LecturesonStochasticProgramming:ModelingandTheory(MPS-SIAMSeriesonOptimization)

Advancedundergraduateorbeginninggraduatestudentsneedaunifiedfoundationfortheirstudyofgeometry,analysis,andalgebra.Forthefirsttimeinatext,thisbookusescategoricalalgebratobuildsuchafoundation,startingfromintuitivedescriptionsofmathematicallyandphysicallycommonphenomenaandadvancingtoaprecisespecificationofthenatureofCategoriesofSets.Settheoryasthealgebraofmappingsisintroducedanddevelopedasaunifyingbasisforadvancedmathematicalsubjectssuchasalgebra,geometry,analysis,andcombinatorics.Theformalstudyevolvesfromgeneralaxiomswhichexpressuniversalpropertiesofsums,products,mappingsets,andnaturalnumberrecursion.ThedistinctivefeaturesofCantorianabstractsets,ascontrastedwiththevariableandcohesivesetsofgeometryandanalysis,aremadeexplicitandtakenasspecialaxioms.Functorcategoriesareintroducedinordertomodelthevariablesetsusedingeometry,andtoillustratethefailureoftheaxiomofchoice.Anappendixprovidesanexplicitintroductiontonecessaryconceptsfromlogic,andanextensiveglossaryprovidesawindowtothemathematicallandscape.