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Nash, John (1928-2015)


John Nash Undergraduate Photo, Princeton University Library

Noted mathematician John Nash, Jr. (1928-2015) received his Ph.D. from Princeton University in 1950. The impact of his 27 page dissertation on the fields of mathematics and economics was tremendous. In 1951 he joined the faculty of the Massachusetts Institute of Technology in Cambridge. His battle with schizophrenia began around 1958, and the struggle with this illness would continue for much of his life. Nash eventually returned to the community of Princeton. He was awarded the Nobel Prize in Economics in 1994. The 2001 film A Beautiful Mind , staring Russell Crowe, was loosely based on the life of Nash.

University Archives

John Nash's Dissertation

Non-cooperative Games, May 1950, is available in PDF format  Non-Cooperative_Games_Nash.pdf . The dissertation is provided for research use only. 

[Note: Chapter 6 of  The Essential John Nash,  edited by Harold W. Kuhn and Sylvia Nasar (Princeton, New Jersey: Princeton University Press, 2001) contains a facsimile of Nash's 1950 Ph.D. dissertation on non-cooperative games.]

In the movie  A Beautiful Mind  there is a scene in which faculty members present their pens to Nash. What is the origin of the pen ceremony? When did it start?

The scene in the movie,  A Beautiful Mind,  in which mathematics professors ritualistically present pens to Nash was completely fabricated in Hollywood. No such custom exists. What it symbolizes is that Nash was accepted and recognized in the mathematics community for his accomplishments. While some movies are based on books, the film A Beautiful Mind states that it was  inspired  by the life of John Nash. There are many discrepancies between the book and the film.

May I see Nash’s graduate school records?

John F. Nash, Jr's records have been digitzed and are available to view here:  Graduate Alumni Records

May I see Nash’s faculty file personnel records?

Personnel files transferred to the archives after 2003 : Files are closed until 100 years after the person's year of birth or 5 years after the person's year of death, whichever is longer. Therefore, John F. Nash's personell files are closed until, June 13, 2028.

May I have a copy of Nash’s 1994 Nobel Prize acceptance speech?

At the Nobel Prize Award ceremony, His Majesty the King of Sweden hands each Laureate a diploma, a medal, and a document confirming the Prize amount. The Laureates do not give acceptance speeches . The scene in the movie A Beautiful Mind in which Nash thanks his wife Alicia for her continued support during his illness is fictional.

Laureates are each invited to give an hour-long lecture; however, the Nobel committee did not ask Nash to do so, due to concerns over his mental health.

Additional Resources

Princetoniana Committee Oral History Project Records , Interview with Harold Kuhn, Part 1, pp. 31-40.  In this part of the interview, Prof. Kuhn discusses his behind the scenes work for John Nash’s Nobel Prize.

Historical Subject Files Collection, 1746-2005 : A Beautiful Mind.

Article:  John Nash automobile accident May 23, 2015, in Monroe Township, New Jersey.

Research Tools for Printed Material (Books, Maps, Prints, etc.)

John Nash: The Genius Who Shaped Game Theory and Economics

  • Publication date January 18, 2024

John Nash’s profound impact on economics and mathematics is a story of intellectual brilliance, personal struggle, and enduring legacy. His journey from a precocious student to a Nobel laureate illuminates the intersection of pure mathematics and economic theory.

Early Life and Education

Born in 1928 in Bluefield, West Virginia, Nash showed early signs of extraordinary intellectual ability. His parents, recognizing his talent, nurtured his academic interests. Nash’s advanced understanding of mathematics led him to take college-level courses while still in high school.

Academic Prodigy at Carnegie and Princeton

Nash’s academic prowess continued to flourish at the Carnegie Institute of Technology, where he completed both a bachelor’s and a master’s degree in mathematics by the age of 19. His remarkable abilities were encapsulated in the succinct recommendation from his adviser, attesting to his “mathematical genius.”

Groundbreaking Work at Princeton

Nash’s time at Princeton University was pivotal. His doctoral dissertation, “Non-Cooperative Games,” was a mere 28 pages but laid the groundwork for modern game theory. The Nash equilibrium, a concept introduced in this work, transformed the understanding of decision-making processes in economics, where multiple actors with conflicting interests are involved.

Professional Life and Mental Health Struggles

Nash’s professional tenure at the Massachusetts Institute of Technology and his work as a codebreaker for the U.S. government were marked by his developing symptoms of mental illness. Diagnosed with paranoid schizophrenia, Nash’s life took a turn as he faced immense personal challenges. Despite these struggles, his intellectual contributions continued, and he resumed his academic career at Princeton.

Nobel Prize and Recognition

In 1994, Nash’s groundbreaking work was recognized with the Nobel Prize in Economics. This accolade brought his contributions to a wider audience, bridging the gap between abstract mathematical theories and practical economic applications.

Cultural Impact and “A Beautiful Mind”

Nash’s life story, marked by both triumph and adversity, was vividly captured in Sylvia Nasar’s biography “A Beautiful Mind” and the subsequent Academy Award-winning film. These works brought Nash’s story and the complexity of his theories to a global audience, making Nash a household name.

A Tragic End and Enduring Legacy

John Nash’s life tragically ended in a taxi crash in 2015, soon after receiving the Abel Prize, one of the highest honors in mathematics. His death marked the loss of one of the most brilliant minds of the 20th century.

Nash’s work continues to influence a wide range of fields, from economics and mathematics to biology and political science. His legacy lives on through the Nash equilibrium, a concept that has become a fundamental tool in understanding complex systems where strategic interactions occur. John Nash remains a symbol of how intellectual prowess can overcome personal challenges and contribute to the broader understanding of our world.

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Read John Nash’s Super Short PhD Thesis with 26 Pages & 2 Citations: The Beauty of Inventing a Field

in Math | June 1st, 2015 1 Comment

nash thesis

Last week  John Nash , the Nobel Prize-win­ning math­e­mati­cian, and sub­ject of the block­buster film A Beau­ti­ful Mind , passed away at the age of 86. He died in a taxi cab acci­dent in New Jer­sey.

