简介:InthispaperL^p-L^qestimatesforthesolutionu(x,t)tothefollowingperturbedhigh-erorderhyperbolicequationareconsidered,(ρπ--a△)(ρπ--b△)u+V(x)u=O,x∈R^n,n≥6,ρ1eu(x,O)=O,ρ^3eu(x,O)=f(x),(j=O,1,2).WeassumethattheotentialV(x)andtheinitialdataf(x)arecompactlysupported,andV(x)issufficientlysmall,thenthesolutionu(x,t)oftheaboveproblemsatisfiesthesameL^p-L^qestimatesasthatoftheunperturbedproblem.
简介:<正>InthispaperitisprovedthatforallcompletelydistributivelatticesL.thecategoryofL-fuzzifyingtopologicalspacescanbewmbeddedinthecategoryofL-topologicalspaces(stratifiedChang-Goguenspaces)asasimultaneouslybireflectiveandbicoreflectivefullsubcategory.
简介:WeconstructaclassofintegrablegeneralizationofTodamechanicswithlong-rangeinteractions.ThesesystemsareassociatedwiththeloopalgebrasL(Cr)andL(Dr)inthesensethattheirLaxmatricescanberealizedintermsofthec=0representationsoftheaffineLiealgebrasCr(1)andDr(1)andtheinteractionspatterninvolvedbearsthetypicalcharactersofthecorrespondingrootsystems.WepresenttheequationsofmotionandtheHamiltonianstructure.Thesegeneralizedsystemscanbeidentifiedunambiguouslybyspecifyingtheunderlyingloopalgebratogetherwithanorderedpairofintegers(n,m).Itturnsoutthatdifferentsystemsassociatedwiththesameunderlyingloopalgebrabutwithdifferentpairsofintegers(n1,m1)and(n2,m2)withn2<n1andm2<m1canberelatedbyanestedHamiltonianreductionprocedure.Forallnontrivialgeneralizations,theextracoordinatesbesidesthestandardTodavariablesarePoissonnon-commute,andwheneithernorm≥3,thePoissonstructurefortheextracoordinatevariablesbecomessomeLiealgebra(i.e.theextravariablesappearlinearlyontheright-handsideofthePoissonbrackets).Inthequantumcase,suchgeneralizationswillbecomesystemswithnoncommutativevariableswithoutspoilingtheintegrability.