Even then it took many years to understand the importance and generality of the ideas involved This one underlying idea can be used to describe the forces and accelerations in Newtonian mechanics and the potential functions of electromagnetism and t ID: 24395
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6|VectorSpaces26.2AxiomsTheprecisedenitionofavectorspaceisgivenbylistingasetofaxioms.Forthispurpose,I'lldenotevectorsbyarrowsoveraletter,andI'lldenotescalarsbyGreekletters.Thesescalarswill,forourpurpose,beeitherrealorcomplexnumbers|itmakesnodi erencewhichfornow.*1Thereisafunction,additionofvectors,denoted+,sothat~v1+~v2isanothervector.2Thereisafunction,multiplicationbyscalars,denotedbyjuxtaposition,sothat ~visavector.3(~v1+~v2)+~v3=~v1+(~v2+~v3)(theassociativelaw).4Thereisazerovector,sothatforeach~v,~v+~O=~v.5Thereisanadditiveinverseforeachvector,sothatforeach~v,thereisanothervector~v0sothat~v+~v0=~O.6Thecommutativelawofadditionholds:~v1+~v2=~v2+~v1.7( +)~v= ~v+~v.8( )~v= (~v).9 (~v1+~v2)= ~v1+ ~v2.101~v=~v.Inaxioms1and2Icalledtheseoperations\functions."Isthattherightuseoftheword?Yes.Withoutgoingintotheprecisedenitionoftheword(seesection12.1),youknowitmeansthatyouhaveoneormoreindependentvariablesandyouhaveasingleoutput.Additionofvectorsandmultiplicationbyscalarscertainlytthatidea.6.3ExamplesofVectorSpacesExamplesofsetssatisfyingtheseaxiomsabound:1Theusualpictureofdirectedlinesegmentsinaplane,usingtheparallelogramlawofaddition.2Thesetofreal-valuedfunctionsofarealvariable,denedonthedomain[axb].Additionisdenedpointwise.Iff1andf2arefunctions,thenthevalueofthefunctionf1+f2atthepointxisthenumberf1(x)+f2(x).Thatis,f1+f2=f3meansf3(x)=f1(x)+f2(x).Similarly,multiplicationbyascalarisdenedas( f)(x)= (f(x)).Noticeasmallconfusionofnotationinthisexpression.Therstmultiplication,( f),multipliesthescalar bythevectorf;thesecondmultipliesthescalar bythenumberf(x).3Likeexample2,butrestrictedtocontinuousfunctions.Theoneobservationbeyondthepreviousexampleisthatthesumoftwocontinuousfunctionsiscontinuous.4Likeexample2,butrestrictedtoboundedfunctions.Theoneobservationbeyondthepreviousexampleisthatthesumoftwoboundedfunctionsisbounded.5Thesetofn-tuplesofrealnumbers:(a1;a2;:::;an)whereadditionandscalarmultiplicationaredenedby(a1;:::;an)+(b1;:::;bn)=(a1+b1;:::;an+bn) (a1;:::;an)=( a1;:::; an)6Thesetofsquare-integrablereal-valuedfunctionsofarealvariableonthedomain[axb].Thatis,restrictexampletwotothosefunctionswithRbadxjf(x)j21.Axiom1istheonlyonerequiringmorethanasecondtocheck.7Thesetofsolutionstotheequation@2 =@x2+@2 =@y2=0inanyxeddomain.(Laplace'sequation) *Foraniceintroductiononlinesee distance-ed.math.tamu.edu/Math640 ,chapterthree. 6|VectorSpaces5Theintegraloftheright-handsideisbyassumptionnite,sothesamemustholdfortheleftside.Thissaysthatthesum(anddi erence)oftwosquare-integrablefunctionsissquare-integrable.Forthisexamplethen,itisn'tverydiculttoshowthatitsatisestheaxiomsforavectorspace,butitrequiresmorethanjustaglance.Thereareafewpropertiesofvectorspacesthatseemtobemissing.Thereisthesomewhatoddnotation~v0fortheadditiveinverseinaxiom5.Isn'tthatjust�~v?Isn'tthezerovectorsimplythenumberzerotimesavector?Yesinbothcases,butthesearetheoremsthatfolloweasilyfromthetenaxiomslisted.Seeproblem 