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Topic: Homotopic

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	PlanetMath: antipodal map on $S^n$ is homotopic to the identity if and only if $n$ is odd
		PlanetMath: antipodal map on $S^n$ is homotopic to the identity if and only if $n$ is odd
		Applying the lemma, we conclude that the antipodal map is homotopic to the identity.
		Cross-references: coordinates, point, maps, degree, reflections, composition, even, odd, identity, homotopic, subspace, identity map, antipodal map, homotopy, field, unit vector
	www.planetmath.org /encyclopedia/AntipodalMapOnSnIsHomotopicToTheIdentityIfAndOnlyIfNIsOdd.html (200 words)

	Homotopy - Wikipedia, the free encyclopedia
		In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
		An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
		Being homotopic is an equivalence relation on the set of all continuous functions from X to Y.
	en.wikipedia.org /wiki/Homotopy (1019 words)

	Homotopy : Homotopic (Site not responding. Last check: 2007-10-19)
		In topology, two continuous functions from one topological spaceto another are called homotopic if one can be "continuously deformed" intothe other, such a deformation being called a homotopy between thetwo functions.
		In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
		It was a startled frightened scream that ended starlight he could make out her head and shoulders disappearing from its hook and flung it aft.
	www.termsdefined.net /ho/homotopic.html (640 words)

	Homotopic grafts of septal neurons combined to polymeric hydrogels placed into a fimbria-fornix lesion cavity attenuate ... (Site not responding. Last check: 2007-10-19)
		Homotopic grafts of septal neurons combined to polymeric hydrogels placed into a fimbria-fornix lesion cavity attenuate locomotor hyperactivity but not mnemonic dysfunctions in rats.
		Nevertheless, in rats with both intraseptal (homotopic) grafts and a hydrogel implant, the locomotor activity did no longer differ from that found in sham-operated controls.
		They further suggest that septal neurons grafted homotopically and/or neurons from the host brain are able to elongate axonal processes through a PHPMA substrate up to the hippocampus.
	www.hvsimage.com /papers/PMID-%2011673668.htm (394 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
		Being homotopic is an equivalence relation on the set of all continuous functions from X to Y.
		No closed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=homotopy_theory (1119 words)

	Homotopic Functions and Homotopy Classes
		Homotopic is reflexive, symmetric, and transitive, and forms an equivalence relation.
		Thus the homotopic functions from R into S form equivalence classes, and these are called homotopy classes.
		A homotopic reparameterization of the domain implies a new function that is homotopic to the original.
	www.mathreference.com /at,topy.html (1409 words)

	MedFriendly.com: Homotopic
		Homotopic means pertaining to or occurring at the same part or place of the body.
		Homotopic comes from the Greek word "homos" meaning "the same," and the Greek word "topos" meaning "place." Put the words together and you get "the same place."
		You may not reprint or redisplay this material for commercial use without the express written consent of MedFriendly.com.
	www.medfriendly.com /homotopic.html (100 words)

	Math Forum Discussions
		> > Is p homotopic to p_a with respect to, relative to { 0,1 }?
		> loop is homotopic to the constant loop rel {0, 1}.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/message.jspa?messageID=5232728&tstart=0 (286 words)

	Homotopic glial regulation of striatal projection neuron differentiation. (Site not responding. Last check: 2007-10-19)
		Homotopic glial regulation of striatal projection neuron differentiation.
		Homotopic glial regulation of striatal projection neuron differentiation.This is not the case when the cells are co-cultured with glia derived from the adjacent telencephalic region, the medial ganglionic eminence.
		Moreover, expression of the striatal projection neuron marker, dopamine- and cAMP-regulated phosphoprotein (DARPP-32) was significantly enhanced in neurons cultured on the homotopic glia.
	www.pdg.cnb.uam.es /UniPub/iHOP/gp/9798037.html (131 words)

	Conference in Goemetric Topology
		The classical Wecken theorem claims that any self-map $f:M\to M$ of a compact manifold of dimension $\ge 3$ is homotopic to a map having exactly $N(f)$ fixed points where $N(f)$ denotes the Nielsen number.
		We prove that every self-map $f:M\to M$ of a compact PL-manifold of dimension $\ge 3$ is homotopic to a map realizing this number i.e.
		In particular (for $NF_n(f)=0$) the map $f$ is homotopic to map with no n-periodic points iff all Nielsen numbers $N(f^k)$, for all $k$ dividing $n$, disappear.
	www.math.uiowa.edu /~wu/gtc/abs/JerzJezi.htm (131 words)

