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John Forbes Nash, Jr.

 
Britannica Concise Encyclopedia:

John Forbes Nash

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(born June 13, 1928, Bluefield, W.Va., U.S.) U.S. mathematician. He earned a doctorate from Princeton University at 22. He began teaching at Massachusetts Institute of Technology in 1951 but left in the late 1950s because of mental illness; thereafter he was informally associated with Princeton. Beginning in the 1950s with his influential thesis "Non-cooperative Games," Nash established the mathematical principles of game theory. His theory, known as the Nash solution or Nash equilibrium, attempted to explain the dynamics of threat and action among competitors. Despite its practical limitations, it was widely applied by business strategists. He shared the 1994 Nobel Prize in Economics with John C. Harsanyi (1920 – 2000) and Reinhard Selten (b. 1930). A film version of his life, A Beautiful Mind (2001), won an Academy Award for best picture.

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Oxford Dictionary of Scientists:

John F. Nash

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American mathematician and economist (1928–)

Nash was born at Bluefield, West Virginia. His father was an electrical engineer and his mother was a teacher. He originally studied chemical engineering at Carnegie Tech. in Pittsburg but moved courses, first to chemistry and then, encouraged by the mathematics faculty, to mathematics.

Nash then entered Princeton on a fellowship as a graduate student. At Carnegie he had taken a course on international economics and this had led to a paper on what he called ‘The Bargaining Problem’. At Princeton, he developed this further using the ideas of game theory first discussed by von Neumann and Morgenstern. The result was Nash's theory of non-cooperative games, which he wrote up for his PhD thesis. The theory, which could be applied to any finite number of players, later found applications in economics.

Nash was not certain that this work would be an acceptable topic for a thesis and, during this period, he also made certain discoveries in pure mathematics concerning manifolds. This work was published later when he was an instructor at the Massachusetts Institute of Technology, a post he took up in 1951. At MIT he also worked on problems in differential geometry, which were relevant to general relativity theory.

At this point Nash seemed set for a brilliant mathematical career but, early in 1959, he began to suffer mental problems. In his own words, it was “the time of my change from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as ‘schizophrenic’ or ‘paranoid schizophrenic’”. He resigned his academic post and spent periods in mental hospitals. After some 25 years Nash appears to have recovered and to have started serious mathematical work again. In 1997 he was awarded the Nobel Prize for economics for the work he had done as a young man many years before on noncooperative game theory.

Gale Encyclopedia of Biography:

John Forbes Nash, Jr.

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Awarded a Nobel Prize in Economics in 1994 for his pioneering work in game theory, John Nash (born 1928) distinguished himself as one of the foremost mathematical researchers and theorists of the twentieth century.

Game theory was the subject of Nash's doctoral dissertation at Princeton University. Expanding upon the initial game theory of John von Neumann and Oskar Morgenstern, published in their The Theory of Games and Economic Behavior, Nash developed what became known as "Nash's equilibrium" to explain how two or more competitors can arrive at a mutually beneficial yet non-cooperative business arrangement. In 1951, he developed the theory that manifolds - objects containing various forms and components - can be described accurately using algebraic equations. He later developed what became known as the Nash-Moser theorem, which explained how it was possible to embed a manifold in a Euclidean space by employing differential calculus instead of algebra and geometry. Nash's subsequent career was diminished by severe mental illness, which was documented in Sylvia Nasar's biography of Nash, A Beautiful Mind, and the film of the same name directed by Ron Howard.

Early Interest in Math

Nash was born on June 13, 1928, in Bluefield, West Virginia, and raised by his parents, John Nash, Sr., an electrical engineer for the Appalachian Power Company, and Margaret Nash, a teacher who retired after her marriage and placed a high value on the education of her two children. As a young man he seemed disinclined to study but displayed a passion for electronics and chemistry experiments that he conducted in his bedroom.

