utm lovasz lpelikan j.vesztergombi k. - discrete mathematics(302s)

utm lovasz lpelikan j.vesztergombi k. - discrete mathematics(302s)

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时间:2018-08-02

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1、PrefaceFormoststudents,thefirstandoftenonlycourseincollegemathematicsiscalculus.Itistruethatcalculusisthesinglemostimportantfieldofmathematics,whoseemergenceintheseventeenthcenturysignaledthebirthofmodernmathematicsandwasthekeytothesuccessfulapplicationsofmathematicsinthesciencesandengineer

2、ing.Butcalculus(oranalysis)isalsoverytechnical.Ittakesalotofworkeventointroduceitsfundamentalnotionslikecontinuityandthederivative(afterall,ittooktwocenturiesjusttodeveloptheproperdefinitionofthesenotions).Togetafeelingforthepowerofitsmethods,saybydescribingoneofitsimportantapplicationsind

3、etail,takesyearsofstudy.Ifyouwanttobecomeamathematician,computerscientist,orengineer,thisinvestmentisnecessary.Butifyourgoalistodevelopafeelingforwhatmathematicsisallabout,wheremathematicalmethodscanbehelpful,andwhatkindsofquestionsdomathematiciansworkon,youmaywanttolookfortheanswerinsome

4、otherfieldsofmathematics.Therearemanysuccessstoriesofappliedmathematicsoutsidecalculus.Arecenthottopicismathematicalcryptography,whichisbasedonnumbertheory(thestudyofthepositiveintegers1,2,3,...),andiswidelyapplied,forexample,incomputersecurityandelectronicbanking.Otherimportantareasinappl

5、iedmathematicsarelinearprogramming,codingtheory,andthetheoryofcomputing.Themathematicalcontentintheseapplicationsiscollectivelycalleddiscretemathematics.(Theword“discrete”isusedinthesenseof“separatedfromeachother,”theoppositeof“continuous;”itisalsooftenusedinthemorerestrictivesenseof“finit

6、e.”Themoreeverydayversionofthisword,meaning“circumspect,”isspelled“discreet.”)viPrefaceTheaimofthisbookisnottocover“discretemathematics”indepth(itshouldbeclearfromthedescriptionabovethatsuchataskwouldbeill-definedandimpossibleanyway).Rather,wediscussanumberofselectedresultsandmethods,mostl

7、yfromtheareasofcombinatoricsandgraphthe-ory,withalittleelementarynumbertheory,probability,andcombinatorialgeometry.Itisimportanttorealizethatthereisnomathematicswithoutproofs.Merelystatingthefacts,withoutsayingsomethingaboutwhythesefactsarevalid,wouldbeterriblyfarfr

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