INFANCY AND CHILDHOOD IN ANCIENT GREEK PHILOSOPHY. Malcolm Schofield

Size: px
Start display at page:

Download "INFANCY AND CHILDHOOD IN ANCIENT GREEK PHILOSOPHY. Malcolm Schofield"

Transcription

1 INFANCYANDCHILDHOODINANCIENTGREEKPHILOSOPHY MalcolmSchofield TheancientGreeksaren tusuallythoughttohavemuchofinteresttotellus aboutchildhoodorinfancy.butthisisinfactasubjectonwhichgreek philosophysaysquitealot,whoseattractionsishallbehopingtorecommendto youinthislecture. 1 Forphilosophers someofthemoresignificantofthem,at anyrate childhoodandinfancyrepresentparadigmsgoodtothinkwithabouta wholerangeofimportanttopics.idon twanttoattemptanysortofsurveyof everythingtheysaidaboutinfancyoraboutchildhood.insteadishalllookat thingsaboutchildrenandinfantsthat,oncetheynoticedthem,gottheir philosophicaladrenalinflowing.whatiamgoingtobehighlightingaresomeof thedifferentparadigmaticrolestheyplayintheirargumentsandtheories.as Greekphilosopherssawit,infantsandchildrenhaveahuge perhapssurprising amounttoteachus. Forthoseofyouwhowouldlikearoadmap,thelecturewillexhibitring composition.weshallstartwithplatoandheraclitus,andreturntothematthe endafteragoodthinkabouttheepicureansandstoics.tobeginwiththefocus willbechildhood,theninhellenisticphilosophyinfancy,whileinoursecond encounterwithplatowewillfindapreoccupationwithbothinfancyand childhood,beforeweendupwithalastlookatchildhoodinheraclitus.the discussionwillproceedatafairlysmart(butihopeacceptable)pace.myaimhas beentopresentinabriefspacesomeintriguingmaterial ratherthantodwell onhowmuchtruththeremightbeinsomeofit:somethingwecanallmakeour ownmindsupabout. 1.Insightandenquiry OneofthemostfamouspassagesinallGreekphilosophicalwritingcomesin Plato sdialoguemeno,atthepointwheremenoquestionssomethingsocrates hasjustproposedaboutenquiryandlearning.it sallrecollection,ἀνάμνησις, Socrateshassaid.ReasonablyenoughMenoaskshimtoexplainwhathemeans bythisstrangeclaim:thatlearningisnotwhatwedo whatwecall learning is actuallyrecollection.wethengetthisstretchofdialogue(82a b): {ΣΩ.}Ἀλλ ἔστιμὲνοὐῥᾴδιον,ὅμωςδὲἐθέλωπροθυμηθῆναισοῦἕνεκα. ἀλλάμοιπροσκάλεσοντῶνπολλῶνἀκολούθωντουτωνὶτῶνσαυτοῦἕνα, ὅντιναβούλει,ἵναἐντούτῳσοιἐπιδείξωμαι. {ΜΕΝ.}Πάνυγε.δεῦροπρόσελθε. {ΣΩ.}Ἕλληνμένἐστικαὶἑλληνίζει; {ΜΕΝ.}Πάνυγεσφόδρα,οἰκογενήςγε. {ΣΩ.}Πρόσεχεδὴτὸννοῦνὁπότερ ἄνσοιφαίνηται,ἢἀναμιμνῃσκόμενοςἢ μανθάνωνπαρ ἐμοῦ. {ΜΕΝ.}Ἀλλὰπροσέξω. {ΣΩ.}Εἰπὲδήμοι,ὦπαῖ,γιγνώσκειςτετράγωνονχωρίονὅτιτοιοῦτόν ἐστιν; 1PresidentialLecture,SocietyforthePromotionofHellenicStudies,11June2011.

2 2 {ΠΑΙ.}Ἔγωγε. Soc.Well,itisn teasy,buti mwillingtotryforyoursake.callformeoneof yourmanyattendantshere,whicheveryoulike,sothaticanperformthe demonstrationonhimforyou. Men.Certainly.Comehere. Soc.IsheaGreek,anddoeshespeakGreek? Men.Yes,indeed;he shome bred. Soc.Thenpayattentionastowhichseemstoyoutobetrueofhim,eitherthat heisrecollectingorthatheislearningfromme. Men.Yes,Iwill. Soc.Tellme,then,boy,areyouawarethatasquarefigureislikethis? BoyIam. Therefollowsthefamousproofthat,thoughtheboyhasneverbeentaughtany geometry(85d e),whenquestionedheisabletomakeoutforhimselfthe answertoachallenginggeometricalproblem.aftersomefailedattempts,he succeedsinspecifyingthelengthofthelinewhichwill,whensquared,generatea figuretwicetheareaofagivensquare,namelythelinewhichtheexperts(οἷ σοφιστάι)callthediagonaloftheoriginalgivenfigure.ifhecouldgetthesame answerwhenquestionedmanytimesandinmanyways,socratescomments,he willhaveknowledge:aknowledgehewillhaverecoveredfromhimself and recoveringknowledgeonehaswithinoneselfissurelyrecollectingit. Buthowlonghasthatknowledgebeenlockedupwithintheboy, forgottenbutwakenedbyquestioning?well,menocanvouchthathe snotbeen taughtitinhispresentlife.so,socratessuggests,itlooksasthoughtruthabout realityisalwaysthereinthesoul:whichmustaccordinglybeimmortal,preexistingourhumanlifeandgoingonafterit.andfromthatitisonlyashortstep towordsworth: Ourbirthisbutasleepandaforgetting: TheSoulthatriseswithus,ourlife sstar, Hathhadelsewhereitssetting, Andcomethfromafar: Notinentireforgetfulness, Andnotinutternakedness, Buttrailingcloudsofglorydowecome FromGod,whoisourhome: Heavenliesaboutusinourinfancy! ButbacktotheMeno,whereinthecool,drymannerofaSocratic conversation,platoisattemptingsomethingclosetoargumentativemelodrama: ademonstrationthattheinnateresourcesofthemindaresuchthatamerechild canwithoutinstructionperformapieceofmathematicalthinkingofsome sophistication,which iftheperformancecanberepeatedsuccessfullyunder variedconditions willconstitutetherecoveryoflatentknowledgeburiedinthe immortalsoul.ofcourse,philosophicalcriticsfindplentytoqueryinthispieceof argumentation,althoughitsbasicanti empiricistthrusthasinrecentdecades cometolooklikesomesortofanticipationofnoamchomsky stheoryofhuman

3 3 languageacquisition.thegreekscholarcanasksomequestionsaboutittoo,as didthelatelamentedbobsharplesinhisfinearis&phillipseditionofthe dialogue(1985),fromwhichiexcerptedthetranslationofthepassageijust quoted.bobpointedoutthatwhensocratessays: Tellmethen,boy,areyou awarethatasquarefigureislikethis?,wecan tbesurethat boy (παῖ)means boy: thegreeks,likethesouthafricans hecomments, calledslaves boy whatevertheirage ;andsocrateshasinvitedmenotopickanyofhisattendants tobetheguinea pig.maybethemessageisnotsomuchthatamerechildcan recoverinnateknowledge,butthatamereuntutoredslavecandoit.atthesame time,asbobobserved: Theyoungertheslave,themorestrikingtheexperiment becomes. 2 Anotherimageofthechild scognitivepowers,insomewaysultimately notthatdifferent,isconveyedbytheearliestgreekthinkerweknowtohave exploitedtheideaofchildhoodforphilosophicalpurposes.quiteanumberofthe sayingsofheraclitusplaywiththeidea.let sstartwithfr.79: ἀ ν ὴ ρ ν ή π ι ο ς ἤ κ ο υ σ ε π ρ ὸ ς δ α ί μ ο ν ο ς ὅ κ ω σ π ε ρ π α ῖ ς π ρ ὸ ς ἀ ν δ ρ ό ς. Maniscalledinfantilebydivinity,justasisthechildbythegrownman. Thegulfbetweendivineunderstandingandtheincomprehensionthatinhumans atlargepassesforthoughtisoneofheraclitus favouritethemes.thechild man comparisongiveshimyetonemorewayofmakingthepoint.wethinkchildren aremuchofthetimeinfantile,silly,childish.butfromthedivineperspective that showwehumanadultsgenerallylook.andnoticethatheraclitusdoesnot endorsetheviewthatchildrenarechildish.thisbecomessignificantwhenin othersayingshedeployscomparisonwiththechildtotwisttheknifestilldeeper intoourself esteem.hereisfr.117: ἀ ν ὴ ρ ὁ κ ό τ α ν μ ε θ υ σ θ ῆ ι, ἄ γ ε τ α ι ὑ π ὸ π α ι δ ὸ ς ἀ ν ή β ο υ σ φ α λ λ ό μ ε ν ο ς, ο ὐ κ ἐ π α ΐ ω ν ὅ κ η β α ί ν ε ι, ὑ γ ρ ὴ ν τ ὴ ν ψ υ χ ὴ ν ἔ χ ω ν. Amanwhenheisdrunkisguidedbyabeardlessboy:stumbling,notknowing whereheisstepping,withhissouldrenched. Thechildcanactuallybemoresensible,morealert,moreincontrolthanamanin thatcondition andthepointisreinforcedbythesprightlinessoftheregular hexameterrhythmofἄ γ ε τ α ι ὑ π ὸ π α ι δ ὸ ς ἀ ν ή β ο υ followedbythe unmetricaltotteringofσ φ α λ λ ό μ ε ν ο ς. Worsestill,peopledon thavetobe insensiblewithdrinktomisstheobvious,eventhoughchildrenhavenoproblem withperceivingit.fr.56instanceshomerhimselfasacaseinpoint: ἐ ξ η π ά τ η ν τ α ι οἱἄ ν θ ρ ω π ο ι π ρ ὸ ς τ ὴ ν γ ν ῶ σ ι ν τ ῶ ν φ α ν ε ρ ῶ ν π α ρ α π λ η σ ί ω ς Ὁ μ ή ρ ω ι, ὃ ς ἐ γ έ ν ε τ ο τ ῶ ν 2R.W.Sharples,Plato:Meno,editedwithtranslationandnotes.Warminster:Aris& Phillips,1985.