When he was a young teenager, Nash read a book by E. T. Bell, Men of Mathematics, to which Nash attributed his eventual passion for number theory. While attending high school and concurrent classes at Bluefield College, Nash collaborated with his father on a paper titled "Sag and Tension Calculations for Cable and Wire Spans Using Catenary Formulas," which was published in a 1945 edition of Electrical Engineering. Nash also entered the George Westinghouse competition, winning one of ten nationally awarded full scholarships, which he used to enroll in the Carnegie Institute of Technology in Pittsburgh.

Mixed Success in Education

Initially aspiring to become an engineer like his father, Nash changed his major to chemistry after performing poorly in mechanical drawing. After he also had trouble with a physical chemistry class, he was convinced by his calculus instructor John Synge to major in mathematics. In 1948, Nash was awarded the John S. Kennedy Fellowship at Princeton University.

At Princeton, Nash was in close proximity to the Institute of Advanced Study, which attracted such notable mathematicians as Albert Einstein, Kurt Godel, Karl Oppenheimer, Hermann Weyl, and John von Neumann. According to Sylvia Nasar, "Princeton in 1948 was to mathematicians what Paris once was to painters and novelists, Vienna to psychoanalysts and architects, and ancient Athens to philosophers and playwrights." In 1949, Nash was awarded an Atomic Energy Commission fellowship to continue his doctoral studies at Princeton. The school's faculty and students admired Nash for his obvious intellect, but his academic career remained undistinguished.

While at Princeton, Nash invented two board games. The first, called "Nash" or "John," was a two-person, zero-sum game, meaning that one player's advantage must result in a proportional disadvantage for the opponent. Unlike other zero-sum games such as chess and tic-tac-toe, however, a tie or draw was impossible in Nash's game. The game had been invented independently from Nash and eventually was marketed in the 1950s as Hex. Nash also collaborated with several students to create the game "So Long, Sucker," a multiple-player game that rewarded the player most skilled at deception.

Battled Mental Illness

After graduating from Princeton, Nash taught mathematics at the Massachusetts Institute of Technology in Cambridge. He had a son with Eleanor Stier before marrying Alicia Larde in 1957, with whom he also fathered a son. Along with teaching at MIT, Nash worked at the RAND Corporation think tank in Santa Monica, California. Nash was fired in 1954 after being arrested for indecent exposure in a public restroom during a Santa Monica police sting against homosexuals. Nasar wrote: "The biggest shock to Nash may not have been the arrest itself, but the subsequent expulsion from RAND." In 1957, he divided his time between the Institute for Advanced Study and the Courant Institute of Mathematical Sciences at New York University.

In early 1959, Nash began exhibiting symptoms of paranoid schizophrenia. After losing his ability to teach and do research, he underwent insulin coma therapy during several stays in psychiatric hospitals, including one where he shared a room with poet Robert Lowell. When not institutionalized, he made several trips to Europe, where he attempted to establish a world government and resign his United States citizenship because he was convinced he was a political prisoner. He also declared himself the emperor of Antarctica and tried to establish a defense fund for what he believed was an impending extra-terrestrial attack.

In 1962, Alicia Nash filed for divorce, and Nash lived with his widowed mother until her death in 1969. He then moved back into the house he shared with Alicia Nash. For the next 15 years, Nash spent much of his time wandering freely on the Princeton campus. In the late 1980s, however, he showed signs of remission from mental illness. He accepted the Nobel Prize for economics in 1994 and spent much of the 1990s attending to his second son's schizophrenia. He and Alicia Nash eventually remarried.

Developed Game Theory

While at RAND, Nash participated in developing new technologies, theories, and strategies for the United States military through a private nonprofit organization that employed many of the nation's most prominent intellectuals. One of the strategies that RAND was beginning to explore for modern warfare was game theory, which expressed itself in such Cold War strategies as mutual deterrence and the arms race. Whereas John von Neumann and Oskar Morgenstern had conceived of game theory as a zero-sum relationship between non-cooperating competitors, Nash argued that some competitors could benefit from an adversarial relationship by seeking an equilibrium point that would either minimize negative repercussions or maximize positive outcomes. Jeremy Bernstein, writing in Commentary, noted: "Part of Nash's contribution was to allow one to relax the assumptions of von Neumann's theorem; the game does not have to be zero-sum or involve only two players. … What he showed was that in a very wide range of such 'games,' there must be at least one such strategy leading to equilibrium, and if there are several, one must decide among them." Assuming that all competitors behave in a rational manner, Nash hypothesized that each party would apply its dominant strategy to yield mutually beneficial results.

