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Topic: Principal homogeneous space

	Principal homogeneous space - Wikipedia, the free encyclopedia
		In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively.
		Another example is the affine space concept: the idea of the affine space A underlying a vector space V can be said succinctly by saying that A is principal homogeneous space for V acting as the additive group of translations.
		The principal homogeneous space concept is a special case of that of principal bundle: it means a principal bundle with base a single point.
	www.wikipedia.org /wiki/Torsor (925 words)

	Affine geometry - Wikipedia, the free encyclopedia
		Affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin 0.
		For any group G there is a notion of principal homogeneous space for G: a set S on which G acts in a way isomorphic to the way it permutes itself by multiplication.
		An affine space A for a vector space V is just such a principal homogeneous space; one then has to recover scalar multiplication on A as a well-defined concept.
	www.wikipedia.org /wiki/Affine_geometry (746 words)

	Homogeneous space (Site not responding. Last check: 2007-11-07)
		A further classical example is the space of lines in projective space of three dimensions (equivalently, the space oftwo-dimensionsional subspaces of a four-dimensional vector space).
		In general, if X is a homogeneous space, and H is the stabilizer of somefixed x in X, the points of X correspond to the cosets G/H. We can assume that H is a closed subgroup of G, for acontinuous action: when it is the identity subgroup {e}, we have a principal homogeneous space.
		It is a finite-dimensional vector space V witha group action of an algebraic group G, such that there is an orbit of G that is open for the Zariski topology (and so, dense).
	www.therfcc.org /homogeneous-space-214701.html (459 words)

	Homogeneous space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional subspaces of a four-dimensional (Click link for more info and facts about vector space) vector space).
		The geometry of the resulting homogeneous space is the (Click link for more info and facts about line geometry) line geometry of (Click link for more info and facts about Julius Plücker) Julius Plücker.
		Since the (Click link for more info and facts about homogeneous coordinates) homogeneous coordinates given by the minors are 6 in number, this means that the latter are not independent of each other.
	www.absoluteastronomy.com /encyclopedia/h/ho/homogeneous_space.htm (465 words)

	Knowledge King - Principal homogeneous space (Site not responding. Last check: 2007-11-07)
		In mathematics, a principal homogeneous space for a group G is a homogeneous space X on which G acts with stabilizer the identity subgroup {e}.
		One way to follow basis-dependence in a linear algebra argument is to track variables x in X. Another linear algebra example is the affine space concept: the idea of the affine space A underlying V can be said succinctly by saying that A means a principal homogeneous space for V as additive group.
		The reason of the interest for Diophantine equations, in the elliptic curve case, is that K may not be algebraically closed.
	www.knowledgeking.net /encyclopedia/p/pr/principal_homogeneous_space.html (545 words)

	homogeneous
		A homogeneous differential equation is usually one of the form Lf = 0, where L is a differential operator, the corresponding inhomogeneous equation being Lf = g with g a given function; it is also used of equation in the form Dy = f(y/x).
		A homogeneous space for a Lie group G, or more general transformation group, is a space X on which G acts transitively and continuously - so equivalently a coset space G/H where H is a closed subgroup.
		Given a colouring of the edges of a complete graph, the term homogeneous applies to a subset of vertices such that all edge connecting two of the subset have the same colour; and in much greater generality in Ramsey theory for colourings of k-element subsets.
	www.fact-library.com /homogeneous.html (257 words)

	METHODOLOGY--Space and Place
		Space in the physical is occurent (geometrical), not centered but homogeneous, a multiplicity of three dimensional points, and measured by distance.
		Is inhabiting a space on-line, to be deceived in?
		Space is shifted and molded like a moebius strip in the virtualization process “a transformation from the interior to the exterior and the exterior to the interior” (Levy, 1998, p.
	www.otal.umd.edu /~paulette/Dissertation/methodology/spaceplace.html (4385 words)