4 4 Ἑ λ λ ή ν ω ν σ ο φ ώ τ ε ρ ο ς π ά ν τ ω ν. ἐ κ ε ῖ ν ό ν τ ε γ ὰ ρ π α ῖ δ ε ς φ θ ε ῖ ρ α ς κ α τ α κ τ ε ί ν ο ν τ ε ς ἐ ξ η π ά τ η σ α ν ε ἰ π ό ν τ ε ς ὅ σ α ε ἴ δ ο μ ε ν κ α ὶ ἐ λ ά β ο μ ε ν, τ α ῦ τ α ἀ π ο λ ε ί π ο μ ε ν, ὅ σ α δ ὲ ο ὔ τ ε ε ἴ δ ο μ ε ν ο ὔ τ ἐ λ ά β ο μ ε ν, τ α ῦ τ α φ έ ρ ο μ ε ν. Peoplearethoroughlydeceivedwhenitcomestorecognitionofwhatis obvious likehomer,whohadprovedwiserthanallthegreeks.foritwas himthatboyswhowerekillinglicethoroughlydeceived,whentheysaid: Whatweseeandcatch,weleavebehind;whatweneitherseenorcatch,we carryaway. Homerwasaparagonofwisdom.Buthecouldnotworkoutariddledevisedby childrenwhoweremakingitobviouswhattheanswermustbebywhatthey weredoinginfrontofhisverynose.hecouldn tconnectlanguagewith behaviourthewaytheycould.onceagaintheintelligenceofchildrenshowsup thestupidityofadults:theiralertnessfunctionsasyardstickagainstwhichwe canmeasurehumanfolly. Childhoodisnowheremoreambitiouslycharacterizedthaninwhat CharlesKahn,inhiseditionandcommentary,rightlydescribesasthe most enigmaticofheracliteanriddles. 3 ThisisFr.52: α ἰ ὼ ν π α ῖ ς ἐ σ τ ι π α ί ζ ω ν, π ε σ σ ε ύ ω ν π α ι δ ὸ ς ἡ β α σ ι λ η ί η. Alifetimeisachildplaying,movingthepieces.Achild sisthekingship. AttheendofthislectureIshallhaveashotatdispellingsomeofthebafflement Fr.52induces.Forthemomentletusnoticesomethingthatisclearaboutit: somethingwhichfurtherdevelopsthepictureofheraclitus thinkingabout childhoodthathasbeenemerging.childhoodisnowseennotjustasunexpected benchmarkforwisdom,butassomehowthekeytoourwholeunderstandingof ourlives andofwhatcontrolsthepatterntheyexhibit. 2.Desireandwhatmotivatesit AtthispointIwanttofastforwardtotheHellenisticage,andtoaverydifferent kindofphilosophicalappealtothechild:whichconcentratesnotonchildren s intelligenceandinsight,butontheirbasicimpulses ontheirconative,nottheir cognitive,powers.theexplorationofhellenisticphilosophyhasbeenoneofthe greatprojectsinscholarshiponancientphilosophyoverthelast30oddyearsor so,andonewithwhichihavehadthegreatgoodfortunetobeassociated.ifone hadtopickasmallselectionofground breakingstudiesthathavehelpedto transformunderstandingofstoics,epicureansandtherest,highoneveryone s listwouldbeapaperbyanhonorarymemberofoursocietywhodiedearlyin 2010,theremarkableJacquesBrunschwig,entitled Thecradleargumentin 3C.H.Kahn,TheArtandThoughtofHeraclitus,aneditionofthefragmentswith translationandcommentary.cambridge:cambridgeuniversitypress,1979.p.227.

5 5 EpicureanismandStoicism,andpublished25yearsbeforeintheproceedings volumeofthethirdsymposiumhellenisticum. 4 Brunschwiglaunchedthepaperwithaquotationthatbeginstoexplain histitle.itcomesfromthefifthbookofcicero sdefinibus,incidentally remindingusthatmuchofourevidenceforthesegreekthinkersisactually availabletousonlyinlatin,andinthepresentcaseprimarilyinthedialoguesof DeFinibus.Thetextinquestionreadsasfollows(Fin.5.55): omnesveteresphilosophi,maximenostri,adincunabulaaccedunt,quodin pueritiafacillimesearbitanturnaturaevoluntatempossecognoscere. Alltheancientphilosophers,inparticularthoseofourschool[i.e.the Peripatetics],turntocradles,becauseitinchildhoodthattheythinktheycan mostreadilyrecognizethewillofnature. AsBrunschwigcomments,thisstatementisperfectlycorrectasasummaryof thebasicapproachtoethicsinhellenisticphilosophy: themoralistsofthe Hellenisticperiod,ofwhateverschool,madefrequentuseofwhatmightbe calledthecradleargument,thatis,aprocedurewhichconsistsfirstindescribing (orinclaimingtodescribe)thebehaviorandpsychologyofthechildinthecradle (usuallyinconjunctionwithyounganimals)andindrawing(orclaimingto draw),moreorlessdirectly,certainconclusionswhich,inonewayoranother, leadtotheformulationandjustificationofamoraldoctrine. 5 ItwasprobablyEpicuruswhowaspioneerinmakingappealtothe experienceofnewbornanimalsthefoundationstoneofhisethicalsystem,which ofcoursewasadistinctiveversionofhedonism.thisishowaccordingtocicero hearticulatedtheappeal(fin.1.30): omneanimalsimulatquenatumsitvoluptatemappetereeaquegaudereut summobono,doloremaspernariutsummummalumetquantumpossitase repellere;idquefacerenondumdepravatum,ipsanatureincorrupteatque integreiudicante. Everyanimalassoonasitisbornseekspleasureandenjoysitassupreme good;itshunspainassupremeevilandresistsitsofarasitisable.thisit doesbeingnotyetdepraved natureitselfdeliversthisverdict,untouchedby corruptionandwithintegrityintact.therefore(hesays[sc.epicurus])thereis noneedforreasoningorargumentationshowingwhypleasureshouldbe soughtandpainshouldbeavoided[sc.morethananythingelseshould]. Epicuruswentontodrawanimmediateconclusion,nodoubtmeanttosurprise andshock(ibid.): 4J.Brunschwig, ThecradleargumentinEpicureanismandStoicism,inTheNormsof Nature:StudiesinHellenisticEthics,ed.M.Schofield&G.Striker.Cambridge:Cambridge UniversityPress,1986.Pp Ibid.p.113.

6 6 itaquenegatopusesserationenequedisputationequamobremvoluptas expetenda,fugiendusdolorest. Therefore(hesays[sc.Epicurus])thereisnoneedforreasoningor argumentationshowingwhypleasureshouldbesoughtandpainshouldbe avoided[sc.morethananythingelseshould]. ThesurpriseisthatEpicurusapparentlythinksthathecanmoveimmediately fromtheobservationaboutnewbornanimals,firsttohisprincipalthesisin ethics:pleasureshouldbesought,painshouldbeavoided;andsecond,tothe claimthattheobservationeliminatesanyneedactuallytoarguefortheposition. Howonearthcanheexpecttojumpallthewayfromnoticingsomethingabout infantsinthecradletofull blownethicalhedonism,andwithoutsupplyingany argumentationdesignedtobridgethegap? AsBrunschwigshowed,theanswercanbegatheredbyfocusingonthe finalcommentepicurusmakesaboutthepsychologicalattitudetopleasureand painrevealedbythebehaviorofthenewbornanimal.thatattituderepresents humanoranimalnature.anditcan tplausiblybemaintainedthatababyin armsgoesforpleasurebecauseitsnaturehasbeencorruptedbysocial influences byhypothesizingwhatnon moralistsmightbemoanas(toquote Brunschwigagain) therelentlessworkingsofawholeseriesofpervertingforces fromnursesandteacherstopoetsandplaywrights,takingturnstofosterataste forsensualindulgenceinthechild,intheadolescent,andfinallytheadult. Observationofthenewbornchildenablesuspreciselytoeliminatethe hypothesisofanysuchdistortion.itdoesn tprovideanargumentativebasisfor hedonism hedonismconsistsinbelievinganddoingwhatcomesnaturally, somethingthatwedon tneedtobearguedinto.whattheobservationdoes achieveissimply theinvalidationofanattempttoinvalidate thethesisthat hedonismtrulyiswhatnaturerecommendstous. 6 PhilosophersfromotherschoolsofcourserejectedtheEpicureans cradle argument.onelineofattackwastoobjectthattherewasafatalambiguityin theirposition.thepleasuretheinfantinarmsisafter,itwassaid,isapositive sensation;thepleasurevaluedbytheepicureansageissomethingquitedifferent freedomfrompainandanxiety(fin ).Whatisofmoreinteresttous, however,isanotherlineofthought,whicheffectivelyagreeswithepicurusin takingitthatethicsneedstoappealtowhathappensinthecradle,butobjects thathehasmisidentifiedwhatthatcradleexperienceconsistsin.wefindcicero presentingthisobjectioninboththeanti EpicureanBook2andtheStoicBook3 ofdefinibus(fin.2.33,3.17),butsincewehaveaversionofitinagreeksource (DiogenesLaertius7.85 6),that stheonei llreproduce: Ὅδὲλέγουσίτινες,πρὸςἡδονὴνγίγνεσθαιτὴνπρώτηνὁρμὴντοῖςζῴοις, ψεῦδοςἀποφαίνουσιν.ἐπιγέννημαγάρφασιν,εἰἄραἔστιν,ἡδονὴνεἶναι ὅταναὐτὴκαθ αὑτὴνἡφύσιςἐπιζητήσασατὰἐναρμόζοντατῇσυστάσει ἀπολάβῃ ὃντρόπονἀφιλαρύνεταιτὰζῷακαὶθάλλειτὰφυτά. 6Ibid.p.121.