In 1950, Nash submitted his equilibrium theory as his Princeton doctoral thesis. It has since become widely used in military and economic strategies, as well as in biology. According to animal behaviorist Peter Hammerstein, quoted by Robert Pool in Science, "The Nash equilibrium turns out to be terribly important in biology. … Such concepts are proving vital in analyzing a range of biological data, from sex ratios to animals' decisions about whether to fight each other for territory or food." The theory earned him the Nobel Prize, shared with fellow game theorists John Harsanyi and Reinhard Selten.

Following his work in game theory, Nash focused on, among other things, manifolds. According to Nasar: "In one dimension, a manifold may be a straight line, in two dimensions a plane, or the surface of a cube, a balloon, or a doughnut." Although the object remains the same, it appears different when viewed from different perspectives. Because of their mutability, manifolds seemingly defied accurate depictions until Nash employed polynomial algebraic equations to describe them in 1950 and 1951.

In September 1951, beginning his tenure at MIT, Nash combined his work with manifolds with an interest in fluid dynamics. Nash applied the results of this research to his next mathematical theory, which asserted that it is possible to embed a Riemannian manifold in a Euclidean space. An eighteenth-century German mathematician, G.F.B. Riemann theorized that previous Euclidean notions of geography were inaccurate, due to the curvature of the earth's surface, therefore making all parallel lines subject to intersection and the sums of any triangle's angles unequal to 180 degrees. Rather than employ geometry or algebra to solve the problem, Nash developed a new method of applying 19th-century differential calculus. Jurgen Moser later applied the breakthrough to celestial mechanics, resulting in its eventual name: the Nash-Moser theorem.

Books

Nasar, Sylvia, A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash, Simon & Schuster, 1998.

Periodicals

Commentary, August 1998.

Forbes, July 3, 1995.

Science, October 21, 1994.

Time, October 24, 1994.

Washington Post, December 18, 2001.

Columbia Encyclopedia:

John Forbes Nash, Jr.

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Nash, John Forbes, Jr., 1928-, American mathematician, b. Bluefield, W.Va., grad. Carnegie Institute of Technology (now Carnegie-Mellon Univ., B.A. and M.A. 1948), Ph.D. Princeton 1950. During a five-year period, beginning with his doctoral thesis in 1949, he established the mathematical principles of modern game theory (see games, theory of). In four papers published between 1950-53 he made seminal contributions to both non-cooperative game theory and to bargaining theory. He began to experience what he termed "mental disturbances" in 1959 and remained in seclusion for the next 30 years, suffering from paranoid schizophrenia, which he blamed on the mental effort expended in resolving contradictions in quantum theory. Nash returned to his academic research once the disease was in remission, and for his landmark work on the mathematics of game theory he shared the 1994 Nobel memorial economics prize with Hungarian-American economist John Harsanyi and German mathematician and economist Reinhard Selten. Nash wrote Essays on Game Theory (1997).

Bibliography

See biography by S. Nasar (1998); H. W. Kuhn, ed., A Celebration of John F. Nash, Jr, (1996).

Wikipedia on Answers.com:

John Forbes Nash, Jr.

