Transcription of Computational & Applied Mathematics - UNIGE
1 Computational & GanderDepartmentofMathematicsandStatisti csMcGillUniversityOnleave at theUniversityofGeneva, 2002/2003 ComputationalandAppliedMathematics Vessel: speed vPolice Boat: speed uyxx0y0 Whatdirectiondoesthepoliceboathave tochoosetoapproachthesuspectvesseluptoa distanceRasquicklyaspossible?Computation alandAppliedMathematics thesametime=)y20+ (vt x0)2= (ut+R) NoteontheOptimalInterceptTimeofVesselsto a NonzeroRange G,SIAMR eview , , ViewSide ViewWhatis themaximumnumberofpiecesonecangetfroma donutwhencuttingit withthreeplanarcuts?Startingwithanapple, how many piecescanweget?ComputationalandAppliedMa thematics viewFirst cutThemostwecangetwitha singleplanarcutis viewFirst and second cutWe getsixpiecesintotal,eachofthefirsttwo is many piecesdowehave now ?ComputationalandAppliedMathematics right octant:1234567891011121314153 piecesUpper top octant:Lower top octant:1 pieceLower left octant:2 piecesUpper left octant:2 pieces1 pieceLower right octant:1 pieceLower bottom octant:2 piecesUpper bottom octant:3 pieces15 Pieceswith3 , themaximumnumberofpiecesonecangetis 13!
2 How ?AnarticlebyMartinGardnerinthe ScientificAmerican containsthegeneralresultfora thepiecesbetweenthecuts,whatis now themaximumnumberofpiecesyoucanget?Comput ationalandAppliedMathematics topushtheredballsothatit bouncesoftherimexactlyonceandthenhitsthe blueball?Is theremorethanonesolution?Computationalan dAppliedMathematics :AlgebraicSolutionPSfragreplacements xy cab1f( ) =(1 +ccos )p(a cos )2+ (b sin )2 (1 acos bsin )p(c+ cos )2+ (sin )2 ComputationalandAppliedMathematics f( ) 13021060240902701203001503301800 PSfragreplacements f( )Aretherealwaysfoursolutions?Computation alandAppliedMathematics em1m2 We needto solutionsandonewith2 (u) = (m2 m2m1)u4+(2m1 2m21+2e2+2m22)u3+6u2m2m1+ ( 2m22+ 2m21+ 2m1 2e2)u m2m1 m2= 0where = 2 arctan(u).Q(u)is a 4thdegreepolynomial, numericalexperiment: :ComputationalandAppliedMathematics (x,y)canbecomputedanalytically(tparamete r):x(t)= ch (1 +c)t6+ 3(1+ 3c)t4+ 3(1 3c)t2+ (1 c) ;y(t)=16hc2t3;h= (1 + 3c+ 2c2)t6+ 3(1+c+ 2c2)t4++3(1 c+ 2c2)t2+ (1 3c+ 2c2)wherecis thepositionofthefixedballonthex ofa coffeemugwitha pointsourceoflightemulatingthecircularbi lliardgame(DrexlerandG,SIAM review , , 1998)ComputationalandAppliedMathematics periodicsignalf(t)canbedecomposedintoits Fouriercomponents:f(t) =1Xk= 1^fkeikt:Discreteversion:forthevectorfof lengthn, wherefj:=f(tj),tj=j t, t= 2 =n, wehavefj=n 1Xk=0^fkeiktj:Whatif thesignalis animage,ora pieceofmusic?
3 ComputationalandAppliedMathematics , Il Volterriano1997-2003 ComputationalandAppliedMathematics considerrabbitsandfoxeslivingina ,theLotkaVolterramodelstates_x=x xy_y= y+xyApproximatingthederivative usingitsdefinition:x(tn+1) x(tn) t=x(tn) x(tn)y(tn)y(tn+1) y(tn) t= y(tn) +x(tn)y(tn)We , SymplecticMethodA verysmallchangeintheoriginalmethod,inste adofx(tn+1) x(tn) t=x(tn) x(tn)y(tn)y(tn+1) y(tn) t= y(tn) +x(tn)y(tn)changinginthesecondlinetntotn +1,x(tn+1) x(tn) t=x(tn) x(tn)y(tn)y(tn+1) y(tn) t= y(tn) +x(tn+1)y(tn)leadstoa methodforwhichonecanprove thattheapproximatesolutionis cycliclike ,like themathematicallyexactor biological ofthenewmethod 25 20 15 10 50510 2 101234 ,inspiteoftheproofofcyclicbehavior, withthezoom!ComputationalandAppliedMathe matics doesnotis verycomplicated:it is a inMontreal@u@t(x; t) =@2u@x2(x; t) +@2u@y2(x; t) +@2u@z2(x; t) +f(x; t)RoomtemperatureinourlivingroominMontre al(outsidetemperatureupto-47degrees) , a TurntableintheMicrowave ?
4 Thephysicalmodelis Maxwell s equationr E= Ht;r H="Et+ a TurntableintheMicrowave ?Thephysicalmodelis Maxwell s equationr E= Ht;r H="Et+ VOLVOS90 Noisesimulationona par-allelcomputertoimprovepassengercomfo rt12124351391511716101468 ComputationalandAppliedMathematics is importanttohave a :B-747inFlightSimulationofa B-747flyingthroughathunderstorm, Fluid Dynamics Radiation Particle Physics Aero DynamicsEngineering: Active Noise Cancellation Cellular Phone Systems Population Dynamics Antibiotics MicromachinesPhysics: Applied MathematicsPure Mathematics : Geometry Asymptotics ExistenceANDC omputationalBiology: Circuits