7 7 Asforwhatsomepeoplesay[sc.theEpicureans],viz.thatpleasureisthe objectofthefirstimpulsetheanimalshave,they[sc.thestoics]representthis asfalse.fortheysaythatpleasure(ifthereactuallyispleasure)isabyproduct,whicharisesonlywhennatureinandofitselfhassoughttheproper requirementsfortheanimal sconstitution.thisishowitcomesaboutthat animalsfrolicandplantsbloom. Inotherwords,if(anditisan if )thebabywhohastakenmilkfromitsmother s breastendsupgurglingcontentedly,thatisnotbecauseit spleasurethatitwas after,butbecauseitsnaturalimpulsewastoacquirethesustenanceit instinctivelyrecognizedtoberequiredinordertopreserveitsconstitution.the pleasureissomekindofspin offfromthesatisfactionofthat primarynatural impulsetoself preservation (D.L.7.85). TheStoicviewwasthatwhatmakesthattheprimaryimpulsemustbe somethingmorefundamentalstill:thenewbornanimal sattachmenttoitselfor identificationwithitselfanditsownconstitution whattheycalledοἰκείωσις somethingwhichinturnrequiresustopositfromtheverybeginningsomekind ofself awareness,howeverinchoateandindefinitethatmaybe(cic.fin.3.16;cf. Sen.Ep ).ThustheStoicsseemtohavereasonedbackfromwhatthey tooktobethephenomenonofself preservingbehaviorcommontoallanimalsas soonastheyareborn,toitspsychologicalpreconditions:theimpulsetomaintain theirownconstitution,whichpresupposesanimpulsetoself love,which presupposesself awareness. Atthispointonemightsay:Iseethatifyoudecidethatababy s behaviourisinthefirstinstancetypicallyself preservingratherthanpleasureseeking,thenstoicοἰκείωσιςtheoryiswhatyoumightneedtoaccountforthe psychologyunderlyingthebehaviour.butdidtheyhaveanynon arbitrarybasis fordecidingtodescribethebehaviouralphenomenaintheirtermsratherthan thosepreferredinepicureanhedonism?chrysippus,themostimportantofall GreekStoics,evidentlythoughthehad.Heinvokedanoverallteleologyinthe realmofnatureatlarge,andonthatbasisofferedapieceofapriorireasoningas towhywemustsupposetheimpulsetoself attachmenttobenatural(d.l.7.85): οὔτεγὰρἀλλοτριῶσαιεἰκὸςἦναὐτὸ<αὑτῷ>τὸζῷον,οὔτεποιήσασαν αὐτό,μήτ ἀλλοτριῶσαιμήτ [οὐκ]οἰκειῶσαι.ἀπολείπεταιτοίνυνλέγειν συστησαμένηναὐτὸοἰκειῶσαιπρὸςἑαυτό. Forthey[sc.theStoics]thoughttherewasnolikelihoodinthealternatives: thatnaturealienatedtheanimalinitself,orthathavingmadeit,nature neitheralienatednotcreatedattachment.sotheonlyoptionremainingisto saythatinconstitutingtheanimalitgaveitattachmenttoitself. FortheStoics(quitedifferentlyfromtheEpicureans),itisbecomingplain, thecradleexperiencecanbeunderstoodonlywithintheframeworkofquitealot ofambitioustheory,includingatonelevelateleologicalunderstandingofthe workingsofnatureatlarge,andatanotherasophisticatedanalysisofthe psychologicalpreconditionsofanimalbehaviour.themoreambitiousthe theory,themoreproblemsitislikelytothrowup;andthesourcesgiveusquitea lotofevidenceaboutwhatinthecaseofstoicismthesewerefelttobein

8 8 antiquity.acrucialissuewasovertheethicalsignificanceofthehumaninfant s self preservingbehaviour seekingandfinding theproperrequirementsforthe animal sconstitution andinparticularitsrelationshiptovirtue,thestoics candidateforthesummumbonum:arelationshipwhichatfirstsightseemsmuch moremysteriousthantheepicureans invocationofcradleexperiencetosupport thethesisthatitispleasurethatisthesummumbonum.butishallfocus attentiononadifferentexampleofthecontroversiesthestoictheoryprovoked. Thiswasovertheboldclaimthateventhenewbornanimalalreadyhasselfawareness,somethingtheStoicsrepresentedasaninevitablepreconditionof οἰκείωσις. WehearofobjectionsraisedagainsttheStoics claimthateventhe youngestanimalshaveself awareness,bothasreportedbysenecaandina fascinatingsectionfromaworkbyhierocles(probablymid secondcenturyad) entitledtheelementsofethics.hierocles,perhapsnotahouseholdname,is preservedfragmentarilyonpapyrus,andisnowavailableinasplendidedition co editedin1992bya.a.long,anewlyappointedhonorarymemberofthe Society(2011). 7 TheobjectionknowntoSenecacomesacrossinhisaccountas rathercrass.how,itisasked(letter121.10),couldaninfantcrediblybe supposedtohaveanyunderstandingofitself,whenthatwouldinvolvea complexandsubtlegraspoftheἡγεμονικόν,theanimal spsychologicaland physiologicalcontrolcentre?wouldn tthatmeanturningallanimalsintoborn logicians?senecahasnodifficultyinpointingoutanignoratioelenchi.thestoics thesisisnotaboutphilosophicalknowledgeorunderstanding,butaboutbasic awareness:quidsitanimal,nescit;animalseessesentit(ibid.11: Whatananimal is,itdoesnotknow;thatitisitselfananimalissomethingitisawareof ).Orif wecancallitanunderstandingofitsownconstitution,thatunderstandingis crude,schematic,andvague(ibid.12). Hierocles,likeSenecaaStoic,forhispartbeginshistreatiseby announcingthathethinksthebeststartingpointforatreatmentoftheelements ofethicsistheλόγοςabouttheπρῶτονοἴκειονofananimal, thefirstthing recognizableasitsown (Elements1.1 2),justthepointfromwhichaccordingto DiogenesLaertiusChrysippusbeganhisworkOnGoals(περὶτελῶν)(D.L.7.85). Andinthisconnectionitisimportanttorecognize,hesays,thatassoonasitis borntheanimalisawareofitself(elements1.35 9).Thisisimportantnotleast becausesomepeopleobjecttotheidea.orrather,asheputsitengagingly,we hadbettersaysomethingtoremindourselvesofitstruth forthesakeofthose whoareratherslow(βραδυτέρων).actually,hegoesonatoncetosay,weneed toputanotherkindofargumentfirst,becausesomepeoplearesoslowandsofar removedfromunderstandingthingsthattheycan tbelievethatanimalshaveany self awarenessmoregenerally(i.e.notjustthattheyhaveitfrombirth).these peoplethinkperceptionorawareness(αἴσθησις)hasbeenbestowedbynature onanimalsforthepurposeofapprehendingexternalobjects,notalsoforselfapprehension(ibid.39 46). SowhatHieroclesgoesontodevelopisquiteacomplicatedlineof thought:firstanextensivesetofconsiderationswhichshowthatthegeneral 7G.Bastianini&A.A.Long,Hierocles:ElementaMoralia,editedwithItaliantranslation andcommentary,incorpusdeipapyrifilosoficigrecielatini,partei:autorinoti, Vol.1**.Firenze:LeoS.OlschkiEditore,1992.Pp

9 9 phenomenonofanimalself awarenessdoesindeedexist,andonlythenadefence ofthemorespecificthesis,seeminglymuchmoredifficulttoestablish,that animalsperceivethemselvesassoonastheyareborn.muchofthefocusinthe firstpartoftheargumentisonanimals self consciousnessabouttheirown powersandlimitations,asinthefollowingsequence(ibid ): Ταύτηικαὶταῦροςμέν,ὁπότεφράττοιτοπρὸςτὴνἐπιβουλήν,τάττειπρὸ παντὸςτοῦλοιποῦσώματοςτὰκέρατα χελώνηδὲσυναισθανομένητινὸς ἐπιθέσεωςτὴνκεφαλὴνκαὶτοὺςπόδαςτῶιὀστρακώδειμέρειἑαυτῆς ὑποστέλλει,τῶισκληρῶικαὶδυσμεταχειρίστωιτὰεὐάλωτα τὸδὲ παραπλήσιονποιεῖκαὶὁκοχλίαςκατειλούμενοςεἰςτὸκερατῶδες,ὁπότε κινδύνουσυναίσθοιτο. Thusthebull,whenhedefendshimselfagainstahostilemanoeuvre,linesup hishornsinfrontofthewholeoftherestofhisbody.thetortoise,whenheis awareofanattack,contractshisheadandhisfeetintohisshell,thepartsthat areeasytogetatintothepartthatishardanddifficulttodealwith. Somethingsimilarisdonebythesnail,whichcontractsitselfintoitsshell whenitisawareofdanger. Noticethat,althoughprovingthatananimal sbasicinstinctisforselfpreservationisn tthemainpointhere,itistheself awarenessrevealedinsuch behaviourthathieroclesproducesinevidence. Thesecondphaseoftheargumentationdividesintotwo:proofthat animalsperceivethemselvescontinuously,andthenatlasttheclinchertowhich thewholesequenceofreasoninghasbeenmoving argumentthattheydoso fromtheverymomentofbirth.hieroclesarguesforcontinuousself awareness principallybyappealtotheideathatthesoulpermeatesthebody,andtherefore anythingthataffectsthebodymustalsoaffectthesoul.therelevanceofthat quicklyemerges.iftheanimalwereevertoloseallself awareness,itwouldhave tobeinsleep.butthereisplentyofevidencethatweareawareofourselves whileasleep:ifpartsofourbodiesgetexposedtothecold,wedragthe bedclothesoverthem;thewine loverkeepsatightgriponthebottle,themiser onhispurse;andsoon. Forthefinaldemonstrationofself awarenessfromthemomentofbirth, wemighthavebeenhopingforsimilarbitsofpurportedempiricalevidence, drawnfromobservationofbabiesandothernewlybornanimals.insteaditnow becomesclearwhyhieroclesinvestedsomucheffortintothemoregeneral considerationshehasbeenpresentingsofar.thereasonsheoffersforbelieving theclaimaboutnewlybornanimalsareallindirect:inferencesofonesortor anotherfromthecontinuityofperception.forexample,ifself perceptionis continuousthroughlife,thatcontinuitymustincludethefirstmoment;forno subsequentmomentlooksabetterbeginningpoint,especiallysincethecapacity forperceptionistherefromthefirstmoment.howeverhierocles prize argument( veryfine,unanswerable(πάνυκαλὴνκαὶἀναντίλεκτον),hecallsit: ibid.5.61)consistsinreasoningquitesimplythatperceptionofexternalthings which,recall,theoriginalreallyslowobjectorsheldtobetheonlyjobnaturehas assignedtoananimal ssensoryapparatus isitselfinherentlyboundupwith self perception.

