Vladimir Arnold

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Vladimir Arnold
Vladimir Arnold-1.jpg
Vladimir Arnold in 2008
Born(1937-06-12)12 June 1937
Odessa, Ukrainian SSR, Soviet Union
Died3 June 2010(2010-06-03) (aged 72)
Paris, France
NationalitySoviet Union, 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 (alternative spelling Arnol'd, Russian: Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010)[3][4][1] was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19, he co-founded two new branches of mathematics—KAM theory, and topological Galois theory (this, with his student Askold Khovanskii).

Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as the famous Mathematical Methods of Classical Mechanics, and Lectures on Partial Differential Equations) 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 anti-Bourbaki.

Biography[edit]

Vladimir Igorevich Arnold was born on 12 June 1937 in Odessa, Soviet Union, his father was Igor Vladimirovich Arnold (1900–1948), a mathematician. His mother was Nina Alexandrovna Arnold (1909–1986, née Isakovich), an art historian;[4] when Arnold was thirteen, an uncle who was an engineer told him about calculus and how it could be used to understand some physical phenomena, this contributed to spark his interest for mathematics, and he started to study by himself the mathematical books his father had left to him, which included some works of Leonhard Euler and Charles Hermite.[7]

While a student of Andrey Kolmogorov at Moscow State University and 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 solving Hilbert's thirteenth problem;[8] this is the Kolmogorov–Arnold representation theorem.

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

He became an academician of the Academy of Sciences of the Soviet Union (Russian Academy of Science since 1991) in 1990.[9] Arnold can be said to have initiated the theory of symplectic topology as a distinct discipline; the Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms and Lagrangian intersections were also a major motivation in the development of Floer homology.

In 1999 he suffered a serious bike accident in Paris, resulting in traumatic brain injury, and though he regained consciousness after a few weeks, he had amnesia and for some time could not even recognize his own wife at the hospital,[10] but he went on to make a good recovery.[11]

Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death; as of 2006 he was reported to have the highest citation index among Russian scientists,[12] and h-index of 40.

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.[13]

Death[edit]

Arnold died of acute pancreatitis[14] on 3 June 2010 in Paris, nine days before his 73rd birthday,[15] his students include Alexander Givental, Victor Goryunov, Sabir Gusein-Zade, Emil Horozov, Boris Khesin, Askold Khovanskii, Nikolay Nekhoroshev, Boris Shapiro, Alexander Varchenko, Victor Vassiliev and Vladimir Zakalyukin.[2]

He was buried on June 15 in Moscow, at the Novodevichy Monastery.[16]

In a telegram to Arnold's family, Russian President Dmitry Medvedev stated:

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.[17]

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, often geometric approach to traditional mathematical topics like ordinary 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 is that his books are meant to teach the subject to "those who truly wish to understand it" (Chicone, 2007).[18]

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 the Bourbaki school in France—initially had a negative impact on French mathematical education, and then later on that of other countries as well.[19][20] Arnold was very interested in the history of mathematics.[21] In an interview,[20] he said he had learned much of what he knew about mathematics through the study of Felix Klein's book Development of Mathematics in the 19th Century —a book he often recommended to his students,[22] he liked to study the classics, most notably the works of Huygens, Newton and Poincaré,[23] and many times he reported to have found in their works ideas that had not been explored yet.[24]

Work[edit]

See also: Arnold conjecture and Stability of the Solar System

Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory.[5]

Hilbert's thirteenth problem[edit]

The problem is the following question: can every continuous function of three variables be expressed as a composition of 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 of Andrey 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.