6.20 .I'lldopart(a)ofthatexerciseasanexamplehere:Theorem:thevector~Oisunique.Proof:assumeitisnot,thentherearetwosuchvectors,~O1and~O2.By[4],~O1+~O2=~O1(~O2isazerovector)By[6],theleftsideis~O2+~O1By[4],thisis~O2(~O1isazerovector)Putthesetogetherand~O1=~O2.Theorem:Ifasubsetofavectorspaceisclosedunderadditionandmultiplicationbyscalars,thenitisitselfavectorspace.Thismeansthatifyouaddtwoelementsofthissubsettoeachothertheyremaininthesubsetandmultiplyinganyelementofthesubsetbyascalarleavesitinthesubset.Itisa\subspace."Proof:theassumptionofthetheoremisthataxioms1and2aresatisedasregardsthesubset.Thataxioms3through10holdfollowsbecausetheelementsofthesubsetinherittheirpropertiesfromthelargervectorspaceofwhichtheyareapart.Isthisallthereistoit?Notquite.Axioms4and5takealittlemorethought,andneedtheresultsoftheproblem 6.20 ,parts(b)and(d).6.4LinearIndependenceAsetofnon-zerovectorsislinearlydependentifoneelementofthesetcanbewrittenasalinearcombinationoftheothers.Thesetislinearlyindependentifthiscannotbedone.Bases,Dimension,ComponentsAbasisforavectorspaceisalinearlyindependentsetofvectorssuchthatanyvectorinthespacecanbewrittenasalinearcombinationofelementsofthisset.Thedimensionofthespaceisthenumberofelementsinthisbasis.Ifyoutaketheusualvectorspaceofarrowsthatstartfromtheoriginandlieinaplane,thecommonbasisisdenoted^{,^|.IfIproposeabasisconsistingof^{;�1 2^{+p 3 2^|;�1 2^{�p 3 2^|thesewillcertainlyspanthespace.Everyvectorcanbewrittenasalinearcombinationofthem.Theyarehowever,redundant;thesumofallthreeiszero,sotheyaren'tlinearlyindependentandaren'tabasis.Ifyouusethemasiftheyareabasis,thecomponentsofagivenvectorwon'tbeunique.Maybethat'so.k.andyouwanttodoit,buteitherbecarefulorlookupthemathematicalsubjectcalled\frames."Beginningwiththemostelementaryproblemsinphysicsandmathematics,itisclearthatthechoiceofanappropriatecoordinatesystemcanprovidegreatcomputationaladvantages.Indealingwiththeusualtwoandthreedimensionalvectorsitisusefultoexpressanarbitraryvectorasasumofunitvectors.Similarly,theuseofFourierseriesfortheanalysisoffunctionsisaverypowerfultoolinanalysis.Thesetwoideasareessentiallythesamethingwhenyoulookatthemasaspectsofvectorspaces.Iftheelementsofthebasisaredenoted~ei,andavector~ais~a=Xiai~ei; 6|VectorSpaces7arefunctions,andassuchtheyareelementsofthevectorspaceofexample2.Allyouneedtodonowistoverifythatthesumoftwosolutionsisasolutionandthataconstanttimesasolutionisasolution.That'swhatthephrase\linear,homogeneous"means.Anothercommondi erentialequationisd2 dt2+g `sin=0Thisdescribesthemotionofanundampedpendulum,andthesetofitssolutionsdonotformavectorspace.Thesumoftwosolutionsisnotasolution.TherstofEqs.( 6.5 )hastwoindependentsolutions,x1(t)=e� tcos!0t;andx2(t)=e� tsin!0t(6:6)where =�b=2mand!0=q k m�b2 4m2.ThisisfromEq.