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paper.' - pnas.org
, "'Itisprovedbyconstructingan"abstract"ma ppingcylinderofXandtran-scribingintoalge braictermstheproofoftheanalogoustheoremo nCW-complexes.*Thisnotearosefromconsulta tionsduringthetenureofaJohnSimonGuggenhe imMemorialFellowshipbyMacLane.'Whitehead , ,"CombinatorialHomotopyIandII," ,55,214-245and453-496(1949).Werefertothe sepapersasCHIandCHII,respectively.'Byaco mplexweshallmeanaconnectedCWcomplex,asde finedin ,S.,"CohomologyTheoryinAbstractGroupsIII ," ,50,736-761(1949),referredtoasCTIII.'An( unpublished) ( )thestrongerhypothesisthatXBcontainsthec enterofA,butalltherelevantdevelopmentsth ereapplyundertheweakerassumption( ).IEilenberg,S.,andMacLane,S.,"Cohomolog yTheoryinAbstractGroupsII," ,48,326-341(1947).8 Eilenberg,S.,andMacLane,S.," ,"thesePROCEEDINGS,32,277-280(1946).9 Blakers, ,"SomeRelationsBetweenHomologyandHomotop yGroups," ,49,428-461(1948), ,requiringthatv-(1)notbecyclic,canberead ilyrealizedbysuitablechoiceofthefreegrou pX,butthishypothesisisnotneededhere( ).
MATHEMATICS: G. POLYA distributionsoverthepurestrategies, thepay-offfunctionsaretheexpecta-tions of the players, thus becoming polylinear forms in the probabilities
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