Dynamical systems[edit]

See also: Kolmogorov–Arnold–Moser theorem and Arnold diffusion

Moser and Arnold expanded the ideas of Kolmogorov (who was inspired by questions of Poincaré) and gave rise to what is now known as Kolmogorov–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.[25]

Singularity theory[edit]

In 1965, Arnold attended René Thom's seminar on catastrophe theory, he later said of it: "I am deeply indebted to Thom, whose singularity seminar at the Institut des Hautes Etudes Scientifiques, which I frequented throughout the year 1965, profoundly changed my mathematical universe."[26] After this event, singularity theory became one of the major interests of Arnold and his students.[27] 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,Ek and Lagrangian singularities".[28][29][30]

Fluid dynamics[edit]

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 the Euler's equations for rotating rigid bodies and the Euler'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.[31][32][33]

Real algebraic geometry[edit]

In the year 1971, Arnold published "On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms",[34] which gave new life to real algebraic geometry. In it, he made major advances in the direction of a solution to Gudkov's conjecture, by finding a connection between it and four-dimensional topology;[35] the conjecture was to be later fully solved by V. A. Rokhlin building on Arnold's work.[36][37]

Symplectic geometry[edit]

The Arnold 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.[38][39]

Topology[edit]

According to Victor 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 the Abel–Ruffini theorem and the initial development of some of the consequent ideas, a work which resulted in the creation of the field of topological Galois theory in the 1960s.[40][41]

Theory of plane curves[edit]

Arnold revolutionized plane curves theory.[42]

Other[edit]

Arnold conjectured the existence of the gömböc.[43]

Honours and awards[edit]

Arnold and Russia's president

The minor planet 10031 Vladarnolda was named after him in 1981 by Lyudmila Georgievna Karachkina.[51]

The Arnold Mathematical Journal, published for the first time in 2015, is named after him.[52]

He was a plenary speaker at both the 1974 and 1983 International Congress of Mathematicians in Vancouver and Warsaw, respectively.[53]

Fields Medal omission[edit]

Even though Arnold was nominated for the 1974 Fields Medal, which was then viewed as the highest honour a mathematician could receive, interference from the Soviet government led to it being withdrawn. Arnold's public opposition to the persecution of dissidents had 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.[54][55]

Selected bibliography[edit]

  • 1966: "Sur la géométrie différentielle des groupes de Lie de dimension infine et ses applications a l'hydrodynamique des fluides parfaits" Annales de l'Institut Fourier 16: 319–361 doi:10.5802/aif.233
  • 1980: Mathematical Methods of Classical Mechanics, Springer-Verlag, ISBN 0-387-96890-3.[56][57]
  • 1985: (with S. M. Gusein-Zade & A. N. Varchenko) Singularities of Differentiable Maps, Volume I: The Classification of Critical Points, Caustics and Wave Fronts. Birkhäuser.
  • 1988: (with S. M. Gusein-Zade & A. N. Varchenko) Singularities of Differentiable Maps, Volume II: Monodromy and Asymptotics of Integrals. Monographs in Mathematics. Birkhäuser.
  • 1988: Geometrical Methods In The Theory Of Ordinary Differential Equations, Springer-Verlag ISBN 0-387-96649-8.
  • 1978; Ordinary Differential Equations, The MIT Press ISBN 0-262-51018-9.
  • 1989: (with A. Avez) Ergodic Problems of Classical Mechanics, Addison-Wesley ISBN 0-201-09406-1.
  • 1990: Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Eric J.F. Primrose translator, Birkhäuser Verlag (1990) ISBN 3-7643-2383-3.[58][59][60]
  • 1995:Topological Invariants of Plane Curves and Caustics,[61] American Mathematical Society (1994) ISBN 978-0-8218-0308-0
  • 1999: (with Valentin Afraimovich) Bifurcation Theory And Catastrophe Theory Springer ISBN 3-540-65379-1
  • 1998: "On the teaching of mathematics" (Russian) Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234; translation in Russian Math. Surveys 53(1): 229–236.
  • 2004: Teoriya Katastrof (Catastrophe Theory,[62] in Russian), 4th ed. Moscow, Editorial-URSS (2004), ISBN 5-354-00674-0.
  • 2001: "Tsepniye Drobi" (Continued Fractions, in Russian), Moscow (2001).
  • 2007; Yesterday and Long Ago, Springer (2007), ISBN 978-3-540-28734-6.
  • 2004: Vladimir I. Arnold, ed. (2004-11-15). Arnold's Problems (2nd ed.). Springer-Verlag. ISBN 978-3-540-20748-1.
  • 2014: V. I. Arnold (2014). Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians. American Mathematical Society. ISBN 978-1-4704-1701-7.
  • Real Algebraic Geometry.
  • Lectures on Partial Differential Equations.
  • 2015: Experimental Mathematics. American Mathematical Society (translated from Russian, 2015).
  • 2015: Lectures and Problems: A Gift to Young Mathematicians, American Math Society, (translated from Russian, 2015)
  • Arnolʹd, Vladimir Igorevich (1991). The Theory of Singularities and Its Applications. Cambridge University Press. ISBN 9780521422802.