Days lat­er, Cliff Pick­over high­light­ed a curi­ous fac­toid: When Nash wrote his Ph.D. the­sis in 1950, “Non Coop­er­a­tive Games” at Prince­ton Uni­ver­si­ty, the dis­ser­ta­tion (you can read it online  here) was brief. It ran only 26 pages. And more par­tic­u­lar­ly, it was light on cita­tions. Nash’s diss cit­ed two texts: One was writ­ten by John von Neu­mann & Oskar Mor­gen­stern, whose book,  The­o­ry of Games and Eco­nom­ic Behav­ior   (1944), essen­tial­ly cre­at­ed game the­o­ry and rev­o­lu­tion­ized the field of eco­nom­ics; the oth­er cit­ed text, “Equi­lib­ri­um Points in n‑Person Games,”  was an arti­cle writ­ten by Nash him­self. And it laid the foun­da­tion for his dis­ser­ta­tion, anoth­er sem­i­nal work in the devel­op­ment of game the­o­ry, for which Nash won the Nobel Prize in Eco­nom­ic Sci­ences in 1994 .

The reward of invent­ing a new field, I guess, is hav­ing a slim bib­li­og­ra­phy.

Relat­ed Con­tent:

A Brilliant Madness — 2002 Film on the Nobel Prize Winning Mathematician" href="" rel="bookmark nofollow">John Nash: A Bril­liant Mad­ness — 2002 Film on the Nobel Prize Win­ning Math­e­mati­cian

The Short­est-Known Paper Pub­lished in a Seri­ous Math Jour­nal: Two Suc­cinct Sen­tences

The World Record for the Short­est Math Arti­cle: 2 Words

Free Online Math Cours­es

by OC | Permalink | Comments (1) |

john nash dissertation

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Comments (1), 1 comment so far.

This was shock­ing to know about the demise of John Nash. I had a chance to view the film “a beau­ti­ful mind” with a close friend, Steve Land­fried in Wis­con­sin-Chica­go where John Nash was a sub­ject of this film. I am glad that Steve made this choice for me since I could see and feel all, that this mag­ni­fi­cient sci­en­tist had gone through.This is still my favorite film because of its sub­ject

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john nash dissertation

John F. Nash

John f nash 20061102 3

J ohn Nash, john harsanyi , and reinhard selten shared the 1994 Nobel Prize in economics “for their pioneering analysis of equilibria in the theory of non-cooperative games.” In other words, Nash received the Nobel prize for his work in game theory .

Except for one course in economics that he took at Carnegie Institute of Technology (now Carnegie Mellon) as an undergraduate in the late 1940s, Nash has no formal training in economics. He earned his Ph.D. in mathematics at Princeton University in 1950. The Nobel Prize he received forty-four years later was mainly for the contributions he made to game theory in his 1950 Ph.D. dissertation.

In this work, Nash introduced the distinction between cooperative and noncooperative games. In cooperative games, players can make enforceable agreements with other players. In noncooperative games, enforceable agreements are impossible; any cooperation that occurs is self-enforced. That is, for cooperation to occur, it must be in each player’s interest to cooperate.

Nash’s major contribution is the concept of equilibrium for noncooperative games, which later came to be called a Nash equilibrium. A Nash equilibrium is a situation in which no player, taking the other players’ strategies as given, can improve his position by choosing an alternative strategy. Nash proved that, for a very broad class of games of any number of players, at least one equilibrium exists as long as mixed strategies are allowed. A mixed strategy is one in which the player does not take one action with certainty but, instead, has a range of actions he might take, each with a positive probability.

A simple example of a Nash equilibrium is the prisoners’ dilemma . Another example is the location problem. Imagine that Budweiser and Miller are trying to decide where to place their beer stands on a beach that is perfectly straight. Assume also that sunbathers are located an equal distance from each other and that they want to minimize the distance they walk to get a beer. Where, then, should Bud locate if Miller has not yet chosen its location? If Bud locates one-quarter of the way along the beach, then Miller can locate next to Bud and have three-quarters of the market. Bud knows this and thus concludes that the best location is right in the middle of the beach. Miller locates just slightly to one side or the other. Neither Bud nor Miller can improve its position by choosing an alternate location. This is a Nash equilibrium.

Nash’s other major contribution is his reasoning about “the bargaining problem.” Before Nash, economists thought that the share of the gains each of two parties to a bargain received was always indeterminate. But Nash got further by asking a different question. Instead of defining a solution directly, Nash asked what conditions the division of gains would have to satisfy. He suggested four conditions and showed mathematically that if these conditions held, a unique solution existed that maximized the product of the participants’ utilities. The bottom line is that how gains are divided depends on how much the deal is worth to each participant and what alternatives each participant has.

As readers of Sylvia Nasar’s biography of Nash, A Beautiful Mind, know, Nash contended with schizophrenia from the late 1950s to the mid-1980s. As Nash put it in his Nobel autobiography, “I later spent time 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.” His productivity suffered accordingly. But he emerged from his mental illness in the late 1980s. In his Nobel lecture, Nash noted his own progress out of mental illness:

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. 1

About the Author

David R. Henderson is the editor of  The Concise Encyclopedia of Economics . He is also an emeritus professor of economics with the Naval Postgraduate School and a research fellow with the Hoover Institution at Stanford University. He earned his Ph.D. in economics at UCLA.

Selected Works

Related links.

Vernon Smith on Markets and Experimental Economics , an EconTalk podcast, May 21, 2007.

Ariel Rubenstein on Game Theory and Behavioral Economics , an EconTalk podcast, April 25, 2011.

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150 Years in the Stacks

Year 91 – 1951: “Non-cooperative Games” by John Nash, in: Annals of Mathematics 54 (2)

Posted on April 7, 2011 in All years , Uncategorized

john nash dissertation

Over the past 60 years, game theory has been one of the most influential theories in the social sciences, pervasive in economics, political science, business administration, and military strategy – the disciplines most consulted by the powers-that-be for “real-world,” high-stakes decisions. But just as there would be no semiconductors or (God forbid) laser pointers if not for the abstruse mathematics of quantum theory, game theory can be traced back to theoretical work by academic mathematicians. In a set of papers in the 1950s, mathematician John Forbes Nash set forth breakthrough ideas that helped transform game theory from an ivory tower abstraction into an indispensable analytical tool used by strategists from Wall Street to the Pentagon.

