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Topic: Homogeneous space

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Torsor

Tangent bundle

Fourier analysis

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Mikio Sato

Uncertainty principle

	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.
	en.wikipedia.org /wiki/Principal_homogeneous_space (844 words)

	Homogeneous space - Wikipedia, the free encyclopedia
		Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups.
		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 vector space).
		Since the homogeneous coordinates given by the minors are 6 in number, this means that the latter are not independent of each other.
	en.wikipedia.org /wiki/Homogeneous_space (490 words)

	PlanetMath: projective space (Site not responding. Last check: 2007-11-06)
		Projective space is defined to be the set of the corresponding equivalence classes.
		Homogeneous coordinates differ from ordinary coordinate systems in that a given element of projective space is labelled by multiple homogeneous ``coordinates''.
		This is version 4 of projective space, born on 2001-12-21, modified 2002-07-24.
	planetmath.org /encyclopedia/ProjectiveSpace.html (326 words)

	PlanetMath: homogeneous space (Site not responding. Last check: 2007-11-06)
		Indeed, the concept of a homogeneous space, is logically equivalent to the concept of a transitive group action.
		Next, we consider the effect of the choice of basepoint on the quotient structure of a homogeneous space.
		This is version 3 of homogeneous space, born on 2003-02-14, modified 2003-02-14.
	planetmath.org /encyclopedia/HomogeneousSpace.html (336 words)

	Homogeneous (Site not responding. Last check: 2007-11-06)
		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; the word homogeneous is also used of equations 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.brainyencyclopedia.com /encyclopedia/h/ho/homogeneous.html (386 words)

	Homogeneous space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		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)

	[No title]
		Thus this X is a locally homogeneous, non-homogeneous space.
		Clearly homogeneity requires not only local homogeneity but also "anti-local" homogeneity: given x and y there exist M and N around them for which X-M and X-N are homeomorphic.
		(Homogeneity will also require a patching condition: we will need to have homeomorphisms on X-M and _the closure of_ M, _and_ the homeomorphisms will have to agree on the boundary of M. This ought to give another, more delicate, family of counterexamples.) You may be interested in a related problem.
	www.math.niu.edu /~rusin/known-math/95/homogen (1110 words)

	CATHOLIC ENCYCLOPEDIA: Space
		The idea of space is one of the most important in the philosophy of the material world; for centuries it has preoccupied and puzzled philosophers and psychologists, and even today the views as to its nature are far from being harmonious.
		Mathematical space is said to be infinite—not a metaphysical infinity, which affirms the positive absence of all limits, and with which the mathematician has no concern, but that mathematical infinity, which signifies that the nature of a reality is such that no limit can be assigned to it.
		Space is therefore as real, as objective, as the corporeal world itself, but in itself it exists apart only in the human mind, seeing that in the reality of existing things it is only the extension of bodies themselves.
	www.newadvent.org /cathen/14167a.htm (2492 words)

	Mijlpalen in de geschiedenis van het menselijk vernuft
		The concept of homogeneous space and the history of the concept (for it has been part of the common stock of philosophical and scientific thought since antiquity) are a wholly different problem, upon which we shall not enter here.
		At first sight this cleavage in space appears to be due to the opposition between an inhabited and organized-hence cosmicized -territory and the unknown space that extends beyond its frontiers; on one side there is a cosmos, on the other a chaos.
		Whether that space appears in the form of a sacred precinct, a ceremonial house, a city, a world, we everywhere find the symbolism of the Center of the World; and it is this symbolism which, in the majority of cases, explains religious behavior in respect to the space in which one lives.
	www.let.leidenuniv.nl /mijlpalen/sacred.html (4036 words)

	[No title]
		The difference is that an active transformation can map a point in configuration space to a new instantiation of that point at a different location in configuration space, whereas a passive transformation leaves all instantiations of the point in the same place in configuration space, but moves a new instantiation of the coordinate frame.
		The input 'modcoords' are over-written by the homogeneous quotient in the output 'modcoords'.
		The superscript denotes the dimensionality of the Euclidean space.
	www.rdg.ac.uk /AcaDepts/si/sisweb7/pop/local/ref/modtransfms (4534 words)