Collected works[edit]

  • 2009: 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.

See also[edit]

References[edit]

  1. ^ a b c Khesin, Boris; Tabachnikov, Sergei (2017). "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. ISSN 0080-4606.
  2. ^ a b Vladimir Arnold at the Mathematics Genealogy Project
  3. ^ Mort d'un grand mathématicien russe, AFP (Le Figaro)
  4. ^ a b Gusein-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. ^ a b O'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. ISBN 9783642136061.
  7. ^ Табачников, С. Л. . "Интервью с В.И.Арнольдом", Квант, 1990, Nº 7, pp. 2–7. (in Russian)
  8. ^ Daniel Robertz (13 October 2014). Formal Algorithmic Elimination for PDEs. Springer. p. 192. ISBN 978-3-319-11445-3.
  9. ^ Great Russian Encyclopedia (2005), Moscow: Bol'shaya Rossiyskaya Enciklopediya Publisher, vol. 2.
  10. ^ Arnold: Yesterday and Long Ago (2010)
  11. ^ Polterovich and Scherbak (2011)
  12. ^ List of Russian Scientists with High Citation Index
  13. ^ "Vladimir Arnold". The Daily Telegraph. London. 12 July 2010.
  14. ^ Kenneth Chang (June 11, 2010). "Vladimir Arnold Dies at 72; Pioneering Mathematician". The New York Times. Retrieved 12 June 2013.
  15. ^ "Number's up as top mathematician Vladimir Arnold dies". Herald Sun. 4 June 2010. Retrieved 2010-06-06.
  16. ^ "From V. I. Arnold's web page". Retrieved 12 June 2013.
  17. ^ "Condolences to the family of Vladimir Arnold". Presidential Press and Information Office. 15 June 2010. Retrieved 1 September 2011.
  18. ^ Carmen Chicone (2007), Book review of "Ordinary Differential Equations", by Vladimir I. Arnold. Springer-Verlag, Berlin, 2006. SIAM Review 49(2):335–336. (Chicone mentions the criticism but does not agree with it.)
  19. ^ See [1] and other essays in [2].
  20. ^ a b An Interview with Vladimir Arnol'd, by S. H. Lui, AMS Notices, 1991.
  21. ^ Oleg Karpenkov. "Vladimir Igorevich Arnold"
  22. ^ B. Khesin and S. Tabachnikov, Tribute to Vladimir Arnold, Notices of the AMS, 59:3 (2012) 378–399.
  23. ^ Goryunov, V.; Zakalyukin, V. (2011), "Vladimir I. Arnold", Moscow Mathematical Journal, 11 (3).
  24. ^ 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".
  25. ^ Szpiro, George G. (2008-07-29). Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles. Penguin. ISBN 9781440634284.
  26. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-22.CS1 maint: Archived copy as title (link)
  27. ^ http://www.ias.ac.in/resonance/Volumes/19/09/0787-0796.pdf
  28. ^ Note: It also appears in another article by him, but in English: Local Normal Forms of Functions, http://www.maths.ed.ac.uk/~aar/papers/arnold15.pdf
  29. ^ Dirk Siersma; Charles Wall; V. Zakalyukin (30 June 2001). New Developments in Singularity Theory. Springer Science & Business Media. p. 29. ISBN 978-0-7923-6996-7.
  30. ^ Landsberg, J. M.; Manivel, L. (2002). "Representation theory and projective geometry". arXiv:math/0203260.
  31. ^ Terence Tao (22 March 2013). Compactness and Contradiction. American Mathematical Soc. pp. 205–206. ISBN 978-0-8218-9492-7.
  32. ^ https://www.theguardian.com/science/2010/aug/19/v-i-arnold-obituary
  33. ^ IAMP News Bulletin, July 2010, pp. 25–26