The foundational game theory work of mathematician John von Neumann and economist Oskar Morgenstern, published in 1944, provided a framework for solutions to zero-sum games, where one player’s win was the other’s loss. Nash, in his dissertation research at Princeton (published in this and three other papers), extended game theory to n -person games in which more than one party can gain, a better reflection of practical situations. Nash demonstrated that “a finite non-cooperative game always has at least one equilibrium point” or stable solution. This result came to be called the “Nash equilibrium,” a situation where no one player can get a better payoff by changing strategies, so long as other players also keep their strategies. Using Nash’s framework, predictions can be made about the outcomes of strategic interactions.

Based on Nash’s advances, game theory developed into one of the pre-eminent tools of economics in the second half of the 20th century. In recognition of his breakthrough work, Nash was joint recipient of the Nobel Prize for Economics in 1994 for “pioneering analysis of equilibria in the theory of non-cooperative games.”

If visions of Russell Crowe have danced in your head while you’ve been reading this post, that’s probably because you remember that Crowe played John Forbes Nash in the 2001 film A Beautiful Mind (at least we hope that’s why). Part of the movie takes place at MIT, portraying Nash’s years as an instructor in mathematics at the Institute, where he worked from 1951 to 1959, until mental illness curtailed his mathematical career.

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*: Thanks to Rebeca Duarte Miguel for this, and to Jeek Midford for spotting some spelling mistakes.

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The Masterpieces of John Forbes Nash Jr.

  • First Online: 24 February 2019

Cite this chapter

john nash dissertation

  • Camillo De Lellis 3 , 4  

Part of the book series: The Abel Prize ((AP))

1736 Accesses

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In this set of notes I follow Nash’s four groundbreaking works on real algebraic manifolds, on isometric embeddings of Riemannian manifolds and on the continuity of solutions to parabolic equations. My aim has been to stay as close as possible to Nash’s original arguments, but at the same time present them with a more modern language and notation. Occasionally I have also provided detailed proofs of the points that Nash leaves to the reader.

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john nash dissertation

Analysis on Noncompact Manifolds and Index Theory: Fredholm Conditions and Pseudodifferential Operators

john nash dissertation

What is the Bochner Technique and Where is it Applied

john nash dissertation

Fred Gehring’s Mathematics

In a short autobiographical note, cf. [ 80 , Ch. 2], Nash states that he made his important discovery while completing his PhD at Princeton. In his own words “…I was fortunate enough, besides developing the idea which led to “NonCooperative Games”, also to make a nice discovery relating manifolds and real algebraic varieties. So, I was prepared actually for the possibility that the game theory work would not be regarded as acceptable as a thesis in the mathematics department and then that I could realize the objective of a Ph.D. thesis with the other results.”

Here we are using the nontrivial fact that in a connected real analytic manifold any pair of points can be joined by a real analytic arc. One simple argument goes as follows: use first Whitney’s theorem to assume, without loss of generality, that Σ is a real analytic submanifold of \(\mathbb R^N\) . Fix two points p and q and use the existence of a real analytic projection in a neighborhood of Σ to reduce our claim to the existence of a real analytic arc connecting any two points inside a connected open subset of the Euclidean space. Finally use the Weierstrass polynomial approximation theorem to show the latter claim.

The projection of an algebraic subvariety is not always an algebraic subvariety: here as well we are taking advantage of the genericity of the projection.

Many thanks to Riccardo Ghiloni for suggesting this argument, which follows closely the proof of [ 55 , Lem. 3.2].

Observe that in this context the closure in the Euclidean topology coincides with the Zariski closure.

Closed manifolds can be C 1 isometrically immersed in lower dimension: already at the time of Nash’s paper this could be shown in \(\mathbb R^{2n-1}\) (for n  > 1!) using Whitney’s immersion theorem. Nowadays we can use Cohen’s solution of the immersion conjecture to lower the dimension to n  −  a ( n ), where a ( n ) is the number of 1’s in the binary expansion of n , cf. [ 17 ].

This is what Nash calls “a stage”, cf. [ 73 , p. 391].

Although the term is nowadays rather common, it was not introduced by Nash, neither in [ 73 ] nor in the subsequent paper [ 75 ].

In his paper Nash claims indeed a much larger K ( n ), cf. [ 73 , bottom of p. 386].

The argument of Nash is slightly different, since it covers the space of positive definite matrices with appropriate simplices.

Nash cites Steenrod’s classical book, [ 94 ].

Nash writes Also they could be obtained by orthogonal propagation , cf. [ 73 , top of p. 387].

The term free was not coined by Nash, but introduced later in the literature by Gromov.

It must be observed that Nash employs this fact without explicitly stating it and he does not prove it neither he gives a reference. He uses it twice, once in the proof of Theorem 29 and once in the proof of Proposition 35 , and although in the first case one could appeal to a more elementary argument, I could not see an easier way in the second.

Indeed Nash does not give any argument and just refers to a similar reasoning that he uses in Proposition 35 below.

Nash suggests an alternative argument which avoids the discussion of the dimensions of \(\mathcal {C} (p, L)\) and \(\mathcal {B}\) . One can apply his result on real algebraic varieties to find an embedding v which realizes v ( Σ ) as a real algebraic submanifold, cf. Theorem 1 . Then any set of coefficients \(A^r_{ij}\) which is algebraically independent over the minimal field \(\mathbb F\) of definition of v ( Σ ) (see Proposition 12 ) belongs to the complement of \(\mathcal {B}\) . Since \(\mathbb F\) is finitely generated over the rationals (see Proposition 12 ), it has countable cardinality and the conclusion follows easily.