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Nobel prize medal.svg John Forbes Nash Jr.
Born June 13, 1928 (1928-06-13) (age 83)
Bluefield, West Virginia, U.S.
Nationality American
Fields Mathematics, Economics
Institutions Massachusetts Institute of Technology
Princeton University
Alma mater Princeton University
Carnegie Institute of Technology (now part of Carnegie Mellon University)
Doctoral advisor Albert W. Tucker
Known for Nash equilibrium
Nash embedding theorem
Algebraic geometry
Partial differential equations
Notable awards Nobel Memorial Prize in Economic Sciences (1994)

John Forbes Nash, Jr. (born June 13, 1928) is an American mathematician whose works in game theory, differential geometry, and partial differential equations have provided insight into the forces that govern chance and events inside complex systems in daily life. His theories are used in market economics, computing, evolutionary biology, artificial intelligence, accounting, politics and military theory. Serving as a Senior Research Mathematician at Princeton University during the latter part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi.

Nash is the subject of the Hollywood movie A Beautiful Mind. The film, loosely based on the biography of the same name, focuses on Nash's mathematical genius and apparent struggle with paranoid schizophrenia.[1][2]

In his own words, he states,

I later spent times of the order of five to eight months in hospitals in New Jersey, always on an involuntary basis and always attempting a legal argument for release. And it did happen that when I had been long enough hospitalized that I would finally renounce delusional hypotheses and revert to thinking of myself as a human of more conventional circumstances and return to mathematical research. In these interludes of, as it were, enforced rationality, I did succeed in doing some respectable mathematical research. Thus there came about the research for "Le problème de Cauchy pour les équations différentielles d'un fluide général"; the idea that Prof. Hironaka called "the Nash blowing-up transformation"; and those of "Arc Structure of Singularities" and "Analyticity of Solutions of Implicit Function Problems with Analytic Data".
But after my return to the dream-like delusional hypotheses in the later 60's I became a person of delusionally influenced thinking but of relatively moderate behavior and thus tended to avoid hospitalization and the direct attention of psychiatrists.
Thus further time passed. Then gradually I began to intellectually reject some of the delusionally influenced lines of thinking which had been characteristic of my orientation. This began, most recognizably, with the rejection of politically-oriented thinking as essentially a hopeless waste of intellectual effort.[3]
Contents

Early life

Nash was born on June 13, 1928, in Bluefield, West Virginia. His father, after whom he is named, was an electrical engineer for the Appalachian Electric Power Company. His mother, Margaret, had been a school teacher prior to marriage. Nash's parents pursued opportunities to supplement their son's education with encyclopedias and even allowed him to take advanced mathematics courses at a local college while still in high school. Nash accepted a scholarship to Carnegie Institute of Technology (now Carnegie Mellon University) and graduated with a Master's Degree in only three years.[3]

Post-graduate life

Nash's advisor and former Carnegie Tech professor, R.J. Duffin, wrote a letter of recommendation consisting of a single sentence: "This man is a genius."[4] Nash was accepted by Harvard University; but the chairman of the mathematics department of Princeton, Solomon Lefschetz, offered him the John S. Kennedy fellowship, which was enough to convince Nash that Harvard valued him less.[5] Thus he went to Princeton where he worked on his equilibrium theory. He earned a doctorate in 1950 with a 28-page dissertation on non-cooperative games.[6] The thesis, which was written under the supervision of Albert W. Tucker, contained the definition and properties of what would later be called the "Nash equilibrium". These studies led to four articles:

Nash did ground-breaking work in the area of real algebraic geometry:

His work in mathematics includes, for example, the Nash embedding theorem, which shows that any abstract Riemannian manifold can be isometrically realized as a submanifold of Euclidean space. He also made significant contributions to the theory of nonlinear parabolic partial differential equations, and to singularity theory.

In the book A Beautiful Mind, author Sylvia Nasar explains that Nash was working on proving a theorem involving elliptic partial differential equations when, in 1956, he suffered a severe disappointment when he learned of an Italian mathematician, Ennio de Giorgi, who had published a proof a couple of months[vague] before Nash achieved his proof. Each took different routes to get to their solutions. The two mathematicians met each other at the Courant Institute of Mathematical Sciences of New York University during the summer of 1956. It has been speculated that if only one of them had solved the problem, he would have been given the Fields Medal for the proof.[3]

Personal life

Per Nash's biography, from 1951 onwards, he had an affair with a nurse, Eleanor Stier. She bore a child named John David Stier. Though Nash had thoughts of marrying her, he later decided against it and left them.