	Articles - Principal bundle (Site not responding. Last check: 2007-11-07)
		In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G.
		Principal G-bundles are G-bundles in the sense that the group G also serves as the structure group of the bundle.
		Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group G determine a unique principal G-bundle from which the original bundle can be reconstructed.
	lastring.com /articles/Principal_bundle?mySession=c2f43531e6199808d3... (919 words)

	Encyclopedia: Orientation (mathematics)
		In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
		In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere.
		In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension.
	www.nationmaster.com /encyclopedia/Orientation-%28mathematics%29 (1538 words)

	Principal homogeneous space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		That is, X is a (Click link for more info and facts about homogeneous space) homogeneous space for G such that the stabilizer of any point is trivial.
		The principal homogeneous space concept is a special case of that of (Click link for more info and facts about principal bundle) principal bundle: it means a principal bundle with base a single point.
		For example a (Click link for more info and facts about differential manifold) differential manifold M has a principal bundle of (One of a series of still transparent photographs on a strip of film used in making movies) frames associated to its (Click link for more info and facts about tangent bundle) tangent bundle.
	www.absoluteastronomy.com /encyclopedia/P/Pr/Principal_homogeneous_space.htm (913 words)

	Station Information - Homogeneous space
		In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts continuously and transitively.
		That is, there is a group action of G on X, respecting the geometric structure of X, and making X into a single G-orbit; here we assume X isn't empty.
		We can parametrize them by line co-ordinates: these are the 2x2 minors of the 2x4 matrix with columns two basis vectors for the subspace.
	www.stationinformation.com /encyclopedia/h/ho/homogeneous_space.html (414 words)

	Homogeneous Space Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		From the point of view of the Erlangen programme, one may understand that "all points are the same", in the geometry of X. This was true of essentially all geometries proposed before Riemannian geometry.
		Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for thier respective symmetry groups.
		The same is true of the models found of non-Euclidean geometry, of constant curvature, such as hyperbolic space.
	www.karr.net /search/encyclopedia/Homogeneous_space (658 words)

	[No title]
		There exists an abelian surface $A$, a principal homogeneous space $Y$ of $A$, and a finite \'etale morphism $f\colon Y \to X$ of degree $n$, that is a torsor under the group scheme $\mu_n$.
		The elements of the Selmer group $\Sel(D,\mu_3)$ are represented by the principal homogeneous spaces $D_a$ defined by $u^3+av^3+a^2u^3=0$, where $a$ is a cube free integer.
		Moreover, $D_a$ is a principal homogeneous space of $D$ of the kind described in Condition (iv) of the theorem.
	www.maths.warwick.ac.uk /%7Emiles/HPFSD75/1.tex (3128 words)

	Geometry of Psychological Space (Site not responding. Last check: 2007-11-07)
		It is important to note that he did not suppose this space to pre-exist as a world of such elements, but rather to come into being through a process of construction by which we create a space in which to place elements as we come to construe them.
		Kelly's `construct' in psychological space is conveniently represented by a pair of disjoint concepts corresponding to the construct poles, both subsumed by a third corresponding to the range of convenience as shown in Figure 4.
		Multiple constructs in psychological space correspond to multiple axes of reference, and the plans representing their distinctions and ranges of convenience intersect to define regions of the space corresponding to conjunction, composition and multiple inheritance in the logic as shown in Figure 6.
	ksi.cpsc.ucalgary.ca /articles/NewPsych92 (3861 words)

	PHILOSOPHY (Site not responding. Last check: 2007-11-07)
		Space has deeply affected philosophy as the study of the truths or principles underlying all knowledge and being (or reality).
		One of the principal advantages of this dualism is the provision of a separate basis for the moral nature of humans; the mind appears as the moral core of the composite, governed by laws that differ from those that control matter.
		That is to say, a space program limited in certain ways may lead to an improvement of the general welfare, but much of suggested space exploration would not necessarily or clearly accomplish that goal.
	www.jsc.nasa.gov /er/seh/philosophy.html (5186 words)

