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Topic: Linear representation

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	Group representation - Wikipedia, the free encyclopedia
		Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces.
		Representation theory is important because it enables many group-theoretic problems to be reduced to problems in linear algebra, which is a very well-understood theory.
		A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V.
	en.wikipedia.org /wiki/Group_representation (1501 words)

	kairatos editions Αντώνης Θωμ. Βασιλάκης. Linear A, Linear B.
		Linear A, Linear B. At first, the Minoans were using a kind of writing which was similar to the hieroglyphics of Egypt.
		Linear writing was invented, because they wanted to have the linear representation of the object and not its image.
		We understood the way of writing the Greek hieroglyphics, Linear A and Linear B, when the Greek newspaper “Kyriakatiki Kathimerini” published an article, which included the text and the translation of the hieroglyphic inscription that is on the smaller ring of the two rings from the Aedonia treasure.
	www.kairatos.com.gr /linear1.htm (1889 words)

	Projective representation - Wikipedia, the free encyclopedia
		This may be described otherwise as a homomorphism to a projective linear group PGL(V).
		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.
		See also linear representation, affine representation, group action.
	en.wikipedia.org /wiki/Projective_representation (222 words)

	Regular representation (Site not responding. Last check: 2007-11-07)
		In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself.
		For G a topological group, the regular representation in the above sense should be replaced by a suitable space of functions on G, with G acting by translation.
		The regular representation of a group ring is such that the left-hand and right-hand regular representations give isomorphic modules (and we often need not distinguish the cases).
	vb.game-host.org /en/Regular_representation.htm (854 words)

	Review: Perception-based Knowledge Representations (Site not responding. Last check: 2007-11-07)
		Linear ordering judgements are made on the relative order of items in a list of items, such as which comes first in the alpahbet L or P. Questions
		Both of these representations are drawn from the oiriginal i.e the example given and only some of the original is maintained in the mind.
		For example with spatial images the positioning of the objects is remebered and with the linear ordering the sequence of the objects is reserved in memory.
	www.scism.sbu.ac.uk /inmandw/review/cogpsy/review/rev6429.html (4338 words)

	Finite Vector Space Representations
		The vector representation of fields, and the matrix representation of linear operators is derived, together with how vectors and matrices are transformed to alternative representations, incuding their eigenrepresentation.
		The linear vector space representation of fields and operators is a powerful component in the toolbox of the mathematician, physicist, engineer and statistician.
		The representation in these examples is often of three cartesian directions of space (the bases), and the vector can be plotted, having a direction and a length.
	www.met.rdg.ac.uk /~ross/DARC/LinearVectorSpaces.html (843 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Conversion of logarithmic printing density to a linear representation requires both a scaling and an anti-log operation.
		When working with 16-bits linear, it is possible to maintain the full printing density range, although the common practice is to put reference white at the maximum code value of 65535.
		Linear code values must be divided by 16 and clipped to 255 before mapping through the 8-bit hardware look-up table.
	www.cinesite.com /CineonTech/GrayScale/Sections/GS-7.0.html (917 words)

	Title page for ETD etd-07072002-002050 ( Browse \| Search ) All Available ETDs
		This representation is based upon the theory of linear spaces and, hence, is referred to as the linear representation.
		The representation is first developed in the classical context, followed by an extension to the reversible paradigm by exploiting the well-developed theory on invertible mappings.
		The consistency of the linear representation is checked against a current existing application in VLSI CAD that exploits the linearity of logic functions for symbolic representation of circuits.
	etd.library.pitt.edu /ETD/available/etd-07072002-002050 (365 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		To illustrate the three alternative representations, we collected a sample of 193 mail messages sent to a faculty member, of which 100 were unwanted and 93 were important.
		Cohen [1996] argues for learning this type of representation: “the greater comprehensibility of the rules may be advantageous in a system that allows users to extend or otherwise modify a learned classier.” Figure 1 presents the set of rules learned with Ripper on all 193 examples of mail messages.
		Prototype representations are more acceptable to users than linear model representations Using word pairs as terms increases the acceptance of profiles (for linear models, prototypes, and rules).
	lcs.www.media.mit.edu /people/lieber/IUI/Pazzani/Pazzani.doc (2904 words)

