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Topic: Affine representation

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	Layered Image Representation
		Affine motions parameters are estimated from the optic flow data by the model estimator for each subregion.
		Affine motion segmentation results from applying these motion models in a classification framework on the motion map.
		Once the affine motions and the corresponding regions are identified, data are collected from all the frames in the sequence and layer components are obtained.
	persci.mit.edu /people/jyawang/demos/garden-layer/layer-demo.html (828 words)

	Layered Image Representation
		Affine motions parameters are estimated from the optic flow data by the model estimator for each subregion.
		Affine motion segmentation results from applying these motion models in a classification framework on the motion map.
		Once the affine motions and the corresponding regions are identified, data are collected from all the frames in the sequence and layer components are obtained.
	www-bcs.mit.edu /people/jyawang/demos/garden-layer/layer-demo.html (828 words)

	Search Encyclopedia.com
		Representation of the People Acts Representation of the People Acts, statutes enacted by the British Parliament to continue the extension of the franchise begun by the Reform Bills (see under Reform Acts).
		representation representation, in government, the term used to designate the means by which a whole population may participate in governing through the device of having a much smaller number of people act on their behalf.
		Representation of the counties and boroughs in the House of Commons had not, except for the effects of parliamentary union with Scotland (1707) and Ireland (1800), been ma...
	www.encyclopedia.com /searchpool.asp?target=Affine+representation (439 words)

	Affine representation - Encyclopedia, History, Geography and Biography
		An affine representation of a topological (Lie) group G is a continuous (smooth) homomorphism from G to the automorphism group of an affine space, A.
		Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation.
		We may ask whether a given affine representation has a fixed point in the given affine space A.
	www.arikah.net /encyclopedia/Affine_representation (180 words)

	Various Geometries
		Affine Geometry is not concerned with the notions of circle, angle and distance.
		Affine transformations preserve collinearity of points: if three points belong to the same straight line, their images under affine transformations also belong to the same line and, in addition, the middle point remains between the other two points.
		Analytically, affine transformations are represented in the matrix form f(x) = Ax + b, where the determinant det(A) of a square matrix A is not 0.
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	Projective representation - Wikipedia, the free encyclopedia
		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.
		See also linear representation, affine representation, group action.
	en.wikipedia.org /wiki/Projective_representation (222 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The second one is the 14-dimensional representation of \gq{} which again has to be enlarged by the trivial representation to give a 15-dimensional irrep of \gqh.
		The 10-dimensional representation is the adjoint representation and its weights are the roots $\{2\a_1+\a_2\,, \,\a_1+\a_2\,, \,\a_1\,, \,\a_2\,, \,0\,, \, 0\,,\,-\a_2\,, \,-\a_1\,, \newline -\a_1-\a_2\,, \,-2\a_1-\a_2\}$.
		The 14-dimensional representation is the adjoint representation with weights equal to the roots $\{2\a_1+3\a_2\,, \,\a_1+3\a_2\,, \,\a_1+2\a_2\,, \,\a_1+\a_2\,, \,\a_1\,, \a_2\,,\, 0\,, 0\,, \,-\a_2\,, \,-\a_1\,, \,-\a_1-\a_2\,, \,-\a_1-2\a_2\,, \, -\a_1-3\a_2\,, \,-2\a_1-3\a_2\}$.
	www.ma.utexas.edu /mp_arc/papers/94-128 (5529 words)

	United States Patent: 4,914,563
		The movement is made in a transformed space where the present (transformed) state of the system is at the center of the space, and the curve approximation is in the form of a power series in the step size.
		affine scaling an initial set of values of said parameters to a domain wherein said parameters are represented by a vector in a vector field including vectors indicative of a steepest approach to a more optimum value of said objective function;
		The affine scaling vector field is a special case of such a generalized steepest descent vector field.
	laniels.org /cache/karmarkar_algorithm.html (8582 words)

	Augmented Reality Home Page - Minimal Calibration
		Affine reprojection (Koenderink and van Doorn 1991; Mundy and Zisserman 1992) is used by the U of R system to create the projection of any other point defined in the affine coordinate frame.
		The use of an affine representation for the points allows the calculation of the reprojection of any affine point without knowing the position or internal calibration parameters for the camera.
		The affine coordinates of any point can be computed from Equation 3 if its projection in at least two views is known and the projections of the affine basis points are also known in those views.
	www.se.rit.edu /~jrv/research/ar/thesis.html (3794 words)

