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Vladimir Arnold

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In this name that followsEastern Slavic naming customs,thepatronymicisIgorevichand thefamily nameisArnold.
Vladimir Arnold
Владимир Арнольд
Arnold in 2008
Born(1937-06-12)12 June 1937
Odesa,Ukrainian SSR,Soviet Union
Died3 June 2010(2010-06-03)(aged 72)
Paris, France
Nationality
  • Soviet
  • Russian
Alma materMoscow State University
Known forADE classification
Arnold's cat map
Arnold conjecture
Arnold diffusion
Arnold's rouble problem
Arnold's spectral sequence
Arnold tongue
ABC flow
Arnold–Givental conjecture
Gömböc
Gudkov's conjecture
Hilbert's thirteenth problem
KAM theorem
Kolmogorov–Arnold theorem
Liouville–Arnold theorem
Topological Galois theory
Mathematical Methods of Classical Mechanics
AwardsShaw Prize(2008)
State Prize of the Russian Federation(2007)
Wolf Prize(2001)
Dannie Heineman Prize for Mathematical Physics(2001)
Harvey Prize(1994)
RAS Lobachevsky Prize(1992)
Crafoord Prize(1982)
Lenin Prize(1965)
Scientific career
FieldsMathematics
InstitutionsParis Dauphine University
Steklov Institute of Mathematics
Independent University of Moscow
Moscow State University
Doctoral advisorAndrey Kolmogorov
Doctoral students

Vladimir Igorevich Arnold(orArnol'd;Russian:Влади́мир И́горевич Арно́льд,IPA:[vlɐˈdʲimʲɪrˈiɡərʲɪvʲɪtɕɐrˈnolʲt];12 June 1937 – 3 June 2010)[1][3][4]was a Soviet and Russian mathematician. He is best known for theKolmogorov–Arnold–Moser theoremregarding thestabilityofintegrable systems,and contributed to several areas, including geometrical theory ofdynamical systems theory,algebra,catastrophe theory,topology,real algebraic geometry,symplectic geometry,symplectic topology,differential equations,classical mechanics,differential geometric approach tohydrodynamics,geometric analysisandsingularity theory,including posing theADE classificationproblem.

His first main result was the solution ofHilbert's thirteenth problemin 1957 at the age of 19. He co-founded three newbranches of mathematics:topological Galois theory(with his studentAskold Khovanskii),symplectic topologyandKAM theory.

Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such asMathematical Methods of Classical Mechanics) and popular mathematics books, he influenced many mathematicians and physicists.[5][6]Many of his books were translated into English. His views on education were particularly opposed to those ofBourbaki.

Biography[edit]

Vladimir Igorevich Arnold was born on 12 June 1937 inOdesa,Soviet Union(nowOdesa,Ukraine). His father was Igor Vladimirovich Arnold (1900–1948), a mathematician. His mother was Nina Alexandrovna Arnold (1909–1986,néeIsakovich), a Jewish art historian.[4]While a school student, Arnold once asked his father on the reason why the multiplication of two negative numbers yielded a positive number, and his father provided an answer involving thefield propertiesof real numbers and the preservation of thedistributive property.Arnold was deeply disappointed with this answer, and developed an aversion to theaxiomatic methodthat lasted through his life.[7]When Arnold was thirteen, his uncle Nikolai B. Zhitkov,[8]who was an engineer, told him aboutcalculusand how it could be used to understand some physical phenomena. This contributed to sparking his interest for mathematics, and he started to study by himself the mathematical books his father had left to him, which included some works ofLeonhard EulerandCharles Hermite.[9]

While a student ofAndrey KolmogorovatMoscow State Universityand still a teenager, Arnold showed in 1957 that any continuous function of several variables can be constructed with a finite number of two-variable functions, thereby solvingHilbert's thirteenth problem.[10]This is theKolmogorov–Arnold representation theorem.

After graduating from Moscow State University in 1959, he worked there until 1986 (a professor since 1965), and then atSteklov Mathematical Institute.

