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Topic: Vector fields on spheres

	A VISUAL TOUR OF CLASSICAL ELECTROMAGNETISM (Site not responding. Last check: 2007-10-20)
		A vector field is a field in which there is a vector associated with every point in space—that is, three numbers instead of only the single number for the scalar field.
		This field is proportional to the electric field of two point charges of opposite signs, with the magnitude of the positive charge three times that of the negative charge.
		Figure 50: The magnetic field of four charges moving in a circle. We show the magnetic field vector directions in only one plane. The “bullet” like icons indicate the direction of the magnetic field at that point in the array spanning the plane.
	evangelion.mit.edu /802TEAL3D/visualizations/guidedtour/Tour.htm (11847 words)

	Electromagnetic Fields (EHC 137, 1992)
		A conservative approach is recommended for pulsed fields where electric and magnetic field strengths are limited to 32 times the values given in Table 34, as averaged over the pulse width, and the power density is limited to a value of 1000 times the corresponding value in Table 34, as averaged over the pulse width.
		The magnitude of the magnetically induced electric fields and current densities is proportional to the radius of the induction loop in the body, to the tissue conductivity, and to the rate of change of magnetic flux density.
		A body is coupled to an electric field in proportion to its capacitance to the ground as one equipotential surface, such that the greater the capacitance the greater the current flow in the body.
	www.inchem.org /documents/ehc/ehc/ehc137.htm (16784 words)

	Sphere FAQ
		Spheres may be viewed as topological spaces, metric spaces, manifolds, and so on.
		Sometimes questions are treated with the tools of algebraic topology -- questions involving homotopy groups of spheres, K-groups (vector bundles), etc. A discussion of the possible vector fields on spheres (and which ones are parallelizable) is relevant to the classification of real division algebras, and so is part of the division-algebra FAQ.
		Generation of random points on the sphere (with a uniform distribution) comes under the purview of probability theory and random processes (where you will find suggestions which are more carefully thought out than the one in the FAQ).
	www.math.niu.edu /~rusin/known-math/index/spheres.html (856 words)

	Search Results for theory
		In 1920 Takagi published his fundamental paper on class field theory in which he built the theory around a remarkable fact which he had discovered, namely that the set of class fields, as defined by Heinrich Weber, over a fixed ground field k is identical to the set of abelian extension fields over k.
		While particle physics and quantum field theory was the work Bell was paid to do, and he made excellent contributions, his great love was for quantum theory, and it is for his work here that he will be remembered.
		This addition of six spheres over the system proposed by Eudoxus increased the accuracy of the theory while preserving the belief that the heavenly bodies had to possess motion based on the circle since that was the 'perfect' path.
	www-history.mcs.st-and.ac.uk /Search/historysearch.cgi?SUGGESTION=theory&CONTEXT=1 (17342 words)

	File: spheres
		The procedure psfilter has been changed in such a manner that the reordering of the vector sequence is done in one step; repeated application is not necessary.
		As an application the construction and plotting of a sphere through four points in the euclidean 3-space is given.
		The circles are represented by 2-dimensional euclidean subspaces of the 5-dimensional pseudo-euclidean vector
	www.mathematik.hu-berlin.de /~sulanke/spheres.html (1447 words)

	Skyscript: JOHANNES KEPLER And the Music of the Spheres by David Plant
		The radius vector is shortest when the planet is at perihelion and longest at aphelion.
		Each sphere was said to correspond to a different note of a grand musical scale.
		His vision of the music of the spheres, however, is based upon the hard facts of astronomical measurement.
	www.skyscript.co.uk /kepler.html (4331 words)

	culture data repository \| VBHT14: Fields
		The field lines in a region point in the same direction as the field vectors, and the strength of the field is reflected by how close together the field lines are.
		The source of the electric field is the charge density, and the source of the magnetic field is the current density.
		The field equation then becomes a very simple equation linking the "divergence" of the Faraday at an event to the amount of 4-current there (actually to fully determine the fields there's another required condition too - that the Faraday tensor is the exterior derivative of a 4-potential, but don't worry about that!).
	www.culturelist.org /cdr/article.cfm?id=22 (1257 words)