	The original VIrtual-Pulse (VIP) time integral methodology of computation structural dynamic problems
		Equation (3.50) is the representation in component form of the original VIP representation for linear structural dynamics.
		In addition, the dimensions of matrices and vectors involved in the VIP representation are only m instead of N, usually, m is less than N (m is the number of modes and N is the total degrees of freedom for the dynamic system at hand).
		The dynamic responses of the elastic and elasto-plastic structure obtained from the VIP representation and the Newmark implicit method are plotted in Figures 27 and 29 respectively.
	www-users.cs.umn.edu /~xiangmin/arc/node36.html (4095 words)

	PWL functions (Site not responding. Last check: 2007-11-07)
		Piecewise linear functions are used in general to ''replace'' a nonlinear function with several local linear functions.
		We have developed a canonical representation using a simplicial partition, which is the first representation valid in an n-dimensional domain.
		After that, we developed the foundations for a completely general canonical PWL representation with the elaboration of an explicit representation for a function on a minimal degenerate intersection (this has been proved to be the basic building block in any PWL representation).
	olympus.ece.jhu.edu /~pjulian/PWL_functions.htm (292 words)

	2.7 Linear system representation
		This function receives as parameters the constant matrices which define a linear system in state-space form or, in the case of system in transfer form, its input must be a rational matrix.
		Note that the transfer function representation of the linear system is itself a tlist.
		Linear systems can be inter-connected , their representation can easily be changed from state-space to transfer function and vice versa.
	scilabsoft.inria.fr /doc/intro/node24.html (662 words)

	Linear representation of Boolean logic
		Suppose we could find a mapping onto a linear Markov model (logic by matrix multiplication), then this would not require any energy to be dumped at all.
		Rather than guessing representations, we use information theoretical ideas to look for a new strategy which might give a 1:1 representation.
		This is a minimal (irreducible) representation, in the sense that it is the smallest representation which satisfies all the conditions.
	www.iu.hio.no /data/QIC/info3/node7.html (848 words)

	'Use-Once' Variables and Linear Objects
		Linear types are better than the "nurseries" of generational garbage collectors; linear objects are "fetal" objects, which have not yet been truly "born" as first-class (shared) objects.
		Linear objects have the property that there is only one access path to them at any given time--i.e., their normal and transitive reference counts are both equal to one.
		Linear objects are much more dynamic than the local finalized variables of C++, however, and can therefore be used to communicate results to a caller as well as act as "envelopes" for messages and "argument lists" in remote procedure calls in distributed computation.
	home.pipeline.com /~hbaker1/Use1Var.html (7056 words)

	Why Spin(0,8)?
		Therefore, although the conventional linear vector representation of D3 is 6-dimensional, you could (and I do) say that D3 has a NONlinear 4-dimensional representation due to conformal transformations, and I see a triality among: two D3 4-dim half-spinor representations and one D3 NONlinear conformal (sort of vector-like) 4-dim representation.
		The 240 elements of the orbit of the permutation group S7 of the 7 imaginaries of the octonion algebra correspond to the discrete octonionic algebra representation of the 240 vertices near the origin of the 8-dimensional E8 spacetime lattice.
		the discrete octonionic algebra representation of a second set of 240 vertices near the origin of the 8-dimensional fermion antiparticle -half-spinor E8 lattice (The Witting polytope is self-dual, and the second set of 240 vertices form another 4-complex-dimensional (8-real-dimensional) Witting polytope that is dual to the first Witting polytope.).
	www.valdostamuseum.org /hamsmith/why8.html (3810 words)

	Matrix Representation of Linear Systems Relationship (Site not responding. Last check: 2007-11-07)
		Considering the 1D linear shift-invariant system for simplicity, we have the input-output relationship given by the convolution integral.
		The limits depend upon causality, the nature of h (IIR, FIR), and whether the convolution desired is linear or circular.
		While causality is an inherent property of physical 1D signal processing systems, it is not always relevant in the 2D case as blurring typically occurs in all directions.
	www.lsi.usp.br /~roseli/pee5830/pee5830_aula08/tsld005.htm (86 words)

	Research Articles - Julianne Geering Rainbolt (Site not responding. Last check: 2007-11-07)
		Abstract: In this paper we explicitly calculate the irreducible representations of the endomorphism algebra of the Gelfand-Graev representation of the unitary group U(3,q).
		One is the regular representation and one is the usual Gelfand-Graev representation.
		The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.
	euler.slu.edu /Dept/Faculty/rainbolt/articles.htm (482 words)

	Joint Mathematics Colloquium (Site not responding. Last check: 2007-11-07)
		Abstract: By a non-linear representation of a finitely generated group G, we mean an action of G on a smooth compact manifold M by volume preserving diffeomorphisms.
		In this work, we consider the question of rigidity of non-linear representations of both higher rank lattices and, more generally, groups with property T of Kazhdan.
		We call a non-linear representation rigid if any perturbation is conjugate back to the original representation.
	www.math.neu.edu /bhmn/fisher.html (119 words)

	E-Example 7.5: Exploring Linear Functions (Site not responding. Last check: 2007-11-07)
		Dynamic graphical representations can suggest mathematical relationships that may be surprising to students and teachers, and exploring such relationships can help students gain a deeper understanding of important concepts.
		With technology, representations can be manipulated rapidly and in ways that would not be practical using paper and a pencil.
		The purpose of this task is for students to better understand linear functions by exploring the relationship between symbolic and graphical representations.
	standards.nctm.org /document/eexamples/chap7/7.5 (603 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		In the strictest sense, linear data representation in imaging systems refers to the representation of original scene luminance values in a linear scale.
		A linear data representation is widely used for computer generated imagery (CGI) because models of illumination, diffusion, and other physical phenomena are defined linearly.
		These operations can be applied to any data representation: linear, film density, gamma-corrected video, or other.
	www.cinesite.com /CineonTech/GrayScale/Sections/GS-9.0.html (286 words)

	In vivo RNA-directed transcription, with template switching, by a mammalian RNA polymerase
		Figure 1 Representation of linear antigenomic RNAs of HDV that were tested for their ability to initiate transcription and accumulation of processed genomic RNA species.
		As indicated at the right side, the electrophoresis conditions were such as to be able to separate circular and linear forms of 1679 nt unit length RNA.
		In each case, the template sequence shown is that at and around the 5' and 3' ends, relative to the rod-like folding of the HDV sequences.
	www.nature.com /emboj/journal/v21/n1/fig_tab/7594236ft.html (841 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		descriptionThe concept of a general signal space based on multidimensional linear spaces is developed and applied to the representation of discrete and continuous signals and to linear transformations.
		The applicability of general signal analysis techniques to practical problems is stressed as is the relationship of these general methods to more familiar approaches to signal analysis and representation.
		To provide sufficient mathematical background in functional analysis to permit the student to apply these techniques to the solution of practical problems and to gain an understanding of how other specialized methods of analysis can be interpreted in terms of a more general approach.
	www.ecn.purdue.edu /courses/data/EE/Archive/646 (302 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Linear Algebra in a Hurry for ODE Students
		This includes definitions of vector spaces and linear transformations, the notion of basis and dimension of a vector space, and representation of a linear transformation by a matrix, in terms of a choice of basis.
		We have a treatment of determinants of square matrices, followed by a discussion of eigenvalues and eigenvectors of a linear transformation, and then of generalized eigenspaces.
	www.math.unc.edu /Faculty/met/linalg.html (138 words)

	matrixApps
		One further advantage of matrix representation of life stages is that the array format, formalized as it is with subscripts denoting specific locations of elements, is easily translated into the repetitive loops characteristic of high level programming languages.
		Statistical models, including simple and multiple linear regression, analysis of variance, and the general linear model may be conveniently expressed in matrix form.
		Representation of these models in matrix form simplifies the problem of remembering the equations for solution and it clarifies the relationships among the variables and the observations.
	www.fw.umn.edu /FW5601/LECTURE/mappls/mappls.html (1189 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		In the case of a Toeplitz matrix rather than nonzero elements we need only to represent the distinct elements, that the Toeplitz matrix can be represented as a sparse matrix.
		Figure 6 — Linear list representation of the Toeplitz matrix of Figure 5 C-matrices An nxn C-matrix is one in which all elements other than those in row 1, row n, and column 1 are zero.
		Similar representations are typically employed for all these variations as well.
	longwood.cs.ucf.edu /courses/cop3530-su02/matrix3.doc (451 words)

	General Approach (Site not responding. Last check: 2007-11-07)
		Given a known value as above and memory objects in a similar representation, it is possible to check whether a memory access will always be in a defined region for which the program has permission to read or write.
		For example, most bitwise logical instructions yield a result of unknown because the simulator cannot create a linear function representation of their output given a linear function representation of their input.
		This works in practice because most memory accesses are done through linear functions (scale-index-base, as seen in the x86 ISA), so the linear values tracked by the simulator contain enough information to reason about most memory addresses, while the other values used by a program aren't usually important for memory access safety.
	www.usenix.org /event/usenix2000/freenix/full_papers/metzsafety/metzsafety_html/node3.html (501 words)

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