Fixed-point theorem - Factbites

Fixed-point theorem - Factbites
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Topic: Fixed-point theorem

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Related Topics

Banach fixed point theorem

Brouwer fixed point theorem

Frobenius theorem

	Fixed-point theorem - Wikipedia, the free encyclopedia
		The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
		In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms.
		Every lambda expression has a fixed point, and a fixed point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression.
	en.wikipedia.org /wiki/Fixed-point_theorem (442 words)

	Brouwer fixed point theorem - Wikipedia, the free encyclopedia
		This is equivalent to the Brouwer fixed point theorem for dimension 2.
		The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis.
		Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, the theorem equally applies if the domain is not the closed unit ball itself but some set homeomorphic to it (and therefore also closed, connected, without holes, etcetera).
	en.wikipedia.org /wiki/Brouwer_fixed_point_theorem (899 words)

	PlanetMath: Banach fixed point theorem
		This is version 17 of Banach fixed point theorem, born on 2002-03-07, modified 2004-02-09.
		Theorem 1 (Banach Fixed Point Theorem) Every contraction has a unique fixed point.
		Cross-references: converges, inequality, sequence, fixed point, estimate, theorem, function, metric space, complete
	planetmath.org /encyclopedia/BanachFixedPointTheorem.html (286 words)

	Fixed Point Theorems
		Therefore, since the assumption of no fixed point leads to a contradiction of the No Retraction Theorem there must be at least one fixed point.
		Fixed points are of interest in themselves but they also provide a way to establish the existence of a solution to a set of equations.
		A physical example of a fixed point of a mapping is the center of a whirlpool in a cup of tea when it is stirred.
	www.applet-magic.com /fixed.htm (1625 words)

	PlanetMath: Brouwer fixed point theorem
		This is version 3 of Brouwer fixed point theorem, born on 2002-06-05, modified 2003-09-05.
		See Also: fixed point, Schauder fixed point theorem
		has the single fixed point at 0; dropping it from the domain yields a map with no fixed points.
	planetmath.org /encyclopedia/BrouwerFixedPointTheorem.html (162 words)

	Brouwer fixed-point theorem
		Brouwer's theorem insists that there must be some point in the coffee that is in exactly the same spot as it was before you started stirring (though it might have moved around in between).
		It states that a continuous function from an n-ball into an n-ball (that is, any way of mapping points in one object that is topologically the same as the filling of an n-dimensional sphere to another such object) must have a fixed point.
		Brouwer's theorem says that there must be at least one point on the top sheet that is in exactly the same position relative the bottom sheet as it was originally.
	www.daviddarling.info /encyclopedia/B/Brouwer_fixed-point_theorem.html (285 words)

	Brouwer fixed point theorem - Wikipedia, the free encyclopedia
		This is equivalent to the Brouwer fixed point theorem for dimension 2.
		The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis.
		This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet right beneath it.
	en.wikipedia.org /wiki/Brouwer_fixed_point_theorem (561 words)

	Brouwer
		One of the major contributors to fixed point theory was L E J Brouwer.
		Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an n-dimensional topological vector space and is extensively employed in proofs of theorems on the existence of solutions of various equations; and the theorem has been generalized to infinite-dimensional topological vector spaces.
		Not coincidentally at this point, we raise to your consciousness that the flow lines of the winds on the earth's surface constitute a continuous mapping of that surface to another point thereon.
	hypatia.math.uri.edu /~kulenm/mth381pr/fixedpoint/fixedpoint.html (1661 words)

	The Brouwer-Kakutani Fixed Point Theorem
		That is, self-replicating molecules are an instantiation of the Fixed Point Theorem where the map is the one determined by the laws of Physics and Chemistry.
		There is a famous Theorem in modern mathematics, called the Fixed Point Theorem, attributed to L. Brouwer, and later clarified by Kakutani.
		Nevertheless, such a Fixed Point does seem to be a point of attraction of the Advance of Civilization, so the best way for an individual to lead society in that direction is by setting an example that is worthy of imitation.
	underground.musenet.org:8080 /utnebury/fixed.point.html (1029 words)

	Fixed-Point Theorems
		One of the oldest fixed-point theorems - Brouwer's - was developed in 1910 and already by 1928, John von Neumann was using it to prove the existence of a "minimax" solution to two-agent games (which translates itself mathematically into the existence of a saddlepoint).
		The existence proofs of Arrow and Debreu (1954), McKenzie (1954) and others gave Kakutani's Fixed Point Theorem a central role.
		Fixed-point theorems are one of the major tools economists use for proving existence, etc.
	cepa.newschool.edu /het/essays/math/fixedpoint.htm (544 words)

