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Topic: Projection operator

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	Projection operator - Wikipedia, the free encyclopedia
		In mathematics, a projection operator P on a vector space is a linear transformation that is idempotent, that is, P
		Such transformations project any vector in the vector space to a vector in the subspace that is the image of the transformation.
		In physics, the term projection operator usually means self-adjoint projection operator.
	en.wikipedia.org /wiki/Projection_operator (181 words)

	PlanetMath: linear involution
		An equivalent definition is that a linear involution is a linear operator that equals its own inverse.
		The next theorem gives a correspondence between involution operators and projection operators.
		Cross-references: characteristic, field, projection, operators, reciprocal polynomial, characteristic polynomial, matrix, complex, eigenvalues, theorem, inverse, equivalent, identity operator, linear operator, vector space
	planetmath.org /encyclopedia/Involution.html (98 words)

	Linear Operators (Site not responding. Last check: 2007-10-16)
		The simplest linear operator is the identity operator
		If the action of a linear operator on the basis vectors is known, then the action on any vector in the vector space is determined.
		The inverse operator of A, denoted by A
	electron6.phys.utk.edu /qm1/modules/m3/operators.htm (286 words)

	765paper
		The algebraic selection operator yields a "horizontal" subset of a given relation-that is, that subset of tuples within the given relation for which a specified predicate is satisfied.
		The projection operator yields a "vertical" subset of a given relation-that is, that subset obtained by selecting specified attributes, in a specified left-to-right order, and then eliminating duplicate tuples within the attributes selected.
		If X denoted the relation to be projected, then the result of the projection has the same qualified attribute-names X. No attributes may be specified more than once in a projection operation.
	www.cs.ndsu.nodak.edu /~patel/765paper.html (2746 words)

	[No title]
		Projection matrices have important symmetry properties and satisfy Pn=P — the projection of a projection remains constant.
		The projection operator looks the same but in the formulas the column vector 'a' is replaced with a matrix 'A' with multiple columns.
		In both cases, p = Pb is the component of b projected into the column space of A. E = b — Pb is the orthogonal error vector.
	uspas.fnal.gov /materials/4_LSQ.doc (1328 words)

	Oblique projection - Wikipedia, the free encyclopedia
		The projectors in oblique projection intersect the projection plane at an oblique angle to produce the projected image, as opposed to the perpendicular angle used in orthographic projection.
		In a general oblique projection, spheres of the space are projected as ellipses on the drawing plane, and not as circles as you would expect them from an orthogonal projection.
		Cabinet Projection, popular in furniture illustrations, is an example of such a technique, wherein the receding axis is scaled to half-size and drawn at an angle of 45-degrees.
	en.wikipedia.org /wiki/Oblique_projection (421 words)

	Locating cross peaks
		Three projection operators are currently supported: D2 (for COSY type spectra), C2v (for NOESY type spectra), and D2d (special case: Works with COSY spectra where the peaks have the same separation constants in the two dimensions).
		The peaks themselves are determined as maxima of the values calculated by the projection operator: For a position to be considered a peak, the value of the projection operator must be greater than Cutoff 2, and the values within Range 2 Hz must be less than the center point.
		The projection operator assumes that you have an anti-phase pattern where the upper right and the lower left subpeaks are positive.
	www.msi.umn.edu /software/pronto/manual/ch7.html (1701 words)

	Projection (linear algebra) - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-16)
		Projections map the whole vector space to a subspace and leave the points in that subspace unchanged.
		The term "orthogonal" is used because the projection of a vector is orthogonal to its complement.
		In symbols, for a projection P and a vector x, the vector Px is the projection of x.
	en.wikipedia.org.cob-web.org:8888 /wiki/Projection_(linear_algebra) (650 words)

	Snuark Particle Physics
		The Pauli spin projection operators all have a trace of 1.
		Among the Dirac algebra, the binons are the projection operators that have a trave of 1.
		For example, the projection operators for spin-1/2 in the +z and -z directions are orthogonal.
	www.snuark.com (579 words)

	Hoon Hong: An Improvement of the Projection Operator in Cylindrical Algebraic Decomposition (Site not responding. Last check: 2007-10-16)
		An important component of the CAD method is the projection operation: given a set $A$ of $r$--variate polynomials, the projection operation produces a certain set $P$ of $(r-1)$--variate polynomials such that from a CAD of $(r-1)$--dimensional space for $P$ one can construct a CAD of $r$--dimensional space for $A$.
		By generalizing a lemma on which the proof of the original projection operation is based we are able to find another projection operation which produces a smaller number of polynomials.
		The number of polynomials produced by the original projection operation is dominated by $m^2 n^3$ whereas the number of polynomials produced by our projection operation is dominated by $m^2 n^2$.
	www4.ncsu.edu /~hong/papers/Hong90a.html (241 words)

