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	PlanetMath: Hopf bundle
		From the long exact sequence of the bundle:
		This is version 2 of Hopf bundle, born on 2002-12-27, modified 2002-12-27.
		The most remarkable fact (for Geometry) is that we can give the coordinates for our calculation to this Bundle as the example of Principal Fibre Bundle.
	planetmath.org /encyclopedia/HopfBundle.html (104 words)

	Heinz Hopf Summary
		Hopf became interested in mathematics while serving in World War I. Later, he studied at the University of Berlin and in 1925 received his Ph.D. He accepted a chair in Zurich, Switzerland, and is remembered for his formula about the integral curvature and for what came to be known as the "Hopf invariant."
		Heinz Hopf (November 19, 1894 – June 3, 1971) was a mathematician born in Gräbschen, Germany (now Grabiszyn, part of Wrocław, Poland).
		Hopf spent the year after his doctorate at Göttingen, where David Hilbert, Richard Courant, Carl Runge, and Emmy Noether were working.
	www.bookrags.com /Heinz_Hopf (567 words)

	Hopf bundle - Wikipedia, the free encyclopedia
		In mathematics, the Hopf bundle (or Hopf fibration) is a particular fiber bundle with base space S
		The Hopf bundle can actually be considered as a principal bundle when the fiber is identified with the circle group.
		As a consequence of Adams' theorem, these are the only fiber bundles with spheres as total space, base space, and fiber.
	en.wikipedia.org /wiki/Hopf_bundle (315 words)

	[No title]
		STABLE GEOMETRIC DIMENSION OF VECTOR BUNDLES OVER EVEN-DIMENSIONAL REAL PROJECTIVE SPACES MARTIN BENDERSKY, DONALD M. Abstract.In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order 2eover RP2n if n is sufficiently large and e 75.
		In this paper, we use the Bendersky-Davis computation of v-11ß*(SO(m)) to determine this geometric dimension for all values of e (still provided that n is sufficiently large).
		References [1]J. Adams, Geometric dimension of vector bundles over RPn, Proc Int Conf on Prospects in Math, Kyoto (1973) 1-14.
	hopf.math.purdue.edu /Bendersky-DavisD-Mahowald/sgd2.txt (4697 words)

	19hairbundle
		Hair bundles are known to contain several isoforms of myosin (11),at least one of which--probably myosin I *(12)--mediatesadaptation of the transduction process.
		Although a hair bundle needs not display sustained spontaneous oscillation, the auditory system might be ex-pected to exploit the local instability in parameter space at the Hopf bifurcation to achieve a highly resonant response.Reduction of the system to two varying free parameters, N Sand *G F#, facilitates the search of parameter space for Hopfbifurcations.
		For threshold stimuli, the amplitude of bundle motion for the active model is 56-fold that for thenonlinear passive formulation and 117-fold that for the linear passive model.
	asterion.rockefeller.edu /marcelo/Reprints/19hairbundle.html (4400 words)

	Hopf link - Encyclopedia, History, Geography and Biography
		In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component.
		For a concrete model take the unit circle in the xy-plane centered at the origin and another unit circle in the yz-plane centered at (0,1,0).
		Depending on the relative orientations of the two components the linking number of the Hopf link is ±1.
	www.arikah.net /encyclopedia/Hopf_link (159 words)

	Chris Wood: Publications
		Finally, it is shown that the Hopf vector fields are the unique global minima of the energy functional restricted to unit vector fields on the 3-sphere.
		The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle.
		In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy.
	www-users.york.ac.uk /~cmw4/publ.html (525 words)