He became an academician of theAcademy of Sciences of the Soviet Union(Russian Academy of Sciencesince 1991) in 1990.[11]Arnold can be said to have initiated the theory ofsymplectic topologyas a distinct discipline. TheArnold conjectureon the number of fixed points ofHamiltonian symplectomorphismsandLagrangian intersectionswas also a motivation in the development ofFloer homology.

In 1999 he suffered a serious bicycle accident in Paris, resulting intraumatic brain injury.He regained consciousness after a few weeks but had amnesia and for some time could not even recognize his own wife at the hospital.[12]He went on to make a good recovery.[13]

Arnold worked at theSteklov Mathematical Institutein Moscow and atParis Dauphine Universityup until his death. His PhD students includeAlexander Givental,Victor Goryunov,Sabir Gusein-Zade,Emil Horozov,Yulij Ilyashenko,Boris Khesin,Askold Khovanskii,Nikolay Nekhoroshev,Boris Shapiro,Alexander Varchenko,Victor VassilievandVladimir Zakalyukin.[2]

To his students and colleagues Arnold was known also for his sense of humour. For example, once at his seminar in Moscow, at the beginning of the school year, when he usually was formulating new problems, he said:

There is a general principle that a stupid man can ask such questions to which one hundred wise men would not be able to answer. In accordance with this principle I shall formulate some problems.[14]

Death[edit]

Arnold died ofacute pancreatitis[15]on 3 June 2010 in Paris, nine days before his 73rd birthday.[16]He was buried on 15 June in Moscow, at theNovodevichy Monastery.[17]

In a telegram to Arnold's family,Russian PresidentDmitry Medvedevstated:

The death of Vladimir Arnold, one of the greatest mathematicians of our time, is an irretrievable loss for world science. It is difficult to overestimate the contribution made by academician Arnold to modern mathematics and the prestige of Russian science.

Teaching had a special place in Vladimir Arnold's life and he had great influence as an enlightened mentor who taught several generations of talented scientists.

The memory of Vladimir Arnold will forever remain in the hearts of his colleagues, friends and students, as well as everyone who knew and admired this brilliant man.[18]

Popular mathematical writings[edit]

Arnold is well known for his lucid writing style, combining mathematical rigour with physical intuition, and an easy conversational style of teaching and education. His writings present a fresh, oftengeometricapproach to traditional mathematical topics likeordinary differential equations,and his many textbooks have proved influential in the development of new areas of mathematics. The standard criticism about Arnold's pedagogy is that his books "are beautiful treatments of their subjects that are appreciated by experts, but too many details are omitted for students to learn the mathematics required to prove the statements that he so effortlessly justifies." His defense was that his books are meant to teach the subject to "those who truly wish to understand it" (Chicone, 2007).[19]

Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. He had very strong opinions on how this approach—which was most popularly implemented by theBourbakischool in France—initially had a negative impact on Frenchmathematical education,and then later on that of other countries as well.[20][21]Arnold was very interested in thehistory of mathematics.[22]In an interview,[21]he said he had learned much of what he knew about mathematics through the study ofFelix Klein's bookDevelopment of Mathematics in the 19th Century—a book he often recommended to his students.[23]He studied the classics, most notably the works ofHuygens,NewtonandPoincaré,[24]and many times he reported to have found in their works ideas that had not been explored yet.[25]

Mathematical work[edit]

See also:Stability of the Solar System

Arnold worked ondynamical systems theory,catastrophe theory,topology,algebraic geometry,symplectic geometry,differential equations,classical mechanics,hydrodynamicsandsingularity theory.[5]Michèle Audindescribed him as "a geometer in the widest possible sense of the word" and said that "he was very fast to make connections between different fields".[26]

Hilbert's thirteenth problem[edit]

See also:Kolmogorov–Arnold representation theorem

The problem is the following question: can every continuous function of three variables be expressed as acompositionof finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student ofAndrey Kolmogorov.Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering the Hilbert's question when posed for the class of continuous functions.[27]

Dynamical systems[edit]