	Banach fixed point theorem - ExampleProblems.com
		The Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points.
		The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions.
		The theorem is named after Stefan Banach (1892-1945), and was first stated by Banach in 1922.
	www.exampleproblems.com /wiki/index.php/Banach_fixed_point_theorem (466 words)

	Reactions to proofs of Gersten's Fixed Point Theorem
		Analyzing the Goldstein-Turner's proof of Gersten's Theorem [GT], we obtain in [3] a bound for the rank of the fixed-point subgroup.
		Reactions to proofs of Gersten's Fixed Point Theorem
		On the rank of the fixed point set of automorphisms of free groups (with W. Imrich and E. Turner), Circles and Rays (G. Hahn et al., eds) Kluwer, Dordrecht 1989.
	www.cse.ogi.edu /~krstic/summary/node6.html (234 words)

	lefschetz.fpt
		If a space is contractible, does a fixed point theorem hold, i.e.
		The Lefschetz fixed-point theorem answers some of these: it implies that if X is a path-connected compact polyhedron for which the homology groups H_n(X) are finite for n > 0, then every continuous f: X --> X has a fixed point.
		Date: Sat, 2 Nov 1996 08:49:53 -0800 x-no-archive: yes This brought up the question: are there some conditions on homology groups for a space such that a fixed point theorem holds?
	www.math.niu.edu /~rusin/known-math/96/lefschetz.fpt (234 words)

	Luitzen Egbertus Jan Brouwer - Open Encyclopedia
		The Brouwer fixed point theorem is named in his honor.
		He proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, the treatment of general continuous mappings.
		Brouwer adhered to an intuitionist philosophy of mathematics, which is sometimes characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning, and wrote books on the subjects mentioned above in which he proceeded accordingly.
	open-encyclopedia.com /L._E._J._Brouwer (215 words)

	Brouwer's Fix Point Theorem
		Theorem 1 Every continuous mapping f of a closed n-ball to itself has a fixed point.
		(This proof is based on the survey paper of P.J.S.G. Ferreira on Fixed point problems - an introduction).
		We want to show that a continuous mapping from a closed triangle T (or circle) to a closed triangle has a fixed point.
	www.scs.carleton.ca /~maheshwa/MAW/MAW/node3.html (637 words)

	Fixed Point Iteration
		The following two theorems establish conditions for the existence of a fixed point and the convergence of the fixed-point iteration process to a fixed point.
		is said to be a repelling fixed point and the iteration exhibits local divergence.
		is a sequence generated by fixed point iteration.
	math.fullerton.edu /mathews/n2003/FixedPointMod.html (240 words)

	Fixed Point Properties of Plane Continua
		Theorem 3: Suppose Q is a continuum and f:T(Q) ® C is a fixed point free map such that f(Q) Then:
		closed curve J, the fixed point index of f on J is defined to be the winding number of the map g(z)
		Q then f has a fixed point in T(Q).
	math.uc.edu /~bellh/FPPOPC/fppopc.htm (2146 words)

	Fixed Point Theorem
		for the existence of a fixed point, and the convergence of the fixed-point iteration process
		A useful application of iteration is for finding fixed points.
		The condition for existence of a fixed pt.
	jewel.morgan.edu /~rpustam/fixed.htm (283 words)

	The Atiyah-Bott fixed point theorem
		Specializing the Atiyah-Bott fixed point theorem to the de Rham complex, one recovers the classical Lefschetz fixed point theorem.
		When applied to other geometrically interesting elliptic complexes, Atiyah and Bott obtained new fixed point theorems, such as a holomorphic Lefschetz fixed point theorem in the complex analytic case and a signature formula in the Riemannian case.
		In the Sixties Atiyah and Bott proved a far-reaching generalization of the Lefschetz fixed point theorem ([42], [44]).
	www.math.harvard.edu /history/bott/bottbio/node18.html (366 words)

	Fixed Points
		For each theorem, X is assumed to be nonempty, f denotes a function from X to X, and F a point-to-set map from X to subsets of X. A fixed point of f is x in X such that f(x)=x.
		The following are fixed point theorems of particular interest in mathematical programming.
		A fixed point of F is x in X such that x is in F(x).
	carbon.cudenver.edu /~hgreenbe/glossary/fixedpts.html (364 words)

