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	Reference.com/Encyclopedia/Quaternionic projective space
		In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H.
		Quaternionic projective space of dimension n is usually denoted by
		Its direct construction is as a special case of the projective space over a division algebra.
	www.reference.com /browse/wiki/Quaternionic_projective_space (280 words)

	Quaternions - FUTEF (Site not responding. Last check: 2007-10-25)
		In mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association for the Advancement of Science.
		Quaternionic projective space of dimension n is usua...
		In mathematics, a quaternion algebra over a field F is a particular kind of central simple algebra A over F, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of F, by ext...
	futef.com /q/cats:[Quaternions] (458 words)

	Projective space - Wikipedia, the free encyclopedia
		Projective spaces are essential to algebraic geometry through the rich field of projective geometry developed in the nineteenth century, but also in the constructions of the modern theory (based on graded algebras).
		Projective spaces and their generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and algebraic groups, and their representation theory.
		The use of projective spaces makes quite rigorous the talk about a 'line at infinity' (where parallel lines meet), or a 'plane at infinity' for three dimensions: a translation of the latter can be made as part of the projective space associated to a four-dimensional real vector space.
	en.wikipedia.org /wiki/Projective_space (807 words)

	Springer Online Reference Works
		is isomorphic to the algebra of quaternions (cf.
		A quaternionic structure on a differentiable manifold is a field of quaternionic structures on the tangent spaces, that is, a subbundle
		A quaternionic Riemannian manifold is the analogue of a Kähler manifold for quaternionic structures.
	eom.springer.de /Q/q076790.htm (626 words)

	Space Case Grinder Soy Bean (Site not responding. Last check: 2007-10-25)
		Quaternionic projective space - In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted by
		Negative space - In art, negative space is the space around the subject of an image.
		Interplanetary space - Interplanetary space is that part of outer space between planets in a solar system and its local star(s), many of which are binaries.
	www.ffeinebean.com /spacecasegrinder.html (556 words)

	[No title]
		Spaces in the genus of infinite quaternionic projective space which admit essential maps from infinite complex projective space are classified.
		Introduction and statement of results In an attempt to understand Lie groups through their classifying spaces, Rector [9] classified the genus of HP1, the infinite projective space over the quaternions, considered as a model for the classifying space BS3.
		The homotopy type of a spaces X is said to be in the genus of HP1, denoted X 2 Genus (HP1), if the p-localizations of X and HP1 are homotopy equivalent for each prime p.
	hopf.math.purdue.edu /YauD/nonexistence_final_2.txt (3593 words)

	[No title]
		Physically, minimal projections correspond to "pure states" - states of affairs in which the answer to some maximally informative question is "yes", like "is the z component of the angular momentum of this spin-1/2 particle equal to 1/2?" Geometrically, the space of minimal projections is just the space of "lines" in our Hilbert space.
		You take the quaternions and complexify them - this amounts to throwing in an extra number i that's a square root of -1 and commutes with the quaternionic I and J - and you get an algebra which is also generated by I, J, and K = iI.
		You take the quaternions and quaternionify them - this amounts to throwing in two square roots of -1, say i and j, which anticommute but which commute with the quaternionic I and J - and you get an algebra which is also generated by I, J, K = iI, and L = jI.
	math.ucr.edu /home/baez/twf_ascii/week106 (4766 words)

	Ph.D. supervision
		Much of my recent work has to do with the immersion problem for lens spaces, as they sit half way between the real version of projective spaces (whose immersion problem is amazingly complicated) and the corresponding complex version (whose immersion problem, although still open, seems to be more accesible).
		The work on axial maps turns out to be closely related to the Topological Complexity of projective and lens spaces, a concept arising naturally in the study of motion planning (robotics) in the space of configurations of the roots of unity in a complex Euclidean space.
		The philosophy behind this is to analyze an extremely hard problem (immersions of projective spaces ---manifolds "built enterely in terms of exact 2-torsion") by approaching it via more accesible variations (immersions of lens spaces ---manifolds "built in terms of higher 2-torsion").
	chucha.math.cinvestav.mx /proyectos.html (3167 words)

	[No title]
		But the experience of the root system theory shows that it is wiser to consider these theories not as daughters of the ordinary linear algebra (corresponding to the special root system A) but rather as its sisters.
		The complex version of this quaternionic geometry fact is the classical result, known to Pontryagin in the thirties: THEOREM The quotient space of the complex projective plane by the conjugation involution is the 4-sphere.
		The quaternionic version has been discovered while trying to study the quaternionic version of the quantum Hall effect and the Berry phase theory, which might be considered as the complex version of my 1972 "Modes and quasi modes" paper on the topological meaning of the Von Neuman-Wigner theorem on the electron terms repulsion.
	www.pdmi.ras.ru /~arnsem/Arnold/rome1999.txt (512 words)

	AMCA: Global structure Hopf hypersurfaces in symmetric spaces of rank one by A. A. Borisenko
		be the complex hyperbolic space of constant holomorphic curvature -4.
		The analogical theorem is true for Hopf hypersurfaces in quaternionic hyperbolic space and it follows.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/f/h/41.htm (509 words)