	Auditory sensitivity provided by self-tuned critical oscillations of hair cells
		By controlling the bundle's propensity to oscillate, this feedback automatically maintains the system in the operating regime where it is most sensitive to sinusoidal stimuli.
		In general, such a system exhibits a Hopf bifurcation [#!stro94!#]: as the value of a control parameter is varied, the behavior abruptly changes from a quiescent state to self-sustained oscillations.
		We introduce the concept of a self-tuned Hopf bifurcation which permits the favorable amplificatory properties of a dynamical instability to be obtained in a robust way.
	www.mpipks-dresden.mpg.de /mpi-doc/julichergruppe/julicher/auditory_sensitivity/auditory_sensitivity.html (949 words)

	Hopf bundle -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-31)
		It was discovered by (Click link for more info and facts about Heinz Hopf) Heinz Hopf in 1931.
		The Hopf bundle can actually be considered as a (Click link for more info and facts about principal bundle) principal bundle when the fiber is identified with the (Click link for more info and facts about circle group) circle group.
		As a consequence of Adams' theorem, these are the only fiber bundles with (A particular environment or walk of life) spheres as total space, base space, and fiber.
	www.absoluteastronomy.com /encyclopedia/H/Ho/Hopf_bundle.htm (515 words)

	Section1.1
		In particular we show that the vector field on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves.
		Hopf bifurcations from time periodic rotating waves to two frequency tori have been studied for a number of years by a variety of authors including Rand and Renardy.
		Discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries.We present a systematic approach to the study of local bifurcation from discrete rotating waves.
	www.math.ist.utl.pt /~edias/MASIE/Section1/Section1.1 (5203 words)

	Two Adaptation Processes in Auditory Hair Cells Together Can Provide an Active Amplifier -- Vilfan and Duke 85 (1): 191 ...
		we have assumed that the bundle is poised precisely at the Hopf
		To achieve this regime the Ca concentration outside the bundle was reduced and the self-tuning mechanism was replaced by a constant Ca flow from the cell body.
		In vivo evidence for a cochlear amplifier in the hair-cell bundle of lizards.
	www.biophysj.org /cgi/content/full/85/1/191 (5943 words)

	Re: Diffeomorphisms
		It's famous because the map from the total space to the base was the first example of a topologically nontrivial map from a sphere to a sphere of lower dimension.
		Anyway, it's the quaternionic version of the Hopf bundle that serves as the inspiration for Milnor's construction of exotic 7-spheres.
		Notice that X(n,m) is the unit sphere bundle of a vector bundle over S^4 whose fiber is the quaternions.
	www.lns.cornell.edu /spr/2000-06/msg0025517.html (2029 words)

	Auditory sensitivity provided by self-tuned critical oscillations of hair cells -- Camalet et al. 97 (7): 3183 -- ...
		Thus the Hopf resonance acts as a sharply tuned high-gain amplifier for weak stimuli and as a low-gain filter for strong stimuli.
		If the Ca concentration C within the cell is artificially high, the hair bundle is initially quiescent, but as Ca ions are pumped from the cell, it gradually begins to oscillate with small amplitude.
		Curves shaded red are the responses of an equivalent passive hair bundle (i.e., a bundle with identical mechanical properties but no force generators).
	www.pnas.org /cgi/content/full/97/7/3183 (4370 words)

	A model for amplification of hair-bundle motion by cyclical binding of Ca2+ to mechanoelectrical-transduction channels ...
		For a hair bundle with 75 active channels (9, 29, 30), the maximal work per cycle would be ~2,000 zJ or ~500 kT.
		For a hair bundle of approximately rectangular cross-section in which the number of stereociliary ranks is conserved, the
		Characteristic frequencies are calculated for points at and beyond the Hopf bifurcations, which occur along the ridge.
	www.pnas.org /cgi/content/full/95/26/15321 (4272 words)

	Math 433 Homepage (Site not responding. Last check: 2007-10-31)
		These are the class notes to Math 433 - The Geometry of Vector Bundles and an Introduction to Gauge Theory - during the spring semester of 1998 at the The University of Illinois at Urbana Champaign.
		The universal bundle for smooth manifolds and vector bundles.
		Constructions for non-compact manifolds and for principal bundles.
	www.math.uiuc.edu /~cwillett/bundles (883 words)

