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Topic: Nash embedding theorem

	Nash embedding theorem
		The Nash embedding theorem (or imbedding theorem) in differential geometry, published in 1965 by John Nash, states that every Riemannian manifold can be isometrically embedded in a Euclidean space R
		A local embedding theorem is much simpler and can be proved using the implicit function theorem[?] of advanced calculus.
		The proof of the global embedding theorem as presented here relies on Nash's far-reaching generalization of the implicit function theorem, the Nash-Moser inverse function theorem[?] and Newton's method with postconditioning see ref.
	ebroadcast.com.au /lookup/encyclopedia/na/Nash_embedding_theorem.html (271 words)

	Embedding
		In mathematics, an embedding is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.
		In general topology: an embedding is a homeomorphism onto its image.
		In other words embedding is diffeomorphism to its image (in particular image of embedding is a submanifold).
	www.guajara.com /wiki/en/wikipedia/e/em/embedding.html (276 words)

	PlanetMath: Nash isometric embedding theorem
		"Nash isometric embedding theorem" is owned by Simone.
		This is version 3 of Nash isometric embedding theorem, born on 2006-01-21, modified 2006-01-22.
		Nash embedding theorem by matte on 2006-01-25 14:41:25
	planetmath.org /encyclopedia/NashIsometricEmbeddingTheorem.html (248 words)

	John Forbes Nash Info - Bored Net - Boredom (Site not responding. Last check: 2007-11-03)
		After a promising start to his mathematical career, Nash began to suffer from schizophrenia around his 30th year, an illness from which he has only recovered some 25 years later.
		John Nash was born in Bluefield, West Virginia as son of John Nash Sr.
		During this time, he proved the Nash embedding theorem, an important result in differential geometry about manifolds.
	www.borednet.com /e/n/encyclopedia/j/jo/john_forbes_nash.html (723 words)

	Nash biography
		Nash won a scholarship in the George Westinghouse Competition and was accepted by the Carnegie Institute of Technology (now Carnegie-Mellon University) which he entered in June 1945 with the intention of taking a degree in chemical engineering.
		His famous theorem, that any compact real manifold is diffeomorphic to a component of a real-algebraic variety, was thought of by Nash as a possible result to fall back on if his work on game theory was not considered suitable for a doctoral thesis.
		In February of 1957 Nash married Alicia; by the autumn of 1958 she was pregnant but, a couple of months later near the end of 1958, Nash's mental state became very disturbed.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Nash.html (3683 words)

	Embedding
		In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.
		An isometric embedding is a smooth embedding f : M → N which preserves the metric in the sense that g is equal to the pullback of h by f, i.e.
		In field theory, an embedding of a field E in a field F is a ring homomorphism σ : E → F.
	articles.gourt.com /?article=embedding (620 words)

	Nasar/Nash2 (Site not responding. Last check: 2007-11-03)
		Nash's return to Princeton initiated his transformation from the local genius of the 50's to the genius loci of the next 2 decades.
		Evidently John Nash differed from most of the world around him in at least 3 respects: (1) He was a genius; (2) He was mad; and (3) He had a sense of humor.
		Nash's paper [ The Bargaining Problem, ] is one of the first to apply the axiomatic method to a problem in the social sciences.
	www.fermentmagazine.org /essays/jnash/fnash2.html (4026 words)

	Nash
		John Nash was born in the small Appalachian town of Bluefield, West Virginia, the son of John Nash Sr., an electrical engineer, and Virginia Martin, a teacher.
		However, encouraged by his wife Alicia, Nash persisted in working in a communitarian setting where his eccentricities were unremarked and developed, among other interests, an interest in the calculation of exact values of large numbers, researches which drove him to Princeton's Information Centers, where he developed computer programs (of high quality) for his work.
		They formed part of the nucleus of a group that contacted the Nobel committee and was able to vouch for Nash's ability to receive the award in recognition of his early work.
	cs-exhibitions.uni-klu.ac.at /index.php?id=348 (960 words)

	Sample Chapter for Nash, J.; Kuhn, H.W. and Nasar, S., eds.: The Essential John Nash.
		Nash’s theory of games—especially his notion of equilibrium for such games (now known as Nash equilibrium)—significantly extended the boundaries of economics as a discipline.
		Nash completed “Real Algebraic Manifolds,” his favorite paper and the only one he concedes is nearly perfect, in the fall of 1951 (see chapter 10).
		Embedding means presenting a given geometric object as a subset of a space of possibly higher dimension, while preserving its essential topological properties.
	www.pupress.princeton.edu /chapters/i7238.html (4380 words)

