简介:A(k;g)-graphisak-regulargraphwithgirthg.A(k;g)-cageisa(k;g)-graphwiththeleastpossiblenumberofvertices.Letf(k;g)denotethenumberofverticesina(k;g)-cage.Thegirthpairofagraphgivesthelengthofashortestoddandashortestevencycle.Afc-regulargraphwithgirthpair(g,h)iscalleda(k;g,h)-graph.A(k;g,h)-cageisa(k;g,h)-graphwiththeleastpossiblenumberofvertices.Letf(k;g,h)denotethenumberofverticesina(k;g,h)-cage.Inthispaper,weprovethefollowingstrictinequalityf(k;h-1,h)
简介:LetZbeatopologicalspaceandmappingA2:Z→B(H)withclosedrangeR(A2)becontinuous.SomenecessaryandsufficientconditionsofthecontinuityofM-PinversesAz+aregivenin[1],[2].ItisoneofthemthatAZ+iscontinuousifanaonlyifAZ+islocallybounded.Inthispaper,wediscussthefollowingproblem:iflimAn=A0inB(H)and||An+||isunbounded(i.e.theabovenecessaryandsufficientconditionfails),whathinHwillmaketheequations:limAm+h=A0+horw-limAn+h=A0+hbetrue.Forthispurposethreetheoremsandanerrorestimationaregiveninthispaper.
简介:1IntroductionLetA∈Cn×n,B∈Cn×n.WesayBisasquarerootofAifA=B×Bi.e.A=B2.Itiswell-knownthatanysymmetricpositivedefinitematrixexistsoneandonlyonesquarerootwhichisasymmetricpositivedefinitematrix,too(e.g.see[5]).Higham[4]studiedcarefullytherelationofarealnonsingularmatrixbetweenitsrealsquarerootsanditseigenvalues.AlefeldandSchneider[1]pointedoutthatforanynonsingularM-ma-trixthereisoneandonlyoneM-matrixasitssquareroot.Inthispaper,westudyona
简介:Weestablishseveralfundamentalidentities,includingrecurrencerelations,degreeelevationformulas,partitionofunityandMarsdenidentity,forquantumBernsteinbasesandquantumBeziercurves.WealsodeveloptwotermrecurrencerelationsforquantumBernsteinbasesandrecursiveevaluationalgorithmsforquantumBeziercurves.Ourproofsusestandardmathematicalinductionandotherelementarytechniques.
简介:ThispapershowsthatforΩ∈H1(Sn-1),MarcinkiewiczintegraloperatorμΩisLp(Rn)boundedfor1<p<∞,whichimprovessomeknownresults.
简介:Inthispaper,westudythehomogeneouspolynomialsorthogonalwiththeweightfunctionh(x(d))=x2k11green.Itissimplerthantheformulain[2],andcanberegardedasanextensionof[1]undertheweightedcase.
简介:Givennon-negativeintegersm,n,handkwithm≥h>1andn≥k>1,an(h,k)-bipartitehypertournamentonm+nverticesisatriple(U,V,A),whereUandVaretwosetsofverticeswith|U|=mand|V|=n,andAisasetof(h+k)-tuplesofvertices,calledarcs,withatmosthverticesfromUandatmostkverticesfromV,suchthatforanyh+ksubsetsU_1∪V_1ofU∪V,Acontainsexactlyoneofthe(h+k)!(h+k)-tupleswhoseentriesbelongtoU_1∪V_1.Necessaryandsufficientconditionsforapairofnon-decreasingsequencesofnon-negativeintegerstobethelosingscorelistsorscorelistsofsome(h,k)-bipartitehypertournamentareobtained.
简介:ForagraphG,P(G,λ)denotesthechromaticpolynomialofG.TwographsGandHaresaidtobechromaticallyequivalent,denotedbyG-H,ifP(G,λ)=p(H,λ).Let[G]={H|H-G}.If[G]={G},thenGissaidtobechromaticallyunique.Foracomplete5-partitegraphGwith5nvertices,defineθ(G)=(a(G,6)-2^n+1-2^n-1+5)/2n-2,wherea(G,6)denotesthenumberof6-independentpartitionsofG.Inthispaper,theauthorsshowthatθ(G)≥0anddetermineallgraphswithθ(G)=0,1,2,5/2,7/2,4,17/4.Byusingtheseresultsthechromaticityof5-partitegraphsoftheformG-Swithθ(G)=0,1,2,5/2,7/2,4,17/4isinvestigated,whereSisasetofedgesofG.Manynewchromaticallyunique5-partitegraphsareobtained.
简介:1IntroductionH-matriceshaveimportantroleinnumericalanalysisandmatrixtheory,etc.Inthispaper,severalpracticalcriteriaforH-matricesareobtained,towhichthematrixisnotnecessarybeadiagonallydominantmatrix,andthenumbersofrowswhicharenotdiago-naldominantcanben-1.Weintroducetheconceptofα-diagonallydominantmatrix,andbyusingtheconcepttoinvestigateH-matrices.
简介:Alooplessgraphonnverticesinwhichverticesareconnectedatleastbyaandatmostbybedgesiscalleda(a,b,n)-graph.A(b,b,n)-graphiscalled(b,n)-graphandisdenotedbyKbn(itisacompletegraph),itscomplementbyKbn.Anonincreasingsequencep=(d1,…,dn)ofnonnegativeintegersissaidtobe(a,b,n)graphicifitisrealizablebyan(a,b,n)-graph.Wesayasimplegraphicsequencep=(d1,…,dn)ispotentiallyK4—K2[K2-graphicifithasaarealizationcontaininganK4—K2[K2asasubgraphwhereK4isacompletegraphonfourverticesandK2[K2isasetofindependentedges.Inthispaper,wefindthesmallestdegreesumsuchthateveryn-termgraphicalsequencecontainsK4—K2[K2assubgraph.
简介:TheauthorsconsidertheextendedHeckegroupsH(λq)generatedbyT(z)=-1/z,S(z)=-1/(z+λq)andR(z)=1/-zwithλq=2cos(π/q)forq≥3aninteger.Inthispaper,theevensubgroupHe(λq),thesecondcommutatorsubgroupH"(λq)andtheprincipalcongruencesubgroupsHp(λq)oftheextendedHeckegroupsH(λq)arestudied.Also,relationsbetweenthemaregiven.