	Re: Typical fiber (Site not responding. Last check: 2007-10-31)
		Specifically, you can't define a continuous field of unit vectors on a 2-sphere (you can't "comb" a sphere), while this is possible on a 1-sphere (circle) or a 3-sphere.
		Hence, the tangent bundle on a sphere is non-trivial.
		This tangent bundle is linked to the first Hopf bundle.
	www.lns.cornell.edu /spr/2000-08/msg0027213.html (354 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		View this as a question in KO-theory: the claim is that H^8 generates the reduced real K-theory \tilde KO(S^8) of the 8-sphere; the bundle H^8 over S^8 is obtained by the standard glueing process along the equator S^7, using the octonion multiplication.
		Now ch(cplx(H^8))=8+x (see the formula in the last paragraph in Husemoller's "Fibre bundles", the chapter on "Bott periodicity and integrality theorems".
		The constant factor is unimportant, so the answer is yes, pi_7(O) is generated by the map S^7---> O which sends a unit octonion A to the map l_A:x --> Ax in SO(8).
	www.math.niu.edu /~rusin/known-math/99/pi_7 (212 words)

	Hopf Notice: Hopf Has Been Moved To A Virtual Website On The Math Department Server Should Be Trans (Site not responding. Last check: 2007-10-31)
		The Hopf name is stamped on a vast number of undistinguished violins, an inestimable quantity of which are unauthentic.
		Kaspar HOPF was born in 1827 in Saxony.
		Hopf Map -- from MathWorld Hopf Map -- from MathWorld The first example discovered of a map from a higher-dimensional sphere to a lower-dimensional sphere which is not null-homotopic.
	www.99hosted.com /names10146.html (461 words)

	Fields Institute - Workshop on Homotopy, Geometry and Physics
		Natural sub-bundles of the tautological (Hopf) bundle and flag varieties were constructed along with natural actions of the unitary group.
		In some cases, the resulting bundles are K(G,1)'s where G is closely related to the mapping class group of an orientable surface.
		The talk will be devoted to differential geometric and holomorphic structures on higher line bundles and their relationship with Deligne cohomology and groups of algebraic cycles.
	www.fields.utoronto.ca /programs/scientific/95-96/homotopy/workshop/abstracts.html (1389 words)

	3-sphere - Wikipedia, the free encyclopedia
		The orbit space of this action is naturally homeomorphic to the two-sphere S
		The resulting map from the 3-sphere to the 2-sphere is known as the Hopf bundle.
		These coordinates are useful in the description of the 3-sphere as the Hopf bundle
	www.wikipedia.org /wiki/3-sphere (2068 words)

	The course of corticofacial projections in the human brainstem -- Urban et al. 124 (9): 1866 -- Brain
		1 = main ventral pyramidal tract; 2 = `aberrant bundle' in a paralemniscal position at the dorsal base of the pons;3 = fibre loop into the ventral medullary region, crossing the midline and ascending in the dorsolateral medullary region to the facial nucleus from below.
		Urban PP, Hopf HC, Connemann B, Hundemer HP, Koehler J. The course of cortico-hypoglossal projections in the human brainstem.
		Urban PP, Hopf HC, Fleischer S, Zorowka PG, Müller-Forell W. Impaired cortico-bulbar tract function in dysarthria due to hemispheric stroke.
	brain.oxfordjournals.org /cgi/content/full/124/9/1866 (3437 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		It is intimately connected to the geometry of fibre bundles.
		The most well-known example of a nontrivial bundle is the M\"obius band, where twisting is done by the 2-element group ${\bf Z}_2$.
		With this language, the mechanism of symmetry breaking can be stated as the case when the twisting of the bundle are by elements of a subgroup $H$ of $G$, and when the connection 1-form takes values in the corresponding Lie subalgebra.
	www.cs.chalmers.se /pub/users/laura/MathTsou.tex (2573 words)