	[No title]
		This result gave the first sign of regularity in the stable homotopy groups of spheres, and their pr* *oof showed that the J-homomorphism is of considerable relevance to geometric topol- ogy.
		Adams discovered these operations after first trying to solve the vector fiel* ds on spheres* problem by use of secondary and higher operations in ordinary cohomology in [Ad62a ], a paper that was obsolete by the time it appeared.
		While Adams was aware of the relationship between the vector fields problem and the study of J, he chose not to discuss this in [Ad62c ]; he published a proof * *of the cited isomorphism in [Ad65a ].
	hopf.math.purdue.edu /May/history.txt (14491 words)

	Nonsolenoidal fields (Site not responding. Last check: 2007-10-20)
		A general vector field (of sufficient smoothness) can be decomposed as follows.
		It is advantageous here to apply the present procedure to all vector fields, whether or not they are solenoidal.
		This is because if a field is both irrotational and solenoidal, so that it has both scaloidal and poloidal representations, the scaloidal representation is usually easier to work with.
	www.ae.su.oz.au /~mcbain/papers/thesis/node144.html (253 words)

	Programa
		It is well-known that for the tangent bundle of a Riemannian manifold equipped with the Sasaki metric, the answer is parallel vector fields.
		vector fields along which the metric remains unchanged) and on curvature collineations (i.e.
		We study to what extent the known results concerning the behaviour of Hopf vector fields, with respect to volume, energy and generalized energy functionals, on the round sphere are still valid for the metrics obtained by performing the canonical variation of the Hopf fibration.
	www.um.es /wgp2004/uk/programa_uk.html (4782 words)

	[No title]
		Adams, "Vector Fields on Spheres", Annals of Math 75 (1962) 603-632.
		The maximum number p of continuous, linearly-independent vector fields on the (n-1)-sphere in R^n (with the usual topology) is p = 2^c - 1 + 8 d where n = (2a+1) 2^(c+4d) and c= 0,1,2,3.
		I guess it's fair to say that this many vector fields were known to exist by computations with "linear" vector fields over R, C, H, and O as far back as Hurwitz; what Adams did was rule out the possibility of having any more.
	www.math.niu.edu /~rusin/known-math/99/vec_fields (732 words)

	res-int-stephen (Site not responding. Last check: 2007-10-20)
		Another aspect of Theriault's work is studying the 2-primary homotopy theory of spheres and Moore spaces, where much less is known than in the odd primary case.
		He has calculated the exact value of the 2-primary homotopy exponent of a large class of Moore spaces (preprint 1999), and recently shown that the degree 2 map on odd dimensional spheres is -- up to an isomorphism -- multiplication by 2 on homotopy groups (preprint 2002).
		The latter problem has a long history and is intimately related to properties of the Whitehead product and vector fields on spheres.
	www.maths.abdn.ac.uk /~ran/rctg/res-int-stephen.html (320 words)

	Earliest Known Uses of Some of the Words of Mathematics (V)
		Vector and scalar also appear in 1846 in a paper "On Symbolical Geometry," in the The Cambridge and Dublin Mathematical Journal vol.
		VECTOR FIELD is found in "Natural Families of Trajectories: Conservative Fields of Force," Edward Kasner, Transactions of the American Mathematical Society, Vol.
		The distribution on the sphere was developed by R. Fisher in 1953 to treat problems in paleomagnetism: Dispersion on a Sphere, Proceedings of the Royal Society, A, 217, 295-305.
	members.aol.com /jeff570/v.html (3510 words)

	The Magnetic Force
		For this device, one sets up electric fields to accelerate ions, then causes those ions to fly through a region in which there is a magnetic field directed perpendicular to the flight path of the ions.
		The magnetic force is understood to be due to the presence of a magnetic field produced by charges in motion.
		The direction of the magnetic field is shown by the white arrows whose rotation depicts the fact that the field is circular around the wire.
	dept.physics.upenn.edu /courses/gladney/mathphys/subsubsection4_1_5_2.html (1816 words)