	Team - Air2 (Site not responding. Last check: 2007-11-03)
		From the methodological point of view, the focus is on robust motion representation techniques, where the motion is locally approximated by an affine field.
		Local motion representation by an affine field, obtained as the tangent plane to the aforementioned surfaces.
		Segmentation of the motion field on the basis of the affine representation.
	www.inria.fr /rapportsactivite/RA2003/air22003/module13.html (383 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		In the image plane, a three dimensional movement of planar rigid object can be represented satisfactory by an affine motion of his boundary (the exterior contour), which we assume to be closed.
		In practice, this means that applying a curve algorithm to a transformed data set is equivalent to applying the curve algorithm to the data and transforming the resultant (invariance of data approximation with respect such class of transformations).
		In the present work, affine case will be developed and illustrated in the coding contours application by the consideration of a set of invariants under affinities.
	www.enic.fr /people/matusiak/articles/axes_ang.html (446 words)

	Definition of proportional representation
		Proportional representation is also used to describe this (intended) effect.
		15:...e is responsible for the phenomenon of pseudoreal representation.
		representation]] of a mathematical structure, such as a [[group...
	www.wordiq.com /search/proportional+representation.html (772 words)

	Abstracts
		We propose the generalized class of quadratic time-frequency representations (QTFRs) that satisfy the scale covariance property, which is important in multiresolution analysis, and the generalized time-shift covariance property, which is important in the analysis of signals propagating through systems with specific dispersive characteristics.
		We discuss a formulation of the generalized class QTFRs in terms of two-dimensional kernel functions, a generalized signal expansion related to the generalized class time-frequency geometry, an important member of the generalized class, a set of desirable QTFR properties and their corresponding kernel constraints, and a "localized-kernel" generalized subclass that is characterized by one-dimensional kernels.
		The affine and hyperbolic classes of quadratic time-frequency representations (QTFRs) are frameworks for multiresolution or constant-Q time-frequency analysis.
	www.eas.asu.edu /~apapand/research/publication/abstract.html (6226 words)

	QMG project: Geometric Objects and Datatypes (Site not responding. Last check: 2007-11-03)
		The second representation is called the chunk representation, because in this representation the brep or simplicial complex occupies a contiguous chunk of memory.
		The third representation of a brep or simplicial complex is as an Ascii string.
		There are three distinguished properties that specify the geometry of the brep; they are affine_coef, which defines the coefficient matrix of the affine hull of the face, affine_rhs, which defines the right-hand side of the affine hull of the face, and point, which defines the point in the relative interior.
	www.cs.cornell.edu /info/people/vavasis/qmg1.1/geom.html (4392 words)

	AffineTransform (Java 2 Platform SE 5.0)
		Affine transformations can be constructed using sequences of translations, scales, flips, rotations, and shears.
		Returns the X coordinate of the translation element (m02) of the 3x3 affine transformation matrix.
		Returns the Y coordinate of the translation element (m12) of the 3x3 affine transformation matrix.
	java.sun.com /j2se/1.5.0/docs/api/java/awt/geom/AffineTransform.html (5137 words)

	Affine representation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-03)
		Since the (Click link for more info and facts about affine group) affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of (Click link for more info and facts about linear representation) linear representation.
		We may ask whether a given affine representation has a (Click link for more info and facts about fixed point) fixed point in the given affine space A.
		See also (Click link for more info and facts about projective representation) projective representation, (Action taken by a group of people) group action.
	www.absoluteastronomy.com /encyclopedia/A/Af/Affine_representation.htm (209 words)

	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.
		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.
		is known, one can upgrade the projective representation to an affine one by applying a transformation which brings the plane at infinity to its canonical position.
	www.cs.unc.edu /~marc/tutorial/node28.html (330 words)

	The Wavelet Digest :: Post a reply
		OK in that case squeezed states are like "wavelet transforms" in the number phase representation of phase space.
		Now whether this type of representation is actually more than a curiosity in quantum mechanics or QFT is beyond me. All I know is that the commutation relation for quantum observables in the affine coherent states representation is not the canonical one [Q,P]=i, but the 'affine' one [Q,P]=iQ or something like that.
		Affine coherent states can be built from square-integrable representations of the affine group (ax+b) in a similar fashion.
	www.wavelet.org /phpBB2/posting.php?mode=quote&p=3927 (912 words)

	Matches for:
		Affine and Related Invariants: 10 Tensors; 11 Invariants; 12 Parallel displacement of a vector around an infinitesimal closed circuit; 13 Covariant differentiation; 14 Alternative methods of covariant differentiation.
		Projective Invariants: 16 Affine representation of projective spaces; 17 Some geometrical interpretations; 18 Projective tensors and invariants; 19 Transformations of the group $\star\mathfrak G$ Conformal Invariants: 20 Fundamental conformal-affine tensor; 21 Affine representation of conformal spaces; 22 Conformal tensors and invariants; 23 Completion of the incomplete covariant derivative.
		Normal Coordinates: 29 Affine normal coordinates; 30 Absolute normal coordinates; 31 Projective normal coordinates; 32 General theory of extension; 33 Some formulae of extension; 34 Scalar differentiation in a space of distant parallelism; 35 Differential invariants defined by means of normal coordinates.
	www.mathaware.org /bookstore?fn=20&arg1=geotopo&item=CHEL-336 (354 words)