	Ph.D. supervision
		Hirsch's theorem claims that the immersion question for a manifold is a problem in homotopy theory: it's just a matter of determining the geometric dimension of the stable normal bundle of the manifold, that is, lifting the corresponding classifying map from BO to a space BO(k) --smallest possible such k.
		This is performed by analizing, in fact, the universal formal group by means of its associated series --iterated formal sums of a single variable.
		Topologically, these associated series can be interpreted as the Euler classes of powers of the Hopf bundle and, therefore, the determination of their algebriac properties is fundamental in the study of the cobordism of lens spaces.
	chucha.math.cinvestav.mx /proyectos.html (2445 words)

	[No title]
		LM is a principal bundle with structure group LG = Maps(S1, G), and if the tangent bundle of M is defined by a representation V of G then the tangent bundle of LM is defined by the representation LV of LG.
		B is a principal bundle with structure group LG, then (motivated by ideas of [14]) we construct a `thickening' of B: Definition 2.5.
		In particular, if P is the refinement of the frame bundle of M via a representa* tion V of G, then the tangent bundle* of LM is defined by the representation LV of LG.
	www.math.purdue.edu /research/atopology/Kitchloo-Morava/Thomprospectra2.txt (4514 words)

	Visualization of the Hopf Bundle and the Dirac Connection -- from Mathematica Information Center
		Visualization of the Hopf Bundle and the Dirac Connection
		Furthermore, we visualize the Dirac connection on this Hopf bundle.
		Hopf bundle, Dirac connection, connection, principal bundle, horizontal lift, holonomy
	library.wolfram.com /infocenter/MathSource/5193 (114 words)

	Re: Diffeomorphisms
		Anyway, this line bundle is better known as the "canonical bundle of the projective plane": the 2-sphere is the same as the space of 1-dimensional subspaces of C^2, and the canonical bundle assigns to each point in the 2-sphere the 1-dimensional subspace that it corresponds to!
		for example, the dual of the canonical bundle has some very obvious holomorphic sections, just staring you in the face after you fight your way through the definitions, and these are all the holomorphic sections!
		A similar trick works for all the powers of the dual of the canonical bundle.
	www.lns.cornell.edu /spr/2000-06/msg0025676.html (425 words)

	Projective Lines
		These bundles play an important role in topology, so it is good to understand them in a number of ways.
		To see how the Hopf invariant is related to linking, we can compute it using homology rather than cohomology.
		A deep study of the Hopf invariant is one way to prove that any division algebra must have dimension 1, 2, 4 or 8.
	math.ucr.edu /home/baez/octonions/node9.html (1085 words)

	[No title]
		http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Dorabiala/transfer Abstract: The goal of this paper is to show that if a smooth fiber bundle has a compact Lie group as structure group, then the transfer map for the algebraic K-theory of spaces satisfies analogs of the Mackey Double coset formula and Feshbach's sum formula.
		The spaces in these spectra are the infinite classical groups and their coset spaces, and their homology was first calculated in the Cartan seminars, but the Hopf ring structure was first determined in the second author's unpublished PhD thesis.
		The Hopf ring viewpoint turns out to be very convenient for understanding the homological effect of various maps between classical groups and fibrations of their connective covers.
	www.lehigh.edu /~dmd1/h1126 (942 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		\enddefinition While the definition of a bundle is a very general one, we will be applying the definition of a bundle to several specialized categories.
		\enddefinition The tangent, normal, and tautological bundles are all vector bundles ({\it cf.
		\enddefinition The Hopf bundle is an example of a principal $S^1$ bundle and the homogeneous bundle $O(n) \to O(n)/O(n-1)$ is a principal $O(n-1)$ bundle.
	www.math.uiuc.edu /~cwillett/bundles/notes-tex/pg2.tex (193 words)

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