsetEd={ry(D)IDEPD.}…     

ON A PROBLEM IN COMPLEX OSCILLATION THEORY OF PERIODIC HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

In this article, the zeros of solutions of differential equation f(k)(z)+A(z)f(z) = 0, (*) are studied, where k 2, A(z) = B(ez), B(ζ) = g1(1/ζ) + g2(ζ), g1 and g2 being entire functions with g2 transcendental and σ(g2) not equal to a positive integer or infinity. It is shown that any linearly independent solutions f1, f2, . . . , fk of Eq.(*) satisfy λe(f1 . . . fk) ≥σ(g2) under the condition that fj(z) and fj(z+ 2πi) (j = 1, . . . , k) are linearly dependent.     

EQUIVALENCE BETWEEN EXACT INTERNAL CONTROLLABILITY OF THE KIRCHHOFF PLATE-LIKE EQUATION AND THEWAVE EQUATION

1.IntroductionConsiderthefollowingillitial-boundaryvalueproblemLetD(A)={vEHI(fl)ItvEL'(fl)},V~Ha(~~),H~L'(fl).Thenthefiniteenergystatespacesofthesystems(1.1),(1.2)are,respectively,uk~D(A)xV,U.~VxH.ConsiderthegeneralizedfunctionspaceH-1~V'DHDV.ForbEC'(fl)anduEH-',wedefinebaEH-1by(bu)(()~u(bo,V(6HJ(n).(1.3)WehaveusedtheSobolevspaces.Wereferthereadersto[1]fortheknowledgeofthosespaces.Definition1.1.Thesystem(1.1)issaidtobeexactlycontrollableinukbyH-1-controlsifforevery(yo,yi)6uk,th…     

THE RIEMANN PROBLEM FOR A TWO-DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS WITH NON-CLASSICAL WEAK SOLUTIONS:SOME ONE-J AND NON-R INITIAL DATA

1IntroductionWeareconcernedwiththefollowing2x2nonstrictly11yperbolicsystemofconservationlawswithinitialdatawhere(m,yi),i=1,2,3,areconstantstates.Wecall(1.1),(1.2)aRiemanll-typeproblelll,.Byaweaksolutionto(1.1),(1.2)wemeanapairofgeneralizedfullctions('IL.l,)(x,y.t)satisfyinglffeceivedFeb.17,1995;revisedApr.23,1996.SuPPortedbyNSFC(YOuth).foreacllof(T,y,f)6C:(R'xR )alld(u,U)(:r,;(/,f)~(it..())(x,y.0)ast-- 0 forallllostevery(x,if/,t)ERZxR .111(1.3).wealwaysassulllethat('(ti))(T,y.f)isals…     

REAL TOMITA-TAKESAKI THEORY

LetMbearealVonNeumannalgebraonarealHilbertspaceH,and(oEHbecyclic-separatingforM.LetK=Mthe,andL=Mthe.Clearly,KnL={o},and(K L)isdenseinH.Hence,wemuststudysuchanondegeneratepair(K,L).Manypropertiesof(K,L)aresimilartothecomplexcase(6l).ThenasatisfactoryrealTondtaTakesakitheoryisobtained(92).Foro-finiterealW*-algebraM,wepointoutthatMf=Mw iMw(63)andetc.Moreover,thereaJToIIiltaTaksakitheoryseemstobeaspecialresultofthecomplexcase.51.NondegeneratePairofClosedLinearSubspacesDefinition1.…     

A REMARK ON EXTINCTION OF A CLASS OF SUPERPROCESSES

1.IntroductionSuperprocesses,i.e.,measure-vauedMarkovprocesses,havebeenanattractivetopicintherecenty6ars,whichcanbedefinedasfollows.Let((,Px)..MbeaMarkovprocesswithstrongcontinuousMarkovsemigroupSt(denotebyAitsinfinitesimaloperator,andbyPt(x,y)theassociatedtransitiollfunction).DenotebyMPthesetofallRadonmeasurespsatisfyingandbypC(t)(resp.pCc(t))thesetofpositivelycontinuousfUnctions(resp.withcompactsupport)int-Thesthcalledsuperprocess(X,P#).6MPwithparameters(andopandtakingvalueinMrisdete…     