	Re: embedding 3 spatial dimensions - why does it require a 6 dimensional (Site not responding. Last check: 2007-11-03)
		When I look it up, the best I can find is N(N+3)/2: Deane Yang Gunther's proof of Nash's isometric embedding theorem http://www.math.poly.edu/~yang/gunther.ps Anyway, this is the sort of result that proved John "Beautiful Mind" Nash was a good mathematician.
		Chris Clarke showed that any spacetime can be embedded isometrically in some R^n that has constant metric tensor, where n is at most 90 and the number of timelike dimensions is at most 3.
		Clarke gives a general result on embedding a manifold with a pseudo-Riemannian metric whose diagonal form has p 1's and q -1's; the result quoted above is for p=1, q=3.
	www.lns.cornell.edu /spr/2003-03/msg0049144.html (751 words)

	John F. Nash, Jr. - Autobiography
		And later, with "heavy analysis", the problem was solved in terms of embeddings with a more proper degree of smoothness.
		And de Giorgi was first actually to achieve the ascent of the summit (of the figuratively described problem) at least for the particularly interesting case of "elliptic equations".
		It seems conceivable that if either de Giorgi or Nash had failed in the attack on this problem (of a priori estimates of Holder continuity) then that the lone climber reaching the peak would have been recognized with mathematics' Fields medal (which has traditionally been restricted to persons less than 40 years old).
	nobelprize.org /nobel_prizes/economics/laureates/1994/nash-autobio.html (2205 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The proof is based on the Cauchy-Kowalewski Theorem (a theorem on the existence of analytic solutions to analytic differential equations), and, in particular, no implicit function theorem is used.
		The first one is the article Pisanelli, Domingos: The proof of the inversion mapping theorem in a Banach scale, in Complex analysis, functional analysis and approximation theory (Campinas 1984), 281-285 (North Holland Math.
		In this paper there is a proof of an analytic inverse function theorem for something called a Banach scale, but this should be more general than one single Banach space.
	www.math.niu.edu /~rusin/known-math/95/embed.analy (714 words)

	Embedding
		In the geometry of manifolds, a manifold M given abstractly is considered as a candidate to be embedded in Euclidean space of given dimension n (at least dim M, naturally: see invariance of domain).
		That means we look for a submanifold of n-dimensional Euclidean space that is at least homeomorphic to M.
		In domain theory, an embedding is a complete partial order F in [X -> Y] is an embedding if
	www.teachersparadise.com /ency/en/wikipedia/e/em/embedding.html (240 words)

	John Forbes Nash - Wikipedia, the free encyclopedia
		On June 13, 1928, John Forbes Nash was born in the small Appalachian city of Bluefield, West Virginia, the son of John Nash Sr., an electrical engineer and graduate of Texas AandM University, and Virginia Martin, a teacher.
		Alicia admitted Nash to a mental hospital in 1959 for schizophrenia; their son John Charles Martin was born soon afterward but remained nameless for a year because she felt that John should have a say in the name.
		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.
	en.wikipedia.org /wiki/John_Forbes_Nash (1583 words)

	Nash-Moser theorem - Wikipedia, the free encyclopedia
		The Nash-Moser theorem, attributed to mathematicians John Forbes Nash and Jurgen Moser is a generalization of the inverse function theorem on Banach spaces to a class of 'tame' Frechet spaces.
		In contrast to the Banach space case, in which the invertibility of the derivative near at a point is sufficient for a map to be locally invertible, the Nash-Moser theorem requires the derivative to be invertible in a neighbourhood.
		The theorem is widely used to prove local uniqueness for non-linear partial differential equations in spaces of smooth functions.
	en.wikipedia.org /wiki/Nash-Moser_theorem (181 words)

	Game Theory - Nash Equilibrium
		When the 21-year old John Nash wrote his 27-page dissertation outlining his "Nash Equilibrium" for strategic non-cooperative games, the impact was enormous.
		perhaps the noncurvature aspect of space-time is expressed in the nash embedding theorem which would say that there is a submanifold of some R^c(5,11), or some such, homeomorphic, or diffeomorphic, to space-time.
		Here's an introductory sketch of Nash Equilibrium that I found by googling on Nash and equilibrium.
	www.physicsforums.com /showthread.php?p=117802 (545 words)

	History-KOF-Beautiful-Mind-excerpt-1998
		The seven-member prize committee for the 1958 Fields awards was headed by Heinz Hopf, the dapper, genial, cigar-smoking geometer from Zurich who showed so much interest in Nash’s embedding theorem, and included another prominent German mathematician, Kurt Friedrichs, formerly of Göttingen, and then at Courant.
		Roth was a shoo-in; he had solved a fundamental problem in number theory that the most senior committee member, Carl Ludwig Siegel, had worked on early in his career.
		He was an outsider, which one person close to the deliberations thought “might have hurt him.” Moser said, “Nash was somebody who didn’t learn the stuff.
	www.friedrichs.us /History-KOF-Beautiful-Mind-excerpt-1998.html (616 words)