10 10 Hereishowthepassagegoes(Elements ): εἰςτίποτ οὖνφέρειοὗτοςὁλόγος;εἰςπάνυκαλὴνκαὶἀναντίλεκτον ὑπόμνησιντοῦπροκειμένου καθόλουγὰροὐσυντελεῖταιτῶνἐκτόςτινος ἀντίληψιςδίχατῆςἑαυτῶναἰσθήσεως.μετὰγὰρτῆςτοῦλευκοῦφέρε εἰπεῖναἰσθήσεωςκαὶἑαυτῶναἰσθανόμεθαλευκαινομένωνκαὶμετὰ<τῆς> τοῦγλυκέωςγλυκαζομένωνκαὶμετὰτῆςτοῦθερμοῦθερμαινομένωνκἀπὶ τῶνἄλλωντἀνάλογον ὥστ ἐπειδὴπάντωςμὲνγεννηθὲνεὐθὺςαἰσθάνεταί τινοςτὸζῶιον,τῆιδ ἑτέρουτινὸςαἰσθήσεισυμπέφυκεν<ἡ>ἑαυτοῦ, φανερὸνὡςἀπ ἀρχῆςαἰσθάνοιτ ἂνἑαυτῶντὰζῶια. Sowhereisthisargumenttakingus?Toaveryfineandunanswerable indicationofthetruthofthethesisweareadvancing.forquitegenerally, apprehensionofanyoftheexternalsisnotachievedwithoutself awareness. Forwiththeperceptionofwhatisbrightcoloured,forexample,weareaware ofourselvesbeingbrightened;withtheperceptionofwhatissweet,of ourselvesbeingsweetened;withtheperceptionofwhatishot,ofourselves beingheated;andinothercasesanalogously[i.e.oftheimpactofbrightness, sweetness,heat,etc.uponus,itakeit].sosinceassuredlytheanimal perceivessomethingassoonasitisborn,andsinceself awarenessis inherentlyboundupwiththeperceptionofsomeotherthing,itisevidentthat fromthebeginninganimalshaveself awareness. ThereareallsortsofquestionsthatHierocles argumenthereislikelytoprompt inreadersbeforetheyarepreparedtosignuptohisconclusion.butitistimeto movebacktoplato. 3.Movement,dance,andplay Idon tthinkplatowouldhavelikedeithertheepicureanorthestoicversionof thecradleargument,thoughthereisnodoubtthatforhimtheepicurean diagnosisofpleasureandpainaswhatprimarilyattractsanddeterschildrenand makesthembehaveastheydomustbeontherightlines.hislastdialogue,the Laws,hasmoretosayaboutchildrenandchildhoodthananyothersurviving Greekphilosophicaltext.Andthatpleasureandpainarethefactorsthatgovern thebehaviourofchildren(andindeedofmostofhumanity)isafundamental thesisofthework.thereisawonderfulpassageatthebeginningofbook2 whichsumsupplato sbasicstoryaboutyounganimalsingeneral(laws2.653d 654a): φησὶνδὲτὸνέονἅπανὡςἔποςεἰπεῖντοῖςτεσώμασικαὶταῖςφωναῖς ἡσυχίανἄγεινοὐδύνασθαι,κινεῖσθαιδὲἀεὶζητεῖνκαὶφθέγγεσθαι,τὰμὲν ἁλλόμενα καὶσκιρτῶντα,οἷονὀρχούμεναμεθ ἡδονῆςκαὶπροσπαίζοντα,τὰδὲ φθεγγόμεναπάσαςφωνάς.τὰμὲνοὖνἄλλαζῷαοὐκἔχειναἴσθησιντῶνἐν ταῖςκινήσεσιντάξεωνοὐδὲἀταξιῶν,οἷςδὴῥυθμὸςὄνομακαὶἁρμονία ἡμῖνδὲοὓςεἴπομεντοὺςθεοὺςσυγχορευτὰςδεδόσθαι,τούτουςεἶναικαὶ τοὺςδεδωκόταςτὴνἔνρυθμόντεκαὶἐναρμόνιοναἴσθησινμεθ ἡδονῆς. Accordingtothestory,prettywelleveryyoungcreatureisincapableof

11 11 keepingquiet,eitherphysicallyorvocally.ithastobetryingtomoveand makesoundsallthetime,nowjumpingandskipping(dancingforpleasure,for example,orplayinggames),nowutteringallkindsofsounds.othercreatures havenoperceptionoforderordisorder whatwecallrhythmandharmony inthesemotions.butinourcase,thegodswesaidhavebeengiventoustobe ourcompanionsinthedance theyarealsotheoneswhohavegivenusthe abilityto take pleasure in the perception of rhythm and harmony. Whenhereprisesthispassagetenorsopageslater(664e),Plato sspeaker the AthenianStranger insertsintohisaccountofthemovementsandsoundsmade byyounganimalsacrucialadverb,inordertosharpenthecontrastimplicitin theoriginalversion.thenatureofallyounganimals,hesays,isfiery intheir restlessnesstheyarealways makingutterancesandleapingaboutindisorder (ἀτάκτως) ;whereasinthespheresofmovementandsoundsalike,itisonly humannaturethathasasenseoforder.inotherwords,humaninfants,likeother younganimals,startlifewithachaoticrepertoireofmovementsandvocalnoises incontrasttotheperceptionofrhythmandharmonyinwhichtheyandonly theywillastheygrowoldertakepleasure. Whyshouldlifeatitsbeginningsbesochaotic?Nowadayswemightlook todevelopmentalpsychologyandpsychologistsforananswertothatkindof question.theanswerplatohadalreadyhimselfoffered,inthetimaeus,is workedoutinthecontextofthatdialogue sspeculativeattempttoexplainwhat isinvolvedinthecreationofacosmosandofthebiospherethatthecosmos sustainsandindeedconstitutes.wherethecreationofthehumanspeciesis concerned,theexplanationfocusesonthenotionofembodiment.wehaveto thinkofhumansasimmortalsoulsimplantedinbodies,justastheslaveboy passageinthemenohadadumbrated.platodevotesalongandcomplexstretch oftexttothistopic:icanonlysummarizethegistandhighlightoneortwokey passages. Thebasicideaisthatuponembodiment,thesoulbecomessubjectto severedisturbance.birthisaterribleshock,andplatoishereexplainingwhy.in andofitselfimmortalsoulispurerationality,itsmotionoractivityperfectly ordered(herepresentsitascircular,exhibitingtheultimateformofregularity). Thebodyofthenewbornanimal,however,isinconstantebbandflow,thanksto itsneedforfrequentintakeofnourishmentwithassociateddigestiveprocesses, andfornolessfrequentexpulsionoffoodwaste.platotalksofitasariverin spate,oragainasatidalwavefloodinginandflowingout.andheaskshimself whattheoutcomemustbeoftheinteractionbetweenthepsychicmotionsofthe soul,nowconfinedwithinabody,andthephysicalebbandflowbroughtabout bytheconsumptionanddigestionofnourishmentandtheejectionofconsequent wastes. HereisthepassagewherePlatosketcheshisreply(Timaeus43a b): αἱδ εἰςποταμὸνἐνδεθεῖσαιπολὺνοὔτ ἐκράτουνοὔτ ἐκρατοῦντο,βίᾳδὲ ἐφέροντοκαὶἔφερον,ὥστετὸμὲνὅλονκινεῖσθαιζῷον,ἀτάκτωςμὴνὅπῃ τύχοιπροϊέναικαὶἀλόγως,τὰςἓξἁπάσαςκινήσειςἔχον εἴςτεγὰρτὸ πρόσθεκαὶὄπισθενκαὶπάλινεἰςδεξιὰκαὶἀριστερὰκάτωτεκαὶἄνωκαὶ πάντῃκατὰτοὺςἓξτόπουςπλανώμεναπροῄειν.

12 12 These[sc.orbitsoftheimmortalsoul],nowboundintoamightyriver,neither masterednorweremasteredbyit,buttosseditviolentlyandwereviolently tossedbyit.consequentlythewholelivinganimalmoved,butitsprogress wasdisordered,withoutanyrationalbasis haphazard,inawaythat involvedallsixformsofmotion.itwouldgoforwardsandbackwards,and againrightandleft,andupwardsanddownwards,wanderingeverywhich wayinthesesixdirections. Whatisbeingsaidhereseemstoadduptosomethinglikethefollowing.Psychic andphysicalmovementsinteractingwithinthebodydoatanyonetimeresolve themselvesintoasinglemovementofthenewbornanimalasawhole.butthere isnostabilitytotheresolution.insteadwegetajerkyoscillation,withasoul motioninforciblecontrolatonepoint,onlytobeovercomebysomebodily motionconnectedtothefeedingandexcretoryprocessesatasubsequent moment. Platothengoesontodiscusswhathedescribesasthestillgreater turbulenceproducedbytheimpactofexternalbodiesontheinfant sownbody. Thesegeneratemotionswithinitthataretransmittedtothesoul:sensations (αἴσθήσεις).sensationscombinewiththedigestiveprocessesto stirandshake violentlytheorbitsofthesoul,mutilatingthemanddisfiguringtheminany everypossibleway.theresultthistimeisdrasticimpairmentofthesoul s cognitivecapacity:itmisidentifiesdifferentthingsasthesame,andtakes identicalthingstobedifferent infact,itisnolongerexercisinganycontrol. All thesedisturbances Platoconcludes, arenodoubtthereasonwhynowaswellas atthebeginning[i.e.ofcreation]soulbecomesunintelligentatfirst,whenitis boundintoamortalbody (Timaeus44a b). Tworeflectionsatthispoint.Thefirstrelatestothecradleargumentwe werediscussingalittlewhileago.thecradleargumenthadofcoursenotyet beeninvented,norconsequentlywasitavailableasfoundationstoneforethics intheepicureanorstoicstyle.butitisnothardtoseewhatplatowouldhave thoughtofitifhehadknownaboutit.forhimcradleargumentationwouldhave tobetotallyandutterlymisconceived.forhimthenewborninfantisina conditionofdysfunctionalshock:natureinmaximaldisturbance.thinkingthat onecouldfindhereanyindicationsfromnatureofthehumanτέλοςwouldonhis premisesbeabizarrestrategicerrorforthemoralphilosopher.mysecond reflectionisreallythereversesideofthatsamecoin.wecanalreadyguesswhat forplatothejobofmoralandpoliticalphilosophywillhavetobe.ifnothingelse, itwillhavetoinvolvethediscoveryandimplementationofaregimeoftraining andeducationthatwillrestoretheimmortalsoultoproperfunctioning sothat itexercisesreliablecontroloverthemovementsofthebody,bringingtheminto rationalorder. TheplacewherePlatoundertakesthisprojectisBook7oftheLaws.His discussionoftrainingandeducationthereextendsto35stephanuspagesoftext, soiwillbeabletoextractonlythreemajorpoints,anddealwiththemvery brieflyatthat.firstandperhapsmostfascinatingishistreatmentoftheearliest stagesofupbringing,whichashearguesshouldincludetheperiodofpregnancy, andwhatintrevorsaunders Penguintranslationisnicelyifover exuberantly describedas theathleticsoftheembryo.plato sstartingpointistheobservation thattheearlieststagesofgrowth inhumansthefirstfiveyears aremuchthe