(4.8).Anysolutionofthisdi erentialequationisalinearcombinationofthesefunctions,andIcanrestatethatfactinthelanguageofthischapterbysayingthatx1andx2formabasisforthevectorspaceofsolutionsofthedampedoscillatorequation.Ithasdimensiontwo.Thesecondequationofthepair( 6.5 )isathirdorderdi erentialequation,andassuchyouwillneedtospecifythreeconditionstodeterminethesolutionandtodetermineallthethreearbitraryconstants.Inotherwords,thedimensionofthesolutionspaceofthisequationisthree.Inchapter4onthesubjectofdi erentialequations,oneofthetopicswassimultaneousdi erentialequations,coupledoscillations.Thesimultaneousdi erentialequations,Eq.(4.45),arem1d2x1 dt2=�k1x1�k3(x1�x2);andm2d2x2 dt2=�k2x2�k3(x2�x1)andhavesolutionsthatarepairsoffunctions.Inthedevelopmentofsection4.10(atleastfortheequalmass,symmetriccase),Ifoundfourpairsoffunctionsthatsatisedtheequations.Nowtranslatethatintothelanguageofthischapter,usingthenotationofcolumnmatricesforthefunctions.Thesolutionisthevectorx1(t)x2(t)andthefourbasisvectorsforthisfour-dimensionalvectorspaceare~e1=ei!1tei!1t;~e2=e�i!1te�i!1t;~e3=ei!2t�ei!2t;~e4=e�i!2t�e�i!2tAnysolutionofthedi erentialequationsisalinearcombinationofthese.Intheoriginalnotation,youhaveEq.(4.52).Inthecurrentnotationyouhavex1x2=A1~e1+A2~e2+A3~e3+A4~e46.5NormsThe\norm"orlengthofavectorisaparticularlyimportanttypeoffunctionthatcanbedenedonavectorspace.Itisafunction,usuallydenotedbykk,andthatsatises1.k~vk0;k~vk=0ifandonlyif~v=~O2.k ~vk=j jk~vk3.k~v1+~v2kk~v1k+k~v2k(thetriangleinequality)Thedistancebetweentwovectors~v1and~v2istakentobek~v1�~v2k. 6|VectorSpaces86.6ScalarProductThescalarproductoftwovectorsisascalarvaluedfunctionoftwovectorvariables.Itcouldbedenotedasf(~u;~v),butastandardnotationforitis ~u;~v .Itmustsatisfytherequirements1. ~w;(~u+~v) = ~w;~u + ~w;~v 2. ~w; ~v = ~w;~v 3. ~u;~v *= ~v;~u 4. ~v;~v 0;and ~v;~v =0ifandonlyif~v=~OWhenascalarproductexistsonaspace,anormnaturallydoestoo:k~vk=q ~v;~v :(6:7)ThatthisisanormwillfollowfromtheCauchy-Schwartzinequality.Notallnormscomefromscalarproducts.ExamplesUsetheexamplesofsection 6.3 toseewhattheseare.Thenumbershererefertothenumbersofthatsection.1Anormistheusualpictureofthelengthofthelinesegment.Ascalarproductistheusualproductoflengthstimesthecosineoftheanglebetweenthevectors. ~u;~v =~u.~v=uvcos#:(6:8)4Anormcanbetakenastheleastupperboundofthemagnitudeofthefunction.Thisisdistinguishedfromthe\maximum"inthatthefunctionmaynotactuallyachieveamaximumvalue.Sinceitisboundedhowever,thereisanupperbound(manyinfact)andwetakethesmallestoftheseasthenorm.On�1x+1,thefunctionjtan�1xjhas=2foritsleastupperbound,thoughitneverequalsthatnumber.5Apossiblescalarproductis (a1;:::;an);(b1;:::;bn) =nXk=1a*kbk:(6:9)Thereareotherscalarproductsforthesamevectorspace,forexample (a1;:::;an);(b1;:::;bn) =nXk=1ka*kbk(6:10)Infactanyotherpositivefunctioncanappearasthecoecientinthesumanditstilldenesavalidscalarproduct.It'ssurprisinghowoftensomethinglikethishappensinrealsituations.Instudyingnormalmodesofoscillationthemassesofdi erentparticleswillappearascoecientsinanaturalscalarproduct.Iusedcomplexconjugationontherstfactorhere,butexample5referredtorealnumbersonly.Thereasonforleavingtheconjugationinplaceisthatwhenyoujumptoexample14youwanttoallowforcomplexnumbers,andit'sharmlesstoputitinfortherealcasebecauseinthatinstanceitleavesthenumberalone. 6|VectorSpaces10saleiswithin1000feetofaschool.Ifyouareanattorneydefendingsomeoneaccusedofthiscrime,whichofthenormsinEq.