简介：LetFbeafieldofcharacteristiczero.Wn=F[t(+1/2),t(+1/2),...,t(+1/n)]δ/δt1+...+F[t(+1/2),t(+1/2),...,t(+1/n)]δ/δtnistheWittalgebraoverF,Wn+=F[t1,t2...,tn]δ/δt+...+F[t1,t2...,tn]δ/δtnisLieshbalgebraofWn.ItiswellknownbothWnandWn+aresimpleinfinitedimensionalLiealgebra.InZhao'spaper,itwasconjecturedthatEnd(Wn^+)-{0}=Aut(Wn^+)anditwasprovedthatthevalidityofthisconjectureimpliesthevalidityofthewell-knownJacobianconjecture.Inthisshortnote,wechecktheconjectureaboveforn=1.WeshowEnd(W1^+)-{0}=Aut(W1^+).

简介：TheLiealgebraofderivationsofrationalfunctionfieldC(t)isC(t)d/dtTheautomorphismgroupofC(t)iswellknownastobeisomorphictotheprojectivelineargroupPGL(2,C).InthisshortnoteweprovethateveryautomorphismofC(t)d/dtcanbeinducedinanaturalwayfromanautomorphismofC(t).

简介：WeputforthDramaTheoryⅡ(DTⅡ)asaformalmultipleparticipantdecisionmakingframeworkthatcanbeusedtosystematicallymodelcomplexsecuritychallenges,andadvancethefieldofSustainableSecuritySystemsEngineering.DTⅡisdefinedasatheoryof'largeworld'pre-gamecommunicationandequilibriumselection.Whilegametheoretictoolshavebeenwidelyappliedtoresolveenvironmentalconflictsandpromoteglobalsecurity,traditionalgametheoryassumesthatdecisionmakers,options,andpreferencesarefixed.Amathematicaltreatmentofkeydramatheoreticconcepts(i.e.positions,intentions,doubtsanddilemmas)isprovided.Thedynamicsofthedramatheoreticprocessarediscussedandtheexpectedequilibriumsetisderived.Thefundamentaltheoremofdramatheoryisprovenandalltheoreticalresultsareappliedtopromotesustainablesecuritysolutions.ItisemphasizedthatDTⅡrepresentsaflexiblesystemsengineeringtechniquetoaddresstime-sensitive,multi-faceted,andcomplexmultipleparticipantnegotiations.