REMARKS ON h-TRANSFORM AND DRIFT

51.h-TransformsInthispaperwearegoingtogiveaformulacharacterizingRevuzmeasuresunderhtrallsform.AssumethatXisarightMarkovprocesswithstatespace(E,B)whichismel,rizable,constructedonthecanonicalspacefiofrightcontinuouspaths,and(Pt)and(Ua)arethesemigroupandresolventofXrespectively.LethbeanexcessivefunctionandletEh:={0相似文献   

LOCAL C-SEMIGROUPS AND LOCAL C-COSINE FUNCTIONS

0IntroductionLetB(X)betheBanachspaceofallboundedlinearoperatorsonaBanachspaceX.CEB(X)isinjective.Alloperatorsarelinear.D(A),Im(A),andP(A)denotethedomain,theimage,andtheresolventsetofoperatorA,respectivelyNandRdenotethesetofallnaturalnumbersandthesetofallrealnumbers.TheconceptoflocalC-sernigroupsalldlocalC-cosinefunctionwereintroducedbyTanakaandOkazawain[1]tF.HuangandT.Huangin[21,respectively.Aspointedoutin[3]and[2]thatnotalllocalC-sendgroups(C--cosinefunctions)canbeextendedtobeC-se…     

ASYMPTOTIC BEHAVIOR OF HARMONIC MAPS FROM COMPLETE MANIFOLDS

51.IntroductionLetMbeanm-dimensionalcompletenoncompactRiemannianmanifold,pEM.P.LiandL.F.Turnintroducedin[8]avolumecomparisoncondition(VC)(seethedefinitionbelow).TheyobtainedsomeimportantanalyticpropertiesifMsatisfies(VC)andtheRiccicurvatureconditionRice(x)2--(m--1)K/(1 r(x))',wherer(x)isthedistancefromptox,K20isaconstant.ItisinterestingtoknowmoreabouttheanalysisonsuchamanifoldM.LetNbeacompleteRiemannianmanifoldwiththesectionalcurvatureKNboundedabovebysomeconstantK20.B.(T)denotestheg…     

POLES OF ZETA FUNCTIONS OF COMPLETE INTERSECTIONS

51.IntroductionLetF.beafinitefieldofqele~swithcharederisticp.LetXbeanThdilnensionalalgebraicsetdeadedoverF..ThezetafunctionofX/F.isdennedbywhereXOdenotesthesetofclosedpointsOfX/F.and#X(F.d)denotesthenUInberof.F.d-rationalpointsonX.ItiseasytoseethatZ(X,T)isapowerserieswithintegercoefficients.Dwork'srationalltytheoremIg]showsthatthezetafunctionZ(X,T)isrationalinT.ThuS,therearealgebraicintegerspll'.3Pr,pl,'3basuchthatThereisagoodreasonthatwemoantheabovezetafunctionbythepower(--1)"-…     

伴随超临界分支的通有同宿轨道分支

本文研究具有非双曲奇点的高维系统在小扰动下的同宿轨道分支问题,通过在未扰同宿轨道邻域建立局部坐标系,导出系统在新坐标系下的Poincare映射,对伴随超临界分支的通有同宿轨道的保存及分支出周期轨道的情况进行了讨论,推广和改进了一些文献的结果.     

Bifurcations of Rough Heteroclinic Loops with Three Saddle Points

In this paper, we study the bifurcation problems of rough heteroclinic loops connecting three saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition, the existence, uniqueness, and incoexistence of the 1-heteroclinic loop with three or two saddle points, 1-homoclinic orbit and 1-periodic orbit near Γ are obtained. Meanwhile, the bifurcation surfaces and existence regions are also given. Moreover, the above bifurcation results are extended to the case for heteroclinic loop with l saddle points. Received January 4, 2001, Accepted July 3, 2001.     

