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  Biographies for the  New Dictionary of Scientific Biography  (2008) Birkhoff, Garrett Chevalley, Claude Dieudonné, Jean Mac Lane, Saunders (co-authored with William Lawvere) Weil, André  Bijvoet zum 80. Geburtstag editedby O. R Frischet al Braunschweig:Vieweg,1959. TheNetherlands. In Fifty .Years of X-RayDiffraction:Dedicated to theInternational Union of Crystallography on the Occasion of the Commemorative MeetinginMunich,.july 1962, editedbyPaul P Ewald.Utrecht:Oosthoek,1962a. Reminiscences. In Fifty .Years of X-RayDiffraction:Dedicated to theInternational Union of Crystallography on the Occasion of theCommemorativeMeetinginMunich, July 1962, editedbyPaul P Ewald.Utrecht:Oosthoek,1962b. As editor,withWG.BurgersandG.Hagg. Early Papers on Diffraction of X-rays byCrystals. 2vols.Utrecht:Oosthoek, 1969 1972. Containsselectedtextfragmentsfromalmost200importantpapersupto1935. OTHER SOURCES Blow D. M. HowBijvoetMadetheDifference:TheGrowingPower of AnomalousScattering. Methodsin Enzymology 374(2003): 3 22.On theimpact of Bijvoet'sanomalousscatteringmethodsonX-raycrystallographyupto2003.Coster,Dirk, K S Kno 'and J Prins. UnterschiedeinderIntensitatderRontgenstrahlenreflextionandenbeiden111FlachenderZinkblende. ZeitschriJtflir Physik 63(1930): 345 369. Ewald,Paul P ed. Fifty } ears of X-Ray Diffi-action: Dedicated to the International Union of Crystallography on the Occasion of theCommemorativeMeetinginMunich, July 1962. Utrecht:Oosthoek,1962.Groenewege,M. P A Peerdeman,andH.   vanSprang. InmemoriamProf.Bijvoet:Markante,onvermoeibareenveeleisendeleermeester. ChemischWeekblad (20March1980):126-127.   and A Peerdeman. JohannesMartinBijvoet.23January 1892 4 March1980.Elected For Mem. R S 1972. BiographicalMemoirs of Fellows of theRoyalSociety 29 (1983): 27 41. Withabibliography of Bijvoet'spublications.Knegtmans,PeterJan. Een kwetsbaarcentrum van de geest. De Universiteit van Amsterdam tussen 1935 en 1950. Amsterdam:AmsterdamUniversityPress,1998.Krom,CornelisJan. Rontgenanalyse langs directen weg. Onderzoek van a-Br-, o l en n-CN-kamfer. PhD diss.,University of Utrecht,3July1946. Le Pair, c and J Volger eds. Physics in the Netherlands: A Selection of DutchContributions to Physics in theFirst Thirty .Years after theSecondWorld Wtl1: 2vols.Utrecht: FOM 1982.Looijenga- Vos Aafje,andJanKroon. J.M.Bijvoet'sDiscovery:ALandmarkintheStructureElucidation of NaturalProducts. Proceedings Koninklijke Nederlandse Akademie vanWetenschappen 100,no. 3 4 (1995): 45 56. MacGillavry,CarolineH.,and A Peerdeman. JohannesMartinBijvoet, 23 January 1892 4 March1980. Acta Crystallographica SectionA36,no.6(1980):837-838.Peerdeman,AntoniusFranciscus. Determination of the Absolute Configuration of Optically Active Compounds by Means of x Rays. PhD diss.,University of Utrecht, 19 December1955.Schenk,Henk. Kristallografie. In De geschiedenisvan de scheikundein Nederland 3: Deontwikkeling van de chemie van 286 Birkhof 1945 tothet beginvan de jaren tachtig edited by Ernst Homburg andLodewijkPalm.Delft:DelftUniversityPress,2004.Snelders,Harry A M. De geschiedenisvande scheikundein Nederland 2: De ontwikkeling van  h~mi en chemischetechnologie in de eerste helft van de twintigste eeuw. Delft:DelftUniversityPress,1997. On thehistory of Dutchchemistry, 1900 1950. Werken aan scheikunde. 