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Topic: Projective representation

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	From projective to affine
		In this case upgrading the geometric structure from projective to affine implies that one first has to find the position of the plane at infinity in the particular projective representation under consideration.
		In Figure 2.1 a projective representation of a cube is given.
		The vanishing points obtained from lines which are parallel in the affine stratum constrain the position of the plane at infinity in the projective representation.
	www.cs.unc.edu /~marc/tutorial/node28.html (330 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 (757 words)

	CS888 Project (Site not responding. Last check: 2007-09-15)
		This representation permits reprojection on the image plane that gives an illusion of 3D world transformation when the view at the center of the sphere is rotated or zoomed.
		The 3D silhouette edges are projected on the sphere with ray-surface intersection between the ray from the center of sphere to the 3D edges points and the sphere surface.
		The 3D silhouette curve on the sphere is projected on the image plane by perspective projection using OpenGL.
	www.cgl.uwaterloo.ca /~efourque/courses/cs888/project.html (2123 words)

	Parity (physics) - Wikipedia, the free encyclopedia
		All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states.
		The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group.
		Projective representations of the rotation group that are not representations are called spinors, and so quantum states may transform not only as tensors but also as spinors.
	en.wikipedia.org /wiki/Parity_(physics) (2044 words)

	Representations (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-09-15)
		Suppose G is a split group of Lie type defined over the field k and r is the least common multiple of the nonzero abelian-group invariants of the coisogeny group of G (see Section Isogeny).
		The adjoint (projective) representation of the group of Lie type G over an extension of its base ring, ie.
		The highest weight (projective) representation with highest weight v of the group of Lie type G over an extension of its base ring.
	www.math.lsu.edu.cob-web.org:8888 /magma/text1057.htm (577 words)

	Projective Geometry for Grouping and Orientation Tasks (Site not responding. Last check: 2007-09-15)
		Projective Geometry has been a succesful research area in Computer Vision within the last decade and has shown to play an important role in image analysis.
		It provides not only a consistent and easy representation of geometric entities such as points, lines and planes, but also for the camera geometry of single and multiple views.
		The introductory tutorial is meant for all researchers and developers who are interested in use of projective geometry in the context of image analysis, especially for points, lines and planes.
	www.ipb.uni-bonn.de /DAGM/Tutorials2001/overview-geometry.html (244 words)

	Moloney (CHArt 1999) (Site not responding. Last check: 2007-09-15)
		Projection fills the gaps; but to arrange the emanations first from drawings to buildings, then from buildings to the experience of the perceiving and moving subject, in such a way as to create in these unstable voids what cannot be displayed in designs - that was where the art lay
		The project was conceived in a physical medium, developed utilizing digital models, appraised via a computer generated physical model and realized by transmitting digital instructions to CNC machines.
		The importance of Bilbao is that it marks a fundamental shift in the relationship between representation and construction.
	www.chart.ac.uk /chart1999/papers/moloney.html (2829 words)

	abstracts page
		Abstract: Recent developments in geometric algebra have shown that by moving from a projective to a conformal representation (5d representation of 3d space), one is able to extend the range of geometrical operations that can be carried out in an efficient and elegant way.
		For example, while in projective space one is able to intersect lines and planes in a simple fashion, in conformal space one is able to intersect and represent spheres, lines, circles and planes.
		In addition, all the operations of Euclidean geometry (dilations, translations, rotations and inversions) are smoothly integrated with the projective representation.
	www.mrao.cam.ac.uk /~clifford/publications/abstracts/ll_surface.html (141 words)

	CMS/CSHPM Summer 2005 Meeting
		There is a well-established classification of the irreducbible representations of a real reductive group in terms of discrete-series representations and their related data.
		The purpose of the talk is to point out a striking analogy between these representations of SL(2,Z) and certain representations of Weyl groups on spaces of characters of semisimple Lie groups, the characters of the Lie group playing the role for the Weyl group which the modular elliptic functions play for SL(2,Z).
		Classifying the irreducible unitary representations of a reductive Lie group may be formulated as the algebraic problem of classifying the irreducible Harish-Chandra modules which admit a positive definite invariant Hermitian form.
	www.cms.math.ca /Events/summer05/abs/rt (1325 words)