13 13 biggestandmostimportant,followedbytheproposalthatit swhenthebodyis gettingmostnourishmentthatitneedsmostexercise,andwhatismore, exercisesthatare well measured (συμμέτρων:laws7.789a).soheadvocates regularwalksforpregnantwomen,andformothersandnursesheprescribes thatbabiesandpre toddlers,andevenforawhiletoddlers(untiltheyarethree yearsold),shouldbecarriedaroundagreatdeal:theirbodiesshouldatthe outsetbemouldedlikewaxwhilestillsupple,andevenwhentheycanwalkthey mustbekeptfromdistortingtheiryounglimbsthroughforcingthemwhenthey putpressureonthem.thereisafamousparagraph,sometimesinvokedin discussionofaristotle sideaoftragiccatharsis,inwhichplatoendorsesthe homeopathicpracticeofnursesand thewomenwhocurecorybantic conditions :mothersputwakefulbabiestosleepbyrockingthemconstantly,not insilencebutsingingtothem,justasbacchicfrenzyiscalmedbydanceandsong (ibid.790d e).asforexpectantmothers,careshouldbetakentoensurethatthey don texperiencefrequentorglosspleasures(orpainseither):somebody expectingababyneedstokeepcheerfulandplacidthroughout. AnybodywhoknowsanythingaboutPlato slawsknowsthatinthe societyheenvisagesinthedialoguebigbrotherisalwayswatchingus.thatisas truefortheregimeoftrainingandeducationprescribedthroughouttherestof childhoodasanyothersphereoflife,andtothisiturnasmysecondtopic. Perhapsthetwokeynotesarethecontrastingpracticesofpunishmentandplay. Ofallwildthings.Platosaysatonejuncture, thechildisthemost unmanageable (ibid.808d).thisisbecauseofchildren sunchanneledpowerof thinking,whichmakesthemscheming,sly,andextremelyinsolent.sotheyneed tobefettered withbridles,asitwere.ontheotherhand,platoisratherkeen onplayasthewaytogetchildren smindsworkingontherightlines.hehasan eloquentpassageonthewaytheegyptiansteachchildrenarithmeticand geometrythroughplayofvarioussorts,andthewaytheysucceedinmaking peoplealertandusefultothemselvesthatway(ibid.819a d). WestartedourlookatwhattheLawssaysaboutchildhoodwiththe passageatthestartofbook2whereplatotalksofthejoyofdance,andthe pleasurehumansaloneamonganimalsareabletotakeintheperceptionof rhythmandharmony.this finally iswhatthedialoguemakestheτέλοςnot onlyofeducationbutofallseriousactivity.anotherfamouspassagepropounds theparadoxthatnothingmuchinthehumansphereshouldberegardedastruly worthtakingseriously andthatplayisinfacttherightwaytolive(ibid.803e): δεῖδὴτὸνκατ εἰρήνηνβίονἕκαστονπλεῖστόντεκαὶἄριστονδιεξελθεῖν.τίς οὖνὀρθότης;παίζοντάἐστινδιαβιωτέοντινὰςδὴπαιδιάς,θύοντακαὶ ᾄδοντακαὶὀρχούμενον,ὥστετοὺςμὲνθεοὺςἵλεωςαὑτῷπαρασκευάζειν δυνατὸνεἶναι,τοὺςδ ἐχθροὺςἀμύνεσθαικαὶνικᾶνμαχόμενον. Soalifespentinpeaceiswhateachofusshouldliveoutasmuchandasbest aswecan.whatthenistherightwayofsoliving?tospendone swholelifein play infactspecificallyinsacrificing,singing,dancing:soastobeableboth toensurethatthegodssmileuponone,andtoresistanddefeatone senemies ifitcomestoafight.

14 14 Playhereseemstoturnwhatistrulyserious ritualworshipofthegods, whetherinsacrificesorinthedailychoralsinginganddancingheprescribesfor allsectorsofthecitizenry intoasortofplay,presumablybecauseofthecalm formalpatternsitexhibits,abstractedfromthemoreordinaryandless predictablebusinessinwhichpeopleengage.heemphasizeswhathethinksit willenablethemtoachieve:harmonywithdivinepowersandtriumphingover enemies.butforuswhatissignificantishowchildhoodandtheplayofthechild haveunexpectedlybecomethemetaphorforthegoodlife. 4. Achild sisthekingship Backnowandlastofall aspromised toheraclitus,andtotheenigmatic wordsoffr.52: α ἰ ὼ ν π α ῖ ς ἐ σ τ ι π α ί ζ ω ν, π ε σ σ ε ύ ω ν π α ι δ ὸ ς ἡ β α σ ι λ η ί η. Alifetimeisachildplaying,movingthepieces.Achild sisthekingship. FiercelydifferingviewsofwhatHeraclitusmightbesayingherecanbefoundin thecommentaries,withkirk 8 andmarcovich 9 resistingthecosmicinterpretation evidentlycommoninlaterantiquity,butwithkahnrevertingtoit. 10 Itis temptingtoreadplato streatmentofchildhoodinthelawsashisownnoncosmicmeditationuponheraclitus words andhisownformofheraclitean paradox.whenhewrites:παίζοντάἐστινδιαβιωτέοντινὰςδὴπαιδιάς,hemight betakentobeconstruingheraclitusasenjoininguponusaviewofwhatliving anentirehumanlifeoughttobelike.itshouldbeplay:theplay(παίζειν)ofa child(παῖς) playingjustisbeingordoingchildhood.itshouldbeplayinthe senseandforthereasonsplatogives.ifweabandonthepursuitsadulthumans usuallybutwronglyregardastheseriousstuffoflife war,makingmoney,for example andifalternativelywesetproperstorebytheorderedandmeasured activityexemplifiedintheplayofachildren sgame(suchastheπεττεία instancedbyheraclitus), 11 thenweshallbeincontrolofourlives,withheaven lineduponoursideandourenemiesunabletotouchus. WhetherHeraclitusreallymeantanythinglikethat,whoknows?ButI m temptedtofindinoneparticularfeatureofhissayingsomethingthatmight chimewiththeplatonicconstrualthatiamconjecturing.lookattherhythmic structureofthefragment.thefirstlimbconsistsalmostentirely(apartfromthe encliticἐστι)oflong,heavysyllables.thesecondbycontrastismuchlighterand morevaried:infactinmetricaltermsaglyconic.perhapsinthefirstlimb Heraclitusissymbolizingthedurationofanentirehumanlife,undifferentiated 8G.S.Kirk,Heraclitus:TheCosmicFragments,editedwithanintroductionand commentary.cambridge:cambridgeuniversitypress, M.Marcovich,Heraclitus,Greektextwithashortcommentary.Mérida:LosAndes UniversityPress, Kahn,TheArtandThoughtofHeraclitus,op.cit.(abovep.4n.3). 11Aboardgameperhapssomethinglikebackgammon.Therewasaparticularthrowof thediceknownastheβ α σ ι λ ι κόν,whichsoundsasthoughitmighthelptoexplain Heraclitus invocationofβ α σ ι λ η ί η, kingship:thethrow presumably thatputsyou incontrolofthegameandsogivesyouβ α σ ι λ η ί η (Plautus,Curcilio359).

15 15 assuch,butintheordereddancerhythmofthesecondthemeasuredactivity thatwillgiveuscontroloverit. 5.Conclusion Isthereanyusefulsumminguptobedone?Perhapsonlythis:Plato,theStoics andtheepicureansagreeonverylittleinwhattheysayaboutinfancyand childhood.butthereisonethingonwhichtheyareunanimous:ifyouthink aboutinfancyandchildhoodproperly,then asheraclitustaught you ll understandhowtothinkandtolive.

. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Friday April 1 ± ǁ

. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Friday April 1 ± ǁ . α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Friday April 1 ± ǁ 1 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant

More information

. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Wednesday March 30 ± ǁ

. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Wednesday March 30 ± ǁ . α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Wednesday March 30 ± ǁ 1 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant

More information

Further Maths A2 (M2FP2D1) Assignment ψ (psi) A Due w/b 19 th March 18

Further Maths A2 (M2FP2D1) Assignment ψ (psi) A Due w/b 19 th March 18 α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω The mathematician s patterns, like the painter s or the poet s, must be beautiful: the ideas, like the colours or the words, must fit together in a harmonious

More information

and in each case give the range of values of x for which the expansion is valid.

and in each case give the range of values of x for which the expansion is valid. α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else P Kraft Further Maths A (MFPD)

More information

4sec 2xtan 2x 1ii C3 Differentiation trig

4sec 2xtan 2x 1ii C3 Differentiation trig A Assignment beta Cover Sheet Name: Question Done Backpack Topic Comment Drill Consolidation i C3 Differentiation trig 4sec xtan x ii C3 Differentiation trig 6cot 3xcosec 3x iii C3 Differentiation trig

More information

ASSIGNMENT COVER SHEET omicron

ASSIGNMENT COVER SHEET omicron ASSIGNMENT COVER SHEET omicron Name Question Done Backpack Ready for test Drill A differentiation Drill B sketches Drill C Partial fractions Drill D integration Drill E differentiation Section A integration

More information

Y1 Double Maths Assignment λ (lambda) Exam Paper to do Core 1 Solomon C on the VLE. Drill

Y1 Double Maths Assignment λ (lambda) Exam Paper to do Core 1 Solomon C on the VLE. Drill α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω Nature is an infinite sphere of which the centre is everywhere and the circumference nowhere Blaise Pascal Y Double Maths Assignment λ (lambda) Tracking

More information

Mathematics Review Exercises. (answers at end)

Mathematics Review Exercises. (answers at end) Brock University Physics 1P21/1P91 Mathematics Review Exercises (answers at end) Work each exercise without using a calculator. 1. Express each number in scientific notation. (a) 437.1 (b) 563, 000 (c)

More information

A2 Assignment zeta Cover Sheet. C3 Differentiation all methods. C3 Sketch and find range. C3 Integration by inspection. C3 Rcos(x-a) max and min

A2 Assignment zeta Cover Sheet. C3 Differentiation all methods. C3 Sketch and find range. C3 Integration by inspection. C3 Rcos(x-a) max and min A Assignment zeta Cover Sheet Name: Question Done Backpack Ready? Topic Comment Drill Consolidation M1 Prac Ch all Aa Ab Ac Ad Ae Af Ag Ah Ba C3 Modulus function Bb C3 Modulus function Bc C3 Modulus function

More information

A2 Assignment lambda Cover Sheet. Ready. Done BP. Question. Aa C4 Integration 1 1. C4 Integration 3

A2 Assignment lambda Cover Sheet. Ready. Done BP. Question. Aa C4 Integration 1 1. C4 Integration 3 A Assignment lambda Cover Sheet Name: Question Done BP Ready Topic Comment Drill Mock Exam Aa C4 Integration sin x+ x+ c 4 Ab C4 Integration e x + c Ac C4 Integration ln x 5 + c Ba C Show root change of

More information

Symbols and dingbats. A 41 Α a 61 α À K cb ➋ à esc. Á g e7 á esc. Â e e5 â. Ã L cc ➌ ã esc ~ Ä esc : ä esc : Å esc * å esc *

Symbols and dingbats. A 41 Α a 61 α À K cb ➋ à esc. Á g e7 á esc. Â e e5 â. Ã L cc ➌ ã esc ~ Ä esc : ä esc : Å esc * å esc * Note: Although every effort ws tken to get complete nd ccurte tble, the uhtor cn not be held responsible for ny errors. Vrious sources hd to be consulted nd MIF hd to be exmined to get s much informtion

More information

MAS114: Exercises. October 26, 2018

MAS114: Exercises. October 26, 2018 MAS114: Exercises October 26, 2018 Note that the challenge problems are intended to be difficult! Doing any of them is an achievement. Please hand them in on a separate piece of paper if you attempt them.