	Projection (Site not responding. Last check: 2007-10-16)
		to be computed by the projection phase is not unique, and there have been several ``projection operators'' proposed for the construction of
		A set of polynomials that is closed under the projection operator is easily seen to define a CAD -- i.e.
		The set of projection factors is constructed by computing the irreducible factors of the
	www.cs.usna.edu /~wcbrown/research/qebycad/Tutorial/node6.html (190 words)

	[No title]
		A projection operator P acts on a vector V to give a new vector PV.
		When the outcome of a probing action appears the vector V suddenly jumps to the vector PV or to P'V, where P is the projection operator associated with the probing action, and P' is the complementary projection operator.
		In this example the probing action is associated with the projection operator P such that the vector PV is directed along A1 and P'V is directed along A2.
	www-physics.lbl.gov /~stapp/Chap5.txt (1203 words)

	Projection-slice theorem - Wikipedia, the free encyclopedia
		In mathematics, the projection-slice theorem in two dimensions states that the Fourier transform of the projection of a two-dimensional function f(r) onto a line is equal to a slice through the origin of the two-dimensional Fourier transform of that function which is parallel to the projection line.
		This theorem is used, for example, in the analysis of medical CAT scans where a "projection" is an x-ray image of an internal organ.
		The projection of f(r) onto the x-axis is the integral of f(r) along lines of sight parallel to the y-axis and is labelled p(x).
	en.wikipedia.org /wiki/Projection-slice_theorem (455 words)

	Projection (Site not responding. Last check: 2007-10-16)
		In mathematics a projection is a linear transformation which remains unchanged if applied twice (''p''(''u'') = p(''p''(''u''))) (in other words, it is idempotent), such as that taking (''x'', y, z) in three dimensions to (''x'', y, 0) in the plane or generalisations of this in other dimensions.
		In cinematography, projection is the display of a movie in a theater using a film projector, which is likely to be replaced by digital projection
		To these personal qualifications must be added a great advantage--and a perfect and intimate knowledge of all the recesses various individuals, whether friendly or hostile, with whom he might come circumstances in which he was placed.
	projection.kiwiki.homeip.net (234 words)

	Projection -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-16)
		A projection is the transformation of points and lines in one
		The branch of geometry dealing with the properties and invariants of geometric figures under projection is called projective geometry.
		The average projected area over all orientations of any ellipsoid is 1/4 the total surface area.
	mathworld.wolfram.com.cob-web.org:8888 /Projection.html (236 words)

	DBAzine.com: Dualities among Relational Algebra Operators (Site not responding. Last check: 2007-10-16)
		They are dual operations in the traditional set theory: unions and intersections in any set identity formula can be interchanged so that we get another identity.
		In the set semantics, Projection operator has the side effect of merging duplicate rows into a single row, while Selection operator doesn’t have any side effects.
		Classic projection reduces the number of columns, so you might wonder how useful is generalized projection that adds extra columns.
	www.dbazine.com /oracle/or-articles/tropashko7 (1106 words)

	Abstracts for Submitted and Published Papers
		McCallum's projection operator for Cylindrical Algebraic Decomposition (CAD) represented a huge step forward for the practical utility of the CAD algorithm.
		The reduced projection has the potential to not simply speed up CAD computation for problems that are currently solvable in practice, but actually increase the scope of problems that can realistically be attacked via CAD's.
		Generalizing the improved projection operator to dimension greater than three is a topic of ongoing research.
	www.cs.usna.edu /~wcbrown/research/abstracts/abstracts/abstracts.html (741 words)

	Projection Operators and Completeness
		A ket vector followed by a bra vector is an example of an operator.
		An operator maps one vector into another vector, so this is an operator.
		We could use this to project out the odd parity states, for example.
	quantummechanics.ucsd.edu /ph130a/130_notes/node185.html (140 words)

	Postulates of Quantum Theory
		Property 2: Different eigenfunctions of a hermitian operator (i.e., eigenfunctions with different eigenvalues) are orthogonal (i.e., the scalar product of two different eigenfunctions is equal to zero).
		Certain operators have a continuous spectrum of eigenvalues.
		For example, the coordinate operator is one such operator since it satisfies the equation
	xbeams.chem.yale.edu /~batista/vvv/node2.html (811 words)

	Projection operator (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-16)
		In mathematics, a projection operator on a vector space is an idempotent linear transformation.
		Such transformations project any point in the vector space to a point in the subspace that is the image of the transformation.
		The condition that M ′ = M says M is a symmetric matrix if all of the entries in M are real.
	projection-operator.peernet.sk.cob-web.org:8888 (143 words)

	Conservative Visibility Preprocessing using Extended Projections
		We introduce novel extended projection operators, which permits efficient occlusion culling with respect to all viewpoints within a cell, and takes into account the combined occlusion effect of multiple occluders.
		We use extended projection of occluders onto a set of projection planes to create extended occlusion maps; we show how to efficiently test occludees against these occlusion maps to determine occlusion with respect to the entire cell.
		We use projection planes at a finite distance from the volumetric cell instead of a projection plane at infinity (direction space) which allows us to treat umbras of finite extent.
	people.csail.mit.edu /fredo/PUBLI/Sig2000/index.htm (536 words)