	James F. Glazebrook Journal Articles
		The construction of a class of harmonic maps to quaternionic projective space, J.
		Harmonic maps between complex projective space, (with D. Barbasch and G. Toth), Geometriae Dedicata, 33, (1990) 37-50.
		On pluriharmonic maps to quaternionic projective space and totally complex immersions, (with G. Ronsse), Tensor, 51, (1992) 48-51.
	www.math.uiuc.edu /~glazebro/pubs.html (773 words)

	Complex projective space - Wikipedia, the free encyclopedia
		Complex projective space is a complex manifold that may be described by n+1 complex coordinates as
		It is a special case of a Grassmannian, and is a homogeneous space for various Lie groups.
		It is BU(1), the classifying space of U(1), in the sense of homotopy theory, and so classifies complex line bundles; equivalently it accounts for the first Chern class.
	en.wikipedia.org /wiki/Complex_projective_space (423 words)

	Quaternions (Site not responding. Last check: 2007-10-25)
		This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to the many illustrative problems.
		Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations.
		The classical quaternions are the case of L the real number field, and A is uniquely defined up to isomorphism by the condition that it is such a quaternion algebra that is not the 2Ã—2 real matrix algebra.
	pr36.mnmhobbies.com /quaternions.html (484 words)

	Math Colloquia
		For example the complex projective space, CP^n is the moduli space of complex lines through the origin in the complex vector space C^{n+1}.
		Closely related to these spaces are spaces of holomorphic maps from Riemann surfaces to other complex manifolds and moduli spaces of of complex vector bundles over Riemann surfaces and other complex manifolds.
		For example the moduli space of linear control systems with $n$ inputs and $m$ outputs is identified with the mapping space Hol(P^1, G_{n,m}), where P^1 is the Riemann sphere, and G_{n,m} is the complex Grassmannian of complex n-planes in C^{n+m}.
	www.math.ucsc.edu /seminars/past/f97.collo.html (1555 words)

	Springer Online Reference Works
		Penrose in the context of twistor theory [a4] but many mathematicians have introduced transforms which may now be viewed in the same framework.
		Since each cycle is a compact complex manifold, the restricted cohomology is a finite-dimensional vector space.
		Supposing that the dimension of this vector space is constant as one varies the cycle, this gives a vector bundle on
	eom.springer.de /p/p120100.htm (607 words)

	Cohomology and Quadrics
		Yang and B. Lee in hep-th/9503204 describe the relationship between the cohomology of the compact Lie algebra of a Lagrangian gauge theory and the BRST cohomology.
		Projective Twistor Space PT is CP3, the set of Complex lines through the origin in C4.
		Kane, The Homology of Hopf Spaces, Elsevier North-Holland (1988)
	www.valdostamuseum.org /hamsmith/coquad.html (4239 words)

	Katrin Leschke: Research (Site not responding. Last check: 2007-10-25)
		A comprehensive introduction to the theory of quaternionic valued functions and the applications to surface theory can be found in the Lecture Notes "Conformal Geometry of Surfaces in S4 and Quaternions" by F. Burstall, D. Ferus, K.Leschke, F.Pedit, U.Pinkall.
		The paper "Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori" by D. Ferus, K.Leschke, F.Pedit, U.Pinkall, outlines jet theory, and proves the quaternionic analogue of the Plücker formula.
		The paper "Willmore surfaces in quaternionic projective space" generalizes the notion of Willmore surfaces to holomorphic curves in quaternionic projective space and shows an analogous result to Bryant's result on Willmore surfaces in S3.
	www.gang.umass.edu /~leschke/Public/research/interests.html (275 words)

	Relatives of the Quotient of the Complex Projective Plane By the Complex Cojugation (ResearchIndex)
		Relatives of the quotient of the complex projective plane by the complex cojugation
		Abstract: It is proved, that the quotient space of the four-dimensional quaternionic projective space by the automorphism group of the quaternionic algebra becomes the 13-dimensional sphere while quotioned the the quaternionic conjugation.
		This fact and its various generalisations are proved using the results of the theory of the hyperbolic partial differential equations, providing also the proof of the theorem (which was, it seems, known to L.S.Pontriagin already in the thirties) claiming that the...
	citeseer.ist.psu.edu /105710.html (238 words)

	How Gauge Bosons See Internal Space
		As Matti Pitkanen has noted, the Quaternions have both a natural Minkowski metric (given by the square root of the real part Re(ZZ) of Z times Z for Quaternion Z) and a natural Euclidean metric (given by the square root of ZZ* of Z times Z conjugate for Quaternion Z).
		Also unlike the Quaternion case, where SU(2) = Spin(3) double-covers the rotation group in 3-space, Aut(O) = G2 is not Spin(7) and is not the double-cover of the rotation group in 7-space, but Spin(7) and G2 are related by Spin(7) / G2 = S7.
		If the space and time axes of the 1-vertex fl hole become connected with the space and time axes of the original spacetime, then the virtual Planck-mass fl hole acts to provide the mass factor (1/MPlanck^2) for the force strength of low-energy effective gravitation in the D4-D5-E6-E7-E8 model.
	www.valdostamuseum.org /hamsmith/See.html (4120 words)