See also:Arnold diffusion

Moserand Arnold expanded the ideas ofKolmogorov(who was inspired by questions ofPoincaré) and gave rise to what is now known asKolmogorov–Arnold–Moser theorem(or "KAM theory" ), which concerns the persistence of some quasi-periodic motions (nearly integrable Hamiltonian systems) when they are perturbed. KAM theory shows that, despite the perturbations, such systems can be stable over an infinite period of time, and specifies what the conditions for this are.[28]

In 1964, Arnold introduced theArnold web,the first example of a stochastic web.[29][30]

Singularity theory[edit]

In 1965, Arnold attendedRené Thom's seminar oncatastrophe theory.He later said of it: "I am deeply indebted to Thom, whose singularity seminar at theInstitut des Hautes Etudes Scientifiques,which I frequented throughout the year 1965, profoundly changed my mathematical universe. "[31]After this event,singularity theorybecame one of the major interests of Arnold and his students.[32]Among his most famous results in this area is his classification of simple singularities, contained in his paper "Normal forms of functions near degenerate critical points, the Weyl groups of Ak,Dk,Ekand Lagrangian singularities ".[33][34][35]

Fluid dynamics[edit]

See also:Arnold–Beltrami–Childress flowandBeltrami vector field § Beltrami fields and complexity in fluid mechanics

In 1966, Arnold published "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits",in which he presented a common geometric interpretation for both theEuler's equations for rotating rigid bodiesand theEuler's equations of fluid dynamics,this effectively linked topics previously thought to be unrelated, and enabled mathematical solutions to many questions related to fluid flows and their turbulence.[36][37][38]

Real algebraic geometry[edit]

In the year 1971, Arnold published "On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smoothmanifolds,and the arithmetic of integral quadratic forms ",[39]which gave new life toreal algebraic geometry.In it, he made major advances in the direction of a solution toGudkov's conjecture,by finding a connection between it andfour-dimensional topology.[40]The conjecture was to be later fully solved byV. A. Rokhlinbuilding on Arnold's work.[41][42]

Symplectic geometry[edit]

TheArnold conjecture,linking the number of fixed points of Hamiltonian symplectomorphisms and the topology of the subjacent manifolds, was the motivating source of many of the pioneer studies in symplectic topology.[43][44]

Topology[edit]

According toVictor Vassiliev,Arnold "worked comparatively little on topology for topology's sake." And he was rather motivated by problems on other areas of mathematics where topology could be of use. His contributions include the invention of a topological form of theAbel–Ruffini theoremand the initial development of some of the consequent ideas, a work which resulted in the creation of the field oftopological Galois theoryin the 1960s.[45][46]

Theory of plane curves[edit]

According toMarcel Berger,Arnold revolutionized plane curves theory.[47]Among his contributions are theArnold invariantsofplane curves.[48]

Other[edit]

Arnold conjectured the existence of thegömböc.[49]

Arnold generalized the results ofIsaac Newton,Pierre-Simon Laplace,andJames Ivoryon theshell theorem,showing it to be applicable to algebraic hypersurfaces.[50]

Honours and awards[edit]

Arnold (left) and Russia's presidentDmitry Medvedev

Theminor planet10031 Vladarnoldawas named after him in 1981 byLyudmila Georgievna Karachkina.[64]

TheArnold Mathematical Journal,published for the first time in 2015, is named after him.[65]

The Arnold Fellowships, of theLondon Instituteare named after him.[66][67]

He was a plenary speaker at both the 1974 and 1983International Congress of Mathematiciansin Vancouver andWarsaw,respectively.[68]

Fields Medal omission[edit]

Even though Arnold was nominated for the 1974Fields Medal,one of the highest honours a mathematician could receive, interference from the Soviet government led to it being withdrawn. Arnold's public opposition to the persecution ofdissidentshad led him into direct conflict with influential Soviet officials, and he suffered persecution himself, including not being allowed to leave the Soviet Union during most of the 1970s and 1980s.[69][70]

Selected bibliography[edit]

Collected works[edit]