In Nash’s paper the operator is called S θ , where θ corresponds to ε −1 . Since it is nowadays rather unusual to parametrize a family of convolutions as Nash does, I have switched to a more modern convention.

Nash does not take advantage of this simple remark and introduces instead a rather unusual notation to keep track of all the estimates for the intermediate norms in the bounds corresponding to ( 59 ), ( 60 ) and ( 61 ).

In fact, De Giorgi’s statement is stronger, since in his theorem ∥ v ∥ ∞ in ( 125 ) is replaced by the L 2 norm of v (note that the power of r should be suitably adjusted: the reader can easily guess the correct exponent using the invariance of the statement under the transformation u r ( x ) =  u ( rx )).

Indeed, it was known that the first partial derivatives of the minimizer satisfy a uniformly elliptic partial differential equation with measurable coefficients. De Giorgi’s stronger version of Theorem 50 would then directly imply the desired Hölder estimate. Nash’s version was also sufficient, because a theorem of Stampacchia guaranteed the local boundedness of the first partial derivatives, cf. [ 93 ].

Nash does not provide any argument nor reference, he only briefly mentions that Theorem 48 follows from Theorem 51 using a regularization scheme and the maximum principle. Note that a derivation of the latter under the weak regularity assumptions of Theorem 48 is, however, not entirely trivial: in Sect. 5.8 we give an alternative argument based on a suitable energy estimate.

In order to simplify the notation we omit the domain of integration when it is the entire space.

The first two equations are the first two equations from [ 77 , p. 487, (1)] whereas the third should correspond to [ 77 , p. 488, (1c)]. The latter is derived by Nash from the third equation in [ 77 , p. 487, (1)], which in turn corresponds to the classical conservation law for the entropy, see, for instance, [ 61 , (49.5)]. The third equation of [ 77 , p. 487, (1)] contains two typos, which disappear in [ 77 , p. 488, (1c)]. The latter however contains another error: Nash has η and ζ in place of \(\frac {\eta }{\rho T S_T}\) and \(\frac {\zeta }{\rho T S_T}\) , but it is easy to see that this would not be consistent with the way he describes its derivation.

Nash’s error has no real consequence for the rest of the note, since he treats the coefficients in front of \(\mathcal {S} (v)_{ij} \mathcal {S} (v)_{ij}\) and (div v ) 2 as arbitrary real analytic functions of ρ and T and the same holds for \(\frac {\eta }{\rho T S_T}\) and \(\frac {\zeta }{\rho T S_T}\) under the assumption S T  ≠ 0. The latter inequality is needed in any case even to treat Nash’s “wrong” equation for T .

Indeed Nash does not mention the positivity of S T , although this is certainly required by his argument when he reduces the existence of solutions of ( 240 ) to the existence of a solutions of a suitable parabolic system, cf. [ 77 , (6) and (7)]: the equation in T is parabolic if and only if \(\frac {\varkappa }{\rho T S_T}\) is positive.

I also have the impression that his argument does not really need the positivity of S and p , although these are quite natural assumptions from the thermodynamical point of view.

In the modern literature it is customary to take an equivalent definition of X through formal power series; we refer to [ 58 ] for the latter and for several important subtleties related to variants of the Nash arc space.

In fact, Nash claims the proposition with any algebraic subset W of V  in place of V s but, although the proposition does hold for W  =  V s , it turns out to be false for a general algebraic subset W ; cf. [ 21 , Ex. 3.7] for a simple explicit counterexample.

S. Akbulut and H. King. On approximating submanifolds by algebraic sets and a solution to the Nash conjecture. Invent. Math. , 107(1):87–98, 1992.

MathSciNet   MATH   Google Scholar  

A. G. Akritas. Sylvester’s forgotten form of the resultant. Fibonacci Quart. , 31(4):325–332, 1993.

A. D. Alexandrov. Intrinsic geometry of convex surfaces . OGIZ, Moscow-Leningrad, 1948.

Google Scholar  

G. E. Andrews, R. Askey, and R. Roy. Special functions. Cambridge: Cambridge University Press, 1999.

MATH   Google Scholar  

D. G. Aronson. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. , 73:890–896, 1967.

D. G. Aronson. Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa (3) , 22:607–694, 1968.

R. F. Bass. Diffusions and elliptic operators . Springer-Verlag, New York, 1998.

R. F. Bass. On Aronson’s upper bounds for heat kernels. Bull. London Math. Soc. , 34(4):415–419, 2002.

J. Bochnak, M. Coste, and M.-F. Roy. Real Algebraic Geometry . Springer-Verlag, Berlin, 1998.

J. F. Borisov. The parallel translation on a smooth surface. IV. Vestnik Leningrad. Univ. , 14(13):83–92, 1959.

J. F. Borisov. C 1, α -isometric immersions of Riemannian spaces. Doklady , 163:869–871, 1965.

T. Buckmaster, C. De Lellis, P. Isett, and L. Székelyhidi, Jr. Anomalous dissipation for 1∕5-Hölder Euler flows. Ann. of Math. (2) , 182(1):127–172, 2015.

T. Buckmaster, C. De Lellis, L. Székelyhidi, Jr., and V. Vicol. Onsager’s conjecture for admissible weak solutions. Comm. Pure Appl. Math. , to appear.

C. Burstin. Ein Beitrag zum Problem der Einbettung der Riemannschen Räume in euklidischen Räumen. Rec. Math. Moscou , 38(3–4):74–85, 1931.

E. Cartan. Sur la possibilité de plonger un espace Riemannien donné dans un espace Euclidien. Ann. Soc. Polon. Math. , 6:1–7, 1928.

H. Cartan. Variétés analytiques réelles et variétés analytiques complexes. Bull. Soc. Math. France , 85:77–99, 1957.

R. L. Cohen. The immersion conjecture for differentiable manifolds. Ann. of Math. (2) , 122(2):237–328, 1985.

S. Cohn-Vossen. Zwei Sätze über die Starrheit der Eiflächen. Nachrichten Göttingen , 1927:125–137, 1927.

S. Conti, C. De Lellis, and L. Székelyhidi, Jr. h -principle and rigidity for C 1, α isometric embeddings. In Nonlinear partial differential equations , volume 7 of Abel Symp. , pages 83–116. Springer, Heidelberg, 2012.