	Henri Bergson: Time and Free Will: Chapter 2: The Multiplicity of Conscious States; The Idead of Duration (Site not responding. Last check: 2007-11-06)
		They are therefore parts of space, and space is, accordingly, the material with which the mind builds up number, the medium in which the mind places it.
		In short, just as nothing will be found homogeneous in duration except a symbolical medium with no duration at all, namely space, in which simultaneities are set out in line, in the same way no homogeneous element will be found in motion except that which least belongs to it, the traversed space, which is motionless.
		It follows from this analysis that space alone is homogeneous, that objects in space form a discrete multiplicity, and that every discrete multiplicity is got by a process of unfolding in space.
	spartan.ac.brocku.ca /~lward/Bergson/Bergson_1910/Bergson_1910_02.html (13080 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)

	What is homogeneous space? - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-11-06)
		It's "homogeneous" because you can specify vector transformations (rotation, skew, scaling, etc.) as well as translations in a single representation (an nxn matrix of rank 1 greater than the vector space).
		Homogeneous in this discussion refers to the fact that the same representation (namely a 4x4 matrix) may represent vector transformation, translation, and perspective transformation.
		It also refers to the homogeneous representations of vectors and points in a vector of 1 degree greater than the dimension of the space.
	www.gamedev.net /community/forums/viewreply.asp?ID=1357314 (1027 words)

	Principal homogeneous space (Site not responding. Last check: 2007-11-06)
		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.sciencedaily.com /encyclopedia/principal_homogeneous_space (597 words)

	Triangle Scan Conversion using 2D Homogeneous Coordinates
		In canonical eye space, the center of projection is at the origin, the direction of projection is aligned down the Z axis, and the field of view is 90 degrees.
		In eye space, this is the tetrahedron with the eye at the apex and the triangle to be rendered as the base.
		One is the reciprocal of the screen space area of the triangle, and well-defined for all rendered triangles; but the other three are for perspective projection of the three vertices, and can be undefined even for visible triangles.
	www.cs.unc.edu /~olano/papers/2dh-tri (4648 words)

	The Nature of Space (Site not responding. Last check: 2007-11-06)
		From the fact that the space in inertial frames is homogeneous and isotropic, we know that it is essentially featureless such that one inertial frame must be internally indistinguishable from another.
		Somehow the space in reference frames fixed in the presence of matter is distorted so that the points in that space are not all the same, and that lack of sameness causes objects in that space to seek new positions.
		Since space in an inertial frame of reference is homogeneous and isotropic, then any object moving at a constant velocity, that is with fixed speed and direction, will pass through a succession of points that are all identical.
	www.mcasco.com /p1ns.html (2521 words)

	List of Publications
		Riemannian manifolds with one-dimensional orbit space, Preprint of Roma University,1991, 32p.,(with A. Alekseevsky), Ann.
		Generalized reductive homogeneous spaces and quasi-graded semisimple Lie algebras (with Spiro),Preprint of University of Ancona,1993,22p, Proceed.
		Homogeneous Ricci-positive 5-manifolds (with I. Dotti-Miatello, C. Ferraris), Preprint Univ. of Cordoba, 1993, n28, 11p.; Pacific Math.
	www.hull.ac.uk /php/masdva/list_of_publications.htm (956 words)

	[No title]
		The homogeneous coordinate (x,y,z,w) is (x/w, y/w, z/w) in non-homogeneous coordinates.
		In the standard graphics pipeline, 4D homogeneous points are used to represent points in 3D space.
		Homogeneous coordinates are useful for many geometric tasks, and show up in graphics and geometry calculations on a regular basis.
	web.mit.edu /thouis/plucker-intro.txt (685 words)