  • 2010: A. B. Givental; B. A. Khesin; J. E. Marsden; A. N. Varchenko; V. A. Vassilev; O. Ya. Viro; V. M. Zakalyukin (editors).Collected Works, Volume I: Representations of Functions, Celestial Mechanics, and KAM Theory (1957–1965).Springer
  • 2013: A. B. Givental; B. A. Khesin; A. N. Varchenko; V. A. Vassilev; O. Ya. Viro; (editors).Collected Works, Volume II: Hydrodynamics, Bifurcation Theory, and Algebraic Geometry (1965–1972).Springer.
  • 2016: Givental, A.B., Khesin, B., Sevryuk, M.B., Vassiliev, V.A., Viro, O.Y. (Eds.).Collected Works, Volume III: Singularity Theory 1972–1979. Springer.
  • 2018: Givental, A.B., Khesin, B., Sevryuk, M.B., Vassiliev, V.A., Viro, O.Y. (Eds.).Collected Works, Volume IV: Singularities in Symplectic and Contact Geometry 1980–1985.Springer.
  • 2023: Alexander B. Givental, Boris A. Khesin, Mikhail B. Sevryuk, Victor A. Vassiliev, Oleg Ya. Viro (Eds.).Collected Works, Volume VI: Dynamics, Combinatorics, and Invariants of Knots, Curves, and Wave Fronts 1992–1995.Springer.

See also[edit]

References[edit]