24 memo ires van hendie de Nederlandsechemie dezeeeuw groot hebben gemaakt. Delft:DelftUniversityPress,1993.Thisbookcontainsthepersonalreminiscences of Bijvoet'sstudentsandcolleaguesCaroline H. MacGillavry,Jan   A Ketelaar,EelkoH.Wiebenga,andAafjeLooijenga- Vos Wiebenga, E H. Bijvoet:ongelooflijkeinteresseenwerkkracht. Chemisch Weekblad 20 March1980):127. ErnstHomburg BIRKHOFF G RRETT  b Princeton,NewJersey,10January1911; d. WaterMill,New York 22November1996), abstractalgebra,computing. Birkhoff was theson of mathematicianGeorgeDavidBirkhoffandMargaretGrafiusBirkhoff.GeorgeBirkhoff,thefather, was thefirstAmericanmathematician to gainwiderespectinEurope.GarrettBirkhoff is morerememberedforpromotingnewconceptionsthanspecifictheorems.Hismostimportantsingleresult was atheoremthatinstitutedaconception,the Birkhoff varietytheorem, srcinatingmodernuniversalalgebra. He showedthepower of deceptivelysimplealgebraicpropertiesandthefeasibility of morecomplexandrealisticappliedmathematics,andhe was amongthefirstmathematicians to rely heavilyoncomputers.Lattices and UniversalAlgebra.EnteringHarvardCollegein1928,Birkhoffaimedatmathematicalphysics.Physicsledhim to partialdifferentialequations,whichinturnledtomoreabstractideas,includingLebesguetheoryandpoint-settopology.Curiosityledhimtofinitegroups.Aftergraduatingin1932,hewenttoCambridgeUniversityforphysics. That July,though,hevisitedMunichandmetConstantinCaratheodory,whopointedhimtowardsalgebraandespeciallyvanderWaerden'sgreatnewtextbook ModerneAlgebra (Berlin:Springer,1930).BackinCambridgeheswitched to algebrawithgrouptheoristPhilipHall.Birkhoffturnedthestudy of subgroups,subrings,and so onintotwobranches of mathematics. The intersection H  K of subgroups of asinglegroupG is also asubgroup of G The union H K) of subgroups of G is generallynotasubgroupbecauseanelement of Hand another of K may NEWDICTIONARY OF SCIENTIFIC BIOGRAPHY ã  Birkhoff combinetogiveone that is in neither. Yet Hand K willgenerateasubgroup H/K which is calledtheir join, defined as thesmallestsubgroup of G thatcontains both Hand 1 , and so is generallylarger than thesettheoreticunion.Thissuggestsadualdefinition:the meet H-Kis thelargestsubgroup of G containedin both Hand K Infact,themeet of subgroups is theirintersection, but other structuresthangroupscanhavemeetsthataresmallerthanintersections.InEnglandBirkhofforganized and generalizedthestudy of suchorderrelationsinto latticetheory. He alsocharacterizedawidearray of structureswhosesubstructuresformlattices and organizedtheir study as universalalgebra. Eachsubjecthadprecedents,notablyinthework of Richard Dedekind and Emmy Noether, but Birkhoffestablished them as subjects.Birkhoffenjoyedtheunity and economy of theabstractidea of an orderrelation on aset. That is anyrelation xByon theelements of thesetsuchthat: 1 xBx forallelements x of theset;2)foranyelements x,y,z, if xBy and yBz then xBz; and3)foranyelements x,y, if xBy and yBx then x = y. The relation xBy is usuallyread  x is less than or equalto y although it mayhave nothing todowithmagnitude. He gavetheexample of logicalpropositionswith xBy definedto mean x implies y, and x = y definedtomean that x is logicallyequivalent   y He defineda lattice as anorderedsetwhereeverytwoele ments x,y haveajoin xly defined as thesmallestelementgreater thanor equal   both x and y, and ameet x-y defined as thelargestelement less than orequalto both x and y withafewfurtherproperties.Logicalpropositionsformalatticewherethejoin xly of propositions is theirdisjunction  x or y, andtheirmeet x-y is theconjunction  x and y. The subgroups of agroup Gform alatticewhen HBKis definedtomean His containedin K The notion of lattice is wideenoughtoincludemanyexamplesyetspecificenough   yield many theorems.Birkhoffalso found furtherabstractconditionscharacterizingvariouskinds of lattice.His1940 book LatticeTheory is stillin print withnewconcepts and resultstriplingitsoriginallength. Not allmathematicalstructuresare as tidy as groups. The substructures of agivenstructuredo not alwaysformalattice.So,whichonesdo?Birkhofffoundanelegantsufficientcondition.A BirkhofJvariety is aclasscontainingallthestructuresdefinedbyagivenset