	Developing Connections Between Algebra and Geometry,
		In general an affine transformation in an invertible linear transformation which preserves collinearity, betweeness, parallelism and ratio of division of segments.
		Finally, the equations describing affine transformations are linear, allowing for the efficient use of matrix algebra in the analysis of these equations and the subsequent use of technology.
		From the prior description, an affine transformation is a function defined by x'=ax+by+h and y'= cx+dy+k where a,b,h,c,d,k are real numbers and ad-bc does not equal zero.
	faculty.millikin.edu /~damiller/affinepaper.htm (1385 words)

	ADE
		I call this graph the representation graph of SU(2) because it is built from the fundamental (natural) representation, R of SU(2) and the equations:
		We start with the trivial representation of dimension = 1; m[i,j] is the multiplicity of the directed edge from node i to node j.
		The columns of the character table of G are eigenvectors and the row of the 2-dimensional representation is the row of eigenvalues.
	math.ucr.edu /home/baez/ADE.html (1245 words)

	From projective or affine to metric
		In some cases it is needed to upgrade the projective or affine representation to metric.
		Once the absolute conic has been identified, the geometry can be upgraded from projective or affine to metric by bringing it to its canonical (metric) position.
		In this case, there must be an affine transformation which brings the absolute conic to its canonical position; or, inversely, from its canonical position to its actual position in the affine representation.
	www.cs.unc.edu /~marc/tutorial/node30.html (318 words)

	Sissa - Geometric Control Group Seminars
		Representation of affine control systems as series of nonlinear power moments and optimality
		The initial point in our analysis is the representation of affine control systems in a neighborhood of the equilibrium as the series of nonlinear power moments.
		This, in turn, gives the asymptotic app roximation of the time optimal problems for affine systems by problems for the homogeneous systems whose series hav e the same principle part.
	www.sissa.it /fa/am/seminars/sklyar.html (180 words)

	[cryst] 2 Affine crystallographic groups
		It is recommended to adopt one representation for affine crystallographic groups, and then to stick to it.
		In the representation by augmented matrices, affine crystallographic groups are infinite matrix groups.
		The connection line is labelled with the number of affine subspaces contained in the lower Wyckoff position that contain a fixed representative affine subspace of the upper Wyckoff position.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/cryst/htm/CHAP002.htm (2609 words)

	M.Konecny: Paper (Site not responding. Last check: 2007-11-03)
		This article considers so-called affine representations where numbers are represented by infinite compositions of affine contracting functions on $I$.
		The first result is that all piecewise affine functions of $n$ variables with rational coefficients are computable by a finite transducer which uses the signed binary representation.
		The second and main result is that any function computable by a finite transducer using an affine representation is affine on any open connected subset of $I^n$ on which it is continuously differentiable.
	www.dcs.ed.ac.uk /home/mkonecny/papers/tcs2002.html (171 words)

	OUP: UK General Catalogue
		The material is suitable for graduate students and research mathematicians interested in representation theory of algebraic groups, Hecke algebras, quantum groups, and combinatorial theory.
		Representations of Lie algebras in positive characteristic, J.
		The representation theory of the Ariki-Koike and cyclotomic $q$-Schur algebras, A.
	www.oup.com /uk/catalogue/?ci=9784931469259 (401 words)

	The affine connection (Site not responding. Last check: 2007-11-03)
		With an affine connection, a path between two points of the manifold establishes a linear transformation between the tangent spaces at those points.
		A metric determines a symmetric affine connection, and a symmetric affine connection together with a sphere at some point of the manifold, determines a metric.
		The affine connection, on the other hand, speaks of no particular coordinate system but is, as any tensor representation, expressed in some coordinate system.
	www.cap-lore.com /MathPhys/Affine.html (479 words)

	{LAL} is square: Representation and expressiveness in light affine logic (Site not responding. Last check: 2007-11-03)
		LAL is square: Representation and expressiveness in light affine logic
		We focus on how the choice of input-output representation has a crucial impact on the expressiveness of so-called “logics of polynomial time.” Our analysis illustrates this dependence in the context of Light Affine Logic (LAL), which is both a restricted version of Linear Logic, and a primitive functional programming language with restricted sharing of arguments.
		By slightly relaxing representation conventions, we derive doubly-exponential expressiveness bounds for this “logic of polynomial time.” We emphasize that squaring is the unifying idea that relates upper bounds on cut elimination for LAL with lower bounds on representation.
	www.linearity.org /turtle/reports/Nee-Mai-LIS-ICC02.html (275 words)

	[No title]
		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.
		Thus we seek to embed g into the algebra of differential operators on the affine space - which is precisely the Heisenberg (aka Weyl) 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)

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