  1. ^abcKhesin, Boris;Tabachnikov, Sergei(2018)."Vladimir Igorevich Arnold. 12 June 1937 – 3 June 2010".Biographical Memoirs of Fellows of the Royal Society.64:7–26.doi:10.1098/rsbm.2017.0016.ISSN0080-4606.
  2. ^abVladimir Arnoldat theMathematics Genealogy Project
  3. ^Mort d'un grand mathématicien russe,AFP (Le Figaro)
  4. ^abGusein-Zade, Sabir M.;Varchenko, Alexander N(December 2010),"Obituary: Vladimir Arnold (12 June 1937 – 3 June 2010)"(PDF),Newsletter of the European Mathematical Society,78:28–29
  5. ^abO'Connor, John J.;Robertson, Edmund F.,"Vladimir Arnold",MacTutor History of Mathematics Archive,University of St Andrews
  6. ^Bartocci, Claudio; Betti, Renato; Guerraggio, Angelo; Lucchetti, Roberto; Williams, Kim (2010).Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles.Springer. p. 211.ISBN9783642136061.
  7. ^Vladimir I. Arnold (2007).Yesterday and Long Ago.Springer. pp. 19–26.ISBN978-3-540-28734-6.
  8. ^Swimming Against the Tide,p. 3
  9. ^Табачников, С. Л.. "Интервью с В.И.Арнольдом",Квант,1990, Nº 7, pp. 2–7. (in Russian)
  10. ^Daniel Robertz (13 October 2014).Formal Algorithmic Elimination for PDEs.Springer. p. 192.ISBN978-3-319-11445-3.
  11. ^Great Russian Encyclopedia(2005), Moscow: Bol'shaya Rossiyskaya Enciklopediya Publisher, vol. 2.
  12. ^Arnold: Yesterday and Long Ago (2010)
  13. ^Polterovich and Scherbak (2011)
  14. ^"Vladimir Arnold".The Daily Telegraph.London. 12 July 2010.
  15. ^Kenneth Chang (11 June 2010)."Vladimir Arnold Dies at 72; Pioneering Mathematician".The New York Times.Retrieved12 June2013.
  16. ^"Number's up as top mathematician Vladimir Arnold dies".Herald Sun.4 June 2010.Retrieved6 June2010.
  17. ^ "From V. I. Arnold's web page".Retrieved12 June2013.
  18. ^"Condolences to the family of Vladimir Arnold".Presidential Press and Information Office.15 June 2010.Retrieved1 September2011.
  19. ^Carmen Chicone (2007), Book review of "Ordinary Differential Equations", by Vladimir I. Arnold. Springer-Verlag, Berlin, 2006.SIAM Review49(2):335–336.(Chicone mentions the criticism but does not agree with it.)
  20. ^See[1](archived from[2]Archived28 April 2017 at theWayback Machine) and other essays in[3].
  21. ^abAn Interview with Vladimir Arnol'd,by S. H. Lui,AMS Notices,1991.
  22. ^Oleg Karpenkov. "Vladimir Igorevich Arnold"
  23. ^B. KhesinandS. Tabachnikov,Tribute to Vladimir Arnold,Notices of the AMS,59:3 (2012) 378–399.
  24. ^Goryunov, V.; Zakalyukin, V. (2011),"Vladimir I. Arnold",Moscow Mathematical Journal,11(3).
  25. ^See for example: Arnold, V. I.; Vasilev, V. A. (1989), "Newton's Principia read 300 years later" and Arnold, V. I. (2006); "Forgotten and neglected theories of Poincaré".
  26. ^"Vladimir Igorevich Arnold and the Invention of Symplectic Topology", chapter I in the bookContact and Symplectic Topology(editors: Frédéric Bourgeois, Vincent Colin, András Stipsicz)
  27. ^Ornes, Stephen (14 January 2021)."Mathematicians Resurrect Hilbert's 13th Problem".Quanta Magazine.
  28. ^Szpiro, George G.(29 July 2008).Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles.Penguin.ISBN9781440634284.
  29. ^Phase Space Crystals, by Lingzhen Guohttps://iopscience.iop.org/book/978-0-7503-3563-8.pdf
  30. ^Zaslavsky web map, by George Zaslavskyhttp:// scholarpedia.org/article/Zaslavsky_web_map
  31. ^"Archived copy"(PDF).Archived fromthe original(PDF)on 14 July 2015.Retrieved22 February2015.{{cite web}}:CS1 maint: archived copy as title (link)
  32. ^"Resonance – Journal of Science Education | Indian Academy of Sciences"(PDF).
  33. ^Note: It also appears in another article by him, but in English:Local Normal Forms of Functions,http:// maths.ed.ac.uk/~aar/papers/arnold15.pdf
  34. ^Dirk Siersma; Charles Wall; V. Zakalyukin (30 June 2001).New Developments in Singularity Theory.Springer Science & Business Media. p. 29.ISBN978-0-7923-6996-7.