	Predmety - Predmety
		The topics studied in the lecture are,e.g., the integer-valued degree of a map and the index of a vector field at its isolated null point.
		Applications: Problem of existence of smooth nonzero vector fields on the spheres.
		Index of a vector field at an isolated zero.
	www.mff.cuni.cz /vnitro/is/sis/predmety/kod.php?kod=MAT009 (223 words)

	[No title]
		The reason is that in the late 1800s there was a big battle between the fans of quaternions and the fans of vectors, and the quaternion crowd lost.
		These spheres are the unit real numbers, the unit complex numbers, the unit quaternions, and the unit octonions, respectively!
		If you know about normed division algebras, it's obvious that these sphere admit the maximum possible number of linear independent vector fields: you can just take a basis of vectors at one point and "left translate" it to get a bunch of linearly independent vector fields.
	math.ucr.edu /home/baez/twf_ascii/week104 (3158 words)

	Vector Bundles & K-Theory Book
		At present about half of the book is in good enough shape to be posted online, approximately 110 pages.
		Part of Chapter 2, introducing K-theory, then proving Bott periodicity in the complex case and Adams' theorem on the Hopf invariant, with its famous applications to division algebras and parallelizability of spheres.
		Not yet written is the proof of Bott Periodicity in the real case, with its application to vector fields on spheres.
	www.math.cornell.edu /~hatcher/VBKT/VBpage.html (289 words)

	Re: Division algebras
		I seem to remember it involves introducing octonion coordinates on a projective space of some dimension to get the relevant vector space.
		And projective spaces, of course, have a "representation" as spheres in a space of higher dimension.
		And I would like to > understand better how the octonions fit into this game: the only > n-spheres that admit n linearly indendent vector fields are for > n = 0,1,3, and 7, corresponding to the reals, complexes, quaternions > and octonions.
	www.lns.cornell.edu /spr/2000-01/msg0020906.html (341 words)

	[No title]
		With R taken as a canonical sphere diagram space, examples include Symmetric spectra, as defined by Jeff Smith.
		In the odd dimensional case the secat is the least power of the Euler class which is trivial.
		In the even dimensional case secat is one when a ceratin homology class in twice the dimension of the sphere is -1 times a square.
	claude.math.wesleyan.edu /~mhovey/archive/letter83 (2120 words)

	[No title]
		Belyi, V. "On Galois extensions of a maximal cyclotomic field", Izv.
		Burgess, C. "Extending self homeomorphisms of a wild sphere describes by Gillman", Duke Math.
		"Conformal field theory and a topological quantum theory of vortices and knots", Phys.
	www.maths.gla.ac.uk /~ajb/btop/knotsbib.txt (9944 words)

	Multiresolution And Wavelets (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties.
		With the aid of the general theory of vector fields on spheres, it is demonstrated that...
		10 Vector fields on spheres (context) - Adams - 1962
	citeseer.ist.psu.edu /611452.html (636 words)

	Carta de Recomendação - Prof. Dold (Site not responding. Last check: 2007-10-20)
		Homotopy theory and its applications to geometric and analytic problems is the field of interest and activity of DG.
		the number of independent vector fields on spheres and other manifolds, the (non-) existence of positive curvatures on certain manifolds, etc.
		DG started (about) 1978 as a "pure" homotopy theorist, dealing with technical tools like cohomology operations, spectral sequences, Hurewicz relations; his results solved publicised open problems.
	www.ime.usp.br /~marcos/dold.html (391 words)

	Martin Raußen: PUBLICATIONS (Site not responding. Last check: 2007-10-20)
		Wiegmann, Reinhard Liftings, homotopy liftings and localization applied to vector field and immersion theory.
		Raussen, Martin Rational homotopy of spaces of maps into spheres and complex projective spaces.
		Smith, Larry A geometric interpretation of sphere bundle boundaries and generalized $J$-homomorphisms with an application to a diagram of I.
	www.math.aau.dk /~raussen/mrpub.html (364 words)