T. de Fernex. Three-dimensional counter-examples to the Nash problem. Compos. Math. , 149(9):1519–1534, 2013.

T. de Fernex. The space of arcs of an algebraic variety. In Algebraic geometry: Salt Lake City 2015 , Proc. Sympos. Pure Math., Vol. 97.1, pp. 169–197, 2015.

E. De Giorgi. Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) , 3:25–43, 1957.

E. De Giorgi. Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Un. Mat. Ital. (4) , 1:135–137, 1968.

E. De Giorgi. Selected papers . Springer-Verlag, Berlin, 2006. Edited by L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda and S. Spagnolo.

C. De Lellis, D. Inauen, and L. Székelyhidi, Jr. A Nash–Kuiper theorem for \(C^{1,\frac {1}{5}-\delta }\) immersions of surfaces in 3 dimensions. Rev. Mat. Iberoam. , 34(3):1119–1152, 2018.

C. De Lellis, H. King, J. Milnor, Nachbar J., L. Székelyhidi, Jr., C. Villani, and J. Weinstein. John Forbes Nash, Jr. 1928–2015. Notices of the AMS , 63(5):492–506, 2017.

C. De Lellis and L. Székelyhidi, Jr. Dissipative continuous Euler flows. Invent. Math. , 193(2):377–407, 2013.

M. Demazure, H. Pinkham, and B. Teissier, editors. Séminaire sur les Singularités des Surfaces , volume 777 of Lecture Notes in Mathematics . Springer, Berlin, 1980. Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977.

Y. Eliashberg and N. Mishachev. Introduction to the h-principle . American Mathematical Society, Providence, RI, 2002.

L. C. Evans. Partial differential equations . American Mathematical Society, Providence (R.I.), 1998.

L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions . CRC Press, Boca Raton, FL, 1992.

E. B. Fabes and D. W. Stroock. A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. , 96(4):327–338, 1986.

E. Feireisl. Dynamics of viscous compressible fluids . Oxford University Press, Oxford, 2004.

J. Fernández de Bobadilla and M. P. Pereira. The Nash problem for surfaces. Ann. of Math. (2) , 176(3):2003–2029, 2012.

A. Friedman. Partial differential equations of parabolic type . Prentice-Hall, 1964.

P. D. González Pérez and B. Teissier. Toric geometry and the Semple-Nash modification. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM , 108(1):1–48, 2014.

M. J. Greenberg. Rational points in Henselian discrete valuation rings. Inst. Hautes Études Sci. Publ. Math. , (31):59–64, 1966.

M. L. Gromov. Isometric imbeddings and immersions. Dokl. Akad. Nauk SSSR , 192:1206–1209, 1970.

M. L. Gromov. Partial differential relations . Springer-Verlag, Berlin, 1986.

M. L. Gromov and V. A. Rohlin. Imbeddings and immersions in Riemannian geometry. Uspehi Mat. Nauk , 25(5 (155)):3–62, 1970.

M. Günther. Zum Einbettungssatz von J. Nash. Math. Nachr. , 144:165–187, 1989.

M. Günther. Isometric embeddings of Riemannian manifolds. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) , pages 1137–1143. Math. Soc. Japan, Tokyo, 1991.

R. S. Hamilton. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) , 7(1):65–222, 1982.

W. Hao, S. Leonardi, and J. Nečas. An example of irregular solution to a nonlinear Euler-Lagrange elliptic system with real analytic coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 23(1):57–67, 1996.

R. Hartshorne. Algebraic geometry . Springer-Verlag, New York-Heidelberg, 1977.

A. Hatcher. Algebraic Topology . Cambridge University Press, Cambridge, 2002.

G. Herglotz. Über die Starrheit der Eiflächen. Abh. Math. Semin. Hansische Univ. , 15:127–129, 1943.

H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) , 79:205–326, 1964.

M. W. Hirsch. Differential Topology . Springer-Verlag, New York-Heidelberg, 1976.

E. Hopf. Zum analytischen Charakter der Lösungen regulärer zweidimensionaler Variationsprobleme. Math. Z. , 30:404–413, 1929.

P. Isett. A Proof of Onsager’s Conjecture. Ann. Math. (2) , 188:871–963, 2018.

S. Ishii and J. Kollár. The Nash problem on arc families of singularities. Duke Math. J. , 120(3):601–620, 2003.

H. Jacobowitz. Implicit function theorems and isometric embeddings. Ann. of Math. (2) , 95:191–225, 1972.

M. Janet. Sur la possibilité de plonger un espace Riemannien donné à n dimensions dans un espace Euclidien à \(\frac {n(n+1)}{2}\) dimensions. C. R. Acad. Sci., Paris , 183:942–943, 1926.

Z. Jelonek. On the extension of real regular embedding. Bull. Lond. Math. Soc. , 40(5):801–806, 2008.

J. M. Johnson and J. Kollár. Arc spaces of cA -type singularities. J. Singul. , 7:238–252, 2013.

A. Källén. Isometric embedding of a smooth compact manifold with a metric of low regularity. Ark. Mat. , 16(1):29–50, 1978.

J. Kollár and A. Némethi. Holomorphic arcs on singularities. Invent. Math. , 200(1):97–147, 2015.

S. G. Krantz and H. R. Parks. A Primer of Real Analytic Functions . Birkhäuser Verlag, Basel, 1992.

N. H. Kuiper. On C 1 -isometric imbeddings. I, II. Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. , 17:545–556, 683–689, 1955.

L. D. Landau and E. M. Lifshitz. Course of theoretical physics. Vol. 6, Fluid dynamics . Pergamon Press, Oxford, second edition, 1987.

H. Lewy. On the existence of a closed convex surface realizing a given Riemannian metric. Proc. Natl. Acad. Sci. USA , 24:104–106, 1938.