	Scientists grow heart tissue in NASA Bioreactor
		The Bioreactor was developed by NASA to simulate the weightless environment of space by putting cells in a growth medium that constantly rotates and keeps the cells in endless free-fall.
		Space exploration will involve slightly increased exposures of crew members to radiation, so what we learn from these cells could help help justify methods of female crew selection, and help manage breast cancer in the national population at the same time."
		Electrophysiological studies conducted using a linear array of extracellular electrodes showed that the peripheral region of constructs exhibited relatively homogeneous electrical properties and sustained macroscopically continuous impulse propagation on a centimeter-size scale.
	science.nasa.gov /NEWHOME/headlines/msad05oct99_1.htm (2029 words)

	Mikhail Borovoi's Publications
		Borovoi, The Manin obstruction to the Hasse principle for homogeneous spaces with connected or abelian stabilizer.
		Borovoi, B. Kunyavskii, On the Hasse principle for homogeneous spaces with finite stabilizers.
		Borovoi, A cohomological obstruction to the Hasse principle for homogeneous spaces.
	www.math.tau.ac.il /~borovoi/publ.html (389 words)

	[No title]
		The space BDiff+(W) is the homotopy quotient of the space of metrics [10, 11] on W by the diffeomorphism group Morava 3 and we can think of morphisms in the (d + 1)-dimensional gravity category as cobordisms between d-manifolds, together with a choice of equivalence class of Riemannian metric on the cobordism.
		In the indefinite case, the manifold Grass-(B) of maximal negative-definite subspaces of B R is a noncompact (contractible) symmetric space defined by a cell of dimension b+ b- in the usual Grassmannian of b- -planes in b-space.
		The graded space Bun *(W) of gauge equivalence classes of connections on SU (2)-bundles over W has components indexed by the second Chern class of the bundle.
	hopf.math.purdue.edu /Morava/PGGravityfinal.txt (3741 words)

	Gauge theory - Wikipedia, the free encyclopedia
		If we have a principal bundle P whose base space is space or spacetime and structure group is a Lie group, then the sections of P form a group called the group of gauge transformations.
		We can define a connection (gauge connection) on this principal bundle, yielding a covariant derivative ∇ in each associated vector bundle.
		If we choose a local frame (a local basis of sections) then we can represent this covariant derivative by the connection form A, a Lie algebra-valued 1-form which is called the gauge potential in physics and which is evidently not an intrinsic but a frame-dependent quantity.
	en.wikipedia.org /wiki/Gauge_field_theory (2365 words)

	Principal Homogeneous Space Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		Looking For principal homogeneous space - Find principal homogeneous space and more at Lycos Search.
		Find principal homogeneous space - Your relevant result is a click away!
		Look for principal homogeneous space - Find principal homogeneous space at one of the best sites the Internet has to offer!
	www.karr.net /search/encyclopedia/Principal_homogeneous_space (1042 words)

	The Ultimate Chiral model - American History Information Guide and Reference
		In nuclear physics, the chiral model is a phenemological model describing mesons in the chiral limit where the masses of the quarks goes to zero (without mentioning quarks at all).
		It's a nonlinear sigma model with the principal homogeneous space of the Lie group SU(N) as its target manifold where N is the number of quark flavors.
		The Riemannian metric of the target manifold is given by a positive constant multiplied by the Killing form acting upon the Maurer-Cartan form of SU(N).
	www.historymania.com /american_history/Chiral_model (166 words)

	Affine geometry: Definition and Links by Encyclopedian.com - All about Affine geometry
		In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry.
		This can be done quickly in terms of a vector space V.
		The general linear group GL(V) isn't the whole affine group: we must allow also translations by vectors v of V.
	www.encyclopedian.com /af/Affine-geometry.html (633 words)

	Principal homogeneous space - Definition up Erdmond.Com
		For example, given a argument is to track variables ''x'' in X. Another linear algebra example is the affine_space concept: the idea of the affine_space A underlying V can be said succinctly by saying that A means a principal homogeneous space for V as additive group.
		The principal homogeneous space concept is a special case of that of associated to its tangent_bundle.
		The reason of the interest for Diophantine_equations, in the elliptic curve case, is that K may not be algebraically_closed.
	www.erdmond.com /Principal_homogeneous_space.html (387 words)

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