of operators and equations.Forexample,a commutativering is aset R withaselectedzeroelement 0 andunit element 1 and addition,subtraction, and multiplication  - . satisfyingequationsfamiliar from arithmetic,such as thezerolaw x 0 = x and the commutative lawforaddition x y = y   x Theseequationsare understood to holdforallelements x,y of R A fieldis acommutativering R meetingafurthermorecom- NEWDICTIONARY OF SCIENTIFIC BIOGRAPHY Birkhoff plexcondition, not anunrestrictedequation but aconditionalequation: If x 7= 0 then x hasan inverse, anelement yin R with x·y = 1 A variety is aclass of structuresdefinablepurelybyoperators and equations, as shownabovefortheclass of commutativerings andnot forthe class of fields.Varietiesenjoyveryspecialpropertiescomparedtootherclasses of structures.Forexample,anytworings R,S havea product   S Anelement of   S is anorderedpair x,u with x and element of Rand u and element of   The zeroelement of   S is thepair of zeros 0 0 the unit is thepair of units1,1. The operationsaredefinedcomponentwise,whichforadditionmeans x,u y,v = x y,u v. The analoguesholdforsubtractionandmultiplication, and   S satisfiesalltheringequationssinceitsatis fies them all ineachcomponent. The samedoes not workforfields.Even if Rand S are both fields,   S is not.Itselement 1 0 is not zerobecausethefirst component is not thezero of R but ithasnoinverseeitherbecausethesec ondcomponent hasnoinversein S The BirkhofJvarietytheorem listsafewconstructionssuch as products,andproves that aclass of structures is avariety if andonly if it is closed under theseoperations.Givenanyclass of structures,nomatter how itwasoriginallydefined:itcanbedefinedpurelybyequations if and onlyif,theselistedconstructionsapply to itandalwaysyieldresultsin that class.Becausefieldsdo not haveproductstherecannotbeanyway to characterizefieldspurelybyequations.Theseconstructionsimplythatthesubstructures of anystructureinavarietyformalattice. The theoremcreated modern universalalgebra defined as thestudy of BirkhofJvarieties. Earliermoresweepinguniversaltheories of algebrawere not soproductive as Birkhoff s.ACareer at Harvard. In 1933,Birkhoffreturnedto Har- vard as a member of theSociety of Fellows, and in 1936he joined themathematicsdepartment. He neverearnedadoctorate. He married Ruth Collinsin1938 and eventually had twodaughters and ason Ruth,John,andNancy). He beganteachingthenewabstractalgebra,whichSaundersMacLanealsotaughtthere. Their 1941 Survey of ModernAlgebra wasthefirsteffectiveEnglishlanguageintroductiontothematerial of vanderWaerden s ModerneAlgebra and wasanimmediatesuccess. It wasaugmentedbythe  967 Algebra withtheorder of theauthors namesreversed and moreemphasis on categorytheory.Thesetwobooks had ahugeimpact on mathemat ics studentsforfiftyyears and continuedtoshapethestandardU.S.algebracurriculumintheearlytwenty-firstcentury. During WorldWarII,Birkhoffworked on fluiddynamics,includingtheexplosion of bazookachargesand 287   irkhoff problems of air-launchedmissilesenteringwater. Chap- ters of his1950book Hydrodynamics: A StudyinLogic Fact and Similitude werenamedforvarious paradoxes whereeitherthemodelsidealizephenomenainunrealistic ways orbasicallyplausiblemodelsgivesomebizarreresults.Birkhoffemphasizedgrouptheoryforhandlingsymmetriesinhydrodynamics,althoughoneparadoxwasthebreakdown of symmetriesinsomerealistichydrodynamicsituations. He urgedinnovativenumericalmethodsandhislatermorespecializedhydrodynamicsreliedmoreheavily on computing. He consulted on reactordesignfortheBettisAtomicPowerLaboratoryfrom1955to1961,working on numericalsolutionsforpartialdifferentialequationsbyrepeatedlyimprovingsuccessiveapproximations.Startingin1959heconsultedforGeneralMotorsonnumericaldescription of