Problems from the Discrete to the Continuous: Probability, Number Theory, Graph Theory, and Combinatorics PDF
Preview Problems from the Discrete to the Continuous: Probability, Number Theory, Graph Theory, and Combinatorics
Universitext Ross G. Pinsky Problems from the Discrete to the Continuous Probability, Number Theory, Graph Theory, and Combinatorics Universitext Universitext SeriesEditors: SheldonAxler SanFranciscoStateUniversity VincenzoCapasso UniversitàdegliStudidiMilano CarlesCasacuberta UniversitatdeBarcelona AngusMacIntyre QueenMaryUniversityofLondon KennethRibet UniversityofCalifornia,Berkeley ClaudeSabbah CNRS,ÉcolePolytechnique,Paris EndreSüli UniversityofOxford WojborA.Woyczynski CaseWesternReserveUniversity,Cleveland,OH Universitextisaseriesoftextbooksthatpresentsmaterialfromawidevarietyofmathematical disciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass-testedbytheirauthor, mayhaveaninformal,personalevenexperimentalapproachtotheirsubjectmatter.Someof themostsuccessfulandestablishedbooksintheserieshaveevolvedthroughseveraleditions, alwaysfollowingtheevolutionofteachingcurricula,toverypolishedtexts. Thusasresearchtopicstrickledownintograduate-levelteaching,firsttextbookswrittenfor new,cutting-edgecoursesmaymaketheirwayintoUniversitext. Forfurthervolumes: http://www.springer.com/series/223 Ross G. Pinsky Problems from the Discrete to the Continuous Probability, Number Theory, Graph Theory, and Combinatorics 123 RossG.Pinsky DepartmentofMathematics Technion-IsraelInstituteofTechnology Haifa,Israel ISSN0172-5939 ISSN2191-6675(electronic) ISBN978-3-319-07964-6 ISBN978-3-319-07965-3(eBook) DOI10.1007/978-3-319-07965-3 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014942157 MathematicsSubjectClassification(2010):05A,05C,05D,11N,60 ©SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Inmostsciencesonegenerationtearsdown whatanotherhasbuilt andwhatonehas establishedanotherundoes.InMathematics aloneeachgenerationbuildsanewstoryto theoldstructure. —HermannHankel Apeculiarbeautyreignsintherealmof mathematics,abeautywhichresemblesnot somuchthebeautyofartasthebeautyof natureandwhichaffectsthereflectivemind, whichhasacquiredanappreciationofit, verymuchlikethelatter. —ErnstKummer ToJeanette andto E.A.P. Y.U.P. L.A.T-P. M.D.P. Preface Itisoftenaverredthattwocontrastingculturescoexistinmathematics—thetheory- buildingcultureandtheproblem-solvingculture.Thepresentvolumewascertainly spawned by the latter. This book takes an array of specific problems and solves them,withtheneededtoolsdevelopedalongthewayinthecontextoftheparticular problems. Thebookisanunusualhybrid.Ittreatsamélangeoftopicsfromcombinatorial probabilitytheory,multiplicativenumbertheory,randomgraphtheory,andcombi- natorics. Objectively, what the problems in this book have in common is that they involve the asymptotic analysis of a discrete construct, as some natural parameter ofthesystemtendstoinfinity.Subjectively,whattheseproblemshaveincommon isthatboththeirstatementsandtheirsolutionsresonateaestheticallywithme. Theresultsinthisbooklendthemselvestothetitle“ProblemsfromtheFiniteto the Infinite”; however, with regard to the methods of proof, the chosen appellation is the more apt. In particular, generating functions in their various guises are a fundamental bridge “from the discrete to the continuous,” as the book’s title would have it; such functions work their magic often in these pages. Besides bridgingdiscretemathematicsandmathematicalanalysis,thebookmakesamodest attempt at bridging disciplines—probability, number theory, graph theory, and combinatorics. In addition to the considerations mentioned above, the problems were selected withaneyetowardaccessibilitytoawideaudience,includingadvancedundergrad- uatestudents.Thetechnicalprerequisitesforthebookareagoodgroundinginbasic undergraduateanalysis,atouchoffamiliaritywithcombinatorics,andalittlebasic probabilitytheory.Oneappendixprovidesthenecessaryprobabilisticbackground, and another appendix provides a warm-up for dealing with generating functions. That said, a moderate dose of the elusive quality known as mathematical maturity willcertainlybehelpfulthroughoutthetextandwillbenecessaryonoccasion. Theprimaryintentofthebookistointroduceanumberofbeautifulproblemsin avarietyofsubjectsquickly,pithily,andcompletelyrigorouslytograduatestudents and advanced undergraduates. The book could be used for a seminar/capstone courseinwhichstudentspresentthelectures.Itishopedthatthebookmightalsobe ix
The list of books you might like