	Introduction (Site not responding. Last check: 2007-09-15)
		A representation of another class of modular lattices was given by Baer [1942] when he developed a unified theory of projective spaces and finite Abelian groups.
		Besides the algebraic representation of projective geometries by unitary modules a further subject is of eminent interest.
		This is the generalization of the fundamental theorem of projective geometry* and hence, the representation of mappings between submodule lattices.
	www1.elsevier.com /homepage/saj/504595/21a.htm (1128 words)

	[No title]
		The purpose of this paper is to improve the bounds in \cite{sz} for the case of orthogonal groups and to provide estimates for the gap between the smallest representations for orthogonal and unitary groups.
		However the results of \cite{gt} for linear groups and those of \cite{tz} for representations in characteristic zero suggest that the bounds of \cite {sz} can be improved.
		This representation is a factor of the reduction modulo $r$ of the smallest irreducible projective representation in characteristic 0.
	www.msri.org /people/members/choffman/public_html/pap/dimnew.tex~ (3306 words)

	[No title] (Site not responding. Last check: 2007-09-15)
		In other words the ntaural representations of quantum mechanics are often naturally projective representations, since the space of states is the projective space of H and this is where the physics occurs.
		Another manifestation of projectivity in quantum mechanics is the Heisenberg uncertainty principle, as encoded in the Heisenberg algebra.
		This arises in representation theory, where one wishes to consider highest weight representations (more generally "category O") as a natural habitat for representation theorists, where one can work like in finite dimensions.
	www.ma.utexas.edu /~benzvi/math/Langlands5 (1667 words)

	[No title] (Site not responding. Last check: 2007-09-15)
		Projective representation theory can be regarded as a natural generalization of ordinary representation theory.
		The techniques involved are quite diverse, and include ordinary cohomology theory, K-theory, vector bundle theory, Hodge duality, representation theory, Lie theory, as well as topics from the combinatorics of binomial coefficients and quadratic residues from number theory.
		Each infinitesimally faithful representation of a reductive complex connected algebraic group $G$ induces a dominant morphism $\Phi$ from the group to its Lie algebra $\g$ by orthogonal projection in the endomorphism ring of the representation space.
	www.math.ucalgary.ca /~nikolaev/talks2002.html (1270 words)

	Wolfgang Förstner: Uncertainty and Projective Geometry (Site not responding. Last check: 2007-09-15)
		The great potential of both, projective geometry and statistics, can be integrated easily for propagating uncertainty through reasoning chains, for making decisions on uncertain spatial relations and for optimally estimating geometric entities or transformations.
		This is achieved by (1) exploiting the potential of statistical estimation and testing theory and by (2) choosing a representation of projective entities and relations which supports this integration.
		The redundancy of the representation of geometric entities with homogeneous vectors and matrices requires a discussion on the equivalence of uncertain projective entities.
	www.ipb.uni-bonn.de /papers/2004/foerstner04.uncertainty.html (190 words)

	Projective representation - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-09-15)
		The interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to a conventional linear group representation.
		This need not come down to a coboundary: that is, projective representations may not lift.
		From Schur's lemma, it follows that the irreducible representations of central extensions of G, and the projective representations of G, describe essentially the same questions of representation theory
	en.wikipedia.org.cob-web.org:8888 /wiki/Projective_representation (306 words)

	[No title]
		If $\sigma_t$ is the 2-group cocycle corresponding to the projective unitary representation $\pi_t$, then $\Cal A_t$ is isomorphic ([Ra]) to $\Cal L(\G,\sigma_t)\otimes B(K)$, where $K$ is an infinite dimensional separable Hilbert space and $\Cal L(\G,\sigma_t)$ is the twisted group algebra of $\G$.
		Assume $D$ is the projection $\langle \cdot,\zeta\rangle\zeta$, where $\zeta$ is a vector of norm 1.
		By considering the cyclic space genetared by $\zeta$ it follows that it is sufficient then to prove the result under the assumption that the Hilbert space $H$ has dimension 1.
	www.math.uiowa.edu /~radulesc/paperNestFlorin (2132 words)

	week81
		A projective unitary representation of a group H can also be thought of as a representation of a bigger group H~ called a "central extension" of H.
		The idea is that this bigger group has a bunch of phases built into it to absorb the phase ambiguities in the projective representation of H. Let U(1) be the unit circle in the complex plane, a group under multiplication.
		For this reason, if we are doing quantum theory and we don't like projective representations, it's okay as long as we understand the central extensions of our group of symmetries.
	math.ucr.edu /home/baez/week81.html (2274 words)