In 1951, Nash went to the Massachusetts Institute of Technology as a C. L. E. Moore Instructor in the mathematics faculty. There, he met Alicia Lopez-Harrison de Lardé (born January 1, 1933), a physics student from El Salvador, whom he married in February 1957. She admitted Nash to a mental hospital in 1959 for schizophrenia; their son, John Charles Martin Nash, was born soon afterward, but remained nameless for a year because his mother felt that her husband should have a say in the name.

Nash and de Lardé divorced in 1963, though after his final hospital discharge in 1970 Nash lived in de Lardé's house. They were remarried in 2001.

Nash has been a longtime resident of West Windsor Township, New Jersey.[7]

Schizophrenia

Nash began to show signs of extreme paranoia and his wife later described his behavior as erratic, as he began speaking of characters like Charles Herman and William Parcher who were putting him in danger. Nash seemed to believe that all men who wore red ties were part of a communist conspiracy against him. Nash mailed letters to embassies in Washington, D.C., declaring that they were establishing a government.[8][9]

He was admitted to the McLean Hospital, April–May 1959, where he was diagnosed with paranoid schizophrenia. The clinical picture is dominated by relatively stable, often paranoid, fixed beliefs that are either false, over-imaginative or unrealistic, usually accompanied by experiences of seemingly real perception of something not actually present — particularly auditory and perceptional disturbances, a lack of motivation for life, and mild clinical depression.[10] Upon his release, Nash resigned from MIT, withdrew his pension, and went to Europe, unsuccessfully seeking political asylum in France and East Germany. He tried to renounce his U.S. citizenship. After a problematic stay in Paris and Geneva, he was arrested by the French police and deported back to the United States at the request of the U.S. government.

In 1961, Nash was committed to the New Jersey State Hospital at Trenton. Over the next nine years, he spent periods in psychiatric hospitals, where, aside from receiving antipsychotic medications, he was administered insulin shock therapy.[10][11][12]

Although he sometimes took prescribed medication, Nash later wrote that he only ever did so under pressure. After 1970, he was never committed to the hospital again and he refused any medication. According to Nash, the film A Beautiful Mind inaccurately implied that he was taking the new atypical antipsychotics during this period. He attributed the depiction to the screenwriter (whose mother, he notes, was a psychiatrist), who was worried about encouraging people with the disorder to stop taking their medication.[13] Others, however, have questioned whether the fabrication obscured a key question as to whether recovery from problems like Nash's can actually be hindered by such drugs,[14] and Nash has said they are overrated and that the adverse effects are not given enough consideration once someone is deemed mentally ill.[15][16][17] According to Sylvia Nasar, author of the book A Beautiful Mind, on which the movie was based, Nash recovered gradually with the passage of time. Encouraged by his then former wife, de Lardé, Nash worked in a communitarian setting where his eccentricities were accepted. De Lardé said of Nash, "it's just a question of living a quiet life".[9]

Nash dates the start of what he terms "mental disturbances" to the early months of 1959 when his wife was pregnant. He has described a process of change "from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic'"[18] including seeing himself as a messenger or having a special function in some way, and with supporters and opponents and hidden schemers, and a feeling of being persecuted, and looking for signs representing divine revelation.[19] Nash has suggested his delusional thinking was related to his unhappiness and his striving to feel important and be recognized, and to his characteristic way of thinking such that "I wouldn't have had good scientific ideas if I had thought more normally." He has said, "If I felt completely pressureless I don't think I would have gone in this pattern".[20] He does not see a categorical distinction between terms such as schizophrenia and bipolar disorder.[21] Nash reports that he did not hear voices until around 1964, later engaging in a process of rejecting them.[22] He reports that he was always taken to hospitals against his will, and only temporarily renounced his "dream-like delusional hypotheses" after being in hospital long enough to decide to superficially conform - to behave normally or to experience "enforced rationality". Only gradually on his own did he "intellectually reject" some of the "delusionally influenced" and "politically-oriented" thinking as a waste of effort. However, by 1995, although he was "thinking rationally again in the style that is characteristic of scientists," he says he also felt more limited.[18][23]

Nash in November 2006 at a game theory conference in Cologne.