	Division algebras (Site not responding. Last check: 2007-10-20)
		There are various slightly different proofs, but as far as I know, they all involve understanding how many linearly independent vector fields you can put on a sphere, which in turn requires some heavy-duty algebraic topology.
		And I would like to understand better how the octonions fit into this game: the only n-spheres that admit n linearly indendent vector fields are for n = 0,1,3, and 7, corresponding to the reals, complexes, quaternions and octonions.
		The importance of this for string theory is fairly well-known, but I think there are still mysteries lurking here.
	www.lns.cornell.edu /spr/2000-01/msg0020880.html (197 words)

	Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields
		Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields: SIAM Journal on Applied Dynamical Systems Vol.
		Many applications give rise to dynamical systems in the form of a vector field with a phase space of moderate dimension.
		In this paper we present an algorithm to compute the k-dimensional unstable manifold of an equilibrium or periodic orbit (or a more general normally hyperbolic invariant manifold) of a vector field with an n-dimensional phase space, where 1< k < n.
	epubs.siam.org /sam-bin/dbq/article/60018 (518 words)

	Solenoidal fields (Site not responding. Last check: 2007-10-20)
		be a vector field possessing partial derivatives of order up to two which are Hölder continuous on
		It has been shown by Backus (1986) that the lemma can be extended from spherical annuli to spheres.
		is the surface Laplacian for the unit sphere (Aris 1989, pp.
	www.ae.su.oz.au /~mcbain/papers/thesis/node143.html (84 words)

	MA408 Algebraic Topology
		This result means that we can combine the theoretical power of singular homology and the computational power of simplicial homology to get many applications.
		These applications will include the Brouwer fixed point theorem, the Lefschetz fixed point theorem and applications to the study of vector fields on spheres.
		Aims: To introduce homology groups for simplicial complexes; to extend these to the singular homology groups of topological spaces; to prove the topological and homotopy invariance of homology; to give applications to some classical topological problems.
	www.maths.warwick.ac.uk /pydc/mauve/mauve-MA408.html (401 words)

	[No title]
		R(Y; X)X. If Mm is a local 2-point homogeneous space, then the local isometries of Mm a* ct transitively on the bundle of unit tangent vectors* S(Mm) so that the eigenvalu* *es of J(X) are constant as X varies in S(Mm).
		C* hi's proof is a lovely blend of algebraic topology and differential geometry and use s in an essential fashion work of Adams [1] concerning vector* fields on spheres.
		Thus C is uniformly bounded away from 0 for s 2 (s0; b] and thus C does not tend to 0 as s # s0.
	hopf.math.purdue.edu /Gilkey-Leahy-Sadofsky/GLSeigen.txt (3994 words)

	Department of Mathematics and Statistics - General - Directory
		Projective Stiefel manifolds and skew linear vector fields.
		The vector field problem: a survey with emphasis on specific manifolds.
		On sectioning tangent bundles and other vector bundles.
	www.math.ucalgary.ca /general/directory.php?personid=71&showyears=All (306 words)

	CURRICULUM VITAE (Site not responding. Last check: 2007-10-20)
		Alexander and J. Yorke, Parameterized functions, bifurcation, and vector fields on spheres, Prob.
		A number of examples are presented that involve fixed points, zeroes of maps, singularities of vector fields, and bifurcation.
		As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem.
	www.glue.umd.edu /~yorke/CV.html (5991 words)

	Adams_Frank (Site not responding. Last check: 2007-10-20)
		By good luck, moreover, my new methods were sufficiently powerful to answer one of the classical problems of my subject, that proposed by H Hopf in 1935.
		The conjecture that Adams solved was the famous conjecture about the existence of H-structures on spheres.
		After spending further time in Princeton, Adams took up a post at Manchester as a Reader in 1962, being appointed to Newman's chair when he retired in 1964.
	www-history.mcs.st-and.ac.uk /history/Mathematicians/Adams_Frank.html (1391 words)

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