	FAST TRANSFORMS IN JACOBIAN OF GENUS 2 HYPERELLIPTIC CURVES IN PROJECTIVE COORDINATES
		Main operators for scalar multiplication of reduced representation divisors (hereinafter referred to as divisors) are the addition and doubling of divisors.
		In this connection in the method described it is suggested to apply the projective representation of divisors.
		In accordance with the paper objective, there was developed a method of arithmetic transforms in Jacobian genus 2 HEC in projective coordinates which provides a lower complexity if compared to the existent methods [4, 6] and, thus, allowing for increase in the efficiency of scalar multiplication.
	www.qarea.com /knowledge_base_pages/knowledge2/knowledge-details.php (2969 words)

	Citations: Qualitative Spatial Reasoning using a N-Dimensional+ Projective Representation - Pais, Pinto-Ferreira ... (Site not responding. Last check: 2007-09-15)
		Geometrical Concepts Ground Projective Level The ground level de nes the projective representation foundations that are based on both kinds of concepts the topological like region and the geometrical like projective axis, projective region vertex and projective axis vertex
		A region results from a topological transformation of the original shape of a body into a bounding box with edges parallel of all projective axis.
		One important purpose of this level is to provide a decoding system from verbal knowledge (symbolic de nitions) into projective geometrical concepts and vice versa.
	citeseer.ist.psu.edu.cob-web.org:8888 /context/1348480/0 (258 words)

	ECCV 2004 Tutorial T1
		The redundancy of the representation of geometric entities with homogeneous vectors and matrices, the non-linearity of the relations and the need for multiple constraints requires a careful analysis of singularities of the uncertainty representation, of the bias of constructing new geometric entities and of the estimation techniques.
		The introductory tutorial is meant for all researchers and developers who are interested in the analysis of uncertain geometric entities in 2D and 3D, especially in the context of image analysis.
		Basic knowledge in linear algebra is recommended, basic knowledge of statistics and projective geometry is useful.
	cmp.felk.cvut.cz /eccv2004/tutorials/eccv2004-t1.htm (384 words)

	Citebase - Projective representation of k-Galilei group (Site not responding. Last check: 2007-09-15)
		Authors: Gonera, C. Kosinski, P. Maslanka, P. Tarlini, M. The projective representations of k-Galilei group G
		are found by contracting the relevant representations of k-Poincare group.
		It is shown that it is not possible to replace the projective representations of G
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9806053 (94 words)

	On The Projective Invariant Representation of Conics in Computer Graphics (Abstract) (Site not responding. Last check: 2007-09-15)
		On The Projective Invariant Representation of Conics in Computer Graphics
		A general formulation for conics and conic arcs for the purpose of computer graphics is given, based on principles and theorems of projective geometry.
		This approach allows the approximation of these curves by line segments to be postponed in the graphics output pipeline; it results in a more compact storage, faster approximation algorithms and smoother outlook of the curves.
	www.eg.org /EG/CGF/volume8/issue4/v08i4pp301-314_abstract.html (91 words)

	Cryptology ePrint Archive (Site not responding. Last check: 2007-09-15)
		Denoting by $P=[k]G$ the elliptic-curve double-and-add multiplication of a public base point $G$ by a secret $k$, we show that allowing an adversary access to the projective representation of $P$ results in information being revealed about $k$.
		Such access might be granted to an adversary by a poor software implementation that does not erase the $Z$ coordinate of $P$ from the computer's memory or by a computationally-constrained secure token that sub-contracts the affine conversion of $P$ to the external world.
		From a wider perspective, our result proves that the choice of representation of elliptic curve points {\sl can reveal} information about their underlying discrete logarithms, hence casting potential doubt on the appropriateness of blindly modelling elliptic-curves as generic groups.
	eprint.iacr.org /2003/191 (158 words)

	SAL- Mathematics - Misc - Symmetrica (Site not responding. Last check: 2007-09-15)
		Symmetrica is a collection of routines, written in C, to handle the following topics:
		ordinary representation theory of the symmetric group and related groups
		Here you may look on a part of the manual.
	gd.tuwien.ac.at:8050 /A/0/SYMMETRICA.html (43 words)

	Hexapedia - Projective space (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-09-15)
		It generalises the projective plane that may be constructed from a three-dimensional vector space, over any field.
		While the theory of projective planes has an aspect that belongs to combinatorics too, that is absent in the general case.
		Projective space is basic in 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).
	www.hexafind.com.cob-web.org:8888 /encyclopedia/projective_space (487 words)

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