	CGAL Developers' Manual (2.3): (Site not responding. Last check: 2007-11-06)
		Homogeneous representation is advantageous over Cartesian representation whenever systems of linear equations with integral coefficients are to be solved.
		Homogeneous representation has the disadvantage that predicates become more complicated.
		While in the internal homogeneous representation, an integral number type is sufficient, rational numbers must sometimes be used outside the internal representation, for example, when the squared length of a vector is computed.
	www.cgal.org /DManual/html/Chapter_kernels.html (996 words)

	thesis71
		In seeking explicit representations of this group one is naturally led to the idea of choosing different carrier spaces for the wave functions and, hence, to homogeneous spaces since Minkowski spacetime is a space of this sort.
		Turning now to the problem of finding suitable restrictions on the elements f of the representation space, the irreducibility requirement is to be met by requiring that the elements f be simultaneous eigenfunctions of a complete set of mutually commuting operators.
		A third approach [8] has been suggested: namely, to accept the breakdown of microcausality as a fact of life for field theories on the homogeneous spaces in question, and to leave the structure functions as undetermined smearing operators whose presence is responsible for the nonlocality of the resulting field theories.
	www.draken.com /drakenr/Paper01.htm (3845 words)

	Xah: The Shapes of Space
		A plane or sphere is a homogeneous space since the geometry is everywhere the same, while a torus or monkey saddle surface is not.
		A closed one-dimentional space of a circle is different from a knot extrinsically, but are identical when considered intrinsically.
		Among these 2d spaces, there's the idea of orientability, which is a intriguing property: that it is possible to move about in the space and become mirror image.
	xahlee.org /Periodic_dosage_dir/20031225_shape_space.html (2644 words)

	[No title]
		Firstly, the Hano-Kobayashi fibration of a compact complex homogeneous space with invariant volume (we might also call it the Ricci form reduction) is holomorphic and coincides with the anticanonical fibration (see \cite{DG1}).
		We also note that every compact complex homogeneous space $M$ with a 2-cohomology class such that its top power is nonzero in the top cohomology group, admits a transitive real Lie transformation group $G$, which acts on $M$ by holomorphic transforms and preserves a volume form.
		He proved that if $G/H$ is a compact complex homogeneous space with a $G$-invariant volume and if $G$ is semisimple, then $G/H$ is a holomorphic fiber bundle over a projective rational homogeneous space, the typical fiber being a complex parallelizable homogeneous space of a reductive complex Lie group.
	www.math.psu.edu /era-mirror/1997-01-013/1997-01-013.tex.html (1518 words)

	Homogeneous Space Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-06)
		Looking For homogeneous space - Find homogeneous space and more at Lycos Search.
		Find homogeneous space - Your relevant result is a click away!
		Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for thier respective symmetry groups.
	www.karr.net /search/encyclopedia/Homogeneous_space (635 words)

	U.S. Pregrant 20040145589 - Method and programmable device for triangle interpolation in homogeneous space (Site not responding. Last check: 2007-11-06)
		After obtaining the vertices of a triangle, the world space coordinates and the attribute of each vertex are transformed to coordinates and an attribute in viewer space.
		Then a set of homogenous coefficients of each vertex is computed based on the viewer space coordinates, and the viewer space coordinates of each vertex are projected to coordinates in screen space.
		Pixels in the screen space that are affected by the triangle are determined based on the screen space coordinates.
	cxp.paterra.com /uspregrant20040145589.html (154 words)

	Color image converting apparatus and method for determining when a homogeneous color space is located outside of a ...
		Especially, in case of a synthetic image formed on a computer screen, if a color which should originally be different is outputted as the same color, it is remarkable, and the outputted image gives an impression that the image is different from that of the computer screen.
		An input color image signal inputted through an image input section 101 is converted in the color converting section 102 to a color in an Lab* space which is a homogeneous color space, and is written in the image memory section 103.
		In the region conversion deciding section 108, image data within a region decided as a unit region to be displaced inward in the radial direction is subjected to color conversion by means of mapping between regions by the map/color converting section 109, and then is sent to the Lab* to CMY converting section 110.
	www.freepatentsonline.com /6016359.html (2375 words)

	Inflation
		The effect of this is to dilute away all the inhomogeneities in space, such as monopoles predicted by many high energy theories (one of the early motivations), or any other sort of fluctuations.
		For example, the Cosmic Microwave background is homogeneous on scales which were not in causal contact when the signal was created, and thus suggests we need something like inflation to explain its homogeneity.
		This change from being the energy of space to kinetic energy of particles arises very naturally when particles are treated as the quantum fields that they are.
	astron.berkeley.edu /~jcohn/inflation.html (1147 words)

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