P.-L. Lions. Mathematical topics in fluid mechanics. Vol. 2 . The Clarendon Press, Oxford University Press, New York, 1998. Compressible models, Oxford Science Publications.

J. W. Milnor and J. D. Stasheff. Characteristic classes . Princeton University Press, Princeton, N. J., 1974.

C. Mooney and O. Savin. Some Singular Minimizers in Low Dimensions in the Calculus of Variations. Arch. Ration. Mech. Anal. , 221(1):1–22, 2016.

C. B. Morrey, Jr. On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. , 43:126–166, 1938.

C. B. Morrey, Jr. Second-order elliptic systems of differential equations. In Contributions to the theory of partial differential equations , Annals of Mathematics Studies, no. 33, pages 101–159. Princeton University Press, Princeton, N. J., 1954.

J. Moser. A new technique for the construction of solutions of non-linear differential equations. Proc. Nat. Acad. Sci. USA , 47:1824–1831, 1961.

J. Moser. On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. , 14:577–591, 1961.

J. Moser. A rapidly convergent iteration method and non-linear differential equations. II. Ann. Scuola Norm. Sup. Pisa (3) , 20:499–535, 1966.

J. Moser. A rapidly convergent iteration method and non-linear partial differential equations. I. Ann. Scuola Norm. Sup. Pisa (3) , 20:265–315, 1966.

J. Nash. Real algebraic manifolds. Ann. of Math. (2) , 56:405–421, 1952.

J. Nash. C 1 isometric imbeddings. Ann. of Math. (2) , 60:383–396, 1954.

J. Nash. A path space and the Stiefel-Whitney classes. Proc. Nat. Acad. Sci. U.S.A. , 41:320–321, 1955.

J. Nash. The imbedding problem for Riemannian manifolds. Ann. of Math. (2) , 63:20–63, 1956.

J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. , 80:931–954, 1958.

J. Nash. Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France , 90:487–497, 1962.

J. Nash. Analyticity of the solutions of implicit function problems with analytic data. Ann. of Math. (2) , 84:345–355, 1966.

J. Nash. Arc structure of singularities. Duke Math. J. , 81(1):31–38 (1996), 1995.

J. Nash. The essential John Nash . Princeton University Press, Princeton, NJ, 2002. Edited by H. W. Kuhn and S. Nasar.

L. Nirenberg. The determination of a closed convex surface having given line element . ProQuest LLC, Ann Arbor, MI, 1949. Thesis (Ph.D.)–New York University.

L. Nirenberg. The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. , 6:337–394, 1953.

A. Nobile. Some properties of the Nash blowing-up. Pacific J. Math. , 60(1):297–305, 1975.

C. Plénat and M. Spivakovsky. The Nash problem and its solution: a survey. J. Singul. , 13:229–244, 2015.

A. V. Pogorelov. Izgibanie vypuklyh poverhnosteı̆ . Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1951.

A. V. Pogorelov. Extrinsic geometry of convex surfaces . American Mathematical Society, Providence, R.I., 1973.

W. Rudin. Principles of mathematical analysis . McGraw-Hill Book Co., New York-Auckland-Düsseldorf, third edition, 1976.

L. Schläfli. Nota alla memoria del sig. Beltrami, “Sugli spazii di curvatura costante”. Annali di Mat. (2) , 5:178–193, 1871.

J. Schwartz. On Nash’s implicit functional theorem. Comm. Pure Appl. Math. , 13:509–530, 1960.

J. G. Semple. Some investigations in the geometry of curve and surface elements. Proc. London Math. Soc. (3) , 4:24–49, 1954.

M. Shiota. Nash manifolds , volume 1269 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1987.

M. Spivakovsky. Sandwiched singularities and desingularization of surfaces by normalized Nash transformations. Ann. of Math. (2) , 131(3):411–491, 1990.

G. Stampacchia. Sistemi di equazioni di tipo ellittico a derivate parziali del primo ordine e proprietà delle estremali degli integrali multipli. Ricerche Mat. , 1:200–226, 1952.

N. Steenrod. The Topology of Fibre Bundles . Princeton Mathematical Series, vol. 14. Princeton University Press, Princeton, N. J., 1951.

V. Šverák and X. Yan. A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var. Partial Differential Equations , 10(3):213–221, 2000.

V. Šverák and X. Yan. Non-Lipschitz minimizers of smooth uniformly convex functionals. Proc. Natl. Acad. Sci. USA , 99(24):15269–15276, 2002.

L. Székelyhidi, Jr. The h -principle and turbulence. ICM 2014 Proceedings Volume , 2014.

R. Thom. Espaces fibrés en sphères et carrés de Steenrod. Ann. Sci. Ecole Norm. Sup. (3) , 69:109–182, 1952.

A. Tognoli. Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa (3) , 27:167–185, 1973.

L. D. Tráng and B. Teissier. On the mathematical work of Professor Heisuke Hironaka. Publ. Res. Inst. Math. Sci. , 44(2):165–177, 2008.

A. H. Wallace. Algebraic approximation of manifolds. Proc. London Math. Soc. (3) , 7:196–210, 1957.

A. Weil. Foundations of Algebraic Geometry . American Mathematical Society, New York, 1946.

H. Weyl. Über die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement. Zürich. Naturf. Ges. 61, 40–72, 1916.

J. H. C. Whitehead. On C 1 -complexes. Ann. of Math. (2) , 41:809–824, 1940.

H. Whitney. Differentiable manifolds. Ann. of Math. (2) , 37(3):645–680, 1936.

H. Whitney. The self-intersections of a smooth n -manifold in 2 n -space. Ann. of Math. (2) , 45:220–246, 1944.

S.-T. Yau. Open problems in geometry. In Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) , volume 54 of Proc. Sympos. Pure Math. , pages 1–28. Amer. Math. Soc., Providence, RI, 1993.

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I am very grateful to Helge and Ragni for entrusting to me this portion of the Nash volume, a wonderful occasion to deepen my understanding of the mathematics of a true genius, who has had a tremendous influence in my own work.