  35. ^Landsberg, J. M.; Manivel, L. (2002). "Representation theory and projective geometry".arXiv:math/0203260.
  36. ^Terence Tao(22 March 2013).Compactness and Contradiction.American Mathematical Soc. pp. 205–206.ISBN978-0-8218-9492-7.
  37. ^MacKay, Robert Sinclair; Stewart, Ian (19 August 2010)."VI Arnold obituary".The Guardian.
  38. ^IAMP News Bulletin, July 2010, pp. 25–26
  39. ^Note: The paper also appears with other names, as inhttp://perso.univ-rennes1.fr/marie-francoise.roy/cirm07/arnold.pdf
  40. ^A. G. Khovanskii; Aleksandr Nikolaevich Varchenko; V. A. Vasiliev (1997).Topics in Singularity Theory: V. I. Arnold's 60th Anniversary Collection (preface).American Mathematical Soc. p. 10.ISBN978-0-8218-0807-8.
  41. ^Khesin, Boris A.; Tabachnikov, Serge L. (10 September 2014).Arnold: Swimming Against the Tide.American Mathematical Society. p. 159.ISBN9781470416997.
  42. ^Degtyarev, A. I.; Kharlamov, V. M. (2000). "Topological properties of real algebraic varieties: Du coté de chez Rokhlin".Russian Mathematical Surveys.55(4): 735–814.arXiv:math/0004134.Bibcode:2000RuMaS..55..735D.doi:10.1070/RM2000v055n04ABEH000315.S2CID250775854.
  43. ^"Arnold and Symplectic Geometry", byHelmut Hofer
  44. ^"Vladimir Igorevich Arnold and the invention of symplectic topology",byMichèle Audinhttps://web.archive.org/web/20160303175152/http://www-irma.u-strasbg.fr/~maudin/Arnold.pdf
  45. ^"Topology in Arnold's work", byVictor Vassiliev
  46. ^http:// ams.org/journals/bull/2008-45-02/S0273-0979-07-01165-2/S0273-0979-07-01165-2.pdfBulletin (New Series) of The American Mathematical Society Volume 45, Number 2, April 2008, pp. 329–334
  47. ^A Panoramic View of Riemannian Geometry,byMarcel Berger,pp. 24–25
  48. ^Extrema of Arnold's invariants of curves on surfaces, by Vladimir Chernovhttps://math.dartmouth.edu/~chernov-china/
  49. ^Mackenzie, Dana (29 December 2010).What's Happening in the Mathematical Sciences.American Mathematical Soc. p. 104.ISBN9780821849996.
  50. ^Ivan Izmestiev,Serge Tabachnikov."Ivory’s theorem revisited",Journal of Integrable Systems,Volume 2, Issue 1, (2017)https://doi.org/10.1093/integr/xyx006
  51. ^O. Karpenkov, "Vladimir Igorevich Arnold",Internat. Math. Nachrichten,no. 214, pp. 49–57, 2010. (link to arXiv preprint)
  52. ^Harold M. Schmeck Jr. (27 June 1982)."American and Russian Share Prize in Mathematics".The New York Times.
  53. ^https://web.archive.org/web/20160126153013/http:// kva.se/globalassets/priser/crafoord/2014/rattigheter/crafoordprize1982_2014.pdf
  54. ^"Vladimir I. Arnold".nasonline.org.Retrieved14 April2022.
  55. ^"Book of Members, 1780–2010: Chapter A"(PDF).American Academy of Arts and Sciences.Retrieved25 April2011.
  56. ^"APS Member History".search.amphilsoc.org.Retrieved14 April2022.
  57. ^D. B. Anosov, A. A. Bolibrukh,Lyudvig D. Faddeev,A. A. Gonchar,M. L. Gromov,S. M. Gusein-Zade,Yu. S. Il'yashenko,B. A. Khesin,A. G. Khovanskii,M. L. Kontsevich,V. V. Kozlov,Yu. I. Manin,A. I. Neishtadt,S. P. Novikov,Yu. S. Osipov, M. B. Sevryuk,Yakov G. Sinai,A. N. Tyurin, A. N. Varchenko,V. A. Vasil'ev,V. M. Vershik and V. M. Zakalyukin (1997)."Vladimir Igorevich Arnol'd (on his sixtieth birthday)".Russian Mathematical Surveys,Volume 52, Number 5. (translated from the Russian by R. F. Wheeler)
  58. ^https://harveypz.net.technion.ac.il/harvey-prize-laureates/
  59. ^American Physical Society – 2001 Dannie Heineman Prize for Mathematical Physics Recipient
  60. ^The Wolf Foundation – Vladimir I. Arnold Winner of Wolf Prize in Mathematics
  61. ^Названы лауреаты Государственной премии РФKommersant20 May 2008.
  62. ^"The 2008 Prize in Mathematical Sciences".Shaw Prize Foundation. Archived fromthe originalon 7 October 2022.Retrieved7 October2022.
