简介:Inthispaper,wefirstdeterminetherelationsamongthebestboundsAandBoftheg-frame,theg-frameoperatorSandthepre-frameoperatorQandgiveanecessaryandsufficientconditionforag-framewithboundsAandBinacomplexHilbertspace.Wealsointroducethedefinitionofag-framesequenceandobtainanecessaryandsufficientconditionforag-framesequencewithboundsAandBinacomplexHilbertspace.Lastly,weconsiderthestabilityofag-framesequenceforacomplexHilbertspaceunderperturbation.
简介:AgraphGiscalledchromatic-choosableifitschoicenumberisequaltoitschromaticnumber,namelych(G)=χ(G).Ohba’sconjecturestatesthateverygraphGwith2χ(G)+1orfewerverticesischromaticchoosable.ItisclearthatOhba’sconjectureistrueifandonlyifitistrueforcompletemultipartitegraphs.Recently,Kostochka,StiebitzandWoodallshowedthatOhba’sconjectureholdsforcompletemultipartitegraphswithpartitesizeatmostfive.Butthecompletemultipartitegraphswithnorestrictionontheirpartitesize,forwhichOhba’sconjecturehasbeenverifiedarenothingmorethanthegraphsKt+3,2*(k-t-1),1*tbyEnotomoetal.,andKt+2,3,2*(k-t-2),1*tfort≤4byShenetal..Inthispaper,usingtheconceptoff-choosable(orL0-size-choosable)ofgraphs,weshowthatOhba’sconjectureisalsotrueforthegraphsKt+2,3,2*(k-t-2),1*twhent≥5.Thus,Ohba’sconjectureistrueforgraphsKt+2,3,2*(k-t-2),1*tforallintegerst≥1.
简介:<正>1IntroductionTheundergroundbrineresourcesdistributingwidelyinSichuanBasin,Chinahavedrawnworldwideattentionduetotheirunusualelementabundanceandexcellentquality.