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Topic: Brouwer fixed point theorem

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Banach fixed point theorem

Lefschetz fixed point theorem

Fixed point (mathematics)

Combinatorial proof

Fundamental group

List of mathematical proofs

Luitzen Egbertus Jan Brouwer

Frobenius theorem

Nash equilibrium

Number theory

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	PlanetMath: Brouwer fixed point theorem
		The theorem also applies to anything homeomorphic to a closed disk, of course.
		The theorem is not true for an open disk.
		This is version 4 of Brouwer fixed point theorem, born on 2002-06-05, modified 2007-06-24.
	www.planetmath.org /encyclopedia/BrouwerFixedPointTheorem.html (168 words)

	PlanetMath: proof of Brouwer fixed point theorem
		"proof of Brouwer fixed point theorem" is owned by bwebste.
		Cross-references: contradiction, onto, group, isomorphism, contractible, induced, functor, homology, reduced, inclusion map, retraction, boundary, identity, point, well defined, continuous, sphere, line, intersection, ray, fixed point, map, Brouwer fixed point theorem
		This is version 3 of proof of Brouwer fixed point theorem, born on 2002-12-04, modified 2003-09-05.
	www.planetmath.org /encyclopedia/ProofOfBrouwerFixedPointTheorem.html (169 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).
		Brouwer's Theorem made a reapparence in Lionel McKenzie (1959), Hirofumi Uzawa (1962) and, later, in the computational work of Herbert Scarf (1973).
		Obviously, we have a fixed-point at point the intersection of the correspondence with the 45
	cepa.newschool.edu /het/essays/math/fixedpoint.htm (544 words)

	NationMaster - Encyclopedia: Brouwer fixed point theorem (Site not responding. Last check: )
		This is equivalent to the Brouwer fixed point theorem for dimension 2.
		In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some...
		Obviously, we have a fixed-point at point the intersection of the correspondence with the 45
	www.nationmaster.com /encyclopedia/Brouwer-fixed-point-theorem (1259 words)

	NationMaster - Encyclopedia: Fixed point theorem (Site not responding. Last check: )
		By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it does not describe how to find the fixed point (see also Sperner's lemma).
		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.
		Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place.
	www.nationmaster.com /encyclopedia/Fixed_point-theorem (500 words)

	Luitzen Egbertus Jan Brouwer - Definition, explanation
		Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.
		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.
	www.calsky.com /lexikon/en/txt/l/lu/luitzen_egbertus_jan_brouwer.php (371 words)

	Omnipelagos.com ~ article "Brouwer fixed point theorem"
		In mathematics, the Brouwer fixed point theorem is an important fixed point theorem that applies to finite-dimensional spaces and forms the basis for several more general fixed point theorems.
		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, bounded, connected, without holes, etc.).
		The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of D
	www.omnipelagos.com /entry?n=brouwer_fixed_point_theorem (1286 words)

	Fixed Point Theorems
		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.
		Point B represents the point A is mapped to.
		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 (1700 words)

	Brouwer
		One of the major contributors to fixed point theory was L E J Brouwer.
		Brouwer establishes with his theorem that under a continuous mapping of an object to itself there is at least one point where the object inevitably confronts itself (i.e., it contains itself).
		Recently, in the comparison of fixed points of different mappings where the mappings are ordered and increasing for a certain order >=, it has been shown that every fixed point of a lower (higher) mapping has at least one higher (lower) fixed point in a higher (lower) mapping.
	hypatia.math.uri.edu /~kulenm/mth381pr/fixedpoint/fixedpoint.html (1661 words)

	Brouwer fixed point theorem Summary
		Brouwer rejected the validity of his proof for two reasons: first, it is nonconstructive because it does not show how to find any fixed points and second, it relies on the law of the excluded middle.
		Brouwer, and the school of intuitionism, rejected the validity of that law and hence the validity of the above proof.
		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, bounded, connected, without holes, etcetera).
	www.bookrags.com /Brouwer_fixed_point_theorem (2012 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: )
		By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).
		The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology is notable because it gives, in some sense, a way to count fixed points.
		The collage theorem in fractal compression proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=fixed-point_theorem (436 words)

	Springer Online Reference Works
		Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an
		Brouwer's theorem can be generalized to infinite-dimensional topological vector spaces.
		There are effective ways to calculate (approximate) Brouwer fixed points and these techniques are important in a multitude of applications including the calculation of economic equilibria, [a1].
	eom.springer.de /B/b017670.htm (334 words)

	News \| TimesDaily.com \| TimesDaily \| Florence, AL (Site not responding. Last check: )
		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.
		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.
		Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place.
	www.timesdaily.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=fixed-point_theorem (470 words)

	Colloquia and Seminars - UNL - Department of Mathematics (Site not responding. Last check: )
		A point p in M is called a fixed point for f, if f(p)=p.
		Fixed points are very important in many areas: topology, analysis, differential equations, applied mathematics.
		One can say that the theory of fixed points has its origin in the work of the Dutch mathematician L.E.J.Brouwer, at the beginning of the 20th century.
	www.math.unl.edu /pi/colloquia/seminarabstract-schedule-20060204 (303 words)