More information

b e i ga set s oane f dast heco mm on n va ns ing lo c u soft w section

b e i ga set s oane f dast heco mm on n va ns ing lo c u soft w section 66 M M Eq: 66 - -I M - - - -- -- - - - - -- - S I T - I S W q - I S T q ] q T G S W q I ] T G ˆ Gα ˆ ˆ ] H Z ˆ T α 6H ; Z - S G W [6 S q W F G S W F S W S W T - I ] T ˆ T κ G Gα ±G κ α G ˆ + G > H O T

More information

Entities for Symbols and Greek Letters

Entities for Symbols and Greek Letters Entities for Symbols and Greek Letters The following table gives the character entity reference, decimal character reference, and hexadecimal character reference for symbols and Greek letters, as well

More information

K e sub x e sub n s i sub o o f K.. w ich i sub s.. u ra to the power of m i sub fi ed.. a sub t to the power of a

K e sub x e sub n s i sub o o f K.. w ich i sub s.. u ra to the power of m i sub fi ed.. a sub t to the power of a - ; ; ˆ ; q x ; j [ ; ; ˆ ˆ [ ˆ ˆ ˆ - x - - ; x j - - - - - ˆ x j ˆ ˆ ; x ; j κ ˆ - - - ; - - - ; ˆ σ x j ; ˆ [ ; ] q x σ; x - ˆ - ; J -- F - - ; x - -x - - x - - ; ; 9 S j P R S 3 q 47 q F x j x ; [ ]

More information

CSE 1400 Applied Discrete Mathematics Definitions

CSE 1400 Applied Discrete Mathematics Definitions CSE 1400 Applied Discrete Mathematics Definitions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Arithmetic 1 Alphabets, Strings, Languages, & Words 2 Number Systems 3 Machine

More information

MAS114: Solutions to Exercises

MAS114: Solutions to Exercises MAS114: s to Exercises Up to week 8 Note that the challenge problems are intended to be difficult! Doing any of them is an achievement. Please hand them in on a separate piece of paper if you attempt them.

More information

T m / A. Table C2 Submicroscopic Masses [2] Symbol Meaning Best Value Approximate Value

T m / A. Table C2 Submicroscopic Masses [2] Symbol Meaning Best Value Approximate Value APPENDIX C USEFUL INFORMATION 1247 C USEFUL INFORMATION This appendix is broken into several tables. Table C1, Important Constants Table C2, Submicroscopic Masses Table C3, Solar System Data Table C4,

More information

CSSS/STAT/SOC 321 Case-Based Social Statistics I. Levels of Measurement

CSSS/STAT/SOC 321 Case-Based Social Statistics I. Levels of Measurement CSSS/STAT/SOC 321 Case-Based Social Statistics I Levels of Measurement Christopher Adolph Department of Political Science and Center for Statistics and the Social Sciences University of Washington, Seattle

More information

PHIL 50 INTRODUCTION TO LOGIC 1 FREE AND BOUND VARIABLES MARCELLO DI BELLO STANFORD UNIVERSITY DERIVATIONS IN PREDICATE LOGIC WEEK #8

PHIL 50 INTRODUCTION TO LOGIC 1 FREE AND BOUND VARIABLES MARCELLO DI BELLO STANFORD UNIVERSITY DERIVATIONS IN PREDICATE LOGIC WEEK #8 PHIL 50 INTRODUCTION TO LOGIC MARCELLO DI BELLO STANFORD UNIVERSITY DERIVATIONS IN PREDICATE LOGIC WEEK #8 1 FREE AND BOUND VARIABLES Before discussing the derivation rules for predicate logic, we should

More information

PRELIMINARIES FOR GENERAL TOPOLOGY. Contents

PRELIMINARIES FOR GENERAL TOPOLOGY. Contents PRELIMINARIES FOR GENERAL TOPOLOGY DAVID G.L. WANG Contents 1. Sets 2 2. Operations on sets 3 3. Maps 5 4. Countability of sets 7 5. Others a mathematician knows 8 6. Remarks 9 Date: April 26, 2018. 2

More information

1 Integration of Rational Functions Using Partial Fractions

1 Integration of Rational Functions Using Partial Fractions MTH Fall 008 Essex County College Division of Mathematics Handout Version 4 September 8, 008 Integration of Rational Functions Using Partial Fractions In the past it was far more usual to simplify or combine

More information

3% 5% 1% 2% d t = 1,, T d i j J i,d,t J i,d,t+ = J i,d,t j j a i,d,t j i t d γ i > 0 i j t d U i,j,d,t = μ i,j,d + β i,j,d,t γ i + ε i,j,d,t ε i,j,d,t t ε i,,d,t μ β μ μ i,j,d d t d, t j μ i,j,d i j d

More information

last name ID 1 c/cmaker/cbreaker 2012 exam version a 6 pages 3 hours 40 marks no electronic devices SHOW ALL WORK

last name ID 1 c/cmaker/cbreaker 2012 exam version a 6 pages 3 hours 40 marks no electronic devices SHOW ALL WORK last name ID 1 c/cmaker/cbreaker 2012 exam version a 6 pages 3 hours 40 marks no electronic devices SHOW ALL WORK 8 a b c d e f g h i j k l m n o p q r s t u v w x y z 1 b c d e f g h i j k l m n o p q

More information

Writing Game Theory in L A TEX

Writing Game Theory in L A TEX Writing Game Theory in L A TEX Thiago Silva First Version: November 22, 2015 This Version: November 13, 2017 List of Figures and Tables 1 2x2 Matrix: Prisoner s ilemma Normal-Form Game............. 3 2

More information

Storm Open Library 3.0

Storm Open Library 3.0 S 50% off! 3 O L Storm Open Library 3.0 Amor Sans, Amor Serif, Andulka, Baskerville, John Sans, Metron, Ozdoby,, Regent, Sebastian, Serapion, Splendid Quartett, Vida & Walbaum. d 50% f summer j sale n

More information

SECOND GENERAL REPORT - APRIL 1954 35 During the same period, France and the Saar increased their hard coal purchases from other countries by 689,000 metric tons, or 14.4 %, with 5,486,000 tons in 1953

More information

A geometric solution of the Kervaire Invariant One problem

A geometric solution of the Kervaire Invariant One problem A geometric solution of the Kervaire Invariant One problem Petr M. Akhmet ev 19 May 2009 Let f : M n 1 R n, n = 4k + 2, n 2 be a smooth generic immersion of a closed manifold of codimension 1. Let g :

More information

Two Mathematical Constants

Two Mathematical Constants 1 Two Mathematical Constants Two of the most important constants in the world of mathematics are π (pi) and e (Euler s number). π = 3.14159265358979323846264338327950288419716939937510... e = 2.71828182845904523536028747135266249775724709369995...

More information

Using Structural Equation Modeling to Conduct Confirmatory Factor Analysis

Using Structural Equation Modeling to Conduct Confirmatory Factor Analysis Using Structural Equation Modeling to Conduct Confirmatory Factor Analysis Advanced Statistics for Researchers Session 3 Dr. Chris Rakes Website: http://csrakes.yolasite.com Email: Rakes@umbc.edu Twitter:

More information

About One way of Encoding Alphanumeric and Symbolic Information

About One way of Encoding Alphanumeric and Symbolic Information Int. J. Open Problems Compt. Math., Vol. 3, No. 4, December 2010 ISSN 1998-6262; Copyright ICSRS Publication, 2010 www.i-csrs.org About One way of Encoding Alphanumeric and Symbolic Information Mohammed

More information

Zone 252, Master Map Normal View

Zone 252, Master Map Normal View Zone Master, Normal View, 10 deg FOV, Master Map Normal View χ1 α ζ ε γ1 γ2 β η1 η3 η2 ψ Fornax ι2 ι1 λ2 ω λ1 φ µ No Map ν π ε π Cetus Sculptor τ -38 00' -36 00' -34 00' -32 00' -30 00' -28 00' -26 00'

More information

Contents. basic algebra. Learning outcomes. Time allocation. 1. Mathematical notation and symbols. 2. Indices. 3. Simplification and factorisation

Contents. basic algebra. Learning outcomes. Time allocation. 1. Mathematical notation and symbols. 2. Indices. 3. Simplification and factorisation basic algebra Contents. Mathematical notation and symbols 2. Indices 3. Simplification and factorisation 4. Arithmetic of algebraic fractions 5. Formulae and transposition Learning outcomes In this workbook

More information

12 th Marcel Grossman Meeting Paris, 17 th July 2009

12 th Marcel Grossman Meeting Paris, 17 th July 2009 Department of Mathematical Analysis, Ghent University (Belgium) 12 th Marcel Grossman Meeting Paris, 17 th July 2009 Outline 1 2 The spin covariant derivative The curvature spinors Bianchi and Ricci identities

More information

Web Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D.

Web Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D. Web Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D. Ruppert A. EMPIRICAL ESTIMATE OF THE KERNEL MIXTURE Here we

More information

. D CR Nomenclature D 1

. D CR Nomenclature D 1 . D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the

More information

R k. t + 1. n E t+1 = ( 1 χ E) W E t+1. c E t+1 = χ E Wt+1 E. Γ E t+1. ) R E t+1q t K t. W E t+1 = ( 1 Γ E t+1. Π t+1 = P t+1 /P t

R k. t + 1. n E t+1 = ( 1 χ E) W E t+1. c E t+1 = χ E Wt+1 E. Γ E t+1. ) R E t+1q t K t. W E t+1 = ( 1 Γ E t+1. Π t+1 = P t+1 /P t R k E 1 χ E Wt E n E t+1 t t + 1 n E t+1 = ( 1 χ E) W E t+1 c E t+1 = χ E Wt+1 E t + 1 q t K t Rt+1 E 1 Γ E t+1 Π t+1 = P t+1 /P t W E t+1 = ( 1 Γ E t+1 ) R E t+1q t K t Π t+1 Γ E t+1 K t q t q t K t j

More information

Homework Hint. Last Time

Homework Hint. Last Time Homework Hint Problem 3.3 Geometric series: ωs τ ħ e s= 0 =? a n ar = For 0< r < 1 n= 0 1 r ωs τ ħ e s= 0 1 = 1 e ħω τ Last Time Boltzmann factor Partition Function Heat Capacity The magic of the partition

More information

Chapter 2: Fundamentals of Statistics Lecture 15: Models and statistics

Chapter 2: Fundamentals of Statistics Lecture 15: Models and statistics Chapter 2: Fundamentals of Statistics Lecture 15: Models and statistics Data from one or a series of random experiments are collected. Planning experiments and collecting data (not discussed here). Analysis:

More information

Outline. Logic. Definition. Theorem (Gödel s Completeness Theorem) Summary of Previous Week. Undecidability. Unification