	Reconstruction (Site not responding. Last check: 2007-10-16)
		This involves two steps; the image is back projected and then filtered with a two dimensional ramp filter.
		This operator represents the accumulation of the projections that pass through the point (x,y).
		The projection slice theorem implies that the Radon transform of the two-dimensional convolution of two functions is equal to the one-dimensional convolution of their Radon transforms.
	www.owlnet.rice.edu /~elec539/Projects97/cult/node3.html (263 words)

	Quantum Logic and Probability Theory (Stanford Encyclopedia of Philosophy)
		It is uncontroversial (though remarkable) that the formal apparatus of quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over by the "quantum logic" of projection operators on a Hilbert space.
		Adhering to the idea that commuting observables -- in particular, projections -- are simultaneously measurable, we conclude that the members of a Boolean "block" (that is, a Boolean sub-ortholattice) of L(H) are simultaneously testable.
		The interpretation of projection operators as representing the properties of a physical system is already explicit in von Neumann's Grundlagen.
	plato.stanford.edu /entries/qt-quantlog/index.html (7987 words)

	casa::Projection class Reference (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-16)
		Geometric parameters needed for a sky projection to a plane.
		In the recent versions, this paper has split into three, and the projection parameters have been reworked into the PV matrix.
		This projection requires no parameters so the printed parameter vector would be of zero length.
	www.aoc.nrao.edu.cob-web.org:8888 /~ddebonis/doxygen/prototype/classcasa_1_1Projection.html (463 words)

	APS - 2005 APS March Meeting - Event - Self-consistent projection operator approach to momentum dependent excitation ... (Site not responding. Last check: 2007-10-16)
		Description of the momentum-dependent excitation spectrum with high resolution in momentum and energy at low temperature is a current issue in the theory of strongly correlated electrons.
		We present such an effective medium approach on the basis of the projection operator technique to the retarded Green function.
		We clarify the reduction of the quasiparticle weight, the relaxation of the quasiparticle band width, and the formation of the Mott-Hubbard band due to nonlocal correlations.
	meetings.aps.org /Meeting/MAR05/Event/22241 (198 words)

	APS - 2005 APS March Meeting - Event - Projection operator method CPA to single-particle excitation spectra (via ... (Site not responding. Last check: 2007-10-16)
		Single-site theories for electron correlations such as the many-body CPA, the dynamical CPA, and the dynamical-mean field theory are useful as a starting point to describe strongly correlated electron systems.
		We propose here the projection operator technique combined with the many-body CPA which allows us to calculate the excitation spectrum directly from the retarded Green function.
		The basic idea is to introduce an energy dependent Liouville operator for the description of the dynamics of correlated electrons.
	meetings.aps.org.cob-web.org:8888 /Meeting/MAR05/Event/28319 (232 words)

	PlanetMath: Drazin inverse
		For example, a projection operator is its own Drazin inverse,
		Cross-references: eigenvalue, eigenvector, nilpotent, regular, Jordan canonical form, matrix, finite, Moore-Penrose pseudoinverse, properties, projection, accumulation point, spectral radius, operator
		(Operator theory :: Other types of operator theory :: Miscellaneous)
	planetmath.org /encyclopedia/DrazinInverse.html (110 words)

	Imdb Glossary letter: B (Site not responding. Last check: 2007-10-16)
		B-films were cheaper for studios because they did not involve the most highly paid actors or costly sets, and were popular with theater owners because they were less expensive to bring into their theaters while still able to draw revenue.
		A photographic technique whereby live action is filmed in front of a screen which the background action is projected on.
		A member of the sound crew who operates the boom microphone.
	www.imdb.com /Glossary/B (1253 words)

	The Role of Projection Operators in the Theory of N-Beam Diffraction and the Inversion of Three-Beam Elastic Scattering ...
		The Role of Projection Operators in the Theory of N-Beam Diffraction and the Inversion of Three-Beam Elastic Scattering Intensities -- from Mathematica Information Center
		The Role of Projection Operators in the Theory of N-Beam Diffraction and the Inversion of Three-Beam Elastic Scattering Intensities
		Commencing from a projection operator description of N-beam diffraction, the mathematical basis for the recovery of phase and amplitude information from a 3-beam convergent beam electron diffraction pattern is given for both the centrosymmetric and non-centro symmetric cases.
	library.wolfram.com /infocenter/Articles/1555 (81 words)

	CS 371 Course Description --- Spring 2005 (Site not responding. Last check: 2007-10-16)
		where op is one of {+,-,.,p}, and print the result of the operation.
		The operations are vector addition, vector subtraction, dot product, and the projection operator from above.
		As far as error-catching is concerned, you should catch illegal operators, as well as handle special cases of the
	www.cs.williams.edu /~lenhart/courses/cs371/assignment1.html (646 words)

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