Recognition and later career

At Princeton, campus legend Nash became "The Phantom of Fine Hall" (Princeton's mathematics center), a shadowy figure who would scribble arcane equations on blackboards in the middle of the night. The legend appears in a work of fiction based on Princeton life, The Mind-Body Problem, by Rebecca Goldstein.

In 1978, Nash was awarded the John von Neumann Theory Prize for his discovery of non-cooperative equilibria, now called Nash equilibria. He won the Leroy P. Steele Prize in 1999.

In 1994, he received the Nobel Memorial Prize in Economic Sciences (along with John Harsanyi and Reinhard Selten) as a result of his game theory work as a Princeton graduate student. In the late 1980s, Nash had begun to use email to gradually link with working mathematicians who realized that he was the John Nash and that his new work had value. They formed part of the nucleus of a group that contacted the Bank of Sweden's Nobel award committee and were able to vouch for Nash's mental health ability to receive the award in recognition of his early work.[citation needed]

As of 2011 Nash's recent work involves ventures in advanced game theory, including partial agency, which show that, as in his early career, he prefers to select his own path and problems. Between 1945 and 1996, he published 23 scientific studies.

Nash has suggested hypotheses on mental illness. He has compared not thinking in an acceptable manner, or being "insane" and not fitting into a usual social function, to being "on strike" from an economic point of view. He has advanced evolutionary psychology views about the value of human diversity and the potential benefits of apparently nonstandard behaviors or roles.[24]

Nash has developed work on the role of money in society. Within the framing theorem that people can be so controlled and motivated by money that they may not be able to reason rationally about it, he has criticized interest groups that promote quasi-doctrines based on Keynesian economics that permit manipulative short-term inflation and debt tactics that ultimately undermine currencies. He has suggested a global "industrial consumption price index" system that would support the development of more "ideal money" that people could trust rather than more unstable "bad money". He notes that some of his thinking parallels economist and political philosopher Friedrich Hayek's thinking regarding money and a nontypical viewpoint of the function of the authorities.[25][26]

Nash received an honorary degree, Doctor of Science and Technology from Carnegie Mellon University in 1999; an honorary degree in economics from the University of Naples Federico II on March 19, 2003;[citation needed] an honorary doctorate in economics from the University of Antwerp in April 2007, and was keynote speaker at a conference on Game Theory. He has also been a prolific guest speaker at a number of world-class events, such as the Warwick Economics Summit in 2005 held at the University of Warwick.

Nash has an Erdős number of 4.[27][28][29]

