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Topic: Contraction mapping

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	Waterloo Fractal Coding and Analysis Page
		Systems of contraction mappings had been considered previously by a number authors for various purposes.
		In the special case the the IFSM maps are dyadic and the wavelets are nonoverlapping Haar wavelets, then S scales subtrees and copies them onto lower subtrees as well as placing a "condensation" tree at the top - what is known as the fractal-wavelet transform.
		The mapping W induces a mapping T in the spatial domain which is equivalent to a local IFSM operator with condensation functions:
	links.uwaterloo.ca /fractals.home.html (3956 words)

	Contraction mapping - Biocrawler (Site not responding. Last check: 2007-10-26)
		In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number k < 1 such that, for all x and y in M,
		Every contraction mapping is Lipschitz continuous and hence uniformly continuous.
		Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))),...
	www.biocrawler.com /encyclopedia/Contraction_mapping (231 words)

	Collage Theorem, Iterated Function Systems
		Functions map points to points but also comprise (function) spaces where each is considered an indivisible one.
		Contraction mappings are defined on metric spaces by the property that they decrease the distance between points.
		In complete spaces, any contraction mapping has a unique fixed point that can be found by (necessarily) convergent iterations.
	www.cut-the-knot.org /ctk/ifs.shtml (1017 words)

	The Naked Hoof - Article: Just What Is Hoof Contraction
		A Contracted hoof is one whose form has been altered so that part or all of the hoof no longer expands on weight bearing or even becomes narrower than before.
		These types of contraction are recognised by the Strasser Method where the unphysiological (contracting) force is acting on the hoof and the region of the corium in an abnormal way and must be freed from this pressure.
		One of the major problems in identifying a contracted hoof is that many farriers and veterinarians have no reference point in being able to say what is a normal or natural hoof.
	www.thenakedhoof.com.au /html/article-WhatIsHoofContraction.htm (596 words)

	PlanetMath: Banach fixed point theorem
		is said to be a contraction mapping if there is a constant
		Theorem 1 (Banach Fixed Point Theorem) ; Every contraction has a unique fixed point.
		There is an estimate to this fixed point that can be useful in applications.
	www.planetmath.org /encyclopedia/BanachFixedPointTheorem.html (286 words)

	PlanetMath: Banach fixed point theorem
		is said to be a contraction mapping if there is a constant
		Theorem 1 (Banach Fixed Point Theorem) ; Every contraction has a unique fixed point.
		There is an estimate to this fixed point that can be useful in applications.
	planetmath.org /encyclopedia/BanachFixedPointTheorem.html (286 words)

	RAND \| Papers \| A Theorem on Contraction Mapping.
		A proof that the conclusion of a well-known theorem of Banach holds more generally from a condition of weakly uniformly strict contraction.
		The following fixpoint theorem is demonstrated: Let (X,d) be a complete metric space and f a mapping of X into itself.
		If, for all e greater than zero, there exists e' greater than zero such that d(x,y) is between e and e + e' and d(f(x),f(y)) is less than e, then f has a unique fixpoint z.
	www.rand.org /pubs/papers/P3993 (304 words)

	Contraction mapping - ExampleProblems.com
		In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number k < 1 such that, for all x and y in M,
		Every contraction mapping is Lipschitz continuous and hence uniformly continuous.
		Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))),...
	www.exampleproblems.com /wiki/index.php?title=Contraction_mapping&printable=yes (221 words)

	Cut The Knot!
		Contraction mappings between metric spaces are defined by the property that they decrease the distance between points.
		In a complete space, any contraction mapping has a unique fixed point that can be found by (necessarily) convergent iterations.
		Two rows of edit controls give you a direct access to the values of a,b,e (the first row) and c,d,f (the second row.) Beware: the coordinate system is the usual one for computer graphics: the Y axis grows downwards.
	www.maa.org /editorial/knot/ifs.html (1043 words)

	Inertial Mass as an Electromagnetic Phenomena
		Poincare proved that for two inertial systems each in motion through the aether, the result of these effects is to render the Lorentz transforms valid for converting between measurements made in them by a pair of inertial observers provided that their axes were aligned and clocks synchronised in the specified manor.
		The normal way of describing the Lorentz contraction is to look at the field at some point in space specified relative to the centre of the charge.
		Since the contraction was derived from equations of potential, it is reasonable to assume that the equipotential surfaces suffer a contraction.
	users.powernet.co.uk /bearsoft/P4ReIn.html (3042 words)

	U N T Resource Magazine: Heart Research
		With the assistance of a group of graduate and undergraduate students, he is studying the interaction of proteins responsible for heart muscle contraction.
		"When we understand the mechanism of muscle contraction and its regulation at the atomic level, it will be possible to have a model that includes the molecular changes known to give rise to this disease," Root explains.
		In Root's research laboratory, he and his assistants are focusing on myosin and actin, two proteins that bind to one another and affect the contraction of the heart.
	www.unt.edu /resource/04heartfeature.htm (1178 words)