	Proceedings of the American Mathematical Society
		In this paper, we shall classify conformal non-superminimal harmonic two-tori in a 2- or 3-dimensional quaternionic projective space, which are not always covered by primitive harmonic two-tori of finite type.
		A. Bahy-El-Dien and J. Wood, The explicit construction of all harmonic two-spheres in quaternionic projective spaces, Proc.
		J. Glazebrook, The construction of a class of harmonic maps to quaternionic projective spaces, J. London Math.
	www.ams.org /proc/1997-125-01/S0002-9939-97-03638-1/home.html (306 words)

	Valley Geometry Seminar
		This work is part of a project where a ``quaternionified'' complex analysis is used to give new results in surface theory.
		Additionally, an important new invariant of the quaternionic holomorphic theory is the Willmore energy of a holomorphic curve.
		(X) is the union of all the linear spaces of a particular arrangement of linear subspaces in a vector space.
	www.math.tamu.edu /~sottile/seminars/UMass/VGS-02W.html (1325 words)

	Weekly Calendar (Site not responding. Last check: 2007-10-25)
		I will then indicate a possible generalization to a flow of curves in the 4-sphere imagined as quaternionic projective space.
		This will involve defining a quaternionic cross ratio and a careful examination of 2-spheres in the 4-sphere via the twistor projection.
		Time and brain permitting it would be nice to see a glimpse of the "spectral curve" and the associated algebraic integrable system for these flows and observe the relationship with "finite type" solutions for a discrete version of the KdV system.
	www.math.umass.edu /Calendar/oldcal.html (286 words)

	Department of Mathematics at MIT \| Graduate Study : Spring 2002 Thesis Defenses
		In particular, we describe spaces in the genus of infinite quaternionic projective space which occur as targets of essential maps from infinite complex projective space, and we compute explicitly the homotopy classes of maps in these cases.
		The most straightforward applications of the methods developed are to enumeration of rational curves with a cusp of specified nature in projective spaces.
		The applications described include enumeration of rational curves with a (3,4)-cusp, genus-two and genus-three curves with complex structure fixed in the two-dimensional complex projective space, and genus-two curves with complex structure fixed in the three-dimensional complex projective space.
	www-math.mit.edu /graduate/thesis-defense-spring-2002.html (1282 words)

	[No title]
		Given a self map f of HPn, the degree of f is the integer deg(f) such that f*(u) = deg(f)u.
		The homotopy classification of self maps of HP1 is well known: self maps are classified by their degrees and the allowable degrees are zero and the odd squa* *re integers [Ms ].
		The situation for finite projective spaces is more complicated.* * It is not true in general that self maps are classified by their degrees [MR ], and e* *ven the set of possible degrees is unknown.
	www.math.purdue.edu /research/atopology/Granja/notehpn.txt (1918 words)

	Document Archive: Research Papers in Pure Mathematics
		A new type of estimate for operators on Hilbert spaces.
		On the existence of harmonic morphisms from symmetric spaces of rank one.
		Approximation of the Spectra of Differential Operators by Projection on Finite Dimensional Subspaces.
	www.maths.lu.se /publications/Pure-Research-Papers.html (1540 words)

	Quaternion References
		M.P. Cayley, Arthur: On the quaternion equation qQ — Qq’ = 0.
		Sketch of the analytical theory of quaternions, 146—159, 1890.
		Pizer, A. On the arithmetic of quaternion algebras I. Acta Arith.
	home.att.net /~t.a.ell/QuatRef.htm (10847 words)

	Stunted projective space - Wikipedia, the free encyclopedia
		This makes a topological space that is no longer a manifold.
		Their properties were therefore linked to the construction of frame fields on spheres.
		In this way the vector fields on spheres question was reduced to a question on stunted projective spaces: for RP
	en.wikipedia.org /wiki/Stunted_projective_space (238 words)

	HOW SURFACES INTERSECT IN SPACE
		The text starts at the most basic level (the intersection of coordinate planes) and gives hands on constructions of the most beautiful examples in topology: the projective plane, Poincare's example of a homology sphere, lens spaces, knotted surfaces, 2-sphere eversions, and higher dimensional manifolds.
		These include the 3-dimensional sphere, lens spaces, and the quaternionic projective space.
		In the final Chapter, higher dimensional spaces are examined from the same elementary point of view.
	www.worldscibooks.com /mathematics/1728.html (423 words)

	Amazon.com: "quaternionic variables": Key Phrase page (Site not responding. Last check: 2007-10-25)
		In the case of several quaternionic variables many new problems arise if one tries to give a notion of H-hyperfunction.
		We consider several questions related to the removability of singu- larities for regular functions of quaternionic variables.
		Let Q be a domain in l x f - R4 x R4 = R8 of two quaternionic variables x = XI +x22 + x3j+ x4k and y = y1 + y2i + y3j + y4k and f =...
	www.amazon.com /phrase/quaternionic-variables (564 words)

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