PDF-De nition1.1(UniqueGame).AuniquegameconsistsofaconstraintgraphG=(V;E),

log1fractionofallconstraintsif1fractionofallconstraintsissatisableRecentlyTrevisan17developedanalgorithmthatsatises1O3p lognfractionofallconstraintsthiscanbeimprovedto1Op logn9. Denition1.ThesizeofanELconceptDisdenedasfollows:–forD2sig(T),s(D)=1;–forD=9r:C,s(D)=s(C)+1wherer2sigR(T)andCisanarbitraryconcept;–forD=C1uC2,s(D)=s(C1)+s(C2)whereC1;C2arearbitraryconc 3Forthetimebeing,thisdenitionissucientandfollowscommonlinguisticusage;however,whenweturntolocallyfreereexives(cf.section5),thetwonotions(anaphorvsreexive)willbedistinguishedalongthelinesproposedby IntervalName 0 PerfectUnison(PU)1 MinorSecond,orhalf-step(m2)2 MajorSecond,orwholestep(M2)3 MinorThird(m3)4 MajorThird(M3)5 PerfectFourth(P4)6 Tritone(TT)7 PerfectFifth(P5)8 MinorSixth(m6)9 MajorSixth (meetingtheminthreecollinearpoints)meet`4inapointQ4notonthelineofthethreecollinearmeetingpointsQi=\`i,thus,isgeneratedbyf`1;`2;`3;`4g.TheexampleabovedoesnotexistiftheeldFisalgebraicallyclosedsincei FixanintervalIintherealline(e.g.,Imightbe(17;19))andletx0beapointinI,i.e.,x02I:Nextconsiderafunction,whosedomainisI,f:I!Randwhosederivativesf(n):I!RexistontheintervalIforn=1;2;3;:::;N.Denition1.TheN Denition1(DisagreementCoefcient) LetHbeahypothesisclass,DbeadistributionoverXf0;1g,andDxbethemarginaldistributionoverX.Leth?beaminimizeroferrD(h).Thedisagreementcoefcientisdef=supr2(0;1)
(B(h?;r) 1.3.Operationsonknots.Muchofwhatisdiscussedhereappliestolinksofmorethanonecomponent,butthesegeneral-isationsshouldbeobvious,anditismoreconvenienttotalkprimarilyaboutknots.Denition1.3.1.Themirror-imag @t=X()(0;x)=x:Denition1.3.IfVisavarifoldinUandX2C1c(U;RN),thentherstvariationofValongXisdenedbyV(X)=d dtt=0M((t)]V);(1.1)wheretistheone-parameterfamilygeneratedbyX.Vhasboundedgeneralizedme ifthereissomelinecontainingallthosepoints.Denition2.Twolinesareparallel iftheynevermeet.Denition3.Whentwolinesmeetinsuchawaythattheadjacentanglesareequal,theequalanglesarecalledrightangles ,andtheli -SMART SMARTFigure1:ThisdiagramillustratesandcontraststheprioritystructuresinducedbytheBiasPropertiesinthedenitionof-SMART(Denition1)andSMART.is,insomesense,oneofthe\smallest"jobsinthesystem.Refer 1.INTRODUCTIONANDEFINITIONSInthisbookletweconsiderthefollowingproblem, Denition1.1.LeastSquaresProblem,alocalminimizerfor aregivenfunctions,and Example1.1.Animportantsourceofleastsquaresproblemsisdat 1.ReductionsDenition1.SAT(Booleansatisable)problem:SATproblemforagivenBooleanformulaetriestoanswer,whetherthereexistsatruthassignmentmakingtheBooleanformulatrue.Inotherwords,canweassignthevariableso 1Bilu{LinialStabilityKonstantinMakarychevkomakary@microsoft.comMicrosoftResearchRedmond,WA,USAYuryMakarychevyury@ttic.eduToyotaTechnologicalInstituteatChicagoChicago,IL,USAThischapterdescribesrecentre 1BiluLinialStabilityKonstantinMakarychevkomakarymicrosoftcomMicrosoftResearchRedmondWAUSAYuryMakarychevyurytticeduToyotaTechnologicalInstituteatChicagoChicagoILUSAThischapterdescribesrecentresultsonBi