Most of the manuscript has been written while I was visiting the CMSA at Harvard and I wish to thank Shing-Tung Yau and the staff at CMSA for giving me the opportunity to carry on my work in such a stimulating environment.

Several friends and colleagues have offered me kind and invaluable help with various portions of this note. In particular I wish to thank Davide Vittone for giving me several precious suggestions with the Sects. 3 and 4 and reading very carefully all the manuscript; Gabriele Di Cerbo, Riccardo Ghiloni and János Kollár for clarifying several important points concerning Sect. 2 and pointing out a few embarassing mistakes; Tommaso de Fernex and János Kollar for kindly reviewing a first rather approximate version of Sect. 6.4 ; Eduard Feireisl for his suggestions on Sect. 6.3 ; Cedric Villani for allowing me to steal a couple of paragraphs from his beautiful review of [ 76 ] in the Nash memorial article [ 26 ]; Francois Costantino for helping me with a delicate topological issue; Jonas Hirsch and Govind Menon for proofreading several portions of the manuscript; Helge Holden for going through all the manuscript with extreme care.

This work has been supported by the grant agreement 154903 of the Swiss National Foundation.

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De Lellis, C. (2019). The Masterpieces of John Forbes Nash Jr.. In: Holden, H., Piene, R. (eds) The Abel Prize 2013-2017. The Abel Prize. Springer, Cham.

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毕业证模板复刻西雅图华盛顿大学没毕业>办理西雅图华盛顿大学毕业证成绩单【微信UW】UW毕业证成绩单UW学历证书UW文凭《UW毕业套号文凭网认证西雅图华盛顿大学毕业证成绩单》《哪里买西雅图华盛顿大学毕业证文凭UW成绩学校快递邮寄信封》《开版西雅图华盛顿大学文凭》UW留信认证本科硕士学历认证 “【微信95270640】办理毕业证成绩单文凭、修改成绩、教育部学历学位认证、假毕业证、假成绩单、假文凭、假学历文凭、毕业证文凭、、文凭毕业证、毕业证认证、留服认证、使馆认证、使馆公证、使馆证明、使馆留学回国人员证明、留学生认证、学历认证、文凭认证、学位认证、留学生学历认证、留学生学位认证、使馆认证(留学回国人员证明)、学生卡(证)、雅思托福成绩单、在读证明、录取通知书offer、驾照等 留学归国服务中心:实体公司,注册经营,行业标杆,精益求精! 为英国、加拿大、澳洲、新西兰、美国、法国、德国、新加坡等国 1.收集客户基本信息! 2.客户支付30%的订金,公司出电子图给客户审核,确保信息无误! 3.根据客户审核后的电子图制作成品再次给客户审核! 4.客户支付完余款,公司把成品邮寄给客户!(国内顺丰 国外DHL) 四:合理推荐业务: 1.如果您只是为了的应付父母亲戚朋友,那么办理一份学位即可 2.如果您是为了回国找工作,只是进私营企业或者外企,那么办理一份学位即可,因为私营企业或者外企是不能查询学位真假的! 3.如果您是要进国企 银行 事业单位 考公务员等就需办理真实教育部学历认证!明办理咨询。基于国内鼓励留学生回国就业、创业的政策,以及大批留学生归国立业之大优势。本公司一直朝着智力密集型的方向转型,建立了一个专业化的由归国留学生组成的专业顾问团队为中心,公司核心部分包括:咨询服务部门、营销部门、运作部、顾问团队共同协作的服务体系。 ★业务选择办理准则★ 一、工作未确定,回国需先给父母、亲戚朋友看下文凭的情况,办理一份就读学校的毕业证文凭即可 二、回国进私企、外企、自己做生意的情况,这些单位是不查询毕业证真伪的,而且国内没有渠道去查询国外文凭的真假,也不需要提供真实教育部认证。鉴于此,办理一份毕业证即可 三、进国企,银行,事业单位,考公务员等等,这些单位是必需要提供真实教育部认证的,办理教育部认证所需资料众多且烦琐,所有材料您都必须提供原件,我们凭借丰富的经验,快捷的绿色通道帮您快速整合材料,让您少走弯路。 ★关于教育部学历认证的小知识: 国外学历学位认证,作为留学生回国后就业、落户、升学必须提交的证明材料,国家虽然没有明文的规定,留学生回国后必须办理,属于自愿行为,不强制要求。但是根据国家部委和国务院学位办的相关规定:留服认证是留学生回国报考公务员,进入国家机关、事业单位,高等教育,大型外企等入职时必须提供的国外文凭的证明材料,不仅关系着留学生回国后的就业,更是影响着落户、升学,甚至留学生往后申请海外高层次人才科研启动基金的有力凭据。

International Transactions on Electrical Energy Systems

Mohammad Sheikh-el-eslami

BMC Genomics

Ricardo Serrão Santos

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European Respiratory Journal

Muhammad Tandru Salam

Journal of Clinical Medicine

Ron Eliashar

Studies in 20th & 21st century literature

Avital Ronell

Procedia - Social and Behavioral Sciences

abbas sadeghi

BMC Public Health

Josephine Birungi


Wojciech Panek

Revista de Investigaciones Veterinarias del Perú

Javier Juárez

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  1. Doctoral Dissertation Help John Nash! Explaining John Nash's

    john nash dissertation

  2. John nash game theory dissertation help

    john nash dissertation

  3. Narrative essay: John nash dissertation

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  4. John Nash: a história do matématico ganhador do Nobel de Economia

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  5. John Nash Doctoral Dissertation Pdf

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  6. John Nash (1928-2015)

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  1. John Nash received the Nobel Prize in Economic Sciences in 1994

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  1. Nash, John (1928-2015)

    Noted mathematician John Nash, Jr. (1928-2015) received his Ph.D. from Princeton University in 1950. The impact of his 27 page dissertation on the fields of mathematics and economics was tremendous. In 1951 he joined the faculty of the Massachusetts Institute of Technology in Cambridge. His battle with schizophrenia began around 1958, and the ...