  34. ^ Note: The paper also appears with other names, as in http://perso.univ-rennes1.fr/marie-francoise.roy/cirm07/arnold.pdf
  35. ^ 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. ISBN 978-0-8218-0807-8.
  36. ^ Khesin, Boris A.; Tabachnikov, Serge L. (2014-09-10). Arnold: Swimming Against the Tide. p. 159. ISBN 9781470416997.
  37. ^ 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. doi:10.1070/RM2000v055n04ABEH000315.
  38. ^ "Arnold and Symplectic Geometry", by Helmut Hofer
  39. ^ "Vladimir Igorevich Arnold and the invention of symplectic topology", by Michèle Audin
  40. ^ "Topology in Arnold's work", by Victor Vassiliev
  41. ^ http://www.ams.org/journals/bull/2008-45-02/S0273-0979-07-01165-2/S0273-0979-07-01165-2.pdf Bulletin (New Series) of The American Mathematical Society Volume 45, Number 2, April 2008, pp. 329–334
  42. ^ A Panoramic View of Riemannian Geometry, by Marcel Berger
  43. ^ Mackenzie, Dana (2010-12-29). What's Happening in the Mathematical Sciences. American Mathematical Soc. p. 104. ISBN 9780821849996.
  44. ^ O. Karpenkov, "Vladimir Igorevich Arnold", Internat. Math. Nachrichten, no. 214, pp. 49–57, 2010. (link to arXiv preprint)
  45. ^ Harold M. Schmeck Jr. (June 27, 1982). "American and Russian Share Prize in Mathematics". New York Times.
  46. ^ "Book of Members, 1780-2010: Chapter A" (PDF). American Academy of Arts and Sciences. Retrieved 25 April 2011.
  47. ^ 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)
  48. ^ American Physical Society – 2001 Dannie Heineman Prize for Mathematical Physics Recipient
  49. ^ The Wolf Foundation – Vladimir I. Arnold Winner of Wolf Prize in Mathematics
  50. ^ Названы лауреаты Государственной премии РФ Kommersant 20 May 2008.
  51. ^ Lutz D. Schmadel (2012-06-10). Dictionary of Minor Planet Names. Springer Science & Business Media. p. 717. ISBN 978-3-642-29718-2.
  52. ^ Editorial (2015), "Journal Description Arnold Mathematical Journal", Arnold Mathematical Journal, 1 (1): 1–3, doi:10.1007/s40598-015-0006-6.
  53. ^ http://www.mathunion.org/db/ICM/Speakers/SortedByLastname.php
  54. ^ Martin L. White (2015). "Vladimir Igorevich Arnold". Encyclopædia Britannica.
  55. ^ Thomas H. Maugh II (June 23, 2010). "Vladimir Arnold, noted Russian mathematician, dies at 72". The Washington Post. Retrieved March 18, 2015.
  56. ^ Review by Ian N. Sneddon (Bulletin of the American Mathematical Society, Vol. 2): http://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14755-2/S0273-0979-1980-14755-2.pdf
  57. ^ Review by R. Broucke (Celestial Mechanics, Vol. 28): Bibcode: 1982CeMec..28..345A.
  58. ^ Kazarinoff, N. (1991-09-01). "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. ISSN 0036-1445.
  59. ^ Thiele, R. (1993-01-01). "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. ISSN 1521-4001.
  60. ^ Heggie, Douglas C. (1991-06-01). "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. ISSN 1464-3839.
  61. ^ Goryunov, V. V. (1996-10-01). "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. ISSN 1464-3839.
  62. ^ Bernfeld, Stephen R. (1985-01-01). "Review of Catastrophe Theory". SIAM Review. 27 (1): 90–91. doi:10.1137/1027019. JSTOR 2031497.

Further reading[edit]

External links[edit]

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