  63. ^"Arnold and Faddeev Receive 2008 Shaw Prize"(PDF).Notices of the American Mathematical Society.55(8): 966. 2008. Archived fromthe original(PDF)on 7 October 2022.Retrieved8 October2022.
  64. ^Lutz D. Schmadel (10 June 2012).Dictionary of Minor Planet Names.Springer Science & Business Media. p. 717.ISBN978-3-642-29718-2.
  65. ^Editorial (2015), "Journal Description Arnold Mathematical Journal",Arnold Mathematical Journal,1(1): 1–3,doi:10.1007/s40598-015-0006-6.
  66. ^"Arnold Fellowships".
  67. ^Fink, Thomas (July 2022)."Britain is rescuing academics from Vladimir Putin's clutches".The Telegraph.
  68. ^"International Mathematical Union (IMU)".Archived fromthe originalon 24 November 2017.Retrieved22 May2015.
  69. ^Martin L. White (2015)."Vladimir Igorevich Arnold".Encyclopædia Britannica.
  70. ^Thomas H. Maugh II (23 June 2010)."Vladimir Arnold, noted Russian mathematician, dies at 72".The Washington Post.Retrieved18 March2015.
  71. ^Sacker, Robert J. (1 August 1975)."Ordinary Differential Equations".Technometrics.17(3): 388–389.doi:10.1080/00401706.1975.10489355.ISSN0040-1706.
  72. ^Kapadia, Devendra A. (March 1995)."Ordinary differential equations, by V. I. Arnold. Pp 334. DM 78. 1992. ISBN 3-540-54813-0 (Springer)".The Mathematical Gazette.79(484): 228–229.doi:10.2307/3620107.ISSN0025-5572.JSTOR3620107.S2CID125723419.
  73. ^Chicone, Carmen (2007)."Review of Ordinary Differential Equations".SIAM Review.49(2): 335–336.ISSN0036-1445.JSTOR20453964.
  74. ^Review by Ian N. Sneddon (Bulletin of the American Mathematical Society,Vol. 2):http:// ams.org/journals/bull/1980-02-02/S0273-0979-1980-14755-2/S0273-0979-1980-14755-2.pdf
  75. ^Review byR. Broucke(Celestial Mechanics,Vol. 28):Bibcode:1982CeMec..28..345A.
  76. ^Kazarinoff, N. (1 September 1991). "Huygens and Barrow, Newton and Hooke: Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals (V. I. Arnol'd)".SIAM Review.33(3): 493–495.doi:10.1137/1033119.ISSN0036-1445.
  77. ^Thiele, R. (1 January 1993). "Arnol'd, V. I., Huygens and Barrow, Newton and Hooke. Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals. Basel etc., Birkhäuser Verlag 1990. 118 pp., sfr 24.00. ISBN 3-7643-2383-3".Journal of Applied Mathematics and Mechanics.73(1): 34.Bibcode:1993ZaMM...73S..34T.doi:10.1002/zamm.19930730109.ISSN1521-4001.
  78. ^Heggie, Douglas C. (1 June 1991)."V. I. Arnol'd, Huygens and Barrow, Newton and Hooke, translated by E. J. F. Primrose (Birkhäuser Verlag, Basel 1990), 118 pp., 3 7643 2383 3, sFr 24".Proceedings of the Edinburgh Mathematical Society.Series 2.34(2): 335–336.doi:10.1017/S0013091500007240.ISSN1464-3839.
  79. ^Goryunov, V. V. (1 October 1996)."V. I. Arnold Topological invariants of plane curves and caustics (University Lecture Series, Vol. 5, American Mathematical Society, Providence, RI, 1995), 60pp., paperback, 0 8218 0308 5, £17.50".Proceedings of the Edinburgh Mathematical Society.Series 2.39(3): 590–591.doi:10.1017/S0013091500023348.ISSN1464-3839.
  80. ^Bernfeld, Stephen R. (1 January 1985). "Review of Catastrophe Theory".SIAM Review.27(1): 90–91.doi:10.1137/1027019.JSTOR2031497.
  81. ^Guenther, Ronald B.; Thomann, Enrique A. (2005). Renardy, Michael; Rogers, Robert C.; Arnold, Vladimir I. (eds.). "Featured Review: Two New Books on Partial Differential Equations".SIAM Review.47(1): 165–168.ISSN0036-1445.JSTOR20453608.
  82. ^Groves, M. (2005). "Book Review: Vladimir I. Arnold, Lectures on Partial Differential Equations. Universitext".Journal of Applied Mathematics and Mechanics.85(4): 304.Bibcode:2005ZaMM...85..304G.doi:10.1002/zamm.200590023.ISSN1521-4001.
  83. ^Review by Fernando Q. Gouvêa ofReal Algebraic Geometryby Arnoldhttps:// maa.org/press/maa-reviews/real-algebraic-geometry

Further reading[edit]

External links[edit]

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