	Contraction mapping (Site not responding. Last check: 2007-10-26)
		Abacci > Abaccipedia > Co > Contraction mapping
		In mathematics, a contraction mapping, or contraction, on a metric space M is a function f from M to itself, with the property that there is some real number k
		Downscaling of remotely sensed soil moisture with a modified fractal interpolation method using contraction mapping and ancillary data
	www.abacci.com /wikipedia/topic.aspx?cur_title=Contraction_mapping (166 words)

	Mapping, mapping standards, mapping program (Site not responding. Last check: 2007-10-26)
		Mapping Hacks is a collection of one hundred simple techniques available to developers and power users who want to draw digital maps.
		Map index designed to assist cities and counties in guarding the public against earthquake-induced ground failure mapping.
		Getmapping is a UK aerial photography company offering a range of aerial maps, digital aerial photos and aerial photo prints for city.
	computer-mind.com /mapping.html (358 words)

	Wong, James Sai-Wing (1964-05-01) Generalizations to the converse of contraction mapping principle. ...
		Wong, James Sai-Wing (1964-05-01) Generalizations to the converse of contraction mapping principle.
		We find sufficient conditions on [...] in order that [...] be contractive on X. In the case when [...] is generated by a finite number of mutually commuting mappings [...], [...],.
		The resulting statement is the following generalization of the converse of contraction mapping principle: Theorem C.
	etd.caltech.edu /etd/available/etd-02102004-095235 (265 words)

	SSRN-Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case? by John Rust, Joseph Traub, ... (Site not responding. Last check: 2007-10-26)
		This paper analyzes the complexity of the contraction fixed point problem: compute an approximation to the fixed point V* = I(V) of a contraction mapping I* that maps a Banach space of continuous functions of variables into itself.
		We focus on quasi linear contractions where I* is a nonlinear functional of a finite number of conditional expectation operators.
		In the absence of further restrictions on the domain of I*, the quasi linear fixed point problem is subject to the curse of dimensionality, i.e., in the worst case the minimal number of function evaluations and arithmetic operations required to compute an approximation to a fixed point increases exponentially in.
	papers.ssrn.com /sol3/papers.cfm?abstract_id=312232 (400 words)

	Warwick: DCS: Reports and Theses
		S.G. Matthews, The Cycle Contraction Mapping Theorem (August 1, 1992).
		The Cycle Contraction Mapping Theorem is both an extension of Wadge's cycle sum theorem for Kahn data flow and a generalisation of Banach's contraction mapping theorem to a class of quasi metric spaces definable using the symmetric Partial Metric distance function.
		This work provides considerable evidence that it is possible after all to construct a metric theory for Scott style partial order domains.
	www.dcs.warwick.ac.uk /reports/228.html (133 words)

	Creating Fractals using Iterated Function Systems
		If a map has a high probability of being chosen, then the region where the contraction mappings maps the point too will be more dense than the other regions.
		The user can specify the color they want for each mappings, but the goal here is to identify what each mapping accomplishes, so it is best to associate a unique color to each mapping wi.
		The mappings used to generate the sierpinski's triangle seems to map points into different region of space, while the two mappings used to create the dragon seems to share a region in space.
	www.cs.mcgill.ca /~ndutil/paper.html (2786 words)

	The GAR-3 Muscarinic Receptor Cooperates With Calcium Signals to Regulate Muscle Contraction in the Caenorhabditis ...
		In contrast, the gpb-2 mutant pharynx (right) begins in a partially open position (the two-headed arrow spans the lumen and is the same size in all four panels).
		The muscles contract and open the lumen further at 99 and 165 msec, but then fail to fully relax.
		The corpus muscles contract at 66 and 198 msec (arrows), expanding the lumen, but relaxation of the muscles leaves the lumen open.
	www.genetics.org /cgi/content/full/167/2/633 (5541 words)

	The Naked Hoof
		In a horse at rest, the tendons of the extensor and flexor apparatus are in an energy neutral balance.
		In the leg, blood is pumped upward by the hooves and the joints.
		Lever forces on a slanted, truncated cone lead to expansion with physiologically correct hoof form, and to contraction with unphysiological hoof form.
	www.thenakedhoof.com.au /html/article-StrasserMethod.htm (305 words)

	Iterated Function Systems (Site not responding. Last check: 2007-10-26)
		Def: A contraction mapping on the metric space (X,d) is a mapping f:X->X from the metric space into itself, such that:
		Contraction mappings are the elementary building blocks of IFSs, but they are un-interesting by themselves (as seen by the above theorem).
		When the transformations of the IFS can be represented as a matrix transformation on a vector plus a translation, there are two fairly simple ways to compute the attractor: the deterministic algorithm, and the random iteration algorithm.
	www.geom.uiuc.edu /java/IFSoft/IFSs (669 words)

	PlanetMath: proof of existence and unicity of Self-similar fractals
		is a contraction on the complete metric space
		Cross-references: Banach fixed point theorem, metric space, contraction, mapping, complete, Hausdorff metric inherits completeness, Hausdorff distance
		This is version 1 of proof of existence and unicity of Self-similar fractals, born on 2006-07-20.
	planetmath.org /encyclopedia/ProofOfExistenceAndUnicityOfSelfSimilarFractals.html (99 words)

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