	Universal Book of Mathematics: list of entries
		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.
		Brower'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).
		This theorem generalizes to higher-dimensional ham sandwiches, when it essentially becomes the Borsuk-Ulam theorem: in n-dimensional space in which there are n globs of positive volume, there is always a hyperplane that cuts all the globs exactly in half.
	www.daviddarling.info /works/Mathematics/mathematics_samples.html (5710 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 Brouwer-Kakutani Fixed Point Theorem
		There is a famous Theorem in modern mathematics, called the Fixed Point Theorem, attributed to L. Brouwer, and later clarified by Kakutani.
		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.
		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)

	Mathematical Programming Glossary Page 2
		The following are fixed point theorems of particular interest in mathematical programming.
		If f is continuous and X is compact and convex, f has a fixed point.
		If f is a contractor, it has a unique fixed point, and successive approximation, x^(k+1)=f(x^k), converges to it from any starting point (x^0).
	orion.math.uwaterloo.ca /~hwolkowi/mirror.d/glossary/second.php?page=fixedpts.html (374 words)

	[No title] (Site not responding. Last check: )
		In higher dimensions, the usual proofs are topological, and are proofs by contradiction (if there were no fixed points, there would be a retraction to the boundary, which is prohibited for topological reasons).
		Brouwer's Intuitionism rejected non-constructive existence proofs, so that his philosophy would force him to dismiss his own mathematics.
		The theorem in 3-dimensional space may be presented imprecisely but viscerally like this: no matter how you stir your coffee, at least one molecule of it has to be returned to its original location.
	www.math.niu.edu /Papers/Rusin/known-math/96/brouwer.fpt (196 words)

	Math Forum Discussions - Re: Fixed point theorem
		fixed point theorem is the easiest of the lot, the one which applies to
		Brouwer fixed point theorem - it applies to convex compact subsets of Banach
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=13&threadID=105267&messageID=540295 (439 words)

	You Are Here - FA+ - Gustavo Aguerre - Ingrid Falk
		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.
		There are effective ways to calculate approximate Brouwer fixed points and these techniques are important in a multitude of applications including the calculation of economic equilibria.
		What Brouwer wants to prove with his theorem (everything in the abstract magic mathematic, that in the end of the chain decides our daily life), is that there is at least one point where an "object" (the mass, you) confronts inevitably it self (it contains it self).
	www.fa-art.pp.se /youarehere.htm (928 words)

	[No title]
		Let me remark about the Theorem that you inquired which is the Brouwer Fixed Point Theorem stating that: (2) Any continuous map f from a closed 2-dimensional disk D into itself has a fixed point.
		Since we assumed that f has no fixed point, f(x) is either to the right of x or f(x) is to the left of x.
		Clearly, the boundary point 0 is such that f(0) is to the right of 0, whereas the boundary point 1 is such that f(1) is to the left of 1.
	www.mathematik.uni-bielefeld.de /~sillke/NEWS/fixed-point-theorem (667 words)

	Brouwer's Fix Point Theorem
		Theorem 1 Every continuous mapping f of a closed n-ball to itself has a fixed point.
		Point x is the limit point of the sequence.
		So the points along this edge of the triangle will be labeled 0 or 1, as required in Sperner's lemma.
	www.scs.carleton.ca /~maheshwa/MAW/MAW/node3.html (637 words)

	Mahalanobis
		One tool he developed, known as the Kakutani fixed-point theorem, was a key step in the original proof of the existence of Nash equilibria, the theorem for which John Forbes Nash received his Nobel Prize.
		Kakutani's theorem is also used to prove a famous 1954 theorem by the economists Kenneth J. Arrow and Gérard Debreu, which says that there are prices for goods that balance supply and demand in a complex economy.
		The most important fixed point theorem is Brouwer's (deals with functions); the extention of this theorem to correspondences is given by Kakutani's fixed point theorem.
	mahalanobis.twoday.net /stories/308405 (625 words)

	Brouwer Fixed Point Theorem -- Mudd Math Fun Facts
		One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem.
		In dimension three, Brouwer's theorem says that if you take a cup of coffee, and slosh it around, then after the sloshing there must be some point in the coffee which is in the exact spot that it was before you did the sloshing (though it might have moved around in between).
		Fixed point theorems are some of the most important theorems in all of mathematics.
	www.math.hmc.edu /funfacts/ffiles/20002.7.shtml (443 words)

	hairy
		There is a theorem in simplicial homology theory which states that any continuous tangent field on a 2-sphere is null at least in a point.
		Mathematically literate hackers tend to associate the term ‘hairy’ with the informal version of this theorem; “You can't comb a hairy ball smooth.
		The adjective ‘long-haired’ is well-attested to have been in slang use among scientists and engineers during the early 1950s; it was equivalent to modern hairy senses 1 and 2, and was very likely ancestral to the hackish use.
	www.catb.org /~esr/jargon/html/H/hairy.html (211 words)

	[No title]
		As an answer to Peter McBurney's question, there are generalizations to Brouwer's fixed point theorem as the following.
		Therefore, by the Lefschetz fixed point theorem, f has a fixed point.
		qed There are generalizations of the Lefschetz fixed point theorem to situations similar to that of Brouwer's theorem of 1960.
	www.lehigh.edu /~dmd1/cp120.txt (315 words)

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