	John Forbes Nash Jr. Biography
		May 20, 1959), remained nameless for a year because Alicia, having just committed Nash to a mental hospital, felt that he should have a say in what to name the baby.
		But, according to Sylvia Nasar's biography of Nash, Alicia referred to him as her "boarder," and they lived "like two distantly related individuals under one roof" until he won the Nobel Prize in 1994, then they renewed their relationship.
		In 1978 he was awarded the John Von Neumann Theory Prize for his invention of non-cooperative equilibriums, now called Nash equilibria.
	www.biographybase.com /biography/Nash_Jr_John_Forbes.html (839 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		This is clearly stated in Theorem I on page 101 of C.B. Allendoerfer and Andre Weil, Amer.
		But their proof involved local embeddings, and hence was not intrinsic.
		The Nash embedding theorem which states that every Riemannian manifold can be found as a submanifold of Euclidean space was proved in the 1950's.
	www.math.purdue.edu /~gottlieb/Papers/AW (279 words)

	List of mathematical proofs - Wikipedia Mirror
		2 Articles devoted to theorems of which a (sketch of a) proof is given
		Theorems of which articles are primarily devoted to proving them
		Articles devoted to theorems of which a (sketch of a) proof is given
	www.wiki-mirror.us /index.php/List_of_mathematical_proofs (181 words)

	57R: Differential topology
		Embedding dimensions of compact smooth manifolds into Euclidean space
		The Nash embedding theorem states that a Riemannian manifold embeds in some R^n isometrically.
		The Morrey-Grauert theorem: any real-analytic manifold admits a real-analytic embedding into some R^n.
	www.math.niu.edu /~rusin/known-math/index/57RXX.html (486 words)

	Differential geometry - ExampleProblems.com
		The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives,integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives.
		These all relate to multivariable calculus; but for the geometric applications must be developed in a way that makes good sense without a preferred coordinate system.
		Certain topological manifolds have no smooth structures at all (see Donaldson's theorem) and others have more than one inequivalent smooth structure (such as exotic spheres).
	www.exampleproblems.com /wiki/index.php/Differential_geometry (1307 words)

	Embedding - Wikipedia, the free encyclopedia
		More explicitly, a map f : X → Y between topological spaces X and Y is an embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y).
		An isometric embedding is a smooth embedding f : M → N which preserves the metric in the sense that g is equal to the pullback of h by f, i.e.
		In field theory, an embedding of a field E in a field F is a ring homomorphism σ : E → F.
	en.wikipedia.org /wiki/Embedding (626 words)

	Differential geometry and topology - Gurupedia
		the extrinsic geometry can be considered as a structure additional to the intrinsic one (see the Nash embedding theorem).
		The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles,
		These all relate to multivariate calculus; but for the geometric applications must be developed in a way that makes good sense without a preferred co-ordinate system.
	www.gurupedia.com /d/di/differential_geometry.htm (938 words)

	Nash embedding theorem - Wikipedia, the free encyclopedia
		The result therefore means that any Riemannian manifold can be visualized as a submanifold of Euclidean space.
		-smooth embeddings and the second for analytic or of class C
		isometrically embedded into an arbitrarily small ball in Euclidean 3-space (from Gauss formula, there is no such C
	en.wikipedia.org /wiki/Nash_embedding_theorem (536 words)

	Nash (print-only)
		In September 1948 Nash entered Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures.
		The outstanding results which Nash had obtained in the course of a few years put him into contention for a 1958 Fields' Medal but since his work on parabolic and elliptic equations was still unpublished when the Committee made their decisions he did not make it.
		The book [2] is highly recommended for its moving account of Nash's mental sufferings.
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Nash.html (3514 words)

	Ralph Howard
		A proof of a folk theorem characterizing the tantrix curves of closed curves is given and extended to higher dimensions and the case of curves symmetric with respect to a group action.(The.ps and.pdf files include a figure not in the.dvi file.)
		This is an edited version of a proof, in the from of exercises with detailed hints, of the classical inverse function and the inverse function theorem for Lipschitz maps between Banach spaces that was given to a graduate class in differential equations as homework.
		A self contained proof of the theorem of E. Hopf that a Riemannian metric on the two dimensional torus without conjugate points is flat.
	www.math.sc.edu /~howard (1489 words)

	The Shape of Madness - - science news articles online technology magazine articles The Shape of Madness (Site not responding. Last check: 2007-11-03)
		In the weeks that followed, I occasionally caught sight of Nash shuffling down the hall in a shabby coat and bright red sneakers or sitting in the cafeteria by himself, staring off into space.
		In the late 1940s and 1950s, John Nash had made discoveries that his peers still use every day—the Nash equilibrium, the Nash embedding theorem—even as they averted their eyes from the man himself.
		Now, four decades after John Nash was lost to mathematics, mathematics itself may hold the key to treating schizophrenia, the mental illness that held his mind hostage.
	discover.com /issues/jan-00/features/featshape (1508 words)

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