  2. John Nash's Super Short PhD Thesis: 26 Pages & 2 Citations

    When John Nash wrote 'Non Cooperative Games,' his Ph.D. dissertation at Princeton in 1950, the text of his thesis (read it online) was brief. It ran only 26 pages. And more particularly, it was light on citations.


    NON-COOPERATIVE GAMES. John F. Nash. Published in Classics in Game Theory 1 September 1951. Mathematics. Classics in Game Theory. we would call cooperative. This theory is based on an analysis of the interrelationships of the various coalitions which can be formed by the players of the game. Our theory, in contradistinction, is based on the ...

  4. PDF The Impact of John Nash on Economics and Game Theory

    Nash's contributions Noncooperative Games, Ph.D. Dissertation Princeton, 1950, and Annals of Mathematics 1951 (announced in PNAS 1950). Introduced the distinction between noncooperative and cooperative game theory. Definedequilibrium point (now called "Nash equilibrium"), the sine qua non for analysis of individual optimizing

  5. John Forbes Nash Jr.

    John Forbes Nash, Jr. (June 13, 1928 - May 23, 2015), known and published as John Nash, was an American mathematician who made fundamental contributions to game theory, real algebraic geometry, ... Nash earned a PhD in 1950 with a 28-page dissertation on non-cooperative games. ...

  6. John Nash: The Genius Who Shaped Game Theory and Economics

    John Nash's profound impact on economics and mathematics is a story of intellectual brilliance, personal struggle, and enduring legacy. ... His doctoral dissertation, "Non-Cooperative Games," was a mere 28 pages but laid the groundwork for modern game theory. The Nash equilibrium, a concept introduced in this work, transformed the ...

  7. PDF Non Cooperative Games Nash

    Joun NASH. (Received October 11, 1950) Introduction. Von Neumann and Morgenstern have developed a very fruitful theory of two-person zero-sum games in their book Theory of Games and Economic Be- havior. This book also contains a theory of n-person games of a type which we would call cooperative. This theory is based on an analysis of the ...

  8. John Nash: Three Phases in the Career of a Beautiful Mind

    John Nash was a real person who lives on Alexander Road in Princeton Junction." Some months later, an acquaintance told me that a woman friend of hers ... His dissertation on game theory was the basis for the Nobel Award in Economics in 1994. A Princeton professor, John von Neumann, an immigrant

  9. JOHN NASH (Received October 11, 1950)

    JOHN NASH (Received October 11, 1950) Introduction Von Neumann and Morgenstern have developed a very fruitful theory of two-person zero-sum games in their book Theory of Games and Economic Be-havior. This book also contains a theory of n-person games of a type which we would call cooperative. This theory is based on an analysis of the interrela-

  10. Commentary: John Nash and evolutionary game theory

    John Nash's work laid the foundations for evolutionary game theory as well as the theory of games with rational agents. The Nash bargaining solution emerges as a natural solution concept in both of these settings. ... His doctoral dissertation takes up less than thirty pages; his demonstration of the existence of Nash equilibrium is ...

  11. Read John Nash's Super Short PhD Thesis with 26 Pages & 2 Citations

    Last week John Nash , the Nobel Prize-win­ning math­e­mati­cian, and sub­ject of the block­buster film A Beau­ti­ful Mind, passed away at the age of 86. He died in a taxi cab acci­dent in New Jer­sey. Days lat­er, Cliff Pick­over high­light­ed a curi­ous fac­toid: When Nash wrote his Ph.D. the­sis in 1950, "Non Coop­er­a ...

  12. John F. Nash

    John F. Nash 1928-2015. SHARE POST: ... Prize he received forty-four years later was mainly for the contributions he made to game theory in his 1950 Ph.D. dissertation. In this work, Nash introduced the distinction between cooperative and noncooperative games. In cooperative games, players can make enforceable agreements with other players.

  13. Year 91

    Nash, in his dissertation research at Princeton (published in this and three other papers), extended game theory to n-person games in which more than one party can gain, a better reflection of practical situations. Nash demonstrated that "a finite non-cooperative game always has at least one equilibrium point" or stable solution.

  14. PDF Author(s): John Nash Source: The Annals of Mathematics, Second Series Non-Cooperative Games Author(s): John Nash Source: The Annals of Mathematics, Second Series, Vol. 54, No. 2, (Sep., 1951), pp. 286-295

  15. Commentary: Nash equilibrium and dynamics

    Abstract. John F. Nash, Jr., submitted his Ph.D. Dissertation entitled Non-cooperative games to Princeton University in 1950. Read it 58 years later, and you will find the germs of various later developments in game theory. Some of these are presented below, followed by a discussion concerning dynamic aspects of equilibrium.

  16. John F Nash PhD

    Project Details. In this page you can find Nash's PhD thesis: Original document. Transcribed into tex/pdf *. *: Thanks to Rebeca Duarte Miguel for this, and to Jeek Midford for spotting some spelling mistakes.

  17. The Masterpieces of John Forbes Nash Jr.

    In 1966 Nash turned again one last time to the isometric embedding problem, addressing the real analytic case. More precisely, his aim was to prove that, if in Theorem 29 we assume that the metric g is real analytic, then there is a real analytic isometric embedding of ( Σ, g) in a sufficiently large Euclidean space.

  18. John Nash

    John Nash (born June 13, 1928, Bluefield, West Virginia, U.S.—died May 23, 2015, near Monroe Township, New Jersey) was an American mathematician who was awarded the 1994 Nobel Prize for Economics for his landmark work, first begun in the 1950s, on the mathematics of game theory.He shared the prize with John C. Harsanyi and Reinhard Selten.In 2015, Nash won (with Louis Nirenberg) the Abel ...

  19. (PDF) PhD Thesis of John Nash

    PhD Thesis of John Nash. PhD Thesis of John Nash. DIBAKAR DATTA. See Full PDF Download PDF. See Full PDF Download PDF. Related Papers. Ciência, Cuidado e Saúde. Análise das investigações em enfermagem e o uso da teoria do cuidado cultural. 2009 • Maria Vera M L Cardoso. Download Free PDF View PDF.