surfaces,toguidenumericallycontrolledmachinerycuttingthediesusedtostamp out automobilebodyparts.Thisledhimtomajorcontributionsto spline methodsfittingsegments of cubicpolynomialstodatapoints.BirkhoffwaselectedtotheAmericanAcademy of Arts   Sciencesin1945,theAmericanPhilosophicalSocietyin1960, and theNationalAcademy of Sciencesin1968. He receivedhonorarydegreesfromtheNationalUniversity of Mexico,theUniversity of Lille,andCaseInstitute of Technology. BIBLIOGRAPHY WORKS BY BIRKHOFF  On the Combination of Subalgebras. Proceedings of theCambridgePhilosophical Society 29 (1933):441-464. Lattice Theory. New York: AmericanMathematicalSociety, 1940. Thirdedition,greatlyexpanded, 1967. WithSaundersMacLane. A Survey of Modern Algebra. New York: Macmillan, 1941. Hydrodynamics: AStudyin Logic Fact andSimilitude Princeton,NJ:PrincetonUniversityPress, 1950. WithGian-CarloRota. OrdinaryDifferential Equations. Boston:Ginn, 1962. MacLane,Saunders,andGarrettBirkhoff. Algebra. New York: Macmillan, 1967. The NumericalSolution of Elliptic Equations. Philadelphia:SocietyforIndustrialandAppliedMathematics, 1971.  CurrentTrendsinAlgebra. The AmericanMathematicalMonthly 80(1973):760-782. WithGerald   AlexandersonandCarrollWilde,  A ConversationwithGarrettBirkhoff. TheTwo Year College MathematicsJournal 14 (1983):126-145. OTHER SOURCES Corry,Leo. Modern Algebra and the Rise of Mathematical Structures. BaselandBoston,MA:Birkha.user, 1996.  88  jerknes Describesthehistoryandinfluence of Birkhoff slatticetheoryanduniversalalgebrainseveralplaces;seetheindex.MacLane,Saunders. GarrettBirkhoff (10 January 1911-22 November 1996). Proceedings of the American PhilosophicalSociety 142 (1998):646-649. Young,David. GarrettBirkhoffandAppliedMathematics. Notices of the AmericanMathematical Society 44(1997):1446-1450. Colin Mc arty BJERKNES, VIL LM  b Christiania,Nor way 14March1862; d. Oslo,Norway,9April1951), meteorology FortheoriginalarticleonBjerknes see DSB vol. 2. Frequentlycalledthefather of modernmeteorology,Bjerknesreluctantlydevotedhimselftoatmosphericscience.Hisscientificcareer,beginninginthe1890s,revealsanastutescientistwilling to overcomeprofessionalmarginalizationbydevelopingskills as adisciplinaryentrepreneur.Beginning as atheoreticalphysicistdevotedtoamechani cal worldview,hehesitantlyturnedtocreatingaphysics of theatmosphereandoceans.In1897heelaboratedahydrodynamicequationforcirculationinfluidsinwhichdensitycoulddependuponseveralvariables. He soonunderstoodthatmotionsintheatmosphereandoceanscouldbecomprehendedthroughthistheorem:Heultimatelydevelopedaphysicalhydrodynamicsthatbecameabasisfordynamicmeteorologyandoceanography. The capstone of hiscareer,thecreation of theso-calledBergenSchool of meteorology,establishedanewconceptualfoundationforthesciencewhilecreatinginnovativepredictivepracticesthatenabledgreaterintegration of weather as aresourceforagriculture,aviation,andfishery.CareerStrategies.Bjerknes sprofessionaloptionsinphysicswerelimitedbybothhisdispositionand his circumstances.Bjerkneshopedtoachieveamechanicaldepiction of theether,whichwouldvindicatehisfatherCarlAnton shydrodynamicanalogiestoelectromagnetismandserve as anillustration of thecontiguous-actionphysicsproposedbyHeinrichHertz. He thoughtthisresearchwouldremaincentraltoEuropeanphysicsandcouldbehisvehicletoprestigeandauthority,buthe was mistaken.Hefelthelplessintheearly1900swhen, to hismind,German-speakingtheoreticalphysicistsinastate of masspsychosisabandonedthemechanicalworldviewforelectromagneticalternatives.InSwedenthere was littlesympathyfortheoreticalphysics,and as aNorwegianworkinginStockholmatatime of tensionsbetweenthetwonations,hefoundhisoptionslimited. NEW DICTIONARY OF SCIENTIFIC BIOGRAPHY
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