( 6.11 )wouldyouarguefor?Thelegislatorswhowrotethislawdidn'tknowlinearalgebra,sotheydidn'tspecifywhichnormtheyintended.Theprosecutingattorneyarguedfornorm#1,\asthecrow ies,"butthedefensearguedthat\crowsdon'tselldrugs"andhumansmovealongcitystreets,sonorm#2ismoreappropriate.TheNewYorkCourtofAppealsdecidedthatthePythagoreannorm(#1)istheappropriateoneandtheyrejectedtheuseofthepedestriannormthatthedefendantadvocated(#2). www.courts.state.ny.us/ctapps/decisions/nov05/162opn05.pdf 6.7BasesandScalarProductsWhenthereisascalarproduct,amostusefultypeofbasisistheorthonormalone,satisfying ~vi;~vj =ij=1ifi=j0ifi6=j(6:15)ThenotationijrepresentstheveryusefulKroneckerdeltasymbol.IntheexampleofEq.( 6.1 )thebasisvectorsareorthonormalwithrespecttothescalarproductinEq.( 6.9 ).Itisorthogonalwithrespecttotheotherscalarproductmentionedthere,butitisnotinthatcasenormalizedtomagnitudeone.Toseehowthechoiceofevenanorthonormalbasisdependsonthescalarproduct,tryadi erentscalarproductonthisspace.Takethespecialcaseoftwodimensions.Thevectorsarenowpairsofnumbers.Thinkofthevectorsas21matrixcolumnandusethe22matrix2112Takethescalarproductoftwovectorstobe (a1;a2);(b1;b2) =(a*1a*2)2112b1b2=2a*1b1+a*1b2+a*2b1+2a*2b2(6:16)Toshowthatthissatisesallthedenedrequirementsforascalarproducttakesasmallamountoflabor.Thevectorsthatyoumayexpecttobeorthogonal,(10)and(01),arenot.Inexample6,ifweletthedomainofthefunctionsbe�Lx+LandthescalarproductisasinEq.( 6.12 ),thenthesetoftrigonometricfunctionscanbeusedasabasis.sinnx Landcosmx Ln=1;2;3;:::andm=0;1;2;3;::::Thatafunctioncanbewrittenasaseriesf(x)=1X1ansinnx L+1X0bmcosmx L(6:17)onthedomain�Lx+LisjustanexampleofFourierseries,andthecomponentsoffinthisbasisareFouriercoecientsa1;:::;b0;:::.Anequallyvalidandmoresuccinctlystatedbasisisenix=L;n=0;1;2;:::Chapter5onFourierseriesshowsmanyotherchoicesofbases,allorthogonal,butnotnecessarilynormalized. 6|VectorSpaces14Exercises1Determineifthesearevectorspaceswiththeusualrulesforadditionandmultiplicationbyscalars.Ifnot,whichaxiom(s)dotheyviolate?(a)Quadraticpolynomialsoftheformax2+bx(b)Quadraticpolynomialsoftheformax2+bx+1(c)Quadraticpolynomialsax2+bx+cwitha+b+c=0(d)Quadraticpolynomialsax2+bx+cwitha+b+c=12Whatisthedimensionofthevectorspaceof(upto)5thdegreepolynomialshavingadoublerootatx=1?3Startingfromthreedimensionalvectors(thecommondirectedlinesegments)andasinglexedvector~B,isthesetofallvectors~vwith~v.~B=0avectorspace?Ifso,whatisit'sdimension?Isthesetofallvectors~vwith~v~B=0avectorspace?Ifso,whatisit'sdimension?4Thesetofalloddpolynomialswiththeexpectedrulesforadditionandmultiplicationbyscalars.Isitavectorspace?5Thesetofallpolynomialswherethefunction\addition"isdenedtobef3=f2+f1ifthenumberf3(x)=f1(�x)+f2(�x).Isitavectorspace?6Sameasthepreceding,butfor(a)evenpolynomials,(b)oddpolynomials7Thesetofdirectedlinesegmentsintheplanewiththenewruleforaddition:addthevectorsaccordingtotheusualrulethenrotatetheresultby10counterclockwise.Whichvectorspaceaxiomsareobeyedandwhichnot? 6|VectorSpaces16andusetheGram-Schmidtproceduretoconstructanorthonormalbasisstartingfrom~v0.Comparetheseresultstotheresultsofsection4.11.[Thesepolynomialsappearinthestudyofelectricpotentialsandinthestudyofangularmomentuminquantummechanics:Legendrepolynomials.]6.8Repeatthepreviousproblem,butuseadi erentscalarproduct: f;g =Z1�1dxe�x2f(x)*g(x)[Thesepolynomialsappearinthestudyoftheharmonicoscillatorinquantummechanicsandinthestudyofcertainwavesintheupperatmosphere.WithaconventionalnormalizationtheyarecalledHermitepolynomials.]6.9ConsiderthesetofallpolynomialsinxhavingdegreeN.Showthatthisisavectorspaceandnditsdimension.6.10ConsiderthesetofallpolynomialsinxhavingdegreeNandonlyevenpowers.Showthatthisisavectorspaceandnditsdimension.Whataboutoddpowersonly?6.11Whichofthesearevectorspaces?(a)allpolynomialsofdegree3(b)allpolynomialsofdegree3[Isthereadi erencebetween(a)and(b)?](c)allfunctionssuchthatf(1)=2f(2)(d)allfunctionssuchthatf(2)=f(1)+1(e)allfunctionssatisfyingf(x+2)=f(x)(f)allpositivefunctions(g)allpolynomialsofdegree4satisfyingR1�1dxxf(x)=0.(h)allpolynomialsofdegree4wherethecoecientofxiszero.[Isthereadi erencebetween(g)and(h)?]6.12(a)Forthecommonpictureofarrowsinthreedimensions,provethatthesubsetofvectors~vthatsatisfy~A.~v=0forxed~Aformsavectorspace.Sketchit.(b)Whatiftherequirementisthatboth~A.~v=0and~B.~v=0hold.Describethisandsketchit.6.13Ifanormisdenedintermsofascalarproduct,k~vk=q ~v;~v ,itsatisesthe\parallelogramidentity"(forrealscalars),k~u+~vk2+k~u�~vk2=2k~uk2+2k~vk2:(6:29)6.14Ifanormsatisestheparallelogramidentity,thenitcomesfromascalarproduct.Again,assumerealscalars.Considercombinationsofk~u+~vk2,k~u�~vk2andconstructwhatoughttobethescalarproduct.Youthenhavetoprovethefourpropertiesofthescalarproductasstatedatthestartofsection 6.6 .Numbersfourandthreeareeasy.Numberonerequiresthatyoukeeppluggingaway,usingtheparallelogramidentity(fourtimesbymycount).Numbertwoisdownrighttricky;leaveittotheend.Ifyoucanproveitforintegerandrationalvaluesoftheconstant ,consideritajobwelldone.Iusedinductionatonepointintheproof.Thenalstep,extending toallrealvalues,requiressomeargumentsaboutlimits,andistypicallythesortofreasoningyouwillseeinanadvancedcalculusormathematicalanalysiscourse. 6|VectorSpaces18(b)Thenumber0timesanyvectoristhezerovector:0~v=~O.(c)Thevector~v0isunique.(d)(�1)~v=~v0.6.21Forthevectorspaceofpolynomials,arethetwofunctionsf1+x2;x+x3glinearlyindependent?6.22Findthedimensionofthespaceoffunctionsthatarelinearcombinationsoff1;sinx;cosx;sin2x;cos2x;sin4x;cos4x;sin2xcos2xg �2�101�101234�2�1012346.23Amodelvectorspaceisformedbydrawingequidistantparallellinesinaplaneandlabellingadjacentlinesbysuccessiveintegersfrom1to+1.Denemultiplicationbya(real)scalarsothatmultiplicationofthevectorby meansmultiplythedistancebetweenthelinesby1= .Deneadditionoftwovectorsbyndingtheintersectionsofthelinesandconnectingoppositecornersoftheparallelogramstoformanothersetofparallellines.Theresultinglinesarelabelledasthesumofthetwointegersfromthe inter secting lines.(Therearetwochoiceshere,ifoneisaddition,whatistheother?)Showthatthisconstructionsatisesalltherequirementsforavectorspace.Justasadirectedlinesegmentisagoodwaytopicturevelocity,thisconstructionisagoodwaytopicturethegradientofafunction.Inthevectorspaceofdirectedlinesegments,youpinthevectorsdownsothattheyallstartfromasinglepoint.Here,youpinthemdownsothatthelineslabeled\zero"allpassthroughaxedpoint.DidIdenehowtomultiplybyanegativescalar?Ifnot,thenyoushould.Thispictureofvectorsisdevelopedextensivelyinthetext\Gravitation"byMisner,Wheeler,andThorne.6.24Inproblem 6.11 (g),ndabasisforthespace.Ans:1,x,3x�5x3.6.25Whatisthedimensionofthesetofpolynomialsofdegreelessthanorequalto10andwithatriplerootatx=1?6.26VerifythatEq.( 6.16 )doessatisfytherequirementsforascalarproduct.6.27Avariationonproblem 6.15 :f3=f1+f2means(a)f3(x)=Af1(x�a)+Bf2(x�b)forxeda,b,A,B.Forwhatvaluesoftheseconstantsisthisavectorspace?(b)Nowwhataboutf3(x)=f1(x3)+f2(x3)?6.28Determineifthesearevectorspaces:(1)Pairsofnumberswithadditiondenedas(x1;x2)+(y1;y2)=(x1+y2;x2+y1)andmultiplicationbyscalarsasc(x1;x2)=(cx1;cx2).(2)Likeexample2ofsection 6.3 ,butrestrictedtothosefsuchthatf(x)0.(realscalars)(3)Liketheprecedingline,butdeneadditionas(f+g)(x)=f(x)g(x)and(cf)(x)=�f(x)c. 6|VectorSpaces20Nowpickupthesamef1androtateitby90clockwiseaboutthepositivex-axis,againnallyexpressingtheresultintermsofsphericalcoordinates.Callitf3.Ifnowyoutaketheoriginalf1androtateitaboutsomerandomaxisbysomerandomangle,showthattheresultingfunctionf4isalinearcombinationofthethreefunctionsf1,f2,andf3.I.e.,allthesepossiblerotatedfunctionsformathreedimensionalvectorspace.Again,calculationssuchasthesearemucheasiertodemonstrateinrectangularcoordinates.6.37Takethefunctionsf1,f2,andf3fromtheprecedingproblemandsketchtheshapeofthefunctionsre�rf1(; );re�rf2(; );re�rf3(; )Tosketchthese,picturethemasdeningsomesortofdensityinspace,ignoringthefactthattheyaresometimesnegative.Youcanjusttaketheabsolutevalueorthesquareinordertovisualizewheretheyarebigorsmall.Usedarkandlightshadingtopicturewherethefunctionsarebigandsmall.Startbyndingwheretheyhavethelargestandsmallestmagnitudes.Seeifyoucanndsimilarpicturesinanintroductorychemistrytext.Alternately,checkout winter.group.shef.ac.uk/orbitron/ 6.38Usetheresultsofproblem 6.17 andapplyittotheLegendreequationEq.(4.55)todemonstratethattheLegendrepolynomialsobeyR1�1dxPn(x)Pm(x)=0ifn6=m.Note:thefunctionT(x)fromproblem 6.17 iszeroattheseendpoints.Thatdoesnotimplythattherearenoconditionsonthefunctionsy1andy2atthoseendpoints.TheproductofT(x)y01y2hastovanishthere.UsetheresultstatedjustafterEq.(4.59)toshowthatonlytheLegendrepolynomialsandnotthemoregeneralsolutionsofEq.(4.58)work.6.39UsingtheresultoftheprecedingproblemthattheLegendrepolynomialsareorthogonal,showthattheequation(4.62)(a)followsfromEq.(4.62)(e).Squarethatequation(e)andintegrateR1�1dx.Dotheintegralontheleftandthenexpandtheresultinaninniteseriesint.OntherightyouhaveintegralsofproductsofLegendrepolynomials,andonlythesquaredtermsarenon-zero.Equatelikepowersoftandyouwillhavetheresult.6.40UsethescalarproductofEq.( 6.16 )andconstructanorthogonalbasisusingtheGram-Schmidtprocessandstartingfrom10and01.Verifythatyouranswerworksinatleastonespecialcase.6.41Forthedi erentialequationx+x=0,pickasetofindependentsolutionstothedi erentialequation|anyonesyoulike.Usethescalarproduct f;g =R10dxf(x)*g(x)andapplytheGram-Schmidtmethodtondanorthogonalbasisinthisspaceofsolutions.Isthereanotherscalarproductthatwouldmakethisanalysissimpler?Sketchtheorthogonalfunctionsthatyoufound.