See Also

References

  1. ^ "Oscar race scrutinizes movies based on true stories". USA Today. March 6, 2002. http://www.usatoday.com/life/movies/oscar2002/2002-03-06-true-stories.htm. Retrieved January 22, 2008. 
  2. ^ "List of Oscar Winners". USA Today. March 25, 2002. http://www.usatoday.com/life/movies/oscar2002/2002-03-24-winners.htm. Retrieved August 30, 2008. 
  3. ^ a b c "John F. Nash, Jr. - Autobiography". Nobel Foundation. 1994. http://www.nobelprize.org/nobel_prizes/economics/laureates/1994/nash-autobio.html. Retrieved February 5, 2011. 
  4. ^ Kuhn W, Harold; Sylvia Nasar (eds.). "The Essential John Nash" (PDF). Princeton University Press. pp. Introduction, xi. http://press.princeton.edu/chapters/i7238.pdf. Retrieved April 17, 2008. 
  5. ^ Nasar, Sylvia (1998), A Beautiful Mind, Simon & Schuster, pp. 46–7 .
  6. ^ Osborne, MJ (2004), An Introduction to Game Theory, Oxford, ENG: Oxford University Press, p. 23 .
  7. ^ Staff. "John Forbes Nash May Lose N.J. Home", Associated Press, March 14, 2002. Retrieved February 22, 2011. "West Windsor, N.J.: John Forbes Nash, whose life is chronicled in the Oscar-nominated movie A Beautiful Mind, could lose his home if the township picks one of its proposals to replace a nearby bridge."
  8. ^ Nasar, A Beautiful Mind, p. 251.
  9. ^ a b Nasar, Sylvia (November 13, 1994). "The Lost Years of a Nobel Laureate". New York Times. http://www.nytimes.com/1994/11/13/business/the-lost-years-of-a-nobel-laureate.html?pagewanted=print&src=pm. 
  10. ^ a b Nasar, A Beautiful Mind, p. 32.
  11. ^ Ebert, Roger (2002), Roger Ebert's Movie Yearbook 2003, Andrews McMeel Publishing, ISBN 9780740726910, http://books.google.com/?id=HJGZOs4S4_EC, retrieved July 10, 2008 
  12. ^ Beam, Alex (2001), Gracefully Insane: The Rise and Fall of America's Premier Mental Hospital, PublicAffairs, ISBN 9781586481612, http://books.google.com/?id=M2ZrduulEAwC, retrieved July 10, 2008 
  13. ^ John Nash "Interview by Marika Greihsel". Nobel Foundation. September 1, 2004
  14. ^ Whitaker, R. (March 4, 2002) "Mind drugs may hinder recovery". USA Today.
  15. ^ John Nash "PBS Interview: Medication". 2002.
  16. ^ John Nash "PBS Interview: Paths to Recovery". 2002.
  17. ^ John Nash "PBS Interview: How does Recovery Happen?" 2002.
  18. ^ a b John Nash (1995) "Autobiography" from Les Prix Nobel. The Nobel Prizes 1994, Editor Tore Frängsmyr, [Nobel Foundation], Stockholm, 1952.
  19. ^ John Nash "PBS Interview: Delusional Thinking". 2002.
  20. ^ John Nash "PBS Interview: The Downward Spiral" 2002.
  21. ^ John Nash (April 10, 2005) "Glimpsing inside a beautiful mind". Interview by Shane Hegarty. Schizophrenia.com.
  22. ^ John Nash "PBS Interview: Hearing voices". 2002.
  23. ^ John Nash "PBS Interview: My experience with mental illness". 2002.
  24. ^ Neubauer, David (June 1, 2007). "John Nash and a Beautiful Mind on Strike". Yahoo Health. Archived from the original on 2008-04-21. http://web.archive.org/web/20080421192944/http://health.yahoo.com/experts/depression/8207/john-nash-and-a-beautiful-mind-on-strike/. 
  25. ^ John Nash (2002) "Ideal Money" Southern Economic Journal, 69(1), p.4-11.
  26. ^ Zuckerman, Julia (April 27, 2005) "Nobel winner Nash critiques economic theory". The Brown Daily Herald.
  27. ^ Kuhn, Harsanyi, Selten, Weibull, van Damme, Nash, Jr., Hammerstein. "The work of John F. Nash Jr. in game theory: Nobel Seminar, 8 December 1994". Project euclid (2011). Duke Mathematical Journal. Volume 81, Number 1 (1995), p. 1-29. Duke University Press.
  28. ^ Hoffman, A. J. and Kuhn, H. W. (1956). "Systems of distinct representatives and linear programming". The American Mathematical Monthly (2000) (JSTOR (2010)).
  29. ^ Erdös, Paul; Siemion Fajtlowicz, Siemion; Hoffman, Alan J. (1980) "Maximum degree in graphs of diameter 2". Networks. Vol. 10, issue 1, pp 87–90. Spring 1980. Wiley.

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