	Physics Help and Math Help - Physics Forums - topology question
		X is homotopic to Y if the identity map of X factors up to homotopy through Y. Note, it must be both ways.
		Homotopies arise because homotopic maps induce the same maps on chain complexes of the resolutions of spaces, where you might see another definition for homotopic maps.
		I doubt there is any name for spaces where the are maps f,g with fg homotopic to id, and gf not, simply because that does not seem a very restrictive criterion.
	www.physicsforums.com /printthread.php?t=14028 (2301 words)

	[No title]
		But it is "homotopic" to the identity, by which I mean that there is a continuously varying family of continuous functions F_t from D to itself, such that F_0 = fg and F_1 is the identity on D. Simply let F_t be scalar multiplication by t!
		Then we say that two spaces X and Y are homotopic if there are continuous functions f: X -> Y, g: Y -> X which are inverse up to homotopy, i.e., such that gf and fg are homotopic to the identity on X and Y, respectively.
		The main goal in homotopy theory is to understand when functions are homotopic and when spaces are homotopic.
	math.ucr.edu /home/baez/twf_ascii/week54 (2370 words)

	CDC-980402 (Site not responding. Last check: 2007-10-19)
		With suitable conditions on the enabling regions and using a suitable metric, we construct a homotopy on the set of solutions and use the homotopy to form an equivalence relation on the trajectories.
		We show the relationship between region equivalence introduced in [1] and homotopic equivalence.
		The tools needed for studying homotopic equivalence are the same as for obtaining continuity with respect to initial conditions.
	nt1.rsip.lsu.edu /cebopenweb/conferences/cdc98/program/Manuscripts/CDC-980402.htm (154 words)

	homotopic - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "homotopic" is defined.
		Homotopic : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include homotopic: chain homotopic, homotopic algebra
	www.onelook.com /?ls=a&w=homotopic (111 words)

	Science Fair Projects - Simplicial approximation theorem
		It applies to mappings between spaces that are built up from simplices — that is, finite simplicial complexes.
		The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (affine-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision of the simplices of both the domain and range, and (ii) replacement of the actual mapping by a homotopic one.
		This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness).
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Simplicial_approximation_theorem (408 words)

	Math Forum Discussions - Re: Re: Homotopic algorithm to solve a system of equations
		Two such maps are said to be homotopic if one can be
		> not be the same as the roots of a "homotopic" equation f==0.
		Homotopic algorithm to solve a system of equations
	www.mathforum.com /kb/thread.jspa?forumID=79&threadID=1391322&messageID=4762831 (1209 words)

	Morphogenesis of Callosal Arbors in the Parietal Cortex of Hamsters -- Hedin-Pereira et al. 9 (1): 50 -- Cerebral Cortex
		A dense plexus is detected in the homotopic cortex (thick arrow), but scattered fibers are also seen extending into other regions of the contralateral hemisphere.
		The arbor is developing in the infragranular layers of the homotopic cortex while simple collaterals extend to the supragranular laminae (layer 4 is shown for orientation).
		Around P13 (B), the arbor is elaborated in the homotopic cortex (asterisk), while the lateral extension of the axon st ill persists.
	cercor.oxfordjournals.org /cgi/content/full/9/1/50 (8248 words)

	Robert Mettin: Homotopic vector fields (Site not responding. Last check: 2007-10-19)
		It turns out that homotopic vector fields like v(z,a) can generate reasonable interpolations from the parent attractors.
		[7k] to listen to a homotopic switching between different chaotic sound generators.
		Homotopic switching can prevent unwanted jumps (divergencies) of the chaotic oscillators.
	www.physik3.gwdg.de /~robert/hom.html (223 words)

	RSNA 2004 - RSNA Event
		RSNA 2004 > The Role of fMRI in Establishing Homotopic Cortical...
		FMRI is a clinical tool for preoperative mapping of eloquent cortex in patients with brain tumors and other resectable lesions, and has generated excitement in the study of lesion-induced cortical reorganization.
		The exhibit emphasizes the limitations of fMRI in establishing that homotopic cortical reorganization has occurs and that alternative localization methods are required to corroborate fMRI in surgical patients.
	rsna2004.rsna.org /rsna2004/V2004/conference/event_display.cfm?id=66601&p_navID=272&em_id=4415694 (234 words)