	Shizuo Kakutani's fixed point theorem (kottke.org)
		Fixed-point theorems are about continua, and aren't going to make any sense for finite sets.
		Here's another fixed-point theorem, with a nice 2-d example.
		My favorite example of the intermediate value theorem was that you can easily show that there are two antipodal points on earth that have the same temperature right now - in fact, there is a whole band of them.
	www.kottke.org /remainder/04/08/6309.html (504 words)

	Mudd Math Fun Facts: Brouwer Fixed Point Theorem
		Fixed point theorems are some of the most important theorems in all of mathematics.
		One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem.
		There are many proofs of the Brouwer fixed point theorem.
	www.math.hmc.edu /funfacts/ffiles/20002.7.shtml (452 words)

	A MOST FUNDAMENTAL FIXED POINT THEOREM
		(a) is a fixed point of f " is meant that f(f
		cannot be a fixed point of f hence (1) implies that f
		has no fixed point and let p and q be any two distinct ordinal numbers.
	www.math.ucdavis.edu /~suh/abian/abian-themostfixed.html (325 words)

	fixed.pt
		I don't know about necessary conditions, but generally speaking, there is Schauder's theorem that states that: if B is a Banach space (possibly infinite dimentional), if C is convex and compact (and nonempty), if T is a continuous function from C to C, then T has a fixed point.
		Then the theorem says that if T is a contraction and X is a complete metric space then T has a unique fixed point.
		In fact, this theorem is used to prove the existence of nonlinear partial differential equations (in a rather non constructive way): the above T is then a differential operator on some functional space.
	www.math.niu.edu /~rusin/known-math/95/fixed.pt (886 words)

	When Index Equals Content
		Therefore, the theorem claims that any continuous transformation of a triangle into itself has a fixed point.
		Such P is known as a fixed point of f.
		But, if we ran into a point P where v(P)=0 we may stop here since then f(P)-P=0 and P is the fixed point of f.
	www.cut-the-knot.org /do_you_know/poincare.shtml (1359 words)

	Fixed point theorem -- CFD-Wiki, the ultimate CFD reference
		The fixed point theorem is a useful tool for proving existence of solutions and also for convergence of iterative schemes.
		Hence if the matrix is diagonally dominant then the fixed point theorem assures us that the Jacobi iterations will converge.
		We will apply the fixed point theorem to show the convergence of Jacobi iterations for the numerical solution of the linear algebraic system
	www.cfd-online.com /Wiki/Fixed_point_theorem (313 words)

	BanachFPT.html
		The Banach Fixed Point Theorem is a very good example of the sort of theorem that the author of this quote would approve.
		There are any number of important applications of The Banach Fixed Point Theorem.
		Tell us that the fixed point is the limit of a certain computable sequence.
	www.umsl.edu /~siegel/SetTheoryandTopology/BanachFPT.html (208 words)

	Reconstructing Brouwer
		. This is known as Brouwer's fixed-point theorem, and it has been used to establish fundamental existence theorems in many different branches of mathematics (e.g., the theory of differential equations).
		For a simple one-dimensional interval, the fixed-point theorem is fairly obvious, because any continuous function x' = f(x) of an interval, say from 0 to 1, into that same range must somewhere meet the line representing x' = x, as illustrated below.
		According to the constructivist point of view, arguments of the "either/or" are not automatically accepted, because one of the basic tenets of constructivism is the rejection of the free use of the "law of the excluded middle". A well-known example is the "proof" that there exist irrational numbers x and y such that x
	www.mathpages.com /home/kmath262/kmath262.htm (847 words)

	The Fixed Point Theorem
		The point p is a fixed point for f
		Then there exists a point p in [0,1] such that f(p) = p, and p is called a fixed point for f.
		So assume the points 0 and 1 are not fixed points.
	www.geom.uiuc.edu /~hesse/summer95/HTML/GenHTML/fix.html (108 words)

	lefschetz.fpt
		The Lefschetz fixed-point theorem answers some of these: it implies that if X is a path-connected compact polyhedron for which the homology groups H_n(X) are finite for n > 0, then every continuous f: X --> X has a fixed point.
		If a space is contractible, does a fixed point theorem hold, i.e.
		And, does a fixed point theorem hold for the projective plane?
	www.math.niu.edu /~rusin/known-math/96/lefschetz.fpt (166 words)

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