Bifurcations of nontwisted heteroclinic loop

Bifurcations of nontwisted and fine heteroclinic loops are studied for higher dimensional systems. The existence and its associated existing regions are given for the 1-hom orbit and the 1-per orbit, respectively, and bifurcation surfaces of the two-fold periodic orbit are also obtained. At last, these bifurcation results are applied to the fine heteroclinic loop for the planar system, which leads to some new and interesting results.     

Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3

Bifurcations of heterodimensional cycles with highly degenerate conditions are studied by establishing a suitable local coordinate system in three-dimensional vector fields. The existence, coexistence and noncoexistence of the periodic orbit, homoclinic loop, heteroclinic loop and double periodic orbit are obtained under some generic hypotheses. The bifurcation surfaces and the existence regions are located; the number of the bifurcation surfaces exhibits variety and complexity of the bifurcation of degenerate heterodimensional cycles. The corresponding bifurcation graph is also drawn.     

Logistic映射三周期轨道分支值确定

运用分支函数的方法研究了Logistic映射出现稳定三周期点的参数窗口,借助软件Singular 2.0.3运用多元多项式的Grobner基并结合严格局部分析的方法给出Logistic映射出现稳定三周期点时的参数值范围,最后数值模拟方法对所得的结果进行了验证,并指出伪轨现象.     

Bifurcations of Fine 3-point-loop in Higher Dimensional Space

By using the linear independent fundamental solutions of the linear variational equation along the heteroclinic loop to establish a suitable local coordinate system in some small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to study the bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversal conditions and the non–twisted condition, the existence, coexistence and incoexistence of 2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and the number of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regions are given. Lastly, the above bifurcation results are applied to a planar system and an inside stability criterion is obtained. This work is supported by the National Natural Science Foundation of China (10371040), the Shanghai Priority Academic Disciplines and the Scientific Research Foundation of Linyi Teacher’s University     

Bifurcations of heteroclinic loops

By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop Γ to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature concerned with the 1-hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1-per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1-hom and 1-per orbit and the inexistence of the 2-hom and 2-per orbit are also obtained. Project supported by the National Natural Science Foundation of China (Grant No. 19771037) and the National Science Foundation of America # 9357622. This paper was completed when the first author was visiting Northwestern University.     

具有共振特征值的轨道翻转双同宿环分支

该文研究了具有轨道翻转的双同宿环四维系统,在主特征值共振和沿轨道奇点处切方向共振下的两种分支．我们分别在系统奇点小邻域内利用规范型的解构造一个奇异映射,再在双同宿环的管状邻域内引起局部活动坐标架,利用系统线性变分方程的解定义了一个正则映射,通过复合两个映射而得到分支研究中一类重要的Poincaré映射,经过简单的计算最终得到后继函数的精确表达式．对分支方程细致地研究,我们给出了原双同宿环的保存性条件,并证明了“大” 1-同宿环分支曲面,2-重“大”1-周期轨分支曲面,“大”2-同宿环分支曲面的存在性、存在区域和近似表达式,及其分支出的“大”周期轨和“大”同宿轨的存在性区域和数量．     

一类扩张了的软弹簧型Duffing方程的紊动性态

本文用Melnikov函数方法讨论了一类扩张了的软弹簧型Duffing方程(k=1,2,3,…)在周期激励下的紊动现象.给出了出现二阶同宿切的条件.文中所采用的方法对于不能给出并宿轨道的显式的系统的研究是非常有用的.     

Homoclinic Bifurcation of Orbit Flip with Resonant Principal Eigenvalues

Codimension-3 bifurcations of an orbit-flip homoclinic orbit with resonant principal eigenvalues are studied for a four-dimensional system. The existence, number, co-existence and noncoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are obtained. The bifurcation surfaces and existence regions are also given.