	Homotopy
		They are always homotopic, but one loop on a torus may not be deformable into another.
		Two rubber bands circling a broomstick in opposite directions cancel each other out when thus multiplied, and the result is homotopic to a point.
		A to be either homotopic to a point or to a loop about the puncture.
	www.cap-lore.com /MathPhys/Homot/Homotopy.html (374 words)

	Real-Time Replanning in High-Dimensional Configuration Spaces Using Sets of Homotopic Paths - Brock, Khatib ... (Site not responding. Last check: 2007-10-19)
		The path is augmented by a set of paths homotopic to it.
		Effectively, this corresponds to delaying part of the planning operation for the homotopic paths until...
		# In homotopic deformations to preplanned trajectories are computed which enable the robot to circumvent unmodeled obstacles; however this...
	citeseer.lcs.mit.edu /brock00realtime.html (511 words)

	Algebraic Topology: Homotopy
		Two maps f,g from X to Y are called homotopic if there exists a map F from X × I to Y such that F(x,0) = f(x) and F(x,1) = g(x) for all x.
		Given two spaces X,Y, and a map f from X to Y, let [f] denote the homotopy class of f, that is, the set of all maps from X to Y homotopic to f.
		The subspace A is called a deformation retract of X if there is a map r from X onto A that is homotopic to the identity on X relative to A.
	www.win.tue.nl /~aeb/at/algtop-3.html (2011 words)

	The fundamental group
		(c) This is homotopic to a loop path that does not wind around at all.
		All paths that ``wrap around'' an even number of times are homotopic.
		Likewise, all paths that wrap around an odd number of times are homotopic.
	msl.cs.uiuc.edu /planning/node155.html (907 words)

	Homotopy
		We call f nullhomotopic or inessential if it is homotopic to a constant map.
		Deduce that being homotopic is a transitive relation on paths and on loops in any space.
		The following X,Y are homotopically equivalent spaces which are not homeomorphic in the usual topologies (cf.
	www.ma.umist.ac.uk /kd/knots/node4.html (404 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		Also, homotopic mappings are easily adaptable to algebraic grid-generating techniques.
		HOMAR is a computer code which uses this homotopic procedure to produce two-dimensional grids in cross-sectional planes, which are then stacked to produce quasi-three-dimensional grid systems for aerospace configurations.
		Once the input geometry is specified, outer boundaries are defined for each cross-section, and the code proceeds to bridge the gap between the inner and outer boundaries by a family of transition lines produced by homotopic mapping between the surfaces.
	www.nttc.edu /cosmic/abstracts/lar-14756.html (431 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		Here's a favorite counterexample: Let A be a nonempty space whose unreduced suspension SA is not contractible.
		SA (CA is the cone.) Any two maps from X to Y will be "pointwise" homotopic.
		But one map induces a homotopy equivalence of the hocolims and another induces a map homotopic to a constant.
	www.lehigh.edu /~dmd1/tg913.txt (118 words)

	Nielsen Theory by Robert F. Brown
		The customary procedure of Nielsen theory consists of defining an equivalence relation on the set of solutions and then identifying the "essential" equivalence classes in such a way that the Nielsen number, defined as the number of essential classes, is a homotopy invariant lower bound for the minimum number.
		The Nielsen coincidence number is a lower bound for the minimum number of solutions to the equation f'(x) = g'(x) among all maps f' homotopic to f and g' homotopic to g and, if n is not 2, it equals that minimum.
		The Nielsen root number is a lower bound for the number of solutions to g(x) = c for all maps g homotopic to f.
	at.yorku.ca /t/a/i/c/39.htm (851 words)

	Tightening Non-simple Paths and Cycles on Surfaces (Site not responding. Last check: 2007-10-19)
		We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface.
		After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity k in O(gnk) time, or the shortest cycle homotopic to a given cycle of complexity k in O(gnk log (nk)) time.
		We also prove that the recent algorithms of Colin de Verdière and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.
	compgeom.cs.uiuc.edu /~jeffe/pubs/octagons.html (201 words)

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