Outline. Logic. Definition. Theorem (Gödel s Completeness Theorem) Summary of Previous Week. Undecidability. Unification Logic Aart Middeldorp Vincent van Oostrom Franziska Rapp Christian Sternagel Department of Computer Science University of Innsbruck WS 2017/2018 AM (DCS @ UIBK) week 11 2/38 Definitions elimination x φ

More information

Here are the more notable corrections and changes made in the paperback edition of Optically polarized atoms: understanding light atom interactions:

Here are the more notable corrections and changes made in the paperback edition of Optically polarized atoms: understanding light atom interactions: Here are the more notable corrections and changes made in the paperback edition of Optically polarized atoms: understanding light atom interactions: Page v, first paragraph of Acknowledgments, and Andrey

More information

Celeste: Variational inference for a generative model of astronomical images

Celeste: Variational inference for a generative model of astronomical images Celeste: Variational inference for a generative model of astronomical images Jerey Regier Statistics Department UC Berkeley July 9, 2015 Joint work with Jon McAulie (UCB Statistics), Andrew Miller, Ryan

More information

Skeletal 2 - joints. Puzzle 1 bones

Skeletal 2 - joints. Puzzle 1 bones Puzzle 1 bones Listed below are the names of some of the bones that make up your skeletal system. But the names have been encrypted using a secret code. Can you decipher this code to find out what they

More information

Compton Scattering Effect and Physics of Compton Photon Beams. Compton Photon Sources around the World, Present and Future

Compton Scattering Effect and Physics of Compton Photon Beams. Compton Photon Sources around the World, Present and Future !!! #! ! # Compton Scattering Effect and Physics of Compton Photon Beams Compton Photon Sources around the World, Present and Future Compton X-ray Sources: Facilities, Projects and Experiments Compton

More information

COMBINATORIALLY CONVEX MODULI AND AN EXAMPLE OF GÖDEL

COMBINATORIALLY CONVEX MODULI AND AN EXAMPLE OF GÖDEL Pioneer Journal of Algebra, Number Theory and its Applications Volume, Number, 2017, Pages This paper is available online at http://www.pspchv.com/content_pjanta.html COMBINATORIALLY CONVEX MODULI AND

More information

Optimising Distributed Energy Operations in Buildings

Optimising Distributed Energy Operations in Buildings Optimising Distributed Energy Operations in Buildings Markus Groissböck Center for Energy and Innovative Technologies Somayeh Heydari University College London Ana Mera Tecnalia Research & Innovation Eugenio

More information

QCD and Instantons: 12 Years Later. Thomas Schaefer North Carolina State

QCD and Instantons: 12 Years Later. Thomas Schaefer North Carolina State QCD and Instantons: 12 Years Later Thomas Schaefer North Carolina State 1 ESQGP: A man ahead of his time 2 Instanton Liquid: Pre-History 1975 (Polyakov): The instanton solution r 2 2 E + B A a µ(x) = 2

More information

SOME CALKIN ALGEBRAS HAVE OUTER AUTOMORPHISMS

SOME CALKIN ALGEBRAS HAVE OUTER AUTOMORPHISMS SOME CALKIN ALGEBRAS HAVE OUTER AUTOMORPHISMS ILIJAS FARAH, PAUL MCKENNEY, AND ERNEST SCHIMMERLING Abstract. We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert

More information

Sudoku and Matrices. Merciadri Luca. June 28, 2011

Sudoku and Matrices. Merciadri Luca. June 28, 2011 Sudoku and Matrices Merciadri Luca June 28, 2 Outline Introduction 2 onventions 3 Determinant 4 Erroneous Sudoku 5 Eigenvalues Example 6 Transpose Determinant Trace 7 Antisymmetricity 8 Non-Normality 9

More information

Asynchronous Training in Wireless Sensor Networks

Asynchronous Training in Wireless Sensor Networks u t... t. tt. tt. u.. tt tt -t t t - t, t u u t t. t tut t t t t t tt t u t ut. t u, t tt t u t t t t, t tt t t t, t t t t t. t t tt u t t t., t- t ut t t, tt t t tt. 1 tut t t tu ut- tt - t t t tu tt-t

More information

Global Production with Export Platforms

Global Production with Export Platforms Discussion of Global Production with Export Platforms by Felix Tintelnot Oleg Itskhoki Princeton University NBER ITI Summer Institute Boston, July 2013 1 / 6 Introduction Question: Where should firms locate

More information

Ideology and Social Networks in the U.S. Congress

Ideology and Social Networks in the U.S. Congress Ideology and Social Networks in the U.S. Congress James H. Fowler University of California, San Diego July 11, 2007 Improve connectedness scores (Fowler 2005) Cosponsorship is about 2 things: The idea

More information

PERIODICITY OF CERTAIN PIECEWISE AFFINE PLANAR MAPS

PERIODICITY OF CERTAIN PIECEWISE AFFINE PLANAR MAPS PERIODICITY OF CERTAIN PIECEWISE AFFINE PLANAR MAPS SHIGEKI AKIYAMA, HORST BRUNOTTE, ATTILA PETHŐ, AND WOLFGANG STEINER Abstract. We determine periodic and aperiodic points of certain piecewise affine

More information

Wavelets For Computer Graphics

Wavelets For Computer Graphics {f g} := f(x) g(x) dx A collection of linearly independent functions Ψ j spanning W j are called wavelets. i J(x) := 6 x +2 x + x + x Ψ j (x) := Ψ j (2 j x i) i =,..., 2 j Res. Avge. Detail Coef 4 [9 7

More information

The Absorption of Gravitational Radiation by a Dissipative Fluid

The Absorption of Gravitational Radiation by a Dissipative Fluid Commun. math. Phys. 30, 335-340(1973) by Springer-Verlag 1973 The Absorption of Gravitational Radiation by a Dissipative Fluid J. Madore Laboratoire de Physique Theoπque, Tnstitut Henri Pomcare, Paris,

More information

Α Dispersion Relation for Open Spiral Galaxies

Α Dispersion Relation for Open Spiral Galaxies J. Astrophys. Astr. (1980) 1, 79 95 Α Dispersion Relation for Open Spiral Galaxies G. Contopoulos Astronomy Department, University of Athens, Athens, Greece Received 1980 March 20; accepted 1980 April

More information

Rational Homotopy Theory Seminar Week 11: Obstruction theory for rational homotopy equivalences J.D. Quigley

Rational Homotopy Theory Seminar Week 11: Obstruction theory for rational homotopy equivalences J.D. Quigley Rational Homotopy Theory Seminar Week 11: Obstruction theory for rational homotopy equivalences J.D. Quigley Reference. Halperin-Stasheff Obstructions to homotopy equivalences Question. When can a given

More information

Lecture 38: Equations of Rigid-Body Motion

Lecture 38: Equations of Rigid-Body Motion Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can

More information

Applications of higher-dimensional forcing

Applications of higher-dimensional forcing Bernhard Irrgang (Bonn) Logic Colloquium, Bern July 4, 2008 Motivation Suppose we want to construct a ccc forcing P of size ω 1. Then we can proceed as follows: Let σ αβ : P α P β α < β < ω 1 be a continuous,

More information

Introduction Benchmark model Belief-based model Empirical analysis Summary. Riot Networks. Lachlan Deer Michael D. König Fernando Vega-Redondo

Introduction Benchmark model Belief-based model Empirical analysis Summary. Riot Networks. Lachlan Deer Michael D. König Fernando Vega-Redondo Riot Networks Lachlan Deer Michael D. König Fernando Vega-Redondo University of Zurich University of Zurich Bocconi University June 7, 2018 Deer & König &Vega-Redondo Riot Networks June 7, 2018 1 / 23

More information

Development of a Dynamic Model of a Small High-Speed Autonomous Underwater Vehicle

Development of a Dynamic Model of a Small High-Speed Autonomous Underwater Vehicle lp i Ml Sll Hi-Sp Uw Vil Hi N., il J. Silwll Bl p Elil p Eii Viii Pli Ii S Uii Bl, V 1 Eil: {, ilwll }@. P : i p, Wii, KS 777 W L. N p p O Eii Viii Pli Ii S Uii Bl, V 1 Eil: @. i l i lp ll, ip w il. il

More information

The employment effect of reforming a public employment agency

The employment effect of reforming a public employment agency The employment effect of reforming a public employment agency Andrey Launov and Klaus Wälde (European Economic Review, 2016) Presented by Ismael Gálvez March 15, 2017 Motivation Rich literature on the

More information

Strong Markov property of determinatal processes

Strong Markov property of determinatal processes Strong Markov property of determinatal processes Hideki Tanemura Chiba university (Chiba, Japan) (August 2, 2013) Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 1 / 27 Introduction The

More information

Properties for systems with weak invariant manifolds

Properties for systems with weak invariant manifolds Statistical properties for systems with weak invariant manifolds Faculdade de Ciências da Universidade do Porto Joint work with José F. Alves Workshop rare & extreme Gibbs-Markov-Young structure Let M

More information

Parallel KS Block-Step Method. Sverre Aarseth. Institute of Astronomy, Cambridge

Parallel KS Block-Step Method. Sverre Aarseth. Institute of Astronomy, Cambridge Parallel KS Block-Step Method Sverre Aarseth Institute of Astronomy, Cambridge Code Overview Hermite KS Prediction & Correction Iteration Time-Steps Decision-Making Binary Project Code Overview Directories

More information

Principal Type Schemes for Functional Programs with Overloading and Subtyping

Principal Type Schemes for Functional Programs with Overloading and Subtyping Principal Type Schemes for Functional Programs with Overloading and Subtyping Geoffrey S. Smith Cornell University December 1994 Abstract We show how the Hindley/Milner polymorphic type system can be extended

More information

Asymptotic Variance Formulas, Gamma Functions, and Order Statistics

Asymptotic Variance Formulas, Gamma Functions, and Order Statistics Statistical Models and Methods for Lifetime Data, Second Edition by Jerald F. Lawless Copyright 2003 John Wiley & Sons, Inc. APPENDIX Β Asymptotic Variance Formulas, Gamma Functions, and Order Statistics

More information

Scaling exponents for certain 1+1 dimensional directed polymers

Scaling exponents for certain 1+1 dimensional directed polymers Scaling exponents for certain 1+1 dimensional directed polymers Timo Seppäläinen Department of Mathematics University of Wisconsin-Madison 2010 Scaling for a polymer 1/29 1 Introduction 2 KPZ equation

More information

Semiclassical limit of the Schrödinger-Poisson-Landau-Lifshitz-Gilbert system

Semiclassical limit of the Schrödinger-Poisson-Landau-Lifshitz-Gilbert system Semiclassical limit of the Schrödinger-Poisson-Landau-Lifshitz-Gilbert system Lihui Chai Department of Mathematics University of California, Santa Barbara Joint work with Carlos J. García-Cervera, and

More information

Binocular Targets: Omega Centauri is also a good object to view in binoculars.

Binocular Targets: Omega Centauri is also a good object to view in binoculars. Eridanus Optics CC April 2006 Targets in Centaurus The following three targets are selected from the Constellation 'Centaurus' to present a naked eye challenge, as well as telescopic challenges. Binoculars

More information

Engineering. Spring Department of Fluid Mechanics, Budapest University of Technology and Economics. Large-Eddy Simulation in Mechanical

Engineering. Spring Department of Fluid Mechanics, Budapest University of Technology and Economics. Large-Eddy Simulation in Mechanical Outline Department of Fluid Mechanics, Budapest University of Technology and Economics Spring 2011 Outline Outline Part I First Lecture Connection between time and ensemble average Ergodicity1 Ergodicity

More information

Communication with Imperfect Shared Randomness

Communication with Imperfect Shared Randomness Communication with Imperfect Shared Randomness (Joint work with Venkatesan Guruswami (CMU), Raghu Meka (?) and Madhu Sudan (MSR)) Who? Clément Canonne (Columbia University) When? November 19, 2014 1 /

More information

How Far Mathematical Foundations Direct Measurement

How Far Mathematical Foundations Direct Measurement {Abstract A significant amount of mathematics is used in the How Far Away Is It channel video books. Although mathematical equations are identified, they were not the focus. They served to deepen understanding

More information

EllipticTheta2. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

EllipticTheta2. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation EllipticTheta Notations Traditional name Jacobi theta function ϑ Traditional notation ϑ z, Mathematica StandardForm notation EllipticTheta, z, Primary definition 09.0.0.0001.01 ϑ z, cos k 1 z ; 1 Specific

More information

Oblique derivative problems for elliptic and parabolic equations, Lecture II

Oblique derivative problems for elliptic and parabolic equations, Lecture II of the for elliptic and parabolic equations, Lecture II Iowa State University July 22, 2011 of the 1 2 of the of the As a preliminary step in our further we now look at a special situation for elliptic.

More information

Lecture 11: Weak Interactions

Lecture 11: Weak Interactions Lecture 11: Weak Interactions Cross-Section and the W Coupling The Cabibbo Angle and the CKM Matrix Parity Violation Kaons and Mixing CP Violation Useful Sections in Martin & Shaw: Sections 4.51, 8.1,

More information

Next is material on matrix rank. Please see the handout

Next is material on matrix rank. Please see the handout B90.330 / C.005 NOTES for Wednesday 0.APR.7 Suppose that the model is β + ε, but ε does not have the desired variance matrix. Say that ε is normal, but Var(ε) σ W. The form of W is W w 0 0 0 0 0 0 w 0

More information

GARCH Models Estimation and Inference. Eduardo Rossi University of Pavia

GARCH Models Estimation and Inference. Eduardo Rossi University of Pavia GARCH Models Estimation and Inference Eduardo Rossi University of Pavia Likelihood function The procedure most often used in estimating θ 0 in ARCH models involves the maximization of a likelihood function

More information

Holographic Entanglement Entropy for Surface Operators and Defects

Holographic Entanglement Entropy for Surface Operators and Defects Holographic Entanglement Entropy for Surface Operators and Defects Michael Gutperle UCLA) UCSB, January 14th 016 Based on arxiv:1407.569, 1506.0005, 151.04953 with Simon Gentle and Chrysostomos Marasinou

More information

An Evolving Gradient Resampling Method for Machine Learning. Jorge Nocedal

An Evolving Gradient Resampling Method for Machine Learning. Jorge Nocedal An Evolving Gradient Resampling Method for Machine Learning Jorge Nocedal Northwestern University NIPS, Montreal 2015 1 Collaborators Figen Oztoprak Stefan Solntsev Richard Byrd 2 Outline 1. How to improve

More information

Leaving Plato s Cave: Beyond The Simplest Models of Dark Matter

Leaving Plato s Cave: Beyond The Simplest Models of Dark Matter Leaving Plato s Cave: Beyond The Simplest Models of Dark Matter Alexander Natale Korea Institute for Advanced Study Nucl. Phys. B914 201-219 (2017), arxiv:1608.06999. High1 2017 February 9th, 2017 1/30

More information

The next two questions pertain to the situation described below. Consider a parallel plate capacitor with separation d:

The next two questions pertain to the situation described below. Consider a parallel plate capacitor with separation d: PHYS 102 Exams Exam 2 PRINT (A) The next two questions pertain to the situation described below. Consider a parallel plate capacitor with separation d: It is connected to a battery with constant emf V.

More information

License: Creative Commons Attribution Non-commercial No Derivatives

License: Creative Commons Attribution Non-commercial No Derivatives Title Type URL Design in the Time of Policy Problems Article Date 2016 Citation Creators http://ualresearchonline.arts.ac.uk/9542/ Kimbell, Lucy (2016) Design in the Time of Policy Problems. Proceedings

More information

Hybrid inflation with a non-minimally coupled scalar field

Hybrid inflation with a non-minimally coupled scalar field Hybrid inflation with a non-minimally coupled scalar field Seoktae Koh CQUeST, Sogang University, Korea COSMO/CosPA 2010 @Tokyo Univ., JAPAN 27 Sept. - 1 Oct., 2010 work with M. Minamitsuji PART I: Non-minimally

More information

Dynamic Asset Allocation - Identifying Regime Shifts in Financial Time Series to Build Robust Portfolios

Dynamic Asset Allocation - Identifying Regime Shifts in Financial Time Series to Build Robust Portfolios Downloaded from orbit.dtu.dk on: Jan 22, 2019 Dynamic Asset Allocation - Identifying Regime Shifts in Financial Time Series to Build Robust Portfolios Nystrup, Peter Publication date: 2018 Document Version

More information

SHANGHAI JIAO TONG UNIVERSITY LECTURE

SHANGHAI JIAO TONG UNIVERSITY LECTURE Lecture 7 SHANGHAI JIAO TONG UNIVERSITY LECTURE 7 017 Anthony J. Leggett Department of Physics University of Illinois at Urbana-Champaign, USA and Director, Center for Complex Physics Shanghai Jiao Tong

More information

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 14: Formulation of the Stability Problem

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 14: Formulation of the Stability Problem .65, MHD Theory of Fusion Systems Prof. Freidberg Lecture 4: Formulation of the Stability Problem Hierarchy of Formulations of the MHD Stability Problem for Arbitrary 3-D Systems. Linearized equations

More information

Hyperelliptic Lefschetz fibrations and the Dirac braid

Hyperelliptic Lefschetz fibrations and the Dirac braid Hyperelliptic Lefschetz fibrations and the Dirac braid joint work with Seiichi Kamada Hisaaki Endo (Tokyo Institute of Technology) Differential Topology 206 March 20, 206, University of Tsukuba March 20,

More information

A PROOF OF BOREL-WEIL-BOTT THEOREM

A PROOF OF BOREL-WEIL-BOTT THEOREM A PROOF OF BOREL-WEIL-BOTT THEOREM MAN SHUN JOHN MA 1. Introduction In this short note, we prove the Borel-Weil-Bott theorem. Let g be a complex semisimple Lie algebra. One basic question in representation

More information

Decentralized Disturbance Attenuation for Large-Scale Nonlinear Systems with Delayed State Interconnections

Decentralized Disturbance Attenuation for Large-Scale Nonlinear Systems with Delayed State Interconnections Decentralized Disturbance Attenuation for Large-Scale Nonlinear Systems with Delayed State Interconnections Yi Guo Abstract The problem of decentralized disturbance attenuation is considered for a new

More information

PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS

PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS OVIDIU SAVIN Abstract. We consider local minimizers of the Ginzburg-Landau energy functional 2 u 2 + 4 ( u2 ) 2 dx and prove that, if the level set is included

More information

Regularity of flat level sets in phase transitions

Regularity of flat level sets in phase transitions Annals of Mathematics, 69 (2009), 4 78 Regularity of flat level sets in phase transitions By Ovidiu Savin Abstract We consider local minimizers of the Ginzburg-Landau energy functional 2 u 2 + 4 ( u2 )

More information

Chapter 1 Fundamentals in Elasticity

Chapter 1 Fundamentals in Elasticity Fs s ν . Po Dfo ν Ps s - Do o - M os - o oos : o o w Uows o: - ss - - Ds W ows s o qos o so s os. w ows o fo s o oos s os of o os. W w o s s ss: - ss - - Ds - Ross o ows s s q s-s os s-sss os .. Do o ..

More information

Simultaneous equation models with spatially autocorrelated error components

Simultaneous equation models with spatially autocorrelated error components MPRA Munich Personal RePEc Archive Simultaneous equation models with spatially autocorrelated error components Claude Marius AMBA OYON and Taoufiki Mbratana University of Yaounde II, University of Yaounde

More information

A model of alignment interaction for oriented particles with phase transition

A model of alignment interaction for oriented particles with phase transition A model of alignment interaction for oriented particles with phase transition Amic Frouvelle Institut de Mathématiques de Toulouse Joint work with Jian-Guo Liu (Duke Univ.) and Pierre Degond (IMT) Amic

More information

Trade and Inequality: From Theory to Estimation

Trade and Inequality: From Theory to Estimation Trade and Inequality: From Theory to Estimation Elhanan Helpman Oleg Itskhoki Marc Muendler Stephen Redding Harvard Princeton UC San Diego Princeton MEF Italia Dipartimento del Tesoro September 2014 1

More information

BERNOULLI ACTIONS AND INFINITE ENTROPY

BERNOULLI ACTIONS AND INFINITE ENTROPY BERNOULLI ACTIONS AND INFINITE ENTROPY DAVID KERR AND HANFENG LI Abstract. We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every

More information

Many-Body physics meets Quantum Information

Many-Body physics meets Quantum Information Many-Body physics meets Quantum Information Rosario Fazio Scuola Normale Superiore, Pisa & NEST, Istituto di Nanoscienze - CNR, Pisa Quantum Computers Interaction between qubits two-level systems Many-Body

More information

Wave operators with non-lipschitz coefficients: energy and observability estimates

Wave operators with non-lipschitz coefficients: energy and observability estimates Wave operators with non-lipschitz coefficients: energy and observability estimates Institut de Mathématiques de Jussieu-Paris Rive Gauche UNIVERSITÉ PARIS DIDEROT PARIS 7 JOURNÉE JEUNES CONTRÔLEURS 2014

More information

The Relative Proj Construction

The Relative Proj Construction The Relative Proj Construction Daniel Murfet October 5, 2006 Earlier we defined the Proj of a graded ring. In these notes we introduce a